LMFDB - Embedded newform 32.4.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [32,4,Mod(1,32)] 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("32.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(32, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	32.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$1.88806112018$
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	$\chi$	$=$	32.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-8.00000 q^{3} -10.0000 q^{5} -16.0000 q^{7} +37.0000 q^{9} +40.0000 q^{11} -50.0000 q^{13} +80.0000 q^{15} -30.0000 q^{17} -40.0000 q^{19} +128.000 q^{21} -48.0000 q^{23} -25.0000 q^{25} -80.0000 q^{27} -34.0000 q^{29} -320.000 q^{31} -320.000 q^{33} +160.000 q^{35} +310.000 q^{37} +400.000 q^{39} +410.000 q^{41} -152.000 q^{43} -370.000 q^{45} +416.000 q^{47} -87.0000 q^{49} +240.000 q^{51} -410.000 q^{53} -400.000 q^{55} +320.000 q^{57} +200.000 q^{59} +30.0000 q^{61} -592.000 q^{63} +500.000 q^{65} -776.000 q^{67} +384.000 q^{69} -400.000 q^{71} -630.000 q^{73} +200.000 q^{75} -640.000 q^{77} +1120.00 q^{79} -359.000 q^{81} -552.000 q^{83} +300.000 q^{85} +272.000 q^{87} -326.000 q^{89} +800.000 q^{91} +2560.00 q^{93} +400.000 q^{95} -110.000 q^{97} +1480.00 q^{99} +O(q^{100})$

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
$3$	−8.00000	−1.53960	−0.769800	−	0.638285i	$-0.779644\pi$
$3$	−8.00000	−1.53960	−0.769800	+	0.638285i	$0.779644\pi$
$4$	0	0
$5$	−10.0000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−10.0000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	−16.0000	−0.863919	−0.431959	−	0.901893i	$-0.642178\pi$
$7$	−16.0000	−0.863919	−0.431959	+	0.901893i	$0.642178\pi$
$8$	0	0
$9$	37.0000	1.37037
$10$	0	0
$11$	40.0000	1.09640	0.548202	−	0.836346i	$-0.315312\pi$
$11$	40.0000	1.09640	0.548202	+	0.836346i	$0.315312\pi$
$12$	0	0
$13$	−50.0000	−1.06673	−0.533366	−	0.845885i	$-0.679073\pi$
$13$	−50.0000	−1.06673	−0.533366	+	0.845885i	$0.679073\pi$
$14$	0	0
$15$	80.0000	1.37706
$16$	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
$19$	−40.0000	−0.482980	−0.241490	−	0.970403i	$-0.577636\pi$
$19$	−40.0000	−0.482980	−0.241490	+	0.970403i	$0.577636\pi$
$20$	0	0
$21$	128.000	1.33009
$22$	0	0
$23$	−48.0000	−0.435161	−0.217580	−	0.976042i	$-0.569816\pi$
$23$	−48.0000	−0.435161	−0.217580	+	0.976042i	$0.569816\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	0	0
$27$	−80.0000	−0.570222
$28$	0	0
$29$	−34.0000	−0.217712	−0.108856	−	0.994058i	$-0.534719\pi$
$29$	−34.0000	−0.217712	−0.108856	+	0.994058i	$0.534719\pi$
$30$	0	0
$31$	−320.000	−1.85399	−0.926995	−	0.375073i	$-0.877617\pi$
$31$	−320.000	−1.85399	−0.926995	+	0.375073i	$0.877617\pi$
$32$	0	0
$33$	−320.000	−1.68803
$34$	0	0
$35$	160.000	0.772712
$36$	0	0
$37$	310.000	1.37740	0.688698	−	0.725048i	$-0.258182\pi$
$37$	310.000	1.37740	0.688698	+	0.725048i	$0.258182\pi$
$38$	0	0
$39$	400.000	1.64234
$40$	0	0
$41$	410.000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	410.000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−152.000	−0.539065	−0.269532	−	0.962991i	$-0.586869\pi$
$43$	−152.000	−0.539065	−0.269532	+	0.962991i	$0.586869\pi$
$44$	0	0
$45$	−370.000	−1.22570
$46$	0	0
$47$	416.000	1.29106	0.645530	−	0.763735i	$-0.276636\pi$
$47$	416.000	1.29106	0.645530	+	0.763735i	$0.276636\pi$
$48$	0	0
$49$	−87.0000	−0.253644
$50$	0	0
$51$	240.000	0.658955
$52$	0	0
$53$	−410.000	−1.06260	−0.531300	−	0.847184i	$-0.678296\pi$
$53$	−410.000	−1.06260	−0.531300	+	0.847184i	$0.678296\pi$
$54$	0	0
$55$	−400.000	−0.980654
$56$	0	0
$57$	320.000	0.743597
$58$	0	0
$59$	200.000	0.441318	0.220659	−	0.975351i	$-0.429179\pi$
$59$	200.000	0.441318	0.220659	+	0.975351i	$0.429179\pi$
$60$	0	0
$61$	30.0000	0.0629690	0.0314845	−	0.999504i	$-0.489977\pi$
$61$	30.0000	0.0629690	0.0314845	+	0.999504i	$0.489977\pi$
$62$	0	0
$63$	−592.000	−1.18389
$64$	0	0
$65$	500.000	0.954113
$66$	0	0
$67$	−776.000	−1.41498	−0.707489	−	0.706725i	$-0.750172\pi$
$67$	−776.000	−1.41498	−0.707489	+	0.706725i	$0.750172\pi$
$68$	0	0
$69$	384.000	0.669973
$70$	0	0
$71$	−400.000	−0.668609	−0.334305	−	0.942465i	$-0.608501\pi$
$71$	−400.000	−0.668609	−0.334305	+	0.942465i	$0.608501\pi$
$72$	0	0
$73$	−630.000	−1.01008	−0.505041	−	0.863096i	$-0.668522\pi$
$73$	−630.000	−1.01008	−0.505041	+	0.863096i	$0.668522\pi$
$74$	0	0
$75$	200.000	0.307920
$76$	0	0
$77$	−640.000	−0.947205
$78$	0	0
$79$	1120.00	1.59506	0.797531	−	0.603278i	$-0.206139\pi$
