LMFDB - Embedded newform 1280.4.d.k.641.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1280,4,Mod(641,1280)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1280.641"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1280.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,12,0,46,0,0,0,0,0,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$75.5224448073$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			641.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1280.641
Dual form			1280.4.d.k.641.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} +5.00000i q^{5} +6.00000 q^{7} +23.0000 q^{9} +60.0000i q^{11} +50.0000i q^{13} -10.0000 q^{15} -30.0000 q^{17} -40.0000i q^{19} +12.0000i q^{21} +178.000 q^{23} -25.0000 q^{25} +100.000i q^{27} +166.000i q^{29} -20.0000 q^{31} -120.000 q^{33} +30.0000i q^{35} -10.0000i q^{37} -100.000 q^{39} +250.000 q^{41} +142.000i q^{43} +115.000i q^{45} -214.000 q^{47} -307.000 q^{49} -60.0000i q^{51} -490.000i q^{53} -300.000 q^{55} +80.0000 q^{57} -800.000i q^{59} +250.000i q^{61} +138.000 q^{63} -250.000 q^{65} +774.000i q^{67} +356.000i q^{69} +100.000 q^{71} +230.000 q^{73} -50.0000i q^{75} +360.000i q^{77} +1320.00 q^{79} +421.000 q^{81} -982.000i q^{83} -150.000i q^{85} -332.000 q^{87} -874.000 q^{89} +300.000i q^{91} -40.0000i q^{93} +200.000 q^{95} -310.000 q^{97} +1380.00i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 12 q^{7} + 46 q^{9} - 20 q^{15} - 60 q^{17} + 356 q^{23} - 50 q^{25} - 40 q^{31} - 240 q^{33} - 200 q^{39} + 500 q^{41} - 428 q^{47} - 614 q^{49} - 600 q^{55} + 160 q^{57} + 276 q^{63} - 500 q^{65}+ \cdots - 620 q^{97}+O(q^{100}) $ 2 * q + 12 * q^7 + 46 * q^9 - 20 * q^15 - 60 * q^17 + 356 * q^23 - 50 * q^25 - 40 * q^31 - 240 * q^33 - 200 * q^39 + 500 * q^41 - 428 * q^47 - 614 * q^49 - 600 * q^55 + 160 * q^57 + 276 * q^63 - 500 * q^65 + 200 * q^71 + 460 * q^73 + 2640 * q^79 + 842 * q^81 - 664 * q^87 - 1748 * q^89 + 400 * q^95 - 620 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	2.00000i	0.384900i	0.981307	+	0.192450i	$0.0616434\pi$
$3$	2.00000i	0.384900i	−0.981307	+	0.192450i	$0.938357\pi$
$4$	0	0
$5$	5.00000i	0.447214i
$5$	5.00000i	0.447214i
$6$	0	0
$7$	6.00000	0.323970	0.161985	−	0.986793i	$-0.448210\pi$
$7$	6.00000	0.323970	0.161985	+	0.986793i	$0.448210\pi$
$8$	0	0
$9$	23.0000	0.851852
$10$	0	0
$11$	60.0000i	1.64461i	0.569049	+	0.822304i	$0.307311\pi$
$11$	60.0000i	1.64461i	−0.569049	+	0.822304i	$0.692689\pi$
$12$	0	0
$13$	50.0000i	1.06673i	0.845885	+	0.533366i	$0.179073\pi$
$13$	50.0000i	1.06673i	−0.845885	+	0.533366i	$0.820927\pi$
$14$	0	0
$15$	−10.0000	−0.172133
$16$	0	0
$17$	−30.0000	−0.428004	−0.214002	−	0.976833i	$-0.568650\pi$
$17$	−30.0000	−0.428004	−0.214002	+	0.976833i	$0.568650\pi$
$18$	0	0
$19$	− 40.0000i	− 0.482980i	−0.970403	−	0.241490i	$-0.922364\pi$
$19$	− 40.0000i	− 0.482980i	0.970403	−	0.241490i	$-0.0776362\pi$
$20$	0	0
$21$	12.0000i	0.124696i
$22$	0	0
$23$	178.000	1.61372	0.806860	−	0.590743i	$-0.201165\pi$
$23$	178.000	1.61372	0.806860	+	0.590743i	$0.201165\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	0	0
$27$	100.000i	0.712778i
$28$	0	0
$29$	166.000i	1.06295i	0.847075	+	0.531473i	$0.178361\pi$
$29$	166.000i	1.06295i	−0.847075	+	0.531473i	$0.821639\pi$
$30$	0	0
$31$	−20.0000	−0.115874	−0.0579372	−	0.998320i	$-0.518452\pi$
$31$	−20.0000	−0.115874	−0.0579372	+	0.998320i	$0.518452\pi$
$32$	0	0
$33$	−120.000	−0.633010
$34$	0	0
$35$	30.0000i	0.144884i
$36$	0	0
$37$	− 10.0000i	− 0.0444322i	−0.999753	−	0.0222161i	$-0.992928\pi$
$37$	− 10.0000i	− 0.0444322i	0.999753	−	0.0222161i	$-0.00707218\pi$
$38$	0	0
$39$	−100.000	−0.410585
$40$	0	0
$41$	250.000	0.952279	0.476140	−	0.879370i	$-0.342036\pi$
$41$	250.000	0.952279	0.476140	+	0.879370i	$0.342036\pi$
$42$	0	0
$43$	142.000i	0.503600i	0.967779	+	0.251800i	$0.0810225\pi$
$43$	142.000i	0.503600i	−0.967779	+	0.251800i	$0.918977\pi$
$44$	0	0
$45$	115.000i	0.380960i
$46$	0	0
$47$	−214.000	−0.664151	−0.332076	−	0.943253i	$-0.607749\pi$
$47$	−214.000	−0.664151	−0.332076	+	0.943253i	$0.607749\pi$
$48$	0	0
$49$	−307.000	−0.895044
$50$	0	0
$51$	− 60.0000i	− 0.164739i
$52$	0	0
$53$	− 490.000i	− 1.26994i	−0.772538	−	0.634969i	$-0.781013\pi$
$53$	− 490.000i	− 1.26994i	0.772538	−	0.634969i	$-0.218987\pi$
$54$	0	0
$55$	−300.000	−0.735491
$56$	0	0
$57$	80.0000	0.185899
$58$	0	0
$59$	− 800.000i	− 1.76527i	−0.470056	−	0.882637i	$-0.655766\pi$
$59$	− 800.000i	− 1.76527i	0.470056	−	0.882637i	$-0.344234\pi$
$60$	0	0
$61$	250.000i	0.524741i	0.964967	+	0.262371i	$0.0845043\pi$
$61$	250.000i	0.524741i	−0.964967	+	0.262371i	$0.915496\pi$
$62$	0	0
$63$	138.000	0.275974
$64$	0	0
$65$	−250.000	−0.477057
$66$	0	0
$67$	774.000i	1.41133i	0.708545	+	0.705665i	$0.249352\pi$
$67$	774.000i	1.41133i	−0.708545	+	0.705665i	$0.750648\pi$
$68$	0	0
$69$	356.000i	0.621121i
$70$	0	0
$71$	100.000	0.167152	0.0835762	−	0.996501i	$-0.473366\pi$
