LMFDB - Embedded newform 800.4.c.f.449.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [800,4,Mod(449,800)] 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("800.449"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	800.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,46,0,120] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$47.2015280046$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	800.449
Dual form			800.4.c.f.449.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} -6.00000i q^{7} +23.0000 q^{9} +60.0000 q^{11} +50.0000i q^{13} +30.0000i q^{17} -40.0000 q^{19} -12.0000 q^{21} +178.000i q^{23} -100.000i q^{27} -166.000 q^{29} +20.0000 q^{31} -120.000i q^{33} -10.0000i q^{37} +100.000 q^{39} -250.000 q^{41} +142.000i q^{43} -214.000i q^{47} +307.000 q^{49} +60.0000 q^{51} +490.000i q^{53} +80.0000i q^{57} +800.000 q^{59} +250.000 q^{61} -138.000i q^{63} +774.000i q^{67} +356.000 q^{69} +100.000 q^{71} -230.000i q^{73} -360.000i q^{77} +1320.00 q^{79} +421.000 q^{81} +982.000i q^{83} +332.000i q^{87} -874.000 q^{89} +300.000 q^{91} -40.0000i q^{93} +310.000i q^{97} +1380.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 46 q^{9} + 120 q^{11} - 80 q^{19} - 24 q^{21} - 332 q^{29} + 40 q^{31} + 200 q^{39} - 500 q^{41} + 614 q^{49} + 120 q^{51} + 1600 q^{59} + 500 q^{61} + 712 q^{69} + 200 q^{71} + 2640 q^{79} + 842 q^{81}+ \cdots + 2760 q^{99}+O(q^{100}) $ 2 * q + 46 * q^9 + 120 * q^11 - 80 * q^19 - 24 * q^21 - 332 * q^29 + 40 * q^31 + 200 * q^39 - 500 * q^41 + 614 * q^49 + 120 * q^51 + 1600 * q^59 + 500 * q^61 + 712 * q^69 + 200 * q^71 + 2640 * q^79 + 842 * q^81 - 1748 * q^89 + 600 * q^91 + 2760 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$1$	$1$	$-1$

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	$-0.938357\pi$
$3$	− 2.00000i	− 0.384900i	0.981307	−	0.192450i	$-0.0616434\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 6.00000i	− 0.323970i	−0.986793	−	0.161985i	$-0.948210\pi$
$7$	− 6.00000i	− 0.323970i	0.986793	−	0.161985i	$-0.0517895\pi$
$8$	0	0
$9$	23.0000	0.851852
$10$	0	0
$11$	60.0000	1.64461	0.822304	−	0.569049i	$-0.192689\pi$
$11$	60.0000	1.64461	0.822304	+	0.569049i	$0.192689\pi$
$12$	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
$15$	0	0
$16$	0	0
$17$	30.0000i	0.428004i	0.976833	+	0.214002i	$0.0686499\pi$
$17$	30.0000i	0.428004i	−0.976833	+	0.214002i	$0.931350\pi$
$18$	0	0
$19$	−40.0000	−0.482980	−0.241490	−	0.970403i	$-0.577636\pi$
$19$	−40.0000	−0.482980	−0.241490	+	0.970403i	$0.577636\pi$
$20$	0	0
$21$	−12.0000	−0.124696
$22$	0	0
$23$	178.000i	1.61372i	0.590743	+	0.806860i	$0.298835\pi$
$23$	178.000i	1.61372i	−0.590743	+	0.806860i	$0.701165\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 100.000i	− 0.712778i
$28$	0	0
$29$	−166.000	−1.06295	−0.531473	−	0.847075i	$-0.678361\pi$
$29$	−166.000	−1.06295	−0.531473	+	0.847075i	$0.678361\pi$
$30$	0	0
$31$	20.0000	0.115874	0.0579372	−	0.998320i	$-0.481548\pi$
$31$	20.0000	0.115874	0.0579372	+	0.998320i	$0.481548\pi$
$32$	0	0
$33$	− 120.000i	− 0.633010i
$34$	0	0
$35$	0	0
$36$	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	$-0.657964\pi$
$41$	−250.000	−0.952279	−0.476140	+	0.879370i	$0.657964\pi$
$42$	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
$45$	0	0
$46$	0	0
$47$	− 214.000i	− 0.664151i	−0.943253	−	0.332076i	$-0.892251\pi$
$47$	− 214.000i	− 0.664151i	0.943253	−	0.332076i	$-0.107749\pi$
$48$	0	0
$49$	307.000	0.895044
$50$	0	0
$51$	60.0000	0.164739
$52$	0	0
$53$	490.000i	1.26994i	0.772538	+	0.634969i	$0.218987\pi$
$53$	490.000i	1.26994i	−0.772538	+	0.634969i	$0.781013\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	80.0000i	0.185899i
$58$	0	0
$59$	800.000	1.76527	0.882637	−	0.470056i	$-0.155766\pi$
$59$	800.000	1.76527	0.882637	+	0.470056i	$0.155766\pi$
$60$	0	0
$61$	250.000	0.524741	0.262371	−	0.964967i	$-0.415496\pi$
$61$	250.000	0.524741	0.262371	+	0.964967i	$0.415496\pi$
$62$	0	0
$63$	− 138.000i	− 0.275974i
$64$	0	0
$65$	0	0
$66$	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
$69$	356.000	0.621121
$70$	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