$79$	1120.00	1.59506	0.797531	+	0.603278i	$0.206139\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	0	0
$83$	−552.000	−0.729998	−0.364999	−	0.931008i	$-0.618931\pi$
$83$	−552.000	−0.729998	−0.364999	+	0.931008i	$0.618931\pi$
$84$	0	0
$85$	300.000	0.382818
$86$	0	0
$87$	272.000	0.335189
$88$	0	0
$89$	−326.000	−0.388269	−0.194134	−	0.980975i	$-0.562190\pi$
$89$	−326.000	−0.388269	−0.194134	+	0.980975i	$0.562190\pi$
$90$	0	0
$91$	800.000	0.921569
$92$	0	0
$93$	2560.00	2.85440
$94$	0	0
$95$	400.000	0.431991
$96$	0	0
$97$	−110.000	−0.115142	−0.0575712	−	0.998341i	$-0.518336\pi$
$97$	−110.000	−0.115142	−0.0575712	+	0.998341i	$0.518336\pi$
$98$	0	0
$99$	1480.00	1.50248
$100$	0	0
$101$	−1098.00	−1.08173	−0.540867	−	0.841108i	$-0.681904\pi$
$101$	−1098.00	−1.08173	−0.540867	+	0.841108i	$0.681904\pi$
$102$	0	0
$103$	48.0000	0.0459183	0.0229591	−	0.999736i	$-0.492691\pi$
$103$	48.0000	0.0459183	0.0229591	+	0.999736i	$0.492691\pi$
$104$	0	0
$105$	−1280.00	−1.18967
$106$	0	0
$107$	−664.000	−0.599919	−0.299959	−	0.953952i	$-0.596973\pi$
$107$	−664.000	−0.599919	−0.299959	+	0.953952i	$0.596973\pi$
$108$	0	0
$109$	−370.000	−0.325134	−0.162567	−	0.986698i	$-0.551977\pi$
$109$	−370.000	−0.325134	−0.162567	+	0.986698i	$0.551977\pi$
$110$	0	0
$111$	−2480.00	−2.12064
$112$	0	0
$113$	1490.00	1.24042	0.620210	−	0.784436i	$-0.287047\pi$
$113$	1490.00	1.24042	0.620210	+	0.784436i	$0.287047\pi$
$114$	0	0
$115$	480.000	0.389219
$116$	0	0
$117$	−1850.00	−1.46182
$118$	0	0
$119$	480.000	0.369761
$120$	0	0
$121$	269.000	0.202104
$122$	0	0
$123$	−3280.00	−2.40445
$124$	0	0
$125$	1500.00	1.07331
$126$	0	0
$127$	1024.00	0.715475	0.357737	−	0.933822i	$-0.383548\pi$
$127$	1024.00	0.715475	0.357737	+	0.933822i	$0.383548\pi$
$128$	0	0
$129$	1216.00	0.829944
$130$	0	0
$131$	−1160.00	−0.773662	−0.386831	−	0.922151i	$-0.626430\pi$
$131$	−1160.00	−0.773662	−0.386831	+	0.922151i	$0.626430\pi$
$132$	0	0
$133$	640.000	0.417256
$134$	0	0
$135$	800.000	0.510022
$136$	0	0
$137$	570.000	0.355463	0.177731	−	0.984079i	$-0.443124\pi$
$137$	570.000	0.355463	0.177731	+	0.984079i	$0.443124\pi$
$138$	0	0
$139$	1960.00	1.19601	0.598004	−	0.801493i	$-0.295961\pi$
$139$	1960.00	1.19601	0.598004	+	0.801493i	$0.295961\pi$
$140$	0	0
$141$	−3328.00	−1.98772
$142$	0	0
$143$	−2000.00	−1.16957
$144$	0	0
$145$	340.000	0.194727
$146$	0	0
$147$	696.000	0.390511
$148$	0	0
$149$	−2010.00	−1.10514	−0.552569	−	0.833467i	$-0.686352\pi$
$149$	−2010.00	−1.10514	−0.552569	+	0.833467i	$0.686352\pi$
$150$	0	0
$151$	720.000	0.388032	0.194016	−	0.980998i	$-0.437849\pi$
$151$	720.000	0.388032	0.194016	+	0.980998i	$0.437849\pi$
$152$	0	0
$153$	−1110.00	−0.586524
$154$	0	0
$155$	3200.00	1.65826
$156$	0	0
$157$	1790.00	0.909921	0.454960	−	0.890512i	$-0.349653\pi$
$157$	1790.00	0.909921	0.454960	+	0.890512i	$0.349653\pi$
$158$	0	0
$159$	3280.00	1.63598
$160$	0	0
$161$	768.000	0.375943
$162$	0	0
$163$	1208.00	0.580478	0.290239	−	0.956954i	$-0.406265\pi$
$163$	1208.00	0.580478	0.290239	+	0.956954i	$0.406265\pi$
$164$	0	0
$165$	3200.00	1.50982
$166$	0	0
$167$	−2896.00	−1.34191	−0.670956	−	0.741497i	$-0.734116\pi$
$167$	−2896.00	−1.34191	−0.670956	+	0.741497i	$0.734116\pi$
$168$	0	0
$169$	303.000	0.137915
$170$	0	0
$171$	−1480.00	−0.661862
$172$	0	0
$173$	750.000	0.329604	0.164802	−	0.986327i	$-0.447302\pi$
$173$	750.000	0.329604	0.164802	+	0.986327i	$0.447302\pi$
$174$	0	0
$175$	400.000	0.172784
$176$	0	0
$177$	−1600.00	−0.679454
$178$	0	0
$179$	−2280.00	−0.952040	−0.476020	−	0.879434i	$-0.657921\pi$
$179$	−2280.00	−0.952040	−0.476020	+	0.879434i	$0.657921\pi$
$180$	0	0
$181$	−442.000	−0.181512	−0.0907558	−	0.995873i	$-0.528928\pi$
$181$	−442.000	−0.181512	−0.0907558	+	0.995873i	$0.528928\pi$
$182$	0	0
$183$	−240.000	−0.0969471
$184$	0	0
$185$	−3100.00	−1.23198
$186$	0	0
$187$	−1200.00	−0.469266
$188$	0	0
$189$	1280.00	0.492626
$190$	0	0
$191$	1920.00	0.727363	0.363681	−	0.931523i	$-0.381520\pi$
$191$	1920.00	0.727363	0.363681	+	0.931523i	$0.381520\pi$
$192$	0	0
$193$	−5070.00	−1.89091	−0.945457	−	0.325746i	$-0.894385\pi$
$193$	−5070.00	−1.89091	−0.945457	+	0.325746i	$0.894385\pi$
$194$	0	0
$195$	−4000.00	−1.46895
$196$	0	0
$197$	1910.00	0.690771	0.345385	−	0.938461i	$-0.387748\pi$
$197$	1910.00	0.690771	0.345385	+	0.938461i	$0.387748\pi$
$198$	0	0
$199$	−2960.00	−1.05442	−0.527208	−	0.849736i	$-0.676761\pi$
$199$	−2960.00	−1.05442	−0.527208	+	0.849736i	$0.676761\pi$
$200$	0	0
$201$	6208.00	2.17850
$202$	0	0
$203$	544.000	0.188085
$204$	0	0
$205$	−4100.00	−1.39686
$206$	0	0