$71$	100.000	0.167152	0.0835762	+	0.996501i	$0.473366\pi$
$72$	0	0
$73$	230.000	0.368760	0.184380	−	0.982855i	$-0.440972\pi$
$73$	230.000	0.368760	0.184380	+	0.982855i	$0.440972\pi$
$74$	0	0
$75$	− 50.0000i	− 0.0769800i
$76$	0	0
$77$	360.000i	0.532803i
$78$	0	0
$79$	1320.00	1.87989	0.939947	−	0.341321i	$-0.110874\pi$
$79$	1320.00	1.87989	0.939947	+	0.341321i	$0.110874\pi$
$80$	0	0
$81$	421.000	0.577503
$82$	0	0
$83$	− 982.000i	− 1.29866i	−0.760508	−	0.649328i	$-0.775050\pi$
$83$	− 982.000i	− 1.29866i	0.760508	−	0.649328i	$-0.224950\pi$
$84$	0	0
$85$	− 150.000i	− 0.191409i
$86$	0	0
$87$	−332.000	−0.409128
$88$	0	0
$89$	−874.000	−1.04094	−0.520471	−	0.853879i	$-0.674244\pi$
$89$	−874.000	−1.04094	−0.520471	+	0.853879i	$0.674244\pi$
$90$	0	0
$91$	300.000i	0.345588i
$92$	0	0
$93$	− 40.0000i	− 0.0446001i
$94$	0	0
$95$	200.000	0.215995
$96$	0	0
$97$	−310.000	−0.324492	−0.162246	−	0.986750i	$-0.551874\pi$
$97$	−310.000	−0.324492	−0.162246	+	0.986750i	$0.551874\pi$
$98$	0	0
$99$	1380.00i	1.40096i
$100$	0	0
$101$	1498.00i	1.47581i	0.674906	+	0.737904i	$0.264184\pi$
$101$	1498.00i	1.47581i	−0.674906	+	0.737904i	$0.735816\pi$
$102$	0	0
$103$	1402.00	1.34120	0.670598	−	0.741821i	$-0.266038\pi$
$103$	1402.00	1.34120	0.670598	+	0.741821i	$0.266038\pi$
$104$	0	0
$105$	−60.0000	−0.0557657
$106$	0	0
$107$	1194.00i	1.07877i	0.842059	+	0.539385i	$0.181343\pi$
$107$	1194.00i	1.07877i	−0.842059	+	0.539385i	$0.818657\pi$
$108$	0	0
$109$	650.000i	0.571181i	0.958352	+	0.285590i	$0.0921897\pi$
$109$	650.000i	0.571181i	−0.958352	+	0.285590i	$0.907810\pi$
$110$	0	0
$111$	20.0000	0.0171019
$112$	0	0
$113$	−1510.00	−1.25707	−0.628535	−	0.777782i	$-0.716345\pi$
$113$	−1510.00	−1.25707	−0.628535	+	0.777782i	$0.716345\pi$
$114$	0	0
$115$	890.000i	0.721678i
$116$	0	0
$117$	1150.00i	0.908697i
$118$	0	0
$119$	−180.000	−0.138660
$120$	0	0
$121$	−2269.00	−1.70473
$122$	0	0
$123$	500.000i	0.366532i
$124$	0	0
$125$	− 125.000i	− 0.0894427i
$126$	0	0
$127$	−1246.00	−0.870588	−0.435294	−	0.900288i	$-0.643355\pi$
$127$	−1246.00	−0.870588	−0.435294	+	0.900288i	$0.643355\pi$
$128$	0	0
$129$	−284.000	−0.193836
$130$	0	0
$131$	− 2660.00i	− 1.77409i	−0.461687	−	0.887043i	$-0.652756\pi$
$131$	− 2660.00i	− 1.77409i	0.461687	−	0.887043i	$-0.347244\pi$
$132$	0	0
$133$	− 240.000i	− 0.156471i
$134$	0	0
$135$	−500.000	−0.318764
$136$	0	0
$137$	−2770.00	−1.72742	−0.863712	−	0.503986i	$-0.831866\pi$
$137$	−2770.00	−1.72742	−0.863712	+	0.503986i	$0.831866\pi$
$138$	0	0
$139$	− 560.000i	− 0.341716i	−0.985296	−	0.170858i	$-0.945346\pi$
$139$	− 560.000i	− 0.341716i	0.985296	−	0.170858i	$-0.0546540\pi$
$140$	0	0
$141$	− 428.000i	− 0.255632i
$142$	0	0
$143$	−3000.00	−1.75435
$144$	0	0
$145$	−830.000	−0.475364
$146$	0	0
$147$	− 614.000i	− 0.344502i
$148$	0	0
$149$	2350.00i	1.29208i	0.763305	+	0.646039i	$0.223576\pi$
$149$	2350.00i	1.29208i	−0.763305	+	0.646039i	$0.776424\pi$
$150$	0	0
$151$	580.000	0.312581	0.156290	−	0.987711i	$-0.450046\pi$
$151$	580.000	0.312581	0.156290	+	0.987711i	$0.450046\pi$
$152$	0	0
$153$	−690.000	−0.364596
$154$	0	0
$155$	− 100.000i	− 0.0518206i
$156$	0	0
$157$	− 1310.00i	− 0.665920i	−0.942941	−	0.332960i	$-0.891953\pi$
$157$	− 1310.00i	− 0.665920i	0.942941	−	0.332960i	$-0.108047\pi$
$158$	0	0
$159$	980.000	0.488799
$160$	0	0
$161$	1068.00	0.522796
$162$	0	0
$163$	− 1862.00i	− 0.894743i	−0.894348	−	0.447371i	$-0.852360\pi$
$163$	− 1862.00i	− 0.894743i	0.894348	−	0.447371i	$-0.147640\pi$
$164$	0	0
$165$	− 600.000i	− 0.283091i
$166$	0	0
$167$	726.000	0.336405	0.168202	−	0.985752i	$-0.446204\pi$
$167$	726.000	0.336405	0.168202	+	0.985752i	$0.446204\pi$
$168$	0	0
$169$	−303.000	−0.137915
$170$	0	0
$171$	− 920.000i	− 0.411428i
$172$	0	0
$173$	3250.00i	1.42828i	0.700001	+	0.714141i	$0.253183\pi$
$173$	3250.00i	1.42828i	−0.700001	+	0.714141i	$0.746817\pi$
$174$	0	0
$175$	−150.000	−0.0647939
$176$	0	0
$177$	1600.00	0.679454
$178$	0	0
$179$	1120.00i	0.467669i	0.972276	+	0.233834i	$0.0751274\pi$
$179$	1120.00i	0.467669i	−0.972276	+	0.233834i	$0.924873\pi$
$180$	0	0
$181$	2842.00i	1.16710i	0.812079	+	0.583548i	$0.198336\pi$
$181$	2842.00i	1.16710i	−0.812079	+	0.583548i	$0.801664\pi$
$182$	0	0
$183$	−500.000	−0.201973
$184$	0	0
$185$	50.0000	0.0198707
$186$	0	0
$187$	− 1800.00i	− 0.703899i
$188$	0	0
$189$	600.000i	0.230918i
$190$	0	0
$191$	−3180.00	−1.20469	−0.602347	−	0.798234i	$-0.705768\pi$
$191$	−3180.00	−1.20469	−0.602347	+	0.798234i	$0.705768\pi$
$192$	0	0
$193$	−4670.00	−1.74173	−0.870865	−	0.491522i	$-0.836441\pi$
$193$	−4670.00	−1.74173	−0.870865	+	0.491522i	$0.836441\pi$
$194$	0	0
$195$	− 500.000i	− 0.183619i
$196$	0	0
$197$	2990.00i	1.08136i	0.841227	+	0.540682i	$0.181834\pi$
$197$	2990.00i	1.08136i	−0.841227	+	0.540682i	$0.818166\pi$
$198$	0	0
$199$	−4240.00	−1.51038	−0.755190	−	0.655506i	$-0.772455\pi$
$199$	−4240.00	−1.51038	−0.755190	+	0.655506i	$0.772455\pi$