$73$	− 230.000i	− 0.368760i	−0.982855	−	0.184380i	$-0.940972\pi$
$73$	− 230.000i	− 0.368760i	0.982855	−	0.184380i	$-0.0590277\pi$
$74$	0	0
$75$	0	0
$76$	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	$0.224950\pi$
$83$	982.000i	1.29866i	−0.760508	+	0.649328i	$0.775050\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	332.000i	0.409128i
$88$	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
$91$	300.000	0.345588
$92$	0	0
$93$	− 40.0000i	− 0.0446001i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	310.000i	0.324492i	0.986750	+	0.162246i	$0.0518738\pi$
$97$	310.000i	0.324492i	−0.986750	+	0.162246i	$0.948126\pi$
$98$	0	0
$99$	1380.00	1.40096
$100$	0	0
$101$	−1498.00	−1.47581	−0.737904	−	0.674906i	$-0.764184\pi$
$101$	−1498.00	−1.47581	−0.737904	+	0.674906i	$0.764184\pi$
$102$	0	0
$103$	1402.00i	1.34120i	0.741821	+	0.670598i	$0.233962\pi$
$103$	1402.00i	1.34120i	−0.741821	+	0.670598i	$0.766038\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1194.00i	− 1.07877i	−0.842059	−	0.539385i	$-0.818657\pi$
$107$	− 1194.00i	− 1.07877i	0.842059	−	0.539385i	$-0.181343\pi$
$108$	0	0
$109$	−650.000	−0.571181	−0.285590	−	0.958352i	$-0.592190\pi$
$109$	−650.000	−0.571181	−0.285590	+	0.958352i	$0.592190\pi$
$110$	0	0
$111$	−20.0000	−0.0171019
$112$	0	0
$113$	− 1510.00i	− 1.25707i	−0.777782	−	0.628535i	$-0.783655\pi$
$113$	− 1510.00i	− 1.25707i	0.777782	−	0.628535i	$-0.216345\pi$
$114$	0	0
$115$	0	0
$116$	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
$125$	0	0
$126$	0	0
$127$	− 1246.00i	− 0.870588i	−0.900288	−	0.435294i	$-0.856645\pi$
$127$	− 1246.00i	− 0.870588i	0.900288	−	0.435294i	$-0.143355\pi$
$128$	0	0
$129$	284.000	0.193836
$130$	0	0
$131$	2660.00	1.77409	0.887043	−	0.461687i	$-0.152756\pi$
$131$	2660.00	1.77409	0.887043	+	0.461687i	$0.152756\pi$
$132$	0	0
$133$	240.000i	0.156471i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2770.00i	− 1.72742i	−0.503986	−	0.863712i	$-0.668134\pi$
$137$	− 2770.00i	− 1.72742i	0.503986	−	0.863712i	$-0.331866\pi$
$138$	0	0
$139$	560.000	0.341716	0.170858	−	0.985296i	$-0.445346\pi$
$139$	560.000	0.341716	0.170858	+	0.985296i	$0.445346\pi$
$140$	0	0
$141$	−428.000	−0.255632
$142$	0	0
$143$	3000.00i	1.75435i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 614.000i	− 0.344502i
$148$	0	0
$149$	2350.00	1.29208	0.646039	−	0.763305i	$-0.276424\pi$
$149$	2350.00	1.29208	0.646039	+	0.763305i	$0.276424\pi$
$150$	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
$153$	690.000i	0.364596i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1310.00i	0.665920i	0.942941	+	0.332960i	$0.108047\pi$
$157$	1310.00i	0.665920i	−0.942941	+	0.332960i	$0.891953\pi$
$158$	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	$0.147640\pi$
$163$	1862.00i	0.894743i	−0.894348	+	0.447371i	$0.852360\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 726.000i	− 0.336405i	−0.985752	−	0.168202i	$-0.946204\pi$
$167$	− 726.000i	− 0.336405i	0.985752	−	0.168202i	$-0.0537962\pi$
$168$	0	0
$169$	−303.000	−0.137915
$170$	0	0
$171$	−920.000	−0.411428
$172$	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
$175$	0	0
$176$	0	0
$177$	− 1600.00i	− 0.679454i
$178$	0	0
$179$	1120.00	0.467669	0.233834	−	0.972276i	$-0.424873\pi$
$179$	1120.00	0.467669	0.233834	+	0.972276i	$0.424873\pi$
$180$	0	0
$181$	−2842.00	−1.16710	−0.583548	−	0.812079i	$-0.698336\pi$
$181$	−2842.00	−1.16710	−0.583548	+	0.812079i	$0.698336\pi$
$182$	0	0
$183$	− 500.000i	− 0.201973i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1800.00i	0.703899i
$188$	0	0
$189$	−600.000	−0.230918
$190$	0	0
$191$	3180.00	1.20469	0.602347	−	0.798234i	$-0.294232\pi$
$191$	3180.00	1.20469	0.602347	+	0.798234i	$0.294232\pi$
$192$	0	0
$193$	− 4670.00i	− 1.74173i	−0.491522	−	0.870865i	$-0.663559\pi$
$193$	− 4670.00i	− 1.74173i	0.491522	−	0.870865i	$-0.336441\pi$
$194$	0	0
$195$	0	0
$196$	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	$-0.227545\pi$
$199$	4240.00	1.51038	0.755190	+	0.655506i	$0.227545\pi$
$200$	0	0
$201$	1548.00	0.543221
$202$	0	0
$203$	996.000i	0.344362i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4094.00i	1.37465i