$207$	−1776.00	−0.596331
$208$	0	0
$209$	−1600.00	−0.529542
$210$	0	0
$211$	−40.0000	−0.0130508	−0.00652539	−	0.999979i	$-0.502077\pi$
$211$	−40.0000	−0.0130508	−0.00652539	+	0.999979i	$0.502077\pi$
$212$	0	0
$213$	3200.00	1.02939
$214$	0	0
$215$	1520.00	0.482154
$216$	0	0
$217$	5120.00	1.60170
$218$	0	0
$219$	5040.00	1.55512
$220$	0	0
$221$	1500.00	0.456565
$222$	0	0
$223$	−4288.00	−1.28765	−0.643824	−	0.765173i	$-0.722653\pi$
$223$	−4288.00	−1.28765	−0.643824	+	0.765173i	$0.722653\pi$
$224$	0	0
$225$	−925.000	−0.274074
$226$	0	0
$227$	6456.00	1.88766	0.943832	−	0.330425i	$-0.107192\pi$
$227$	6456.00	1.88766	0.943832	+	0.330425i	$0.107192\pi$
$228$	0	0
$229$	−1066.00	−0.307613	−0.153806	−	0.988101i	$-0.549153\pi$
$229$	−1066.00	−0.307613	−0.153806	+	0.988101i	$0.549153\pi$
$230$	0	0
$231$	5120.00	1.45832
$232$	0	0
$233$	−5910.00	−1.66170	−0.830852	−	0.556494i	$-0.812146\pi$
$233$	−5910.00	−1.66170	−0.830852	+	0.556494i	$0.812146\pi$
$234$	0	0
$235$	−4160.00	−1.15476
$236$	0	0
$237$	−8960.00	−2.45576
$238$	0	0
$239$	3360.00	0.909374	0.454687	−	0.890651i	$-0.349751\pi$
$239$	3360.00	0.909374	0.454687	+	0.890651i	$0.349751\pi$
$240$	0	0
$241$	3970.00	1.06112	0.530561	−	0.847647i	$-0.321981\pi$
$241$	3970.00	1.06112	0.530561	+	0.847647i	$0.321981\pi$
$242$	0	0
$243$	5032.00	1.32841
$244$	0	0
$245$	870.000	0.226866
$246$	0	0
$247$	2000.00	0.515210
$248$	0	0
$249$	4416.00	1.12391
$250$	0	0
$251$	−6840.00	−1.72007	−0.860034	−	0.510237i	$-0.829558\pi$
$251$	−6840.00	−1.72007	−0.860034	+	0.510237i	$0.829558\pi$
$252$	0	0
$253$	−1920.00	−0.477112
$254$	0	0
$255$	−2400.00	−0.589388
$256$	0	0
$257$	4610.00	1.11893	0.559463	−	0.828855i	$-0.311007\pi$
$257$	4610.00	1.11893	0.559463	+	0.828855i	$0.311007\pi$
$258$	0	0
$259$	−4960.00	−1.18996
$260$	0	0
$261$	−1258.00	−0.298346
$262$	0	0
$263$	4848.00	1.13666	0.568328	−	0.822802i	$-0.307591\pi$
$263$	4848.00	1.13666	0.568328	+	0.822802i	$0.307591\pi$
$264$	0	0
$265$	4100.00	0.950419
$266$	0	0
$267$	2608.00	0.597779
$268$	0	0
$269$	5550.00	1.25795	0.628977	−	0.777424i	$-0.283474\pi$
$269$	5550.00	1.25795	0.628977	+	0.777424i	$0.283474\pi$
$270$	0	0
$271$	480.000	0.107594	0.0537969	−	0.998552i	$-0.482868\pi$
$271$	480.000	0.107594	0.0537969	+	0.998552i	$0.482868\pi$
$272$	0	0
$273$	−6400.00	−1.41885
$274$	0	0
$275$	−1000.00	−0.219281
$276$	0	0
$277$	1030.00	0.223418	0.111709	−	0.993741i	$-0.464368\pi$
$277$	1030.00	0.223418	0.111709	+	0.993741i	$0.464368\pi$
$278$	0	0
$279$	−11840.0	−2.54065
$280$	0	0
$281$	−3270.00	−0.694206	−0.347103	−	0.937827i	$-0.612835\pi$
$281$	−3270.00	−0.694206	−0.347103	+	0.937827i	$0.612835\pi$
$282$	0	0
$283$	−2168.00	−0.455386	−0.227693	−	0.973733i	$-0.573118\pi$
$283$	−2168.00	−0.455386	−0.227693	+	0.973733i	$0.573118\pi$
$284$	0	0
$285$	−3200.00	−0.665093
$286$	0	0
$287$	−6560.00	−1.34921
$288$	0	0
$289$	−4013.00	−0.816813
$290$	0	0
$291$	880.000	0.177273
$292$	0	0
$293$	2070.00	0.412733	0.206366	−	0.978475i	$-0.433836\pi$
$293$	2070.00	0.412733	0.206366	+	0.978475i	$0.433836\pi$
$294$	0	0
$295$	−2000.00	−0.394727
$296$	0	0
$297$	−3200.00	−0.625195
$298$	0	0
$299$	2400.00	0.464199
$300$	0	0
$301$	2432.00	0.465708
$302$	0	0
$303$	8784.00	1.66544
$304$	0	0
$305$	−300.000	−0.0563211
$306$	0	0
$307$	−1896.00	−0.352477	−0.176238	−	0.984347i	$-0.556393\pi$
$307$	−1896.00	−0.352477	−0.176238	+	0.984347i	$0.556393\pi$
$308$	0	0
$309$	−384.000	−0.0706958
$310$	0	0
$311$	1680.00	0.306315	0.153158	−	0.988202i	$-0.451056\pi$
$311$	1680.00	0.306315	0.153158	+	0.988202i	$0.451056\pi$
$312$	0	0
$313$	970.000	0.175168	0.0875841	−	0.996157i	$-0.472085\pi$
$313$	970.000	0.175168	0.0875841	+	0.996157i	$0.472085\pi$
$314$	0	0
$315$	5920.00	1.05890
$316$	0	0
$317$	7230.00	1.28100	0.640500	−	0.767958i	$-0.278727\pi$
$317$	7230.00	1.28100	0.640500	+	0.767958i	$0.278727\pi$
$318$	0	0
$319$	−1360.00	−0.238700
$320$	0	0
$321$	5312.00	0.923635
$322$	0	0
$323$	1200.00	0.206718
$324$	0	0
$325$	1250.00	0.213346
$326$	0	0
$327$	2960.00	0.500576
$328$	0	0
$329$	−6656.00	−1.11537
$330$	0	0
$331$	5800.00	0.963132	0.481566	−	0.876410i	$-0.340068\pi$
$331$	5800.00	0.963132	0.481566	+	0.876410i	$0.340068\pi$
$332$	0	0
$333$	11470.0	1.88754
$334$	0	0
$335$	7760.00	1.26559
$336$	0	0
$337$	−1870.00	−0.302271	−0.151136	−	0.988513i	$-0.548293\pi$
$337$	−1870.00	−0.302271	−0.151136	+	0.988513i	$0.548293\pi$
$338$	0	0
$339$	−11920.0	−1.90975
$340$	0	0
$341$	−12800.0	−2.03272
$342$	0	0
$343$	6880.00	1.08305
$344$	0	0
$345$	−3840.00	−0.599242