$200$	0	0
$201$	−1548.00	−0.543221
$202$	0	0
$203$	996.000i	0.344362i
$204$	0	0
$205$	1250.00i	0.425872i
$206$	0	0
$207$	4094.00	1.37465
$208$	0	0
$209$	2400.00	0.794313
$210$	0	0
$211$	4060.00i	1.32465i	0.749215	+	0.662327i	$0.230431\pi$
$211$	4060.00i	1.32465i	−0.749215	+	0.662327i	$0.769569\pi$
$212$	0	0
$213$	200.000i	0.0643370i
$214$	0	0
$215$	−710.000	−0.225217
$216$	0	0
$217$	−120.000	−0.0375398
$218$	0	0
$219$	460.000i	0.141936i
$220$	0	0
$221$	− 1500.00i	− 0.456565i
$222$	0	0
$223$	5622.00	1.68824	0.844119	−	0.536156i	$-0.180124\pi$
$223$	5622.00	1.68824	0.844119	+	0.536156i	$0.180124\pi$
$224$	0	0
$225$	−575.000	−0.170370
$226$	0	0
$227$	− 1554.00i	− 0.454373i	−0.973851	−	0.227186i	$-0.927047\pi$
$227$	− 1554.00i	− 0.454373i	0.973851	−	0.227186i	$-0.0729526\pi$
$228$	0	0
$229$	− 1134.00i	− 0.327235i	−0.986524	−	0.163618i	$-0.947684\pi$
$229$	− 1134.00i	− 0.327235i	0.986524	−	0.163618i	$-0.0523163\pi$
$230$	0	0
$231$	−720.000	−0.205076
$232$	0	0
$233$	1710.00	0.480798	0.240399	−	0.970674i	$-0.422722\pi$
$233$	1710.00	0.480798	0.240399	+	0.970674i	$0.422722\pi$
$234$	0	0
$235$	− 1070.00i	− 0.297017i
$236$	0	0
$237$	2640.00i	0.723571i
$238$	0	0
$239$	−4440.00	−1.20167	−0.600836	−	0.799372i	$-0.705166\pi$
$239$	−4440.00	−1.20167	−0.600836	+	0.799372i	$0.705166\pi$
$240$	0	0
$241$	−850.000	−0.227192	−0.113596	−	0.993527i	$-0.536237\pi$
$241$	−850.000	−0.227192	−0.113596	+	0.993527i	$0.536237\pi$
$242$	0	0
$243$	3542.00i	0.935059i
$244$	0	0
$245$	− 1535.00i	− 0.400276i
$246$	0	0
$247$	2000.00	0.515210
$248$	0	0
$249$	1964.00	0.499853
$250$	0	0
$251$	− 660.000i	− 0.165971i	−0.996551	−	0.0829857i	$-0.973554\pi$
$251$	− 660.000i	− 0.165971i	0.996551	−	0.0829857i	$-0.0264456\pi$
$252$	0	0
$253$	10680.0i	2.65394i
$254$	0	0
$255$	300.000	0.0736734
$256$	0	0
$257$	−7590.00	−1.84222	−0.921111	−	0.389299i	$-0.872717\pi$
$257$	−7590.00	−1.84222	−0.921111	+	0.389299i	$0.872717\pi$
$258$	0	0
$259$	− 60.0000i	− 0.0143947i
$260$	0	0
$261$	3818.00i	0.905472i
$262$	0	0
$263$	762.000	0.178658	0.0893288	−	0.996002i	$-0.471528\pi$
$263$	762.000	0.178658	0.0893288	+	0.996002i	$0.471528\pi$
$264$	0	0
$265$	2450.00	0.567933
$266$	0	0
$267$	− 1748.00i	− 0.400659i
$268$	0	0
$269$	− 150.000i	− 0.0339987i	−0.999856	−	0.0169994i	$-0.994589\pi$
$269$	− 150.000i	− 0.0339987i	0.999856	−	0.0169994i	$-0.00541133\pi$
$270$	0	0
$271$	6580.00	1.47493	0.737466	−	0.675384i	$-0.236022\pi$
$271$	6580.00	1.47493	0.737466	+	0.675384i	$0.236022\pi$
$272$	0	0
$273$	−600.000	−0.133017
$274$	0	0
$275$	− 1500.00i	− 0.328921i
$276$	0	0
$277$	− 4530.00i	− 0.982604i	−0.870989	−	0.491302i	$-0.836521\pi$
$277$	− 4530.00i	− 0.982604i	0.870989	−	0.491302i	$-0.163479\pi$
$278$	0	0
$279$	−460.000	−0.0987078
$280$	0	0
$281$	−6950.00	−1.47545	−0.737726	−	0.675100i	$-0.764101\pi$
$281$	−6950.00	−1.47545	−0.737726	+	0.675100i	$0.764101\pi$
$282$	0	0
$283$	− 3882.00i	− 0.815410i	−0.913114	−	0.407705i	$-0.866329\pi$
$283$	− 3882.00i	− 0.815410i	0.913114	−	0.407705i	$-0.133671\pi$
$284$	0	0
$285$	400.000i	0.0831367i
$286$	0	0
$287$	1500.00	0.308509
$288$	0	0
$289$	−4013.00	−0.816813
$290$	0	0
$291$	− 620.000i	− 0.124897i
$292$	0	0
$293$	− 1370.00i	− 0.273161i	−0.990629	−	0.136581i	$-0.956389\pi$
$293$	− 1370.00i	− 0.273161i	0.990629	−	0.136581i	$-0.0436113\pi$
$294$	0	0
$295$	4000.00	0.789454
$296$	0	0
$297$	−6000.00	−1.17224
$298$	0	0
$299$	8900.00i	1.72141i
$300$	0	0
$301$	852.000i	0.163151i
$302$	0	0
$303$	−2996.00	−0.568039
$304$	0	0
$305$	−1250.00	−0.234671
$306$	0	0
$307$	− 4106.00i	− 0.763328i	−0.924301	−	0.381664i	$-0.875351\pi$
$307$	− 4106.00i	− 0.763328i	0.924301	−	0.381664i	$-0.124649\pi$
$308$	0	0
$309$	2804.00i	0.516226i
$310$	0	0
$311$	2220.00	0.404774	0.202387	−	0.979306i	$-0.435130\pi$
$311$	2220.00	0.404774	0.202387	+	0.979306i	$0.435130\pi$
$312$	0	0
$313$	9430.00	1.70292	0.851462	−	0.524417i	$-0.175717\pi$
$313$	9430.00	1.70292	0.851462	+	0.524417i	$0.175717\pi$
$314$	0	0
$315$	690.000i	0.123419i
$316$	0	0
$317$	− 6470.00i	− 1.14635i	−0.819435	−	0.573173i	$-0.805712\pi$
$317$	− 6470.00i	− 1.14635i	0.819435	−	0.573173i	$-0.194288\pi$
$318$	0	0
$319$	−9960.00	−1.74813
$320$	0	0
$321$	−2388.00	−0.415219
$322$	0	0
$323$	1200.00i	0.206718i
$324$	0	0
$325$	− 1250.00i	− 0.213346i
$326$	0	0
$327$	−1300.00	−0.219848
$328$	0	0
$329$	−1284.00	−0.215165
$330$	0	0
$331$	900.000i	0.149452i	0.997204	+	0.0747258i	$0.0238082\pi$
$331$	900.000i	0.149452i	−0.997204	+	0.0747258i	$0.976192\pi$
$332$	0	0
$333$	− 230.000i	− 0.0378496i
$334$	0	0
$335$	−3870.00	−0.631166
$336$	0	0
$337$	530.000	0.0856704	0.0428352	−	0.999082i	$-0.486361\pi$
$337$	530.000	0.0856704	0.0428352	+	0.999082i	$0.486361\pi$
$338$	0	0
$339$	− 3020.00i	− 0.483846i
$340$	0	0
$341$	− 1200.00i	− 0.190568i
$342$	0	0
$343$	−3900.00	−0.613936
$344$	0	0
$345$	−1780.00	−0.277774
$346$	0	0