$208$	0	0
$209$	−2400.00	−0.794313
$210$	0	0
$211$	−4060.00	−1.32465	−0.662327	−	0.749215i	$-0.730431\pi$
$211$	−4060.00	−1.32465	−0.662327	+	0.749215i	$0.730431\pi$
$212$	0	0
$213$	− 200.000i	− 0.0643370i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 120.000i	− 0.0375398i
$218$	0	0
$219$	−460.000	−0.141936
$220$	0	0
$221$	−1500.00	−0.456565
$222$	0	0
$223$	− 5622.00i	− 1.68824i	−0.536156	−	0.844119i	$-0.680124\pi$
$223$	− 5622.00i	− 1.68824i	0.536156	−	0.844119i	$-0.319876\pi$
$224$	0	0
$225$	0	0
$226$	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
$229$	−1134.00	−0.327235	−0.163618	−	0.986524i	$-0.552316\pi$
$229$	−1134.00	−0.327235	−0.163618	+	0.986524i	$0.552316\pi$
$230$	0	0
$231$	−720.000	−0.205076
$232$	0	0
$233$	− 1710.00i	− 0.480798i	−0.970674	−	0.240399i	$-0.922722\pi$
$233$	− 1710.00i	− 0.480798i	0.970674	−	0.240399i	$-0.0772782\pi$
$234$	0	0
$235$	0	0
$236$	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
$245$	0	0
$246$	0	0
$247$	− 2000.00i	− 0.515210i
$248$	0	0
$249$	1964.00	0.499853
$250$	0	0
$251$	−660.000	−0.165971	−0.0829857	−	0.996551i	$-0.526446\pi$
$251$	−660.000	−0.165971	−0.0829857	+	0.996551i	$0.526446\pi$
$252$	0	0
$253$	10680.0i	2.65394i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7590.00i	1.84222i	0.389299	+	0.921111i	$0.372717\pi$
$257$	7590.00i	1.84222i	−0.389299	+	0.921111i	$0.627283\pi$
$258$	0	0
$259$	−60.0000	−0.0143947
$260$	0	0
$261$	−3818.00	−0.905472
$262$	0	0
$263$	762.000i	0.178658i	0.996002	+	0.0893288i	$0.0284722\pi$
$263$	762.000i	0.178658i	−0.996002	+	0.0893288i	$0.971528\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1748.00i	0.400659i
$268$	0	0
$269$	150.000	0.0339987	0.0169994	−	0.999856i	$-0.494589\pi$
$269$	150.000	0.0339987	0.0169994	+	0.999856i	$0.494589\pi$
$270$	0	0
$271$	−6580.00	−1.47493	−0.737466	−	0.675384i	$-0.763978\pi$
$271$	−6580.00	−1.47493	−0.737466	+	0.675384i	$0.763978\pi$
$272$	0	0
$273$	− 600.000i	− 0.133017i
$274$	0	0
$275$	0	0
$276$	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	$-0.235899\pi$
$281$	6950.00	1.47545	0.737726	+	0.675100i	$0.235899\pi$
$282$	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
$285$	0	0
$286$	0	0
$287$	1500.00i	0.308509i
$288$	0	0
$289$	4013.00	0.816813
$290$	0	0
$291$	620.000	0.124897
$292$	0	0
$293$	1370.00i	0.273161i	0.990629	+	0.136581i	$0.0436113\pi$
$293$	1370.00i	0.273161i	−0.990629	+	0.136581i	$0.956389\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 6000.00i	− 1.17224i
$298$	0	0
$299$	−8900.00	−1.72141
$300$	0	0
$301$	852.000	0.163151
$302$	0	0
$303$	2996.00i	0.568039i
$304$	0	0
$305$	0	0
$306$	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
$309$	2804.00	0.516226
$310$	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
$313$	− 9430.00i	− 1.70292i	−0.524417	−	0.851462i	$-0.675717\pi$
$313$	− 9430.00i	− 1.70292i	0.524417	−	0.851462i	$-0.324283\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6470.00i	1.14635i	0.819435	+	0.573173i	$0.194288\pi$
$317$	6470.00i	1.14635i	−0.819435	+	0.573173i	$0.805712\pi$
$318$	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
$325$	0	0
$326$	0	0
$327$	1300.00i	0.219848i
$328$	0	0
$329$	−1284.00	−0.215165
$330$	0	0
$331$	900.000	0.149452	0.0747258	−	0.997204i	$-0.476192\pi$
$331$	900.000	0.149452	0.0747258	+	0.997204i	$0.476192\pi$
$332$	0	0
$333$	− 230.000i	− 0.0378496i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 530.000i	− 0.0856704i	−0.999082	−	0.0428352i	$-0.986361\pi$
$337$	− 530.000i	− 0.0856704i	0.999082	−	0.0428352i	$-0.0136390\pi$
$338$	0	0
$339$	−3020.00	−0.483846
$340$	0	0
$341$	1200.00	0.190568
$342$	0	0
$343$	− 3900.00i	− 0.613936i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	414.000i	0.0640481i	0.999487	+	0.0320240i	$0.0101953\pi$
$347$	414.000i	0.0640481i	−0.999487	+	0.0320240i	$0.989805\pi$
$348$	0	0
$349$	−8614.00	−1.32119	−0.660597	−	0.750741i	$-0.729697\pi$
$349$	−8614.00	−1.32119	−0.660597	+	0.750741i	$0.729697\pi$
$350$	0	0
$351$	5000.00	0.760343
$352$	0	0
$353$	− 2270.00i	− 0.342266i	−0.985248	−	0.171133i	$-0.945257\pi$