$346$	0	0
$347$	−376.000	−0.0581693	−0.0290846	−	0.999577i	$-0.509259\pi$
$347$	−376.000	−0.0581693	−0.0290846	+	0.999577i	$0.509259\pi$
$348$	0	0
$349$	−7586.00	−1.16352	−0.581761	−	0.813360i	$-0.697636\pi$
$349$	−7586.00	−1.16352	−0.581761	+	0.813360i	$0.697636\pi$
$350$	0	0
$351$	4000.00	0.608274
$352$	0	0
$353$	2530.00	0.381468	0.190734	−	0.981642i	$-0.438913\pi$
$353$	2530.00	0.381468	0.190734	+	0.981642i	$0.438913\pi$
$354$	0	0
$355$	4000.00	0.598022
$356$	0	0
$357$	−3840.00	−0.569284
$358$	0	0
$359$	−9680.00	−1.42309	−0.711547	−	0.702638i	$-0.752005\pi$
$359$	−9680.00	−1.42309	−0.711547	+	0.702638i	$0.752005\pi$
$360$	0	0
$361$	−5259.00	−0.766730
$362$	0	0
$363$	−2152.00	−0.311159
$364$	0	0
$365$	6300.00	0.903444
$366$	0	0
$367$	−2784.00	−0.395977	−0.197989	−	0.980204i	$-0.563441\pi$
$367$	−2784.00	−0.395977	−0.197989	+	0.980204i	$0.563441\pi$
$368$	0	0
$369$	15170.0	2.14016
$370$	0	0
$371$	6560.00	0.918001
$372$	0	0
$373$	7910.00	1.09803	0.549014	−	0.835813i	$-0.315003\pi$
$373$	7910.00	1.09803	0.549014	+	0.835813i	$0.315003\pi$
$374$	0	0
$375$	−12000.0	−1.65247
$376$	0	0
$377$	1700.00	0.232240
$378$	0	0
$379$	−1720.00	−0.233115	−0.116557	−	0.993184i	$-0.537186\pi$
$379$	−1720.00	−0.233115	−0.116557	+	0.993184i	$0.537186\pi$
$380$	0	0
$381$	−8192.00	−1.10155
$382$	0	0
$383$	−11008.0	−1.46862	−0.734311	−	0.678813i	$-0.762495\pi$
$383$	−11008.0	−1.46862	−0.734311	+	0.678813i	$0.762495\pi$
$384$	0	0
$385$	6400.00	0.847206
$386$	0	0
$387$	−5624.00	−0.738718
$388$	0	0
$389$	−12330.0	−1.60708	−0.803542	−	0.595248i	$-0.797054\pi$
$389$	−12330.0	−1.60708	−0.803542	+	0.595248i	$0.797054\pi$
$390$	0	0
$391$	1440.00	0.186250
$392$	0	0
$393$	9280.00	1.19113
$394$	0	0
$395$	−11200.0	−1.42667
$396$	0	0
$397$	−4370.00	−0.552453	−0.276227	−	0.961093i	$-0.589084\pi$
$397$	−4370.00	−0.552453	−0.276227	+	0.961093i	$0.589084\pi$
$398$	0	0
$399$	−5120.00	−0.642408
$400$	0	0
$401$	3298.00	0.410709	0.205354	−	0.978688i	$-0.434165\pi$
$401$	3298.00	0.410709	0.205354	+	0.978688i	$0.434165\pi$
$402$	0	0
$403$	16000.0	1.97771
$404$	0	0
$405$	3590.00	0.440466
$406$	0	0
$407$	12400.0	1.51018
$408$	0	0
$409$	−9110.00	−1.10137	−0.550685	−	0.834713i	$-0.685634\pi$
$409$	−9110.00	−1.10137	−0.550685	+	0.834713i	$0.685634\pi$
$410$	0	0
$411$	−4560.00	−0.547271
$412$	0	0
$413$	−3200.00	−0.381263
$414$	0	0
$415$	5520.00	0.652930
$416$	0	0
$417$	−15680.0	−1.84137
$418$	0	0
$419$	−7880.00	−0.918767	−0.459383	−	0.888238i	$-0.651930\pi$
$419$	−7880.00	−0.918767	−0.459383	+	0.888238i	$0.651930\pi$
$420$	0	0
$421$	−5290.00	−0.612396	−0.306198	−	0.951968i	$-0.599057\pi$
$421$	−5290.00	−0.612396	−0.306198	+	0.951968i	$0.599057\pi$
$422$	0	0
$423$	15392.0	1.76923
$424$	0	0
$425$	750.000	0.0856008
$426$	0	0
$427$	−480.000	−0.0544001
$428$	0	0
$429$	16000.0	1.80067
$430$	0	0
$431$	−13920.0	−1.55569	−0.777845	−	0.628456i	$-0.783687\pi$
$431$	−13920.0	−1.55569	−0.777845	+	0.628456i	$0.783687\pi$
$432$	0	0
$433$	4930.00	0.547161	0.273580	−	0.961849i	$-0.411792\pi$
$433$	4930.00	0.547161	0.273580	+	0.961849i	$0.411792\pi$
$434$	0	0
$435$	−2720.00	−0.299802
$436$	0	0
$437$	1920.00	0.210174
$438$	0	0
$439$	10640.0	1.15676	0.578382	−	0.815766i	$-0.303684\pi$
$439$	10640.0	1.15676	0.578382	+	0.815766i	$0.303684\pi$
$440$	0	0
$441$	−3219.00	−0.347587
$442$	0	0
$443$	9288.00	0.996131	0.498066	−	0.867139i	$-0.334044\pi$
$443$	9288.00	0.996131	0.498066	+	0.867139i	$0.334044\pi$
$444$	0	0
$445$	3260.00	0.347278
$446$	0	0
$447$	16080.0	1.70147
$448$	0	0
$449$	12850.0	1.35062	0.675311	−	0.737533i	$-0.264010\pi$
$449$	12850.0	1.35062	0.675311	+	0.737533i	$0.264010\pi$
$450$	0	0
$451$	16400.0	1.71230
$452$	0	0
$453$	−5760.00	−0.597414
$454$	0	0
$455$	−8000.00	−0.824276
$456$	0	0
$457$	10490.0	1.07375	0.536873	−	0.843663i	$-0.319606\pi$
$457$	10490.0	1.07375	0.536873	+	0.843663i	$0.319606\pi$
$458$	0	0
$459$	2400.00	0.244058
$460$	0	0
$461$	11118.0	1.12325	0.561624	−	0.827393i	$-0.310177\pi$
$461$	11118.0	1.12325	0.561624	+	0.827393i	$0.310177\pi$
$462$	0	0
$463$	−5792.00	−0.581376	−0.290688	−	0.956818i	$-0.593884\pi$
$463$	−5792.00	−0.581376	−0.290688	+	0.956818i	$0.593884\pi$
$464$	0	0
$465$	−25600.0	−2.55306
$466$	0	0
$467$	−2216.00	−0.219581	−0.109790	−	0.993955i	$-0.535018\pi$
$467$	−2216.00	−0.219581	−0.109790	+	0.993955i	$0.535018\pi$
$468$	0	0
$469$	12416.0	1.22243
$470$	0	0
$471$	−14320.0	−1.40091
$472$	0	0
$473$	−6080.00	−0.591033
$474$	0	0
$475$	1000.00	0.0965961
$476$	0	0
$477$	−15170.0	−1.45616