$347$	− 414.000i	− 0.0640481i	−0.999487	−	0.0320240i	$-0.989805\pi$
$347$	− 414.000i	− 0.0640481i	0.999487	−	0.0320240i	$-0.0101953\pi$
$348$	0	0
$349$	8614.00i	1.32119i	0.750741	+	0.660597i	$0.229697\pi$
$349$	8614.00i	1.32119i	−0.750741	+	0.660597i	$0.770303\pi$
$350$	0	0
$351$	−5000.00	−0.760343
$352$	0	0
$353$	−2270.00	−0.342266	−0.171133	−	0.985248i	$-0.554743\pi$
$353$	−2270.00	−0.342266	−0.171133	+	0.985248i	$0.554743\pi$
$354$	0	0
$355$	500.000i	0.0747528i
$356$	0	0
$357$	− 360.000i	− 0.0533704i
$358$	0	0
$359$	8080.00	1.18787	0.593936	−	0.804512i	$-0.297573\pi$
$359$	8080.00	1.18787	0.593936	+	0.804512i	$0.297573\pi$
$360$	0	0
$361$	5259.00	0.766730
$362$	0	0
$363$	− 4538.00i	− 0.656152i
$364$	0	0
$365$	1150.00i	0.164914i
$366$	0	0
$367$	−2374.00	−0.337662	−0.168831	−	0.985645i	$-0.553999\pi$
$367$	−2374.00	−0.337662	−0.168831	+	0.985645i	$0.553999\pi$
$368$	0	0
$369$	5750.00	0.811201
$370$	0	0
$371$	− 2940.00i	− 0.411421i
$372$	0	0
$373$	− 1810.00i	− 0.251255i	−0.992077	−	0.125628i	$-0.959906\pi$
$373$	− 1810.00i	− 0.251255i	0.992077	−	0.125628i	$-0.0400945\pi$
$374$	0	0
$375$	250.000	0.0344265
$376$	0	0
$377$	−8300.00	−1.13388
$378$	0	0
$379$	8120.00i	1.10052i	0.834994	+	0.550259i	$0.185471\pi$
$379$	8120.00i	1.10052i	−0.834994	+	0.550259i	$0.814529\pi$
$380$	0	0
$381$	− 2492.00i	− 0.335089i
$382$	0	0
$383$	11782.0	1.57189	0.785943	−	0.618299i	$-0.212178\pi$
$383$	11782.0	1.57189	0.785943	+	0.618299i	$0.212178\pi$
$384$	0	0
$385$	−1800.00	−0.238277
$386$	0	0
$387$	3266.00i	0.428993i
$388$	0	0
$389$	4350.00i	0.566976i	0.958976	+	0.283488i	$0.0914917\pi$
$389$	4350.00i	0.566976i	−0.958976	+	0.283488i	$0.908508\pi$
$390$	0	0
$391$	−5340.00	−0.690679
$392$	0	0
$393$	5320.00	0.682846
$394$	0	0
$395$	6600.00i	0.840714i
$396$	0	0
$397$	− 7470.00i	− 0.944354i	−0.881504	−	0.472177i	$-0.843468\pi$
$397$	− 7470.00i	− 0.944354i	0.881504	−	0.472177i	$-0.156532\pi$
$398$	0	0
$399$	480.000	0.0602257
$400$	0	0
$401$	11698.0	1.45678	0.728392	−	0.685161i	$-0.240268\pi$
$401$	11698.0	1.45678	0.728392	+	0.685161i	$0.240268\pi$
$402$	0	0
$403$	− 1000.00i	− 0.123607i
$404$	0	0
$405$	2105.00i	0.258267i
$406$	0	0
$407$	600.000	0.0730735
$408$	0	0
$409$	3650.00	0.441274	0.220637	−	0.975356i	$-0.429186\pi$
$409$	3650.00	0.441274	0.220637	+	0.975356i	$0.429186\pi$
$410$	0	0
$411$	− 5540.00i	− 0.664886i
$412$	0	0
$413$	− 4800.00i	− 0.571895i
$414$	0	0
$415$	4910.00	0.580777
$416$	0	0
$417$	1120.00	0.131527
$418$	0	0
$419$	1120.00i	0.130586i	0.997866	+	0.0652931i	$0.0207982\pi$
$419$	1120.00i	0.130586i	−0.997866	+	0.0652931i	$0.979202\pi$
$420$	0	0
$421$	− 4850.00i	− 0.561460i	−0.959787	−	0.280730i	$-0.909424\pi$
$421$	− 4850.00i	− 0.561460i	0.959787	−	0.280730i	$-0.0905765\pi$
$422$	0	0
$423$	−4922.00	−0.565758
$424$	0	0
$425$	750.000	0.0856008
$426$	0	0
$427$	1500.00i	0.170000i
$428$	0	0
$429$	− 6000.00i	− 0.675251i
$430$	0	0
$431$	12580.0	1.40593	0.702967	−	0.711223i	$-0.251858\pi$
$431$	12580.0	1.40593	0.702967	+	0.711223i	$0.251858\pi$
$432$	0	0
$433$	13130.0	1.45725	0.728623	−	0.684915i	$-0.240161\pi$
$433$	13130.0	1.45725	0.728623	+	0.684915i	$0.240161\pi$
$434$	0	0
$435$	− 1660.00i	− 0.182968i
$436$	0	0
$437$	− 7120.00i	− 0.779395i
$438$	0	0
$439$	8560.00	0.930630	0.465315	−	0.885145i	$-0.345941\pi$
$439$	8560.00	0.930630	0.465315	+	0.885145i	$0.345941\pi$
$440$	0	0
$441$	−7061.00	−0.762445
$442$	0	0
$443$	− 4258.00i	− 0.456667i	−0.973583	−	0.228334i	$-0.926672\pi$
$443$	− 4258.00i	− 0.456667i	0.973583	−	0.228334i	$-0.0733277\pi$
$444$	0	0
$445$	− 4370.00i	− 0.465523i
$446$	0	0
$447$	−4700.00	−0.497321
$448$	0	0
$449$	2550.00	0.268022	0.134011	−	0.990980i	$-0.457214\pi$
$449$	2550.00	0.268022	0.134011	+	0.990980i	$0.457214\pi$
$450$	0	0
$451$	15000.0i	1.56613i
$452$	0	0
$453$	1160.00i	0.120312i
$454$	0	0
$455$	−1500.00	−0.154552
$456$	0	0
$457$	6710.00	0.686828	0.343414	−	0.939184i	$-0.388417\pi$
$457$	6710.00	0.686828	0.343414	+	0.939184i	$0.388417\pi$
$458$	0	0
$459$	− 3000.00i	− 0.305072i
$460$	0	0
$461$	− 14482.0i	− 1.46311i	−0.681782	−	0.731555i	$-0.738795\pi$
$461$	− 14482.0i	− 1.46311i	0.681782	−	0.731555i	$-0.261205\pi$
$462$	0	0
$463$	−162.000	−0.0162609	−0.00813043	−	0.999967i	$-0.502588\pi$
$463$	−162.000	−0.0162609	−0.00813043	+	0.999967i	$0.502588\pi$
$464$	0	0
$465$	200.000	0.0199458
$466$	0	0
$467$	15974.0i	1.58284i	0.611270	+	0.791422i	$0.290659\pi$
$467$	15974.0i	1.58284i	−0.611270	+	0.791422i	$0.709341\pi$
$468$	0	0
$469$	4644.00i	0.457228i
$470$	0	0
$471$	2620.00	0.256313
$472$	0	0
$473$	−8520.00	−0.828224
$474$	0	0
$475$	1000.00i	0.0965961i
$476$	0	0
$477$	− 11270.0i	− 1.08180i
$478$	0	0
$479$	10760.0	1.02638	0.513191	−	0.858274i	$-0.328463\pi$
$479$	10760.0	1.02638	0.513191	+	0.858274i	$0.328463\pi$
$480$	0	0
$481$	500.000	0.0473972
$482$	0	0
$483$	2136.00i	0.201224i
$484$	0	0
$485$	− 1550.00i	− 0.145117i
$486$	0	0