$353$	− 2270.00i	− 0.342266i	0.985248	−	0.171133i	$-0.0547428\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 360.000i	− 0.0533704i
$358$	0	0
$359$	−8080.00	−1.18787	−0.593936	−	0.804512i	$-0.702427\pi$
$359$	−8080.00	−1.18787	−0.593936	+	0.804512i	$0.702427\pi$
$360$	0	0
$361$	−5259.00	−0.766730
$362$	0	0
$363$	− 4538.00i	− 0.656152i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 2374.00i	− 0.337662i	−0.985645	−	0.168831i	$-0.946001\pi$
$367$	− 2374.00i	− 0.337662i	0.985645	−	0.168831i	$-0.0539991\pi$
$368$	0	0
$369$	−5750.00	−0.811201
$370$	0	0
$371$	2940.00	0.411421
$372$	0	0
$373$	1810.00i	0.251255i	0.992077	+	0.125628i	$0.0400945\pi$
$373$	1810.00i	0.251255i	−0.992077	+	0.125628i	$0.959906\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8300.00i	− 1.13388i
$378$	0	0
$379$	−8120.00	−1.10052	−0.550259	−	0.834994i	$-0.685471\pi$
$379$	−8120.00	−1.10052	−0.550259	+	0.834994i	$0.685471\pi$
$380$	0	0
$381$	−2492.00	−0.335089
$382$	0	0
$383$	− 11782.0i	− 1.57189i	−0.618299	−	0.785943i	$-0.712178\pi$
$383$	− 11782.0i	− 1.57189i	0.618299	−	0.785943i	$-0.287822\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3266.00i	0.428993i
$388$	0	0
$389$	4350.00	0.566976	0.283488	−	0.958976i	$-0.408508\pi$
$389$	4350.00	0.566976	0.283488	+	0.958976i	$0.408508\pi$
$390$	0	0
$391$	−5340.00	−0.690679
$392$	0	0
$393$	− 5320.00i	− 0.682846i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7470.00i	0.944354i	0.881504	+	0.472177i	$0.156532\pi$
$397$	7470.00i	0.944354i	−0.881504	+	0.472177i	$0.843468\pi$
$398$	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
$405$	0	0
$406$	0	0
$407$	− 600.000i	− 0.0730735i
$408$	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
$411$	−5540.00	−0.664886
$412$	0	0
$413$	− 4800.00i	− 0.571895i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1120.00i	− 0.131527i
$418$	0	0
$419$	1120.00	0.130586	0.0652931	−	0.997866i	$-0.479202\pi$
$419$	1120.00	0.130586	0.0652931	+	0.997866i	$0.479202\pi$
$420$	0	0
$421$	4850.00	0.561460	0.280730	−	0.959787i	$-0.409424\pi$
$421$	4850.00	0.561460	0.280730	+	0.959787i	$0.409424\pi$
$422$	0	0
$423$	− 4922.00i	− 0.565758i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1500.00i	− 0.170000i
$428$	0	0
$429$	6000.00	0.675251
$430$	0	0
$431$	−12580.0	−1.40593	−0.702967	−	0.711223i	$-0.748142\pi$
$431$	−12580.0	−1.40593	−0.702967	+	0.711223i	$0.748142\pi$
$432$	0	0
$433$	13130.0i	1.45725i	0.684915	+	0.728623i	$0.259839\pi$
$433$	13130.0i	1.45725i	−0.684915	+	0.728623i	$0.740161\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 7120.00i	− 0.779395i
$438$	0	0
$439$	−8560.00	−0.930630	−0.465315	−	0.885145i	$-0.654059\pi$
$439$	−8560.00	−0.930630	−0.465315	+	0.885145i	$0.654059\pi$
$440$	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
$445$	0	0
$446$	0	0
$447$	− 4700.00i	− 0.497321i
$448$	0	0
$449$	−2550.00	−0.268022	−0.134011	−	0.990980i	$-0.542786\pi$
$449$	−2550.00	−0.268022	−0.134011	+	0.990980i	$0.542786\pi$
$450$	0	0
$451$	−15000.0	−1.56613
$452$	0	0
$453$	− 1160.00i	− 0.120312i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6710.00i	0.686828i	0.939184	+	0.343414i	$0.111583\pi$
$457$	6710.00i	0.686828i	−0.939184	+	0.343414i	$0.888417\pi$
$458$	0	0
$459$	3000.00	0.305072
$460$	0	0
$461$	−14482.0	−1.46311	−0.731555	−	0.681782i	$-0.761205\pi$
$461$	−14482.0	−1.46311	−0.731555	+	0.681782i	$0.761205\pi$
$462$	0	0
$463$	162.000i	0.0162609i	0.999967	+	0.00813043i	$0.00258802\pi$
$463$	162.000i	0.0162609i	−0.999967	+	0.00813043i	$0.997412\pi$
$464$	0	0
$465$	0	0
$466$	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
$469$	4644.00	0.457228
$470$	0	0
$471$	2620.00	0.256313
$472$	0	0
$473$	8520.00i	0.828224i
$474$	0	0
$475$	0	0
$476$	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
$485$	0	0
$486$	0	0
$487$	9266.00i	0.862182i	0.902309	+	0.431091i	$0.141871\pi$
$487$	9266.00i	0.862182i	−0.902309	+	0.431091i	$0.858129\pi$
$488$	0	0
$489$	3724.00	0.344387
$490$	0	0
$491$	2860.00	0.262872	0.131436	−	0.991325i	$-0.458041\pi$
$491$	2860.00	0.262872	0.131436	+	0.991325i	$0.458041\pi$
$492$	0	0