$478$	0	0
$479$	10560.0	1.00730	0.503652	−	0.863907i	$-0.331989\pi$
$479$	10560.0	1.00730	0.503652	+	0.863907i	$0.331989\pi$
$480$	0	0
$481$	−15500.0	−1.46931
$482$	0	0
$483$	−6144.00	−0.578803
$484$	0	0
$485$	1100.00	0.102986
$486$	0	0
$487$	−13264.0	−1.23419	−0.617094	−	0.786890i	$-0.711690\pi$
$487$	−13264.0	−1.23419	−0.617094	+	0.786890i	$0.711690\pi$
$488$	0	0
$489$	−9664.00	−0.893704
$490$	0	0
$491$	4840.00	0.444860	0.222430	−	0.974949i	$-0.428601\pi$
$491$	4840.00	0.444860	0.222430	+	0.974949i	$0.428601\pi$
$492$	0	0
$493$	1020.00	0.0931815
$494$	0	0
$495$	−14800.0	−1.34386
$496$	0	0
$497$	6400.00	0.577624
$498$	0	0
$499$	−19560.0	−1.75476	−0.877381	−	0.479795i	$-0.840711\pi$
$499$	−19560.0	−1.75476	−0.877381	+	0.479795i	$0.840711\pi$
$500$	0	0
$501$	23168.0	2.06601
$502$	0	0
$503$	528.000	0.0468039	0.0234019	−	0.999726i	$-0.492550\pi$
$503$	528.000	0.0468039	0.0234019	+	0.999726i	$0.492550\pi$
$504$	0	0
$505$	10980.0	0.967532
$506$	0	0
$507$	−2424.00	−0.212335
$508$	0	0
$509$	−19554.0	−1.70278	−0.851391	−	0.524532i	$-0.824240\pi$
$509$	−19554.0	−1.70278	−0.851391	+	0.524532i	$0.824240\pi$
$510$	0	0
$511$	10080.0	0.872628
$512$	0	0
$513$	3200.00	0.275406
$514$	0	0
$515$	−480.000	−0.0410705
$516$	0	0
$517$	16640.0	1.41552
$518$	0	0
$519$	−6000.00	−0.507458
$520$	0	0
$521$	15162.0	1.27497	0.637485	−	0.770463i	$-0.279975\pi$
$521$	15162.0	1.27497	0.637485	+	0.770463i	$0.279975\pi$
$522$	0	0
$523$	−10968.0	−0.917012	−0.458506	−	0.888691i	$-0.651615\pi$
$523$	−10968.0	−0.917012	−0.458506	+	0.888691i	$0.651615\pi$
$524$	0	0
$525$	−3200.00	−0.266018
$526$	0	0
$527$	9600.00	0.793515
$528$	0	0
$529$	−9863.00	−0.810635
$530$	0	0
$531$	7400.00	0.604770
$532$	0	0
$533$	−20500.0	−1.66595
$534$	0	0
$535$	6640.00	0.536584
$536$	0	0
$537$	18240.0	1.46576
$538$	0	0
$539$	−3480.00	−0.278097
$540$	0	0
$541$	−6722.00	−0.534198	−0.267099	−	0.963669i	$-0.586065\pi$
$541$	−6722.00	−0.534198	−0.267099	+	0.963669i	$0.586065\pi$
$542$	0	0
$543$	3536.00	0.279455
$544$	0	0
$545$	3700.00	0.290808
$546$	0	0
$547$	−20424.0	−1.59647	−0.798233	−	0.602348i	$-0.794232\pi$
$547$	−20424.0	−1.59647	−0.798233	+	0.602348i	$0.794232\pi$
$548$	0	0
$549$	1110.00	0.0862908
$550$	0	0
$551$	1360.00	0.105151
$552$	0	0
$553$	−17920.0	−1.37800
$554$	0	0
$555$	24800.0	1.89676
$556$	0	0
$557$	−6610.00	−0.502827	−0.251414	−	0.967880i	$-0.580895\pi$
$557$	−6610.00	−0.502827	−0.251414	+	0.967880i	$0.580895\pi$
$558$	0	0
$559$	7600.00	0.575037
$560$	0	0
$561$	9600.00	0.722482
$562$	0	0
$563$	2712.00	0.203015	0.101507	−	0.994835i	$-0.467633\pi$
$563$	2712.00	0.203015	0.101507	+	0.994835i	$0.467633\pi$
$564$	0	0
$565$	−14900.0	−1.10946
$566$	0	0
$567$	5744.00	0.425441
$568$	0	0
$569$	3530.00	0.260080	0.130040	−	0.991509i	$-0.458489\pi$
$569$	3530.00	0.260080	0.130040	+	0.991509i	$0.458489\pi$
$570$	0	0
$571$	13640.0	0.999678	0.499839	−	0.866118i	$-0.333392\pi$
$571$	13640.0	0.999678	0.499839	+	0.866118i	$0.333392\pi$
$572$	0	0
$573$	−15360.0	−1.11985
$574$	0	0
$575$	1200.00	0.0870321
$576$	0	0
$577$	−6270.00	−0.452380	−0.226190	−	0.974083i	$-0.572627\pi$
$577$	−6270.00	−0.452380	−0.226190	+	0.974083i	$0.572627\pi$
$578$	0	0
$579$	40560.0	2.91125
$580$	0	0
$581$	8832.00	0.630659
$582$	0	0
$583$	−16400.0	−1.16504
$584$	0	0
$585$	18500.0	1.30749
$586$	0	0
$587$	8616.00	0.605827	0.302913	−	0.953018i	$-0.402041\pi$
$587$	8616.00	0.605827	0.302913	+	0.953018i	$0.402041\pi$
$588$	0	0
$589$	12800.0	0.895441
$590$	0	0
$591$	−15280.0	−1.06351
$592$	0	0
$593$	5490.00	0.380181	0.190090	−	0.981767i	$-0.439122\pi$
$593$	5490.00	0.380181	0.190090	+	0.981767i	$0.439122\pi$
$594$	0	0
$595$	−4800.00	−0.330724
$596$	0	0
$597$	23680.0	1.62338
$598$	0	0
$599$	15440.0	1.05319	0.526595	−	0.850116i	$-0.323468\pi$
$599$	15440.0	1.05319	0.526595	+	0.850116i	$0.323468\pi$
$600$	0	0
$601$	8890.00	0.603379	0.301689	−	0.953406i	$-0.402449\pi$
$601$	8890.00	0.603379	0.301689	+	0.953406i	$0.402449\pi$
$602$	0	0
$603$	−28712.0	−1.93904
$604$	0	0
$605$	−2690.00	−0.180767
$606$	0	0
$607$	−23744.0	−1.58771	−0.793854	−	0.608108i	$-0.791929\pi$
$607$	−23744.0	−1.58771	−0.793854	+	0.608108i	$0.791929\pi$
$608$	0	0
$609$	−4352.00	−0.289576
$610$	0	0
$611$	−20800.0	−1.37721
$612$	0	0
$613$	−15210.0	−1.00216	−0.501082	−	0.865400i	$-0.667064\pi$
$613$	−15210.0	−1.00216	−0.501082	+	0.865400i	$0.667064\pi$
$614$	0	0
$615$	32800.0	2.15061
$616$	0	0
$617$	−12630.0	−0.824092	−0.412046	−	0.911163i	$-0.635186\pi$
$617$	−12630.0	−0.824092	−0.412046	+	0.911163i	$0.635186\pi$