$487$	−9266.00	−0.862182	−0.431091	−	0.902309i	$-0.641871\pi$
$487$	−9266.00	−0.862182	−0.431091	+	0.902309i	$0.641871\pi$
$488$	0	0
$489$	3724.00	0.344387
$490$	0	0
$491$	2860.00i	0.262872i	0.991325	+	0.131436i	$0.0419587\pi$
$491$	2860.00i	0.262872i	−0.991325	+	0.131436i	$0.958041\pi$
$492$	0	0
$493$	− 4980.00i	− 0.454945i
$494$	0	0
$495$	−6900.00	−0.626529
$496$	0	0
$497$	600.000	0.0541523
$498$	0	0
$499$	− 7160.00i	− 0.642336i	−0.947022	−	0.321168i	$-0.895925\pi$
$499$	− 7160.00i	− 0.642336i	0.947022	−	0.321168i	$-0.104075\pi$
$500$	0	0
$501$	1452.00i	0.129482i
$502$	0	0
$503$	−1398.00	−0.123924	−0.0619620	−	0.998079i	$-0.519736\pi$
$503$	−1398.00	−0.123924	−0.0619620	+	0.998079i	$0.519736\pi$
$504$	0	0
$505$	−7490.00	−0.660001
$506$	0	0
$507$	− 606.000i	− 0.0530836i
$508$	0	0
$509$	7446.00i	0.648405i	0.945988	+	0.324203i	$0.105096\pi$
$509$	7446.00i	0.648405i	−0.945988	+	0.324203i	$0.894904\pi$
$510$	0	0
$511$	1380.00	0.119467
$512$	0	0
$513$	4000.00	0.344258
$514$	0	0
$515$	7010.00i	0.599801i
$516$	0	0
$517$	− 12840.0i	− 1.09227i
$518$	0	0
$519$	−6500.00	−0.549746
$520$	0	0
$521$	16438.0	1.38227	0.691134	−	0.722726i	$-0.257111\pi$
$521$	16438.0	1.38227	0.691134	+	0.722726i	$0.257111\pi$
$522$	0	0
$523$	− 7322.00i	− 0.612177i	−0.952003	−	0.306089i	$-0.900980\pi$
$523$	− 7322.00i	− 0.612177i	0.952003	−	0.306089i	$-0.0990204\pi$
$524$	0	0
$525$	− 300.000i	− 0.0249392i
$526$	0	0
$527$	600.000	0.0495947
$528$	0	0
$529$	19517.0	1.60409
$530$	0	0
$531$	− 18400.0i	− 1.50375i
$532$	0	0
$533$	12500.0i	1.01583i
$534$	0	0
$535$	−5970.00	−0.482440
$536$	0	0
$537$	−2240.00	−0.180006
$538$	0	0
$539$	− 18420.0i	− 1.47200i
$540$	0	0
$541$	10878.0i	0.864476i	0.901759	+	0.432238i	$0.142276\pi$
$541$	10878.0i	0.864476i	−0.901759	+	0.432238i	$0.857724\pi$
$542$	0	0
$543$	−5684.00	−0.449215
$544$	0	0
$545$	−3250.00	−0.255440
$546$	0	0
$547$	− 16114.0i	− 1.25957i	−0.776769	−	0.629785i	$-0.783143\pi$
$547$	− 16114.0i	− 1.25957i	0.776769	−	0.629785i	$-0.216857\pi$
$548$	0	0
$549$	5750.00i	0.447002i
$550$	0	0
$551$	6640.00	0.513382
$552$	0	0
$553$	7920.00	0.609028
$554$	0	0
$555$	100.000i	0.00764822i
$556$	0	0
$557$	3690.00i	0.280701i	0.990102	+	0.140350i	$0.0448229\pi$
$557$	3690.00i	0.280701i	−0.990102	+	0.140350i	$0.955177\pi$
$558$	0	0
$559$	−7100.00	−0.537206
$560$	0	0
$561$	3600.00	0.270931
$562$	0	0
$563$	2562.00i	0.191786i	0.995392	+	0.0958929i	$0.0305706\pi$
$563$	2562.00i	0.191786i	−0.995392	+	0.0958929i	$0.969429\pi$
$564$	0	0
$565$	− 7550.00i	− 0.562179i
$566$	0	0
$567$	2526.00	0.187094
$568$	0	0
$569$	6050.00	0.445746	0.222873	−	0.974848i	$-0.428457\pi$
$569$	6050.00	0.445746	0.222873	+	0.974848i	$0.428457\pi$
$570$	0	0
$571$	8260.00i	0.605377i	0.953090	+	0.302688i	$0.0978842\pi$
$571$	8260.00i	0.605377i	−0.953090	+	0.302688i	$0.902116\pi$
$572$	0	0
$573$	− 6360.00i	− 0.463687i
$574$	0	0
$575$	−4450.00	−0.322744
$576$	0	0
$577$	−16870.0	−1.21717	−0.608585	−	0.793489i	$-0.708263\pi$
$577$	−16870.0	−1.21717	−0.608585	+	0.793489i	$0.708263\pi$
$578$	0	0
$579$	− 9340.00i	− 0.670392i
$580$	0	0
$581$	− 5892.00i	− 0.420725i
$582$	0	0
$583$	29400.0	2.08855
$584$	0	0
$585$	−5750.00	−0.406382
$586$	0	0
$587$	− 966.000i	− 0.0679235i	−0.999423	−	0.0339617i	$-0.989188\pi$
$587$	− 966.000i	− 0.0679235i	0.999423	−	0.0339617i	$-0.0108124\pi$
$588$	0	0
$589$	800.000i	0.0559651i
$590$	0	0
$591$	−5980.00	−0.416217
$592$	0	0
$593$	26290.0	1.82057	0.910287	−	0.413977i	$-0.135861\pi$
$593$	26290.0	1.82057	0.910287	+	0.413977i	$0.135861\pi$
$594$	0	0
$595$	− 900.000i	− 0.0620108i
$596$	0	0
$597$	− 8480.00i	− 0.581346i
$598$	0	0
$599$	−11640.0	−0.793986	−0.396993	−	0.917822i	$-0.629946\pi$
$599$	−11640.0	−0.793986	−0.396993	+	0.917822i	$0.629946\pi$
$600$	0	0
$601$	25450.0	1.72733	0.863667	−	0.504064i	$-0.168162\pi$
$601$	25450.0	1.72733	0.863667	+	0.504064i	$0.168162\pi$
$602$	0	0
$603$	17802.0i	1.20224i
$604$	0	0
$605$	− 11345.0i	− 0.762380i
$606$	0	0
$607$	−16694.0	−1.11629	−0.558145	−	0.829743i	$-0.688487\pi$
$607$	−16694.0	−1.11629	−0.558145	+	0.829743i	$0.688487\pi$
$608$	0	0
$609$	−1992.00	−0.132545
$610$	0	0
$611$	− 10700.0i	− 0.708471i
$612$	0	0
$613$	− 15890.0i	− 1.04697i	−0.852036	−	0.523484i	$-0.824632\pi$
$613$	− 15890.0i	− 1.04697i	0.852036	−	0.523484i	$-0.175368\pi$
$614$	0	0
$615$	−2500.00	−0.163918
$616$	0	0
$617$	1230.00	0.0802560	0.0401280	−	0.999195i	$-0.487223\pi$
$617$	1230.00	0.0802560	0.0401280	+	0.999195i	$0.487223\pi$
$618$	0	0
$619$	− 10840.0i	− 0.703871i	−0.936024	−	0.351936i	$-0.885524\pi$
$619$	− 10840.0i	− 0.703871i	0.936024	−	0.351936i	$-0.114476\pi$
$620$	0	0
$621$	17800.0i	1.15022i
$622$	0	0
$623$	−5244.00	−0.337233
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	4800.00i	0.305731i
$628$	0	0
$629$	300.000i	0.0190171i
$630$	0	0
$631$	−14060.0	−0.887036	−0.443518	−	0.896265i	$-0.646270\pi$
$631$	−14060.0	−0.887036	−0.443518	+	0.896265i	$0.646270\pi$