$493$	− 4980.00i	− 0.454945i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 600.000i	− 0.0541523i
$498$	0	0
$499$	−7160.00	−0.642336	−0.321168	−	0.947022i	$-0.604075\pi$
$499$	−7160.00	−0.642336	−0.321168	+	0.947022i	$0.604075\pi$
$500$	0	0
$501$	−1452.00	−0.129482
$502$	0	0
$503$	− 1398.00i	− 0.123924i	−0.998079	−	0.0619620i	$-0.980264\pi$
$503$	− 1398.00i	− 0.123924i	0.998079	−	0.0619620i	$-0.0197357\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	606.000i	0.0530836i
$508$	0	0
$509$	−7446.00	−0.648405	−0.324203	−	0.945988i	$-0.605096\pi$
$509$	−7446.00	−0.648405	−0.324203	+	0.945988i	$0.605096\pi$
$510$	0	0
$511$	−1380.00	−0.119467
$512$	0	0
$513$	4000.00i	0.344258i
$514$	0	0
$515$	0	0
$516$	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	$-0.742889\pi$
$521$	−16438.0	−1.38227	−0.691134	+	0.722726i	$0.742889\pi$
$522$	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
$525$	0	0
$526$	0	0
$527$	600.000i	0.0495947i
$528$	0	0
$529$	−19517.0	−1.60409
$530$	0	0
$531$	18400.0	1.50375
$532$	0	0
$533$	− 12500.0i	− 1.01583i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 2240.00i	− 0.180006i
$538$	0	0
$539$	18420.0	1.47200
$540$	0	0
$541$	10878.0	0.864476	0.432238	−	0.901759i	$-0.357724\pi$
$541$	10878.0	0.864476	0.432238	+	0.901759i	$0.357724\pi$
$542$	0	0
$543$	5684.00i	0.449215i
$544$	0	0
$545$	0	0
$546$	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
$549$	5750.00	0.447002
$550$	0	0
$551$	6640.00	0.513382
$552$	0	0
$553$	− 7920.00i	− 0.609028i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 3690.00i	− 0.280701i	−0.990102	−	0.140350i	$-0.955177\pi$
$557$	− 3690.00i	− 0.280701i	0.990102	−	0.140350i	$-0.0448229\pi$
$558$	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	$-0.969429\pi$
$563$	− 2562.00i	− 0.191786i	0.995392	−	0.0958929i	$-0.0305706\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2526.00i	− 0.187094i
$568$	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
$571$	8260.00	0.605377	0.302688	−	0.953090i	$-0.402116\pi$
$571$	8260.00	0.605377	0.302688	+	0.953090i	$0.402116\pi$
$572$	0	0
$573$	− 6360.00i	− 0.463687i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16870.0i	1.21717i	0.793489	+	0.608585i	$0.208263\pi$
$577$	16870.0i	1.21717i	−0.793489	+	0.608585i	$0.791737\pi$
$578$	0	0
$579$	−9340.00	−0.670392
$580$	0	0
$581$	5892.00	0.420725
$582$	0	0
$583$	29400.0i	2.08855i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	966.000i	0.0679235i	0.999423	+	0.0339617i	$0.0108124\pi$
$587$	966.000i	0.0679235i	−0.999423	+	0.0339617i	$0.989188\pi$
$588$	0	0
$589$	−800.000	−0.0559651
$590$	0	0
$591$	5980.00	0.416217
$592$	0	0
$593$	26290.0i	1.82057i	0.413977	+	0.910287i	$0.364139\pi$
$593$	26290.0i	1.82057i	−0.413977	+	0.910287i	$0.635861\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 8480.00i	− 0.581346i
$598$	0	0
$599$	11640.0	0.793986	0.396993	−	0.917822i	$-0.370054\pi$
$599$	11640.0	0.793986	0.396993	+	0.917822i	$0.370054\pi$
$600$	0	0
$601$	−25450.0	−1.72733	−0.863667	−	0.504064i	$-0.831838\pi$
$601$	−25450.0	−1.72733	−0.863667	+	0.504064i	$0.831838\pi$
$602$	0	0
$603$	17802.0i	1.20224i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 16694.0i	− 1.11629i	−0.829743	−	0.558145i	$-0.811513\pi$
$607$	− 16694.0i	− 1.11629i	0.829743	−	0.558145i	$-0.188487\pi$
$608$	0	0
$609$	1992.00	0.132545
$610$	0	0
$611$	10700.0	0.708471
$612$	0	0
$613$	15890.0i	1.04697i	0.852036	+	0.523484i	$0.175368\pi$
$613$	15890.0i	1.04697i	−0.852036	+	0.523484i	$0.824632\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1230.00i	0.0802560i	0.999195	+	0.0401280i	$0.0127766\pi$
$617$	1230.00i	0.0802560i	−0.999195	+	0.0401280i	$0.987223\pi$
$618$	0	0
$619$	10840.0	0.703871	0.351936	−	0.936024i	$-0.385524\pi$
$619$	10840.0	0.703871	0.351936	+	0.936024i	$0.385524\pi$
$620$	0	0
$621$	17800.0	1.15022
$622$	0	0
$623$	5244.00i	0.337233i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4800.00i	0.305731i
$628$	0	0
$629$	300.000	0.0190171
$630$	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
$633$	8120.00i	0.509859i
$634$	0	0
$635$	0	0
$636$	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	$0.316832\pi$