$618$	0	0
$619$	−11160.0	−0.724650	−0.362325	−	0.932052i	$-0.618017\pi$
$619$	−11160.0	−0.724650	−0.362325	+	0.932052i	$0.618017\pi$
$620$	0	0
$621$	3840.00	0.248138
$622$	0	0
$623$	5216.00	0.335433
$624$	0	0
$625$	−11875.0	−0.760000
$626$	0	0
$627$	12800.0	0.815284
$628$	0	0
$629$	−9300.00	−0.589531
$630$	0	0
$631$	−13040.0	−0.822685	−0.411342	−	0.911481i	$-0.634940\pi$
$631$	−13040.0	−0.822685	−0.411342	+	0.911481i	$0.634940\pi$
$632$	0	0
$633$	320.000	0.0200930
$634$	0	0
$635$	−10240.0	−0.639940
$636$	0	0
$637$	4350.00	0.270570
$638$	0	0
$639$	−14800.0	−0.916242
$640$	0	0
$641$	−16910.0	−1.04197	−0.520987	−	0.853565i	$-0.674436\pi$
$641$	−16910.0	−1.04197	−0.520987	+	0.853565i	$0.674436\pi$
$642$	0	0
$643$	−4488.00	−0.275256	−0.137628	−	0.990484i	$-0.543948\pi$
$643$	−4488.00	−0.275256	−0.137628	+	0.990484i	$0.543948\pi$
$644$	0	0
$645$	−12160.0	−0.742325
$646$	0	0
$647$	−2064.00	−0.125416	−0.0627080	−	0.998032i	$-0.519974\pi$
$647$	−2064.00	−0.125416	−0.0627080	+	0.998032i	$0.519974\pi$
$648$	0	0
$649$	8000.00	0.483864
$650$	0	0
$651$	−40960.0	−2.46597
$652$	0	0
$653$	4270.00	0.255893	0.127946	−	0.991781i	$-0.459161\pi$
$653$	4270.00	0.255893	0.127946	+	0.991781i	$0.459161\pi$
$654$	0	0
$655$	11600.0	0.691984
$656$	0	0
$657$	−23310.0	−1.38419
$658$	0	0
$659$	19800.0	1.17041	0.585204	−	0.810886i	$-0.301015\pi$
$659$	19800.0	1.17041	0.585204	+	0.810886i	$0.301015\pi$
$660$	0	0
$661$	27110.0	1.59524	0.797622	−	0.603157i	$-0.206091\pi$
$661$	27110.0	1.59524	0.797622	+	0.603157i	$0.206091\pi$
$662$	0	0
$663$	−12000.0	−0.702928
$664$	0	0
$665$	−6400.00	−0.373205
$666$	0	0
$667$	1632.00	0.0947396
$668$	0	0
$669$	34304.0	1.98247
$670$	0	0
$671$	1200.00	0.0690395
$672$	0	0
$673$	32210.0	1.84488	0.922440	−	0.386140i	$-0.126192\pi$
$673$	32210.0	1.84488	0.922440	+	0.386140i	$0.126192\pi$
$674$	0	0
$675$	2000.00	0.114044
$676$	0	0
$677$	27190.0	1.54357	0.771785	−	0.635884i	$-0.219364\pi$
$677$	27190.0	1.54357	0.771785	+	0.635884i	$0.219364\pi$
$678$	0	0
$679$	1760.00	0.0994736
$680$	0	0
$681$	−51648.0	−2.90625
$682$	0	0
$683$	20328.0	1.13884	0.569421	−	0.822046i	$-0.307167\pi$
$683$	20328.0	1.13884	0.569421	+	0.822046i	$0.307167\pi$
$684$	0	0
$685$	−5700.00	−0.317935
$686$	0	0
$687$	8528.00	0.473600
$688$	0	0
$689$	20500.0	1.13351
$690$	0	0
$691$	−12520.0	−0.689267	−0.344633	−	0.938737i	$-0.611997\pi$
$691$	−12520.0	−0.689267	−0.344633	+	0.938737i	$0.611997\pi$
$692$	0	0
$693$	−23680.0	−1.29802
$694$	0	0
$695$	−19600.0	−1.06974
$696$	0	0
$697$	−12300.0	−0.668430
$698$	0	0
$699$	47280.0	2.55836
$700$	0	0
$701$	11550.0	0.622307	0.311154	−	0.950360i	$-0.399285\pi$
$701$	11550.0	0.622307	0.311154	+	0.950360i	$0.399285\pi$
$702$	0	0
$703$	−12400.0	−0.665256
$704$	0	0
$705$	33280.0	1.77787
$706$	0	0
$707$	17568.0	0.934530
$708$	0	0
$709$	−34154.0	−1.80914	−0.904570	−	0.426325i	$-0.859808\pi$
$709$	−34154.0	−1.80914	−0.904570	+	0.426325i	$0.859808\pi$
$710$	0	0
$711$	41440.0	2.18582
$712$	0	0
$713$	15360.0	0.806783
$714$	0	0
$715$	20000.0	1.04609
$716$	0	0
$717$	−26880.0	−1.40007
$718$	0	0
$719$	22880.0	1.18676	0.593380	−	0.804923i	$-0.297793\pi$
$719$	22880.0	1.18676	0.593380	+	0.804923i	$0.297793\pi$
$720$	0	0
$721$	−768.000	−0.0396696
$722$	0	0
$723$	−31760.0	−1.63370
$724$	0	0
$725$	850.000	0.0435424
$726$	0	0
$727$	−10416.0	−0.531373	−0.265686	−	0.964060i	$-0.585599\pi$
$727$	−10416.0	−0.531373	−0.265686	+	0.964060i	$0.585599\pi$
$728$	0	0
$729$	−30563.0	−1.55276
$730$	0	0
$731$	4560.00	0.230722
$732$	0	0
$733$	14750.0	0.743252	0.371626	−	0.928383i	$-0.378800\pi$
$733$	14750.0	0.743252	0.371626	+	0.928383i	$0.378800\pi$
$734$	0	0
$735$	−6960.00	−0.349284
$736$	0	0
$737$	−31040.0	−1.55139
$738$	0	0
$739$	2360.00	0.117475	0.0587375	−	0.998273i	$-0.481293\pi$
$739$	2360.00	0.117475	0.0587375	+	0.998273i	$0.481293\pi$
$740$	0	0
$741$	−16000.0	−0.793218
$742$	0	0
$743$	−32208.0	−1.59031	−0.795153	−	0.606409i	$-0.792609\pi$
$743$	−32208.0	−1.59031	−0.795153	+	0.606409i	$0.792609\pi$
$744$	0	0
$745$	20100.0	0.988466
$746$	0	0
$747$	−20424.0	−1.00037
$748$	0	0
$749$	10624.0	0.518281
$750$	0	0
$751$	36640.0	1.78031	0.890155	−	0.455658i	$-0.150596\pi$
$751$	36640.0	1.78031	0.890155	+	0.455658i	$0.150596\pi$
$752$	0	0
$753$	54720.0	2.64822
$754$	0	0
$755$	−7200.00	−0.347066
$756$	0	0
$757$	−12090.0	−0.580474	−0.290237	−	0.956955i	$-0.593734\pi$
$757$	−12090.0	−0.580474	−0.290237	+	0.956955i	$0.593734\pi$
$758$	0	0
$759$	15360.0	0.734562
$760$	0	0