$632$	0	0
$633$	−8120.00	−0.509859
$634$	0	0
$635$	− 6230.00i	− 0.389339i
$636$	0	0
$637$	− 15350.0i	− 0.954771i
$638$	0	0
$639$	2300.00	0.142389
$640$	0	0
$641$	−17650.0	−1.08757	−0.543786	−	0.839224i	$-0.683010\pi$
$641$	−17650.0	−1.08757	−0.543786	+	0.839224i	$0.683010\pi$
$642$	0	0
$643$	− 27358.0i	− 1.67791i	−0.544203	−	0.838953i	$-0.683168\pi$
$643$	− 27358.0i	− 1.67791i	0.544203	−	0.838953i	$-0.316832\pi$
$644$	0	0
$645$	− 1420.00i	− 0.0866860i
$646$	0	0
$647$	−6786.00	−0.412342	−0.206171	−	0.978516i	$-0.566100\pi$
$647$	−6786.00	−0.412342	−0.206171	+	0.978516i	$0.566100\pi$
$648$	0	0
$649$	48000.0	2.90318
$650$	0	0
$651$	− 240.000i	− 0.0144491i
$652$	0	0
$653$	− 9030.00i	− 0.541150i	−0.962699	−	0.270575i	$-0.912786\pi$
$653$	− 9030.00i	− 0.541150i	0.962699	−	0.270575i	$-0.0872139\pi$
$654$	0	0
$655$	13300.0	0.793395
$656$	0	0
$657$	5290.00	0.314129
$658$	0	0
$659$	15600.0i	0.922139i	0.887364	+	0.461070i	$0.152534\pi$
$659$	15600.0i	0.922139i	−0.887364	+	0.461070i	$0.847466\pi$
$660$	0	0
$661$	− 16850.0i	− 0.991511i	−0.868462	−	0.495756i	$-0.834891\pi$
$661$	− 16850.0i	− 0.991511i	0.868462	−	0.495756i	$-0.165109\pi$
$662$	0	0
$663$	3000.00	0.175732
$664$	0	0
$665$	1200.00	0.0699759
$666$	0	0
$667$	29548.0i	1.71530i
$668$	0	0
$669$	11244.0i	0.649803i
$670$	0	0
$671$	−15000.0	−0.862993
$672$	0	0
$673$	−7990.00	−0.457640	−0.228820	−	0.973469i	$-0.573487\pi$
$673$	−7990.00	−0.457640	−0.228820	+	0.973469i	$0.573487\pi$
$674$	0	0
$675$	− 2500.00i	− 0.142556i
$676$	0	0
$677$	− 18690.0i	− 1.06103i	−0.847677	−	0.530513i	$-0.821999\pi$
$677$	− 18690.0i	− 1.06103i	0.847677	−	0.530513i	$-0.178001\pi$
$678$	0	0
$679$	−1860.00	−0.105126
$680$	0	0
$681$	3108.00	0.174888
$682$	0	0
$683$	19182.0i	1.07464i	0.843379	+	0.537320i	$0.180563\pi$
$683$	19182.0i	1.07464i	−0.843379	+	0.537320i	$0.819437\pi$
$684$	0	0
$685$	− 13850.0i	− 0.772527i
$686$	0	0
$687$	2268.00	0.125953
$688$	0	0
$689$	24500.0	1.35468
$690$	0	0
$691$	23380.0i	1.28714i	0.765385	+	0.643572i	$0.222548\pi$
$691$	23380.0i	1.28714i	−0.765385	+	0.643572i	$0.777452\pi$
$692$	0	0
$693$	8280.00i	0.453869i
$694$	0	0
$695$	2800.00	0.152820
$696$	0	0
$697$	−7500.00	−0.407579
$698$	0	0
$699$	3420.00i	0.185059i
$700$	0	0
$701$	11850.0i	0.638471i	0.947675	+	0.319236i	$0.103426\pi$
$701$	11850.0i	0.638471i	−0.947675	+	0.319236i	$0.896574\pi$
$702$	0	0
$703$	−400.000	−0.0214599
$704$	0	0
$705$	2140.00	0.114322
$706$	0	0
$707$	8988.00i	0.478117i
$708$	0	0
$709$	− 25646.0i	− 1.35847i	−0.733921	−	0.679235i	$-0.762312\pi$
$709$	− 25646.0i	− 1.35847i	0.733921	−	0.679235i	$-0.237688\pi$
$710$	0	0
$711$	30360.0	1.60139
$712$	0	0
$713$	−3560.00	−0.186989
$714$	0	0
$715$	− 15000.0i	− 0.784571i
$716$	0	0
$717$	− 8880.00i	− 0.462524i
$718$	0	0
$719$	30280.0	1.57059	0.785294	−	0.619122i	$-0.212512\pi$
$719$	30280.0	1.57059	0.785294	+	0.619122i	$0.212512\pi$
$720$	0	0
$721$	8412.00	0.434507
$722$	0	0
$723$	− 1700.00i	− 0.0874463i
$724$	0	0
$725$	− 4150.00i	− 0.212589i
$726$	0	0
$727$	17446.0	0.890009	0.445004	−	0.895528i	$-0.353202\pi$
$727$	17446.0	0.890009	0.445004	+	0.895528i	$0.353202\pi$
$728$	0	0
$729$	4283.00	0.217599
$730$	0	0
$731$	− 4260.00i	− 0.215543i
$732$	0	0
$733$	− 16750.0i	− 0.844032i	−0.906588	−	0.422016i	$-0.861323\pi$
$733$	− 16750.0i	− 0.844032i	0.906588	−	0.422016i	$-0.138677\pi$
$734$	0	0
$735$	3070.00	0.154066
$736$	0	0
$737$	−46440.0	−2.32108
$738$	0	0
$739$	36560.0i	1.81987i	0.414755	+	0.909933i	$0.363867\pi$
$739$	36560.0i	1.81987i	−0.414755	+	0.909933i	$0.636133\pi$
$740$	0	0
$741$	4000.00i	0.198305i
$742$	0	0
$743$	−30142.0	−1.48829	−0.744147	−	0.668016i	$-0.767144\pi$
$743$	−30142.0	−1.48829	−0.744147	+	0.668016i	$0.767144\pi$
$744$	0	0
$745$	−11750.0	−0.577834
$746$	0	0
$747$	− 22586.0i	− 1.10626i
$748$	0	0
$749$	7164.00i	0.349488i
$750$	0	0
$751$	−11860.0	−0.576268	−0.288134	−	0.957590i	$-0.593035\pi$
$751$	−11860.0	−0.576268	−0.288134	+	0.957590i	$0.593035\pi$
$752$	0	0
$753$	1320.00	0.0638824
$754$	0	0
$755$	2900.00i	0.139790i
$756$	0	0
$757$	− 37010.0i	− 1.77695i	−0.458925	−	0.888475i	$-0.651765\pi$
$757$	− 37010.0i	− 1.77695i	0.458925	−	0.888475i	$-0.348235\pi$
$758$	0	0
$759$	−21360.0	−1.02150
$760$	0	0
$761$	11718.0	0.558183	0.279091	−	0.960265i	$-0.409967\pi$
$761$	11718.0	0.558183	0.279091	+	0.960265i	$0.409967\pi$
$762$	0	0
$763$	3900.00i	0.185045i
$764$	0	0
$765$	− 3450.00i	− 0.163052i
$766$	0	0
$767$	40000.0	1.88307
$768$	0	0
$769$	4706.00	0.220680	0.110340	−	0.993894i	$-0.464806\pi$
$769$	4706.00	0.220680	0.110340	+	0.993894i	$0.464806\pi$
$770$	0	0
$771$	− 15180.0i	− 0.709072i
$772$	0	0
$773$	28670.0i	1.33401i	0.745054	+	0.667004i	$0.232424\pi$
$773$	28670.0i	1.33401i	−0.745054	+	0.667004i	$0.767576\pi$
$774$	0	0
$775$	500.000	0.0231749
$776$	0	0
$777$	120.000	0.00554051
$778$	0	0
$779$	− 10000.0i	− 0.459932i
$780$	0	0
$781$	6000.00i	0.274900i
$782$	0	0
$783$	−16600.0	−0.757644