$643$	27358.0i	1.67791i	−0.544203	+	0.838953i	$0.683168\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6786.00i	0.412342i	0.978516	+	0.206171i	$0.0661003\pi$
$647$	6786.00i	0.412342i	−0.978516	+	0.206171i	$0.933900\pi$
$648$	0	0
$649$	48000.0	2.90318
$650$	0	0
$651$	−240.000	−0.0144491
$652$	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
$655$	0	0
$656$	0	0
$657$	− 5290.00i	− 0.314129i
$658$	0	0
$659$	15600.0	0.922139	0.461070	−	0.887364i	$-0.347466\pi$
$659$	15600.0	0.922139	0.461070	+	0.887364i	$0.347466\pi$
$660$	0	0
$661$	16850.0	0.991511	0.495756	−	0.868462i	$-0.334891\pi$
$661$	16850.0	0.991511	0.495756	+	0.868462i	$0.334891\pi$
$662$	0	0
$663$	3000.00i	0.175732i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 29548.0i	− 1.71530i
$668$	0	0
$669$	−11244.0	−0.649803
$670$	0	0
$671$	15000.0	0.862993
$672$	0	0
$673$	− 7990.00i	− 0.457640i	−0.973469	−	0.228820i	$-0.926513\pi$
$673$	− 7990.00i	− 0.457640i	0.973469	−	0.228820i	$-0.0734868\pi$
$674$	0	0
$675$	0	0
$676$	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
$685$	0	0
$686$	0	0
$687$	2268.00i	0.125953i
$688$	0	0
$689$	−24500.0	−1.35468
$690$	0	0
$691$	−23380.0	−1.28714	−0.643572	−	0.765385i	$-0.722548\pi$
$691$	−23380.0	−1.28714	−0.643572	+	0.765385i	$0.722548\pi$
$692$	0	0
$693$	− 8280.00i	− 0.453869i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 7500.00i	− 0.407579i
$698$	0	0
$699$	−3420.00	−0.185059
$700$	0	0
$701$	11850.0	0.638471	0.319236	−	0.947675i	$-0.396574\pi$
$701$	11850.0	0.638471	0.319236	+	0.947675i	$0.396574\pi$
$702$	0	0
$703$	400.000i	0.0214599i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8988.00i	0.478117i
$708$	0	0
$709$	−25646.0	−1.35847	−0.679235	−	0.733921i	$-0.737688\pi$
$709$	−25646.0	−1.35847	−0.679235	+	0.733921i	$0.737688\pi$
$710$	0	0
$711$	30360.0	1.60139
$712$	0	0
$713$	3560.00i	0.186989i
$714$	0	0
$715$	0	0
$716$	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
$725$	0	0
$726$	0	0
$727$	− 17446.0i	− 0.890009i	−0.895528	−	0.445004i	$-0.853202\pi$
$727$	− 17446.0i	− 0.890009i	0.895528	−	0.445004i	$-0.146798\pi$
$728$	0	0
$729$	4283.00	0.217599
$730$	0	0
$731$	−4260.00	−0.215543
$732$	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
$735$	0	0
$736$	0	0
$737$	46440.0i	2.32108i
$738$	0	0
$739$	36560.0	1.81987	0.909933	−	0.414755i	$-0.136133\pi$
$739$	36560.0	1.81987	0.909933	+	0.414755i	$0.136133\pi$
$740$	0	0
$741$	−4000.00	−0.198305
$742$	0	0
$743$	− 30142.0i	− 1.48829i	−0.668016	−	0.744147i	$-0.732856\pi$
$743$	− 30142.0i	− 1.48829i	0.668016	−	0.744147i	$-0.267144\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	22586.0i	1.10626i
$748$	0	0
$749$	−7164.00	−0.349488
$750$	0	0
$751$	11860.0	0.576268	0.288134	−	0.957590i	$-0.406965\pi$
$751$	11860.0	0.576268	0.288134	+	0.957590i	$0.406965\pi$
$752$	0	0
$753$	1320.00i	0.0638824i
$754$	0	0
$755$	0	0
$756$	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	$-0.590033\pi$
$761$	−11718.0	−0.558183	−0.279091	+	0.960265i	$0.590033\pi$
$762$	0	0
$763$	3900.00i	0.185045i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	40000.0i	1.88307i
$768$	0	0
$769$	−4706.00	−0.220680	−0.110340	−	0.993894i	$-0.535194\pi$
$769$	−4706.00	−0.220680	−0.110340	+	0.993894i	$0.535194\pi$
$770$	0	0
$771$	15180.0	0.709072
$772$	0	0
$773$	− 28670.0i	− 1.33401i	−0.745054	−	0.667004i	$-0.767576\pi$
$773$	− 28670.0i	− 1.33401i	0.745054	−	0.667004i	$-0.232424\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	120.000i	0.00554051i
$778$	0	0
$779$	10000.0	0.459932
$780$	0	0
$781$	6000.00	0.274900
$782$	0	0
$783$	16600.0i	0.757644i
$784$	0	0
$785$	0	0
$786$	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
$789$	1524.00	0.0687653
$790$	0	0
$791$	−9060.00	−0.407252
$792$	0	0
$793$	12500.0i	0.559758i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 3930.00i	− 0.174665i	−0.996179	−	0.0873323i	$-0.972166\pi$
$797$	− 3930.00i	− 0.174665i	0.996179	−	0.0873323i	$-0.0278342\pi$
$798$	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
$805$	0	0
$806$	0	0
$807$	− 300.000i	− 0.0130861i