$761$	−3318.00	−0.158052	−0.0790259	−	0.996873i	$-0.525181\pi$
$761$	−3318.00	−0.158052	−0.0790259	+	0.996873i	$0.525181\pi$
$762$	0	0
$763$	5920.00	0.280889
$764$	0	0
$765$	11100.0	0.524603
$766$	0	0
$767$	−10000.0	−0.470768
$768$	0	0
$769$	11506.0	0.539554	0.269777	−	0.962923i	$-0.413050\pi$
$769$	11506.0	0.539554	0.269777	+	0.962923i	$0.413050\pi$
$770$	0	0
$771$	−36880.0	−1.72270
$772$	0	0
$773$	22230.0	1.03436	0.517178	−	0.855878i	$-0.326982\pi$
$773$	22230.0	1.03436	0.517178	+	0.855878i	$0.326982\pi$
$774$	0	0
$775$	8000.00	0.370798
$776$	0	0
$777$	39680.0	1.83206
$778$	0	0
$779$	−16400.0	−0.754289
$780$	0	0
$781$	−16000.0	−0.733067
$782$	0	0
$783$	2720.00	0.124144
$784$	0	0
$785$	−17900.0	−0.813858
$786$	0	0
$787$	21336.0	0.966387	0.483193	−	0.875514i	$-0.339477\pi$
$787$	21336.0	0.966387	0.483193	+	0.875514i	$0.339477\pi$
$788$	0	0
$789$	−38784.0	−1.75000
$790$	0	0
$791$	−23840.0	−1.07162
$792$	0	0
$793$	−1500.00	−0.0671709
$794$	0	0
$795$	−32800.0	−1.46327
$796$	0	0
$797$	−7170.00	−0.318663	−0.159332	−	0.987225i	$-0.550934\pi$
$797$	−7170.00	−0.318663	−0.159332	+	0.987225i	$0.550934\pi$
$798$	0	0
$799$	−12480.0	−0.552579
$800$	0	0
$801$	−12062.0	−0.532072
$802$	0	0
$803$	−25200.0	−1.10746
$804$	0	0
$805$	−7680.00	−0.336254
$806$	0	0
$807$	−44400.0	−1.93675
$808$	0	0
$809$	−23654.0	−1.02797	−0.513987	−	0.857798i	$-0.671832\pi$
$809$	−23654.0	−1.02797	−0.513987	+	0.857798i	$0.671832\pi$
$810$	0	0
$811$	30440.0	1.31799	0.658997	−	0.752146i	$-0.270981\pi$
$811$	30440.0	1.31799	0.658997	+	0.752146i	$0.270981\pi$
$812$	0	0
$813$	−3840.00	−0.165652
$814$	0	0
$815$	−12080.0	−0.519195
$816$	0	0
$817$	6080.00	0.260358
$818$	0	0
$819$	29600.0	1.26289
$820$	0	0
$821$	−19930.0	−0.847213	−0.423606	−	0.905846i	$-0.639236\pi$
$821$	−19930.0	−0.847213	−0.423606	+	0.905846i	$0.639236\pi$
$822$	0	0
$823$	9872.00	0.418124	0.209062	−	0.977902i	$-0.432959\pi$
$823$	9872.00	0.418124	0.209062	+	0.977902i	$0.432959\pi$
$824$	0	0
$825$	8000.00	0.337605
$826$	0	0
$827$	5704.00	0.239840	0.119920	−	0.992784i	$-0.461736\pi$
$827$	5704.00	0.239840	0.119920	+	0.992784i	$0.461736\pi$
$828$	0	0
$829$	27230.0	1.14082	0.570408	−	0.821361i	$-0.306785\pi$
$829$	27230.0	1.14082	0.570408	+	0.821361i	$0.306785\pi$
$830$	0	0
$831$	−8240.00	−0.343974
$832$	0	0
$833$	2610.00	0.108561
$834$	0	0
$835$	28960.0	1.20024
$836$	0	0
$837$	25600.0	1.05719
$838$	0	0
$839$	18800.0	0.773597	0.386799	−	0.922164i	$-0.373581\pi$
$839$	18800.0	0.773597	0.386799	+	0.922164i	$0.373581\pi$
$840$	0	0
$841$	−23233.0	−0.952602
$842$	0	0
$843$	26160.0	1.06880
$844$	0	0
$845$	−3030.00	−0.123355
$846$	0	0
$847$	−4304.00	−0.174601
$848$	0	0
$849$	17344.0	0.701113
$850$	0	0
$851$	−14880.0	−0.599389
$852$	0	0
$853$	−12090.0	−0.485292	−0.242646	−	0.970115i	$-0.578015\pi$
$853$	−12090.0	−0.485292	−0.242646	+	0.970115i	$0.578015\pi$
$854$	0	0
$855$	14800.0	0.591988
$856$	0	0
$857$	−470.000	−0.0187338	−0.00936692	−	0.999956i	$-0.502982\pi$
$857$	−470.000	−0.0187338	−0.00936692	+	0.999956i	$0.502982\pi$
$858$	0	0
$859$	−24440.0	−0.970759	−0.485380	−	0.874304i	$-0.661319\pi$
$859$	−24440.0	−0.970759	−0.485380	+	0.874304i	$0.661319\pi$
$860$	0	0
$861$	52480.0	2.07725
$862$	0	0
$863$	22592.0	0.891125	0.445562	−	0.895251i	$-0.353004\pi$
$863$	22592.0	0.891125	0.445562	+	0.895251i	$0.353004\pi$
$864$	0	0
$865$	−7500.00	−0.294807
$866$	0	0
$867$	32104.0	1.25757
$868$	0	0
$869$	44800.0	1.74883
$870$	0	0
$871$	38800.0	1.50940
$872$	0	0
$873$	−4070.00	−0.157788
$874$	0	0
$875$	−24000.0	−0.927255
$876$	0	0
$877$	−17330.0	−0.667266	−0.333633	−	0.942703i	$-0.608275\pi$
$877$	−17330.0	−0.667266	−0.333633	+	0.942703i	$0.608275\pi$
$878$	0	0
$879$	−16560.0	−0.635444
$880$	0	0
$881$	−31470.0	−1.20346	−0.601732	−	0.798698i	$-0.705522\pi$
$881$	−31470.0	−1.20346	−0.601732	+	0.798698i	$0.705522\pi$
$882$	0	0
$883$	3352.00	0.127751	0.0638753	−	0.997958i	$-0.479654\pi$
$883$	3352.00	0.127751	0.0638753	+	0.997958i	$0.479654\pi$
$884$	0	0
$885$	16000.0	0.607722
$886$	0	0
$887$	48144.0	1.82245	0.911227	−	0.411904i	$-0.135136\pi$
$887$	48144.0	1.82245	0.911227	+	0.411904i	$0.135136\pi$
$888$	0	0
$889$	−16384.0	−0.618112
$890$	0	0
$891$	−14360.0	−0.539931
$892$	0	0
$893$	−16640.0	−0.623557
$894$	0	0
$895$	22800.0	0.851531
$896$	0	0
$897$	−19200.0	−0.714682
$898$	0	0
$899$	10880.0	0.403636
$900$	0	0
$901$	12300.0	0.454797
$902$	0	0
$903$	−19456.0	−0.717005
$904$	0	0
$905$	4420.00	0.162349
$906$	0	0
$907$	−16216.0	−0.593653	−0.296827	−	0.954931i	$-0.595928\pi$