$784$	0	0
$785$	6550.00	0.297808
$786$	0	0
$787$	− 20434.0i	− 0.925532i	−0.886481	−	0.462766i	$-0.846857\pi$
$787$	− 20434.0i	− 0.925532i	0.886481	−	0.462766i	$-0.153143\pi$
$788$	0	0
$789$	1524.00i	0.0687653i
$790$	0	0
$791$	−9060.00	−0.407252
$792$	0	0
$793$	−12500.0	−0.559758
$794$	0	0
$795$	4900.00i	0.218598i
$796$	0	0
$797$	3930.00i	0.174665i	0.996179	+	0.0873323i	$0.0278342\pi$
$797$	3930.00i	0.174665i	−0.996179	+	0.0873323i	$0.972166\pi$
$798$	0	0
$799$	6420.00	0.284259
$800$	0	0
$801$	−20102.0	−0.886728
$802$	0	0
$803$	13800.0i	0.606465i
$804$	0	0
$805$	5340.00i	0.233802i
$806$	0	0
$807$	300.000	0.0130861
$808$	0	0
$809$	4854.00	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	4854.00	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	− 13140.0i	− 0.568937i	−0.958685	−	0.284468i	$-0.908183\pi$
$811$	− 13140.0i	− 0.568937i	0.958685	−	0.284468i	$-0.0918171\pi$
$812$	0	0
$813$	13160.0i	0.567702i
$814$	0	0
$815$	9310.00	0.400141
$816$	0	0
$817$	5680.00	0.243229
$818$	0	0
$819$	6900.00i	0.294390i
$820$	0	0
$821$	− 22050.0i	− 0.937333i	−0.883375	−	0.468666i	$-0.844735\pi$
$821$	− 22050.0i	− 0.937333i	0.883375	−	0.468666i	$-0.155265\pi$
$822$	0	0
$823$	14578.0	0.617445	0.308722	−	0.951152i	$-0.400099\pi$
$823$	14578.0	0.617445	0.308722	+	0.951152i	$0.400099\pi$
$824$	0	0
$825$	3000.00	0.126602
$826$	0	0
$827$	− 37054.0i	− 1.55803i	−0.627003	−	0.779017i	$-0.715719\pi$
$827$	− 37054.0i	− 1.55803i	0.627003	−	0.779017i	$-0.284281\pi$
$828$	0	0
$829$	− 6150.00i	− 0.257658i	−0.991667	−	0.128829i	$-0.958878\pi$
$829$	− 6150.00i	− 0.257658i	0.991667	−	0.128829i	$-0.0411218\pi$
$830$	0	0
$831$	9060.00	0.378204
$832$	0	0
$833$	9210.00	0.383082
$834$	0	0
$835$	3630.00i	0.150445i
$836$	0	0
$837$	− 2000.00i	− 0.0825927i
$838$	0	0
$839$	−8200.00	−0.337420	−0.168710	−	0.985666i	$-0.553960\pi$
$839$	−8200.00	−0.337420	−0.168710	+	0.985666i	$0.553960\pi$
$840$	0	0
$841$	−3167.00	−0.129854
$842$	0	0
$843$	− 13900.0i	− 0.567902i
$844$	0	0
$845$	− 1515.00i	− 0.0616776i
$846$	0	0
$847$	−13614.0	−0.552282
$848$	0	0
$849$	7764.00	0.313851
$850$	0	0
$851$	− 1780.00i	− 0.0717011i
$852$	0	0
$853$	42990.0i	1.72561i	0.505533	+	0.862807i	$0.331296\pi$
$853$	42990.0i	1.72561i	−0.505533	+	0.862807i	$0.668704\pi$
$854$	0	0
$855$	4600.00	0.183996
$856$	0	0
$857$	−32130.0	−1.28068	−0.640338	−	0.768093i	$-0.721206\pi$
$857$	−32130.0	−1.28068	−0.640338	+	0.768093i	$0.721206\pi$
$858$	0	0
$859$	15440.0i	0.613278i	0.951826	+	0.306639i	$0.0992045\pi$
$859$	15440.0i	0.613278i	−0.951826	+	0.306639i	$0.900796\pi$
$860$	0	0
$861$	3000.00i	0.118745i
$862$	0	0
$863$	−46938.0	−1.85143	−0.925717	−	0.378216i	$-0.876538\pi$
$863$	−46938.0	−1.85143	−0.925717	+	0.378216i	$0.876538\pi$
$864$	0	0
$865$	−16250.0	−0.638747
$866$	0	0
$867$	− 8026.00i	− 0.314391i
$868$	0	0
$869$	79200.0i	3.09169i
$870$	0	0
$871$	−38700.0	−1.50551
$872$	0	0
$873$	−7130.00	−0.276419
$874$	0	0
$875$	− 750.000i	− 0.0289767i
$876$	0	0
$877$	− 31230.0i	− 1.20247i	−0.799074	−	0.601233i	$-0.794676\pi$
$877$	− 31230.0i	− 1.20247i	0.799074	−	0.601233i	$-0.205324\pi$
$878$	0	0
$879$	2740.00	0.105140
$880$	0	0
$881$	25550.0	0.977073	0.488537	−	0.872543i	$-0.337531\pi$
$881$	25550.0	0.977073	0.488537	+	0.872543i	$0.337531\pi$
$882$	0	0
$883$	− 4318.00i	− 0.164567i	−0.996609	−	0.0822833i	$-0.973779\pi$
$883$	− 4318.00i	− 0.164567i	0.996609	−	0.0822833i	$-0.0262212\pi$
$884$	0	0
$885$	8000.00i	0.303861i
$886$	0	0
$887$	1766.00	0.0668506	0.0334253	−	0.999441i	$-0.489358\pi$
$887$	1766.00	0.0668506	0.0334253	+	0.999441i	$0.489358\pi$
$888$	0	0
$889$	−7476.00	−0.282044
$890$	0	0
$891$	25260.0i	0.949766i
$892$	0	0
$893$	8560.00i	0.320772i
$894$	0	0
$895$	−5600.00	−0.209148
$896$	0	0
$897$	−17800.0	−0.662569
$898$	0	0
$899$	− 3320.00i	− 0.123168i
$900$	0	0
$901$	14700.0i	0.543538i
$902$	0	0
$903$	−1704.00	−0.0627969
$904$	0	0
$905$	−14210.0	−0.521941
$906$	0	0
$907$	41906.0i	1.53414i	0.641563	+	0.767071i	$0.278286\pi$
$907$	41906.0i	1.53414i	−0.641563	+	0.767071i	$0.721714\pi$
$908$	0	0
$909$	34454.0i	1.25717i
$910$	0	0
$911$	−25140.0	−0.914298	−0.457149	−	0.889390i	$-0.651129\pi$
$911$	−25140.0	−0.914298	−0.457149	+	0.889390i	$0.651129\pi$
$912$	0	0
$913$	58920.0	2.13578
$914$	0	0
$915$	− 2500.00i	− 0.0903251i
$916$	0	0
$917$	− 15960.0i	− 0.574750i
$918$	0	0
$919$	32920.0	1.18164	0.590822	−	0.806802i	$-0.298804\pi$
$919$	32920.0	1.18164	0.590822	+	0.806802i	$0.298804\pi$
$920$	0	0
$921$	8212.00	0.293805
$922$	0	0
$923$	5000.00i	0.178307i
$924$	0	0
$925$	250.000i	0.00888643i
$926$	0	0
$927$	32246.0	1.14250
$928$	0	0
$929$	10150.0	0.358461	0.179231	−	0.983807i	$-0.442639\pi$
$929$	10150.0	0.358461	0.179231	+	0.983807i	$0.442639\pi$
$930$	0	0
$931$	12280.0i	0.432289i
$932$	0	0
$933$	4440.00i	0.155798i
$934$	0	0
$935$	9000.00	0.314793
$936$	0	0
$937$	−28530.0	−0.994701	−0.497350	−	0.867550i	$-0.665694\pi$