$808$	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
$811$	−13140.0	−0.568937	−0.284468	−	0.958685i	$-0.591817\pi$
$811$	−13140.0	−0.568937	−0.284468	+	0.958685i	$0.591817\pi$
$812$	0	0
$813$	13160.0i	0.567702i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 5680.00i	− 0.243229i
$818$	0	0
$819$	6900.00	0.294390
$820$	0	0
$821$	22050.0	0.937333	0.468666	−	0.883375i	$-0.344735\pi$
$821$	22050.0	0.937333	0.468666	+	0.883375i	$0.344735\pi$
$822$	0	0
$823$	14578.0i	0.617445i	0.951152	+	0.308722i	$0.0999014\pi$
$823$	14578.0i	0.617445i	−0.951152	+	0.308722i	$0.900099\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37054.0i	1.55803i	0.627003	+	0.779017i	$0.284281\pi$
$827$	37054.0i	1.55803i	−0.627003	+	0.779017i	$0.715719\pi$
$828$	0	0
$829$	6150.00	0.257658	0.128829	−	0.991667i	$-0.458878\pi$
$829$	6150.00	0.257658	0.128829	+	0.991667i	$0.458878\pi$
$830$	0	0
$831$	−9060.00	−0.378204
$832$	0	0
$833$	9210.00i	0.383082i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2000.00i	− 0.0825927i
$838$	0	0
$839$	8200.00	0.337420	0.168710	−	0.985666i	$-0.446040\pi$
$839$	8200.00	0.337420	0.168710	+	0.985666i	$0.446040\pi$
$840$	0	0
$841$	3167.00	0.129854
$842$	0	0
$843$	− 13900.0i	− 0.567902i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 13614.0i	− 0.552282i
$848$	0	0
$849$	−7764.00	−0.313851
$850$	0	0
$851$	1780.00	0.0717011
$852$	0	0
$853$	− 42990.0i	− 1.72561i	−0.505533	−	0.862807i	$-0.668704\pi$
$853$	− 42990.0i	− 1.72561i	0.505533	−	0.862807i	$-0.331296\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 32130.0i	− 1.28068i	−0.768093	−	0.640338i	$-0.778794\pi$
$857$	− 32130.0i	− 1.28068i	0.768093	−	0.640338i	$-0.221206\pi$
$858$	0	0
$859$	−15440.0	−0.613278	−0.306639	−	0.951826i	$-0.599204\pi$
$859$	−15440.0	−0.613278	−0.306639	+	0.951826i	$0.599204\pi$
$860$	0	0
$861$	3000.00	0.118745
$862$	0	0
$863$	46938.0i	1.85143i	0.378216	+	0.925717i	$0.376538\pi$
$863$	46938.0i	1.85143i	−0.378216	+	0.925717i	$0.623462\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 8026.00i	− 0.314391i
$868$	0	0
$869$	79200.0	3.09169
$870$	0	0
$871$	−38700.0	−1.50551
$872$	0	0
$873$	7130.00i	0.276419i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	31230.0i	1.20247i	0.799074	+	0.601233i	$0.205324\pi$
$877$	31230.0i	1.20247i	−0.799074	+	0.601233i	$0.794676\pi$
$878$	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	$0.0262212\pi$
$883$	4318.00i	0.164567i	−0.996609	+	0.0822833i	$0.973779\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1766.00i	− 0.0668506i	−0.999441	−	0.0334253i	$-0.989358\pi$
$887$	− 1766.00i	− 0.0668506i	0.999441	−	0.0334253i	$-0.0106416\pi$
$888$	0	0
$889$	−7476.00	−0.282044
$890$	0	0
$891$	25260.0	0.949766
$892$	0	0
$893$	8560.00i	0.320772i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	17800.0i	0.662569i
$898$	0	0
$899$	−3320.00	−0.123168
$900$	0	0
$901$	−14700.0	−0.543538
$902$	0	0
$903$	− 1704.00i	− 0.0627969i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 41906.0i	− 1.53414i	−0.641563	−	0.767071i	$-0.721714\pi$
$907$	− 41906.0i	− 1.53414i	0.641563	−	0.767071i	$-0.278286\pi$
$908$	0	0
$909$	−34454.0	−1.25717
$910$	0	0
$911$	25140.0	0.914298	0.457149	−	0.889390i	$-0.348871\pi$
$911$	25140.0	0.914298	0.457149	+	0.889390i	$0.348871\pi$
$912$	0	0
$913$	58920.0i	2.13578i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 15960.0i	− 0.574750i
$918$	0	0
$919$	−32920.0	−1.18164	−0.590822	−	0.806802i	$-0.701196\pi$
$919$	−32920.0	−1.18164	−0.590822	+	0.806802i	$0.701196\pi$
$920$	0	0
$921$	−8212.00	−0.293805
$922$	0	0
$923$	5000.00i	0.178307i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	32246.0i	1.14250i
$928$	0	0
$929$	−10150.0	−0.358461	−0.179231	−	0.983807i	$-0.557361\pi$
$929$	−10150.0	−0.358461	−0.179231	+	0.983807i	$0.557361\pi$
$930$	0	0
$931$	−12280.0	−0.432289
$932$	0	0
$933$	− 4440.00i	− 0.155798i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 28530.0i	− 0.994701i	−0.867550	−	0.497350i	$-0.834306\pi$
$937$	− 28530.0i	− 0.994701i	0.867550	−	0.497350i	$-0.165694\pi$
$938$	0	0
$939$	−18860.0	−0.655456
$940$	0	0
$941$	9678.00	0.335275	0.167638	−	0.985849i	$-0.446386\pi$