$907$	−16216.0	−0.593653	−0.296827	+	0.954931i	$0.595928\pi$
$908$	0	0
$909$	−40626.0	−1.48238
$910$	0	0
$911$	−49440.0	−1.79805	−0.899023	−	0.437901i	$-0.855722\pi$
$911$	−49440.0	−1.79805	−0.899023	+	0.437901i	$0.855722\pi$
$912$	0	0
$913$	−22080.0	−0.800374
$914$	0	0
$915$	2400.00	0.0867121
$916$	0	0
$917$	18560.0	0.668381
$918$	0	0
$919$	16080.0	0.577182	0.288591	−	0.957452i	$-0.406813\pi$
$919$	16080.0	0.577182	0.288591	+	0.957452i	$0.406813\pi$
$920$	0	0
$921$	15168.0	0.542674
$922$	0	0
$923$	20000.0	0.713226
$924$	0	0
$925$	−7750.00	−0.275479
$926$	0	0
$927$	1776.00	0.0629250
$928$	0	0
$929$	−11310.0	−0.399428	−0.199714	−	0.979854i	$-0.564001\pi$
$929$	−11310.0	−0.399428	−0.199714	+	0.979854i	$0.564001\pi$
$930$	0	0
$931$	3480.00	0.122505
$932$	0	0
$933$	−13440.0	−0.471603
$934$	0	0
$935$	12000.0	0.419724
$936$	0	0
$937$	25130.0	0.876159	0.438080	−	0.898936i	$-0.355659\pi$
$937$	25130.0	0.876159	0.438080	+	0.898936i	$0.355659\pi$
$938$	0	0
$939$	−7760.00	−0.269689
$940$	0	0
$941$	−22322.0	−0.773301	−0.386651	−	0.922226i	$-0.626368\pi$
$941$	−22322.0	−0.773301	−0.386651	+	0.922226i	$0.626368\pi$
$942$	0	0
$943$	−19680.0	−0.679607
$944$	0	0
$945$	−12800.0	−0.440618
$946$	0	0
$947$	−36456.0	−1.25096	−0.625481	−	0.780239i	$-0.715097\pi$
$947$	−36456.0	−1.25096	−0.625481	+	0.780239i	$0.715097\pi$
$948$	0	0
$949$	31500.0	1.07749
$950$	0	0
$951$	−57840.0	−1.97223
$952$	0	0
$953$	40650.0	1.38172	0.690862	−	0.722987i	$-0.257231\pi$
$953$	40650.0	1.38172	0.690862	+	0.722987i	$0.257231\pi$
$954$	0	0
$955$	−19200.0	−0.650573
$956$	0	0
$957$	10880.0	0.367503
$958$	0	0
$959$	−9120.00	−0.307091
$960$	0	0
$961$	72609.0	2.43728
$962$	0	0
$963$	−24568.0	−0.822111
$964$	0	0
$965$	50700.0	1.69129
$966$	0	0
$967$	−34704.0	−1.15409	−0.577045	−	0.816712i	$-0.695794\pi$
$967$	−34704.0	−1.15409	−0.577045	+	0.816712i	$0.695794\pi$
$968$	0	0
$969$	−9600.00	−0.318263
$970$	0	0
$971$	30760.0	1.01662	0.508309	−	0.861175i	$-0.330271\pi$
$971$	30760.0	1.01662	0.508309	+	0.861175i	$0.330271\pi$
$972$	0	0
$973$	−31360.0	−1.03325
$974$	0	0
$975$	−10000.0	−0.328468
$976$	0	0
$977$	−38110.0	−1.24795	−0.623975	−	0.781444i	$-0.714483\pi$
$977$	−38110.0	−1.24795	−0.623975	+	0.781444i	$0.714483\pi$
$978$	0	0
$979$	−13040.0	−0.425700
$980$	0	0
$981$	−13690.0	−0.445554
$982$	0	0
$983$	−19632.0	−0.636992	−0.318496	−	0.947924i	$-0.603178\pi$
$983$	−19632.0	−0.636992	−0.318496	+	0.947924i	$0.603178\pi$
$984$	0	0
$985$	−19100.0	−0.617844
$986$	0	0
$987$	53248.0	1.71723
$988$	0	0
$989$	7296.00	0.234580
$990$	0	0
$991$	47680.0	1.52836	0.764180	−	0.645003i	$-0.223144\pi$
$991$	47680.0	1.52836	0.764180	+	0.645003i	$0.223144\pi$
$992$	0	0
$993$	−46400.0	−1.48284
$994$	0	0
$995$	29600.0	0.943099
$996$	0	0
$997$	−39690.0	−1.26078	−0.630389	−	0.776280i	$-0.717104\pi$
$997$	−39690.0	−1.26078	−0.630389	+	0.776280i	$0.717104\pi$
$998$	0	0
$999$	−24800.0	−0.785423

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.4.a.a.1.1	✓	1
3.2	odd	2		288.4.a.h.1.1		1
4.3	odd	2		32.4.a.c.1.1	yes	1
5.2	odd	4		800.4.c.b.449.2		2
5.3	odd	4		800.4.c.b.449.1		2
5.4	even	2		800.4.a.k.1.1		1
7.6	odd	2		1568.4.a.o.1.1		1
8.3	odd	2		64.4.a.a.1.1		1
8.5	even	2		64.4.a.e.1.1		1
12.11	even	2		288.4.a.i.1.1		1
16.3	odd	4		256.4.b.c.129.2		2
16.5	even	4		256.4.b.e.129.2		2
16.11	odd	4		256.4.b.c.129.1		2
16.13	even	4		256.4.b.e.129.1		2
20.3	even	4		800.4.c.a.449.2		2
20.7	even	4		800.4.c.a.449.1		2
20.19	odd	2		800.4.a.a.1.1		1
24.5	odd	2		576.4.a.g.1.1		1
24.11	even	2		576.4.a.h.1.1		1
28.27	even	2		1568.4.a.c.1.1		1
40.19	odd	2		1600.4.a.bw.1.1		1
40.29	even	2		1600.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.4.a.a.1.1	✓	1	1.1	even	1	trivial
32.4.a.c.1.1	yes	1	4.3	odd	2
64.4.a.a.1.1		1	8.3	odd	2
64.4.a.e.1.1		1	8.5	even	2
256.4.b.c.129.1		2	16.11	odd	4
256.4.b.c.129.2		2	16.3	odd	4
256.4.b.e.129.1		2	16.13	even	4
256.4.b.e.129.2		2	16.5	even	4
288.4.a.h.1.1		1	3.2	odd	2
288.4.a.i.1.1		1	12.11	even	2
576.4.a.g.1.1		1	24.5	odd	2
576.4.a.h.1.1		1	24.11	even	2
800.4.a.a.1.1		1	20.19	odd	2
800.4.a.k.1.1		1	5.4	even	2
800.4.c.a.449.1		2	20.7	even	4
800.4.c.a.449.2		2	20.3	even	4
800.4.c.b.449.1		2	5.3	odd	4
800.4.c.b.449.2		2	5.2	odd	4
1568.4.a.c.1.1		1	28.27	even	2
1568.4.a.o.1.1		1	7.6	odd	2
1600.4.a.e.1.1		1	40.29	even	2
1600.4.a.bw.1.1		1	40.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	32.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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