$937$	−28530.0	−0.994701	−0.497350	+	0.867550i	$0.665694\pi$
$938$	0	0
$939$	18860.0i	0.655456i
$940$	0	0
$941$	9678.00i	0.335275i	0.985849	+	0.167638i	$0.0536138\pi$
$941$	9678.00i	0.335275i	−0.985849	+	0.167638i	$0.946386\pi$
$942$	0	0
$943$	44500.0	1.53671
$944$	0	0
$945$	−3000.00	−0.103270
$946$	0	0
$947$	− 36986.0i	− 1.26915i	−0.772862	−	0.634574i	$-0.781176\pi$
$947$	− 36986.0i	− 1.26915i	0.772862	−	0.634574i	$-0.218824\pi$
$948$	0	0
$949$	11500.0i	0.393368i
$950$	0	0
$951$	12940.0	0.441228
$952$	0	0
$953$	3350.00	0.113869	0.0569345	−	0.998378i	$-0.481867\pi$
$953$	3350.00	0.113869	0.0569345	+	0.998378i	$0.481867\pi$
$954$	0	0
$955$	− 15900.0i	− 0.538756i
$956$	0	0
$957$	− 19920.0i	− 0.672855i
$958$	0	0
$959$	−16620.0	−0.559633
$960$	0	0
$961$	−29391.0	−0.986573
$962$	0	0
$963$	27462.0i	0.918952i
$964$	0	0
$965$	− 23350.0i	− 0.778925i
$966$	0	0
$967$	43774.0	1.45572	0.727858	−	0.685728i	$-0.240516\pi$
$967$	43774.0	1.45572	0.727858	+	0.685728i	$0.240516\pi$
$968$	0	0
$969$	−2400.00	−0.0795656
$970$	0	0
$971$	8740.00i	0.288857i	0.989515	+	0.144428i	$0.0461343\pi$
$971$	8740.00i	0.288857i	−0.989515	+	0.144428i	$0.953866\pi$
$972$	0	0
$973$	− 3360.00i	− 0.110706i
$974$	0	0
$975$	2500.00	0.0821170
$976$	0	0
$977$	−48310.0	−1.58196	−0.790979	−	0.611843i	$-0.790429\pi$
$977$	−48310.0	−1.58196	−0.790979	+	0.611843i	$0.790429\pi$
$978$	0	0
$979$	− 52440.0i	− 1.71194i
$980$	0	0
$981$	14950.0i	0.486561i
$982$	0	0
$983$	2282.00	0.0740432	0.0370216	−	0.999314i	$-0.488213\pi$
$983$	2282.00	0.0740432	0.0370216	+	0.999314i	$0.488213\pi$
$984$	0	0
$985$	−14950.0	−0.483601
$986$	0	0
$987$	− 2568.00i	− 0.0828170i
$988$	0	0
$989$	25276.0i	0.812669i
$990$	0	0
$991$	31580.0	1.01228	0.506141	−	0.862451i	$-0.331071\pi$
$991$	31580.0	1.01228	0.506141	+	0.862451i	$0.331071\pi$
$992$	0	0
$993$	−1800.00	−0.0575239
$994$	0	0
$995$	− 21200.0i	− 0.675462i
$996$	0	0
$997$	2790.00i	0.0886261i	0.999018	+	0.0443130i	$0.0141099\pi$
$997$	2790.00i	0.0886261i	−0.999018	+	0.0443130i	$0.985890\pi$
$998$	0	0
$999$	1000.00	0.0316703

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.4.d.k.641.2		2
4.3	odd	2		1280.4.d.f.641.1		2
8.3	odd	2		1280.4.d.f.641.2		2
8.5	even	2	inner	1280.4.d.k.641.1		2
16.3	odd	4		320.4.a.i.1.1		1
16.5	even	4		160.4.a.b.1.1	yes	1
16.11	odd	4		160.4.a.a.1.1	✓	1
16.13	even	4		320.4.a.f.1.1		1
48.5	odd	4		1440.4.a.n.1.1		1
48.11	even	4		1440.4.a.o.1.1		1
80.19	odd	4		1600.4.a.r.1.1		1
80.27	even	4		800.4.c.f.449.2		2
80.29	even	4		1600.4.a.bj.1.1		1
80.37	odd	4		800.4.c.e.449.1		2
80.43	even	4		800.4.c.f.449.1		2
80.53	odd	4		800.4.c.e.449.2		2
80.59	odd	4		800.4.a.h.1.1		1
80.69	even	4		800.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.a.a.1.1	✓	1	16.11	odd	4
160.4.a.b.1.1	yes	1	16.5	even	4
320.4.a.f.1.1		1	16.13	even	4
320.4.a.i.1.1		1	16.3	odd	4
800.4.a.d.1.1		1	80.69	even	4
800.4.a.h.1.1		1	80.59	odd	4
800.4.c.e.449.1		2	80.37	odd	4
800.4.c.e.449.2		2	80.53	odd	4
800.4.c.f.449.1		2	80.43	even	4
800.4.c.f.449.2		2	80.27	even	4
1280.4.d.f.641.1		2	4.3	odd	2
1280.4.d.f.641.2		2	8.3	odd	2
1280.4.d.k.641.1		2	8.5	even	2	inner
1280.4.d.k.641.2		2	1.1	even	1	trivial
1440.4.a.n.1.1		1	48.5	odd	4
1440.4.a.o.1.1		1	48.11	even	4
1600.4.a.r.1.1		1	80.19	odd	4
1600.4.a.bj.1.1		1	80.29	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1280.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} +5.00000i q^{5} +6.00000 q^{7} +23.0000 q^{9} +60.0000i q^{11} +50.0000i q^{13} -10.0000 q^{15} -30.0000 q^{17} -40.0000i q^{19} +12.0000i q^{21} +178.000 q^{23} -25.0000 q^{25} +100.000i q^{27} +166.000i q^{29} -20.0000 q^{31} -120.000 q^{33} +30.0000i q^{35} -10.0000i q^{37} -100.000 q^{39} +250.000 q^{41} +142.000i q^{43} +115.000i q^{45} -214.000 q^{47} -307.000 q^{49} -60.0000i q^{51} -490.000i q^{53} -300.000 q^{55} +80.0000 q^{57} -800.000i q^{59} +250.000i q^{61} +138.000 q^{63} -250.000 q^{65} +774.000i q^{67} +356.000i q^{69} +100.000 q^{71} +230.000 q^{73} -50.0000i q^{75} +360.000i q^{77} +1320.00 q^{79} +421.000 q^{81} -982.000i q^{83} -150.000i q^{85} -332.000 q^{87} -874.000 q^{89} +300.000i q^{91} -40.0000i q^{93} +200.000 q^{95} -310.000 q^{97} +1380.00i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 12 q^{7} + 46 q^{9} - 20 q^{15} - 60 q^{17} + 356 q^{23} - 50 q^{25} - 40 q^{31} - 240 q^{33} - 200 q^{39} + 500 q^{41} - 428 q^{47} - 614 q^{49} - 600 q^{55} + 160 q^{57} + 276 q^{63} - 500 q^{65}+ \cdots - 620 q^{97}+O(q^{100}) \) 2 * q + 12 * q^7 + 46 * q^9 - 20 * q^15 - 60 * q^17 + 356 * q^23 - 50 * q^25 - 40 * q^31 - 240 * q^33 - 200 * q^39 + 500 * q^41 - 428 * q^47 - 614 * q^49 - 600 * q^55 + 160 * q^57 + 276 * q^63 - 500 * q^65 + 200 * q^71 + 460 * q^73 + 2640 * q^79 + 842 * q^81 - 664 * q^87 - 1748 * q^89 + 400 * q^95 - 620 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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