$941$	9678.00	0.335275	0.167638	+	0.985849i	$0.446386\pi$
$942$	0	0
$943$	− 44500.0i	− 1.53671i
$944$	0	0
$945$	0	0
$946$	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
$949$	11500.0	0.393368
$950$	0	0
$951$	12940.0	0.441228
$952$	0	0
$953$	− 3350.00i	− 0.113869i	−0.998378	−	0.0569345i	$-0.981867\pi$
$953$	− 3350.00i	− 0.113869i	0.998378	−	0.0569345i	$-0.0181326\pi$
$954$	0	0
$955$	0	0
$956$	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
$965$	0	0
$966$	0	0
$967$	− 43774.0i	− 1.45572i	−0.685728	−	0.727858i	$-0.740516\pi$
$967$	− 43774.0i	− 1.45572i	0.685728	−	0.727858i	$-0.259484\pi$
$968$	0	0
$969$	−2400.00	−0.0795656
$970$	0	0
$971$	8740.00	0.288857	0.144428	−	0.989515i	$-0.453866\pi$
$971$	8740.00	0.288857	0.144428	+	0.989515i	$0.453866\pi$
$972$	0	0
$973$	− 3360.00i	− 0.110706i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	48310.0i	1.58196i	0.611843	+	0.790979i	$0.290429\pi$
$977$	48310.0i	1.58196i	−0.611843	+	0.790979i	$0.709571\pi$
$978$	0	0
$979$	−52440.0	−1.71194
$980$	0	0
$981$	−14950.0	−0.486561
$982$	0	0
$983$	2282.00i	0.0740432i	0.999314	+	0.0370216i	$0.0117870\pi$
$983$	2282.00i	0.0740432i	−0.999314	+	0.0370216i	$0.988213\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2568.00i	0.0828170i
$988$	0	0
$989$	−25276.0	−0.812669
$990$	0	0
$991$	−31580.0	−1.01228	−0.506141	−	0.862451i	$-0.668929\pi$
$991$	−31580.0	−1.01228	−0.506141	+	0.862451i	$0.668929\pi$
$992$	0	0
$993$	− 1800.00i	− 0.0575239i
$994$	0	0
$995$	0	0
$996$	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	800.4.c.f.449.1		2
4.3	odd	2		800.4.c.e.449.2		2
5.2	odd	4		160.4.a.a.1.1	✓	1
5.3	odd	4		800.4.a.h.1.1		1
5.4	even	2	inner	800.4.c.f.449.2		2
15.2	even	4		1440.4.a.o.1.1		1
20.3	even	4		800.4.a.d.1.1		1
20.7	even	4		160.4.a.b.1.1	yes	1
20.19	odd	2		800.4.c.e.449.1		2
40.3	even	4		1600.4.a.bj.1.1		1
40.13	odd	4		1600.4.a.r.1.1		1
40.27	even	4		320.4.a.f.1.1		1
40.37	odd	4		320.4.a.i.1.1		1
60.47	odd	4		1440.4.a.n.1.1		1
80.27	even	4		1280.4.d.k.641.1		2
80.37	odd	4		1280.4.d.f.641.2		2
80.67	even	4		1280.4.d.k.641.2		2
80.77	odd	4		1280.4.d.f.641.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.a.a.1.1	✓	1	5.2	odd	4
160.4.a.b.1.1	yes	1	20.7	even	4
320.4.a.f.1.1		1	40.27	even	4
320.4.a.i.1.1		1	40.37	odd	4
800.4.a.d.1.1		1	20.3	even	4
800.4.a.h.1.1		1	5.3	odd	4
800.4.c.e.449.1		2	20.19	odd	2
800.4.c.e.449.2		2	4.3	odd	2
800.4.c.f.449.1		2	1.1	even	1	trivial
800.4.c.f.449.2		2	5.4	even	2	inner
1280.4.d.f.641.1		2	80.77	odd	4
1280.4.d.f.641.2		2	80.37	odd	4
1280.4.d.k.641.1		2	80.27	even	4
1280.4.d.k.641.2		2	80.67	even	4
1440.4.a.n.1.1		1	60.47	odd	4
1440.4.a.o.1.1		1	15.2	even	4
1600.4.a.r.1.1		1	40.13	odd	4
1600.4.a.bj.1.1		1	40.3	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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} -6.00000i q^{7} +23.0000 q^{9} +60.0000 q^{11} +50.0000i q^{13} +30.0000i q^{17} -40.0000 q^{19} -12.0000 q^{21} +178.000i q^{23} -100.000i q^{27} -166.000 q^{29} +20.0000 q^{31} -120.000i q^{33} -10.0000i q^{37} +100.000 q^{39} -250.000 q^{41} +142.000i q^{43} -214.000i q^{47} +307.000 q^{49} +60.0000 q^{51} +490.000i q^{53} +80.0000i q^{57} +800.000 q^{59} +250.000 q^{61} -138.000i q^{63} +774.000i q^{67} +356.000 q^{69} +100.000 q^{71} -230.000i q^{73} -360.000i q^{77} +1320.00 q^{79} +421.000 q^{81} +982.000i q^{83} +332.000i q^{87} -874.000 q^{89} +300.000 q^{91} -40.0000i q^{93} +310.000i q^{97} +1380.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 46 q^{9} + 120 q^{11} - 80 q^{19} - 24 q^{21} - 332 q^{29} + 40 q^{31} + 200 q^{39} - 500 q^{41} + 614 q^{49} + 120 q^{51} + 1600 q^{59} + 500 q^{61} + 712 q^{69} + 200 q^{71} + 2640 q^{79} + 842 q^{81}+ \cdots + 2760 q^{99}+O(q^{100}) \) 2 * q + 46 * q^9 + 120 * q^11 - 80 * q^19 - 24 * q^21 - 332 * q^29 + 40 * q^31 + 200 * q^39 - 500 * q^41 + 614 * q^49 + 120 * q^51 + 1600 * q^59 + 500 * q^61 + 712 * q^69 + 200 * q^71 + 2640 * q^79 + 842 * q^81 - 1748 * q^89 + 600 * q^91 + 2760 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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