LMFDB - Embedded newform 1568.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,4,Mod(1,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1568.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1568.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$92.5149948890$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1568.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-8.00000 q^{3} +10.0000 q^{5} +37.0000 q^{9} +O(q^{10})$

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

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	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	$-0.352416\pi$
$5$	10.0000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	37.0000	1.37037
$10$	0	0
$11$	−40.0000	−1.09640	−0.548202	−	0.836346i	$-0.684688\pi$
$11$	−40.0000	−1.09640	−0.548202	+	0.836346i	$0.684688\pi$
$12$	0	0
$13$	50.0000	1.06673	0.533366	−	0.845885i	$-0.320927\pi$
$13$	50.0000	1.06673	0.533366	+	0.845885i	$0.320927\pi$
$14$	0	0
$15$	−80.0000	−1.37706
$16$	0	0
$17$	30.0000	0.428004	0.214002	−	0.976833i	$-0.431350\pi$
$17$	30.0000	0.428004	0.214002	+	0.976833i	$0.431350\pi$
$18$	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
$21$	0	0
$22$	0	0
$23$	48.0000	0.435161	0.217580	−	0.976042i	$-0.430184\pi$
$23$	48.0000	0.435161	0.217580	+	0.976042i	$0.430184\pi$
$24$	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
$35$	0	0
$36$	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	$-0.785223\pi$
$41$	−410.000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	152.000	0.539065	0.269532	−	0.962991i	$-0.413131\pi$
$43$	152.000	0.539065	0.269532	+	0.962991i	$0.413131\pi$
$44$	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
$49$	0	0
$50$	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	$-0.510023\pi$
$61$	−30.0000	−0.0629690	−0.0314845	+	0.999504i	$0.510023\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	500.000	0.954113
$66$	0	0
$67$	776.000	1.41498	0.707489	−	0.706725i	$-0.249828\pi$
$67$	776.000	1.41498	0.707489	+	0.706725i	$0.249828\pi$
$68$	0	0
$69$	−384.000	−0.669973
$70$	0	0
$71$	400.000	0.668609	0.334305	−	0.942465i	$-0.391499\pi$
$71$	400.000	0.668609	0.334305	+	0.942465i	$0.391499\pi$
$72$	0	0
$73$	630.000	1.01008	0.505041	−	0.863096i	$-0.331478\pi$
$73$	630.000	1.01008	0.505041	+	0.863096i	$0.331478\pi$
$74$	0	0
$75$	200.000	0.307920
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1120.00	−1.59506	−0.797531	−	0.603278i	$-0.793861\pi$
$79$	−1120.00	−1.59506	−0.797531	+	0.603278i	$0.793861\pi$
$80$	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	$-0.437810\pi$
$89$	326.000	0.388269	0.194134	+	0.980975i	$0.437810\pi$
$90$	0	0
$91$	0	0
$92$	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	$-0.481664\pi$
$97$	110.000	0.115142	0.0575712	+	0.998341i	$0.481664\pi$
$98$	0	0
$99$	−1480.00	−1.50248
$100$	0	0
$101$	1098.00	1.08173	0.540867	−	0.841108i	$-0.318096\pi$
$101$	1098.00	1.08173	0.540867	+	0.841108i	$0.318096\pi$
$102$	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
$105$	0	0
$106$	0	0
$107$	664.000	0.599919	0.299959	−	0.953952i	$-0.403027\pi$
$107$	664.000	0.599919	0.299959	+	0.953952i	$0.403027\pi$
$108$	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
$119$	0	0
$120$	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	$-0.616452\pi$
$127$	−1024.00	−0.715475	−0.357737	+	0.933822i	$0.616452\pi$
$128$	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
$133$	0	0
$134$	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
$147$	0	0
$148$	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	$-0.562151\pi$
$151$	−720.000	−0.388032	−0.194016	+	0.980998i	$0.562151\pi$
$152$	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	$-0.650347\pi$
$157$	−1790.00	−0.909921	−0.454960	+	0.890512i	$0.650347\pi$
$158$	0	0
$159$	3280.00	1.63598
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−1208.00	−0.580478	−0.290239	−	0.956954i	$-0.593735\pi$
$163$	−1208.00	−0.580478	−0.290239	+	0.956954i	$0.593735\pi$
$164$	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	$-0.552698\pi$
$173$	−750.000	−0.329604	−0.164802	+	0.986327i	$0.552698\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1600.00	−0.679454
$178$	0	0
$179$	2280.00	0.952040	0.476020	−	0.879434i	$-0.342079\pi$
$179$	2280.00	0.952040	0.476020	+	0.879434i	$0.342079\pi$
$180$	0	0
$181$	442.000	0.181512	0.0907558	−	0.995873i	$-0.471072\pi$
$181$	442.000	0.181512	0.0907558	+	0.995873i	$0.471072\pi$
$182$	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
$189$	0	0
$190$	0	0
$191$	−1920.00	−0.727363	−0.363681	−	0.931523i	$-0.618480\pi$
$191$	−1920.00	−0.727363	−0.363681	+	0.931523i	$0.618480\pi$
$192$	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
$203$	0	0
$204$	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	$-0.497923\pi$
$211$	40.0000	0.0130508	0.00652539	+	0.999979i	$0.497923\pi$
$212$	0	0
$213$	−3200.00	−1.02939
$214$	0	0
$215$	1520.00	0.482154
$216$	0	0
$217$	0	0
$218$	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	$-0.450847\pi$
$229$	1066.00	0.307613	0.153806	+	0.988101i	$0.450847\pi$
$230$	0	0
$231$	0	0
$232$	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	$-0.650249\pi$
$239$	−3360.00	−0.909374	−0.454687	+	0.890651i	$0.650249\pi$
$240$	0	0
$241$	−3970.00	−1.06112	−0.530561	−	0.847647i	$-0.678019\pi$
$241$	−3970.00	−1.06112	−0.530561	+	0.847647i	$0.678019\pi$
$242$	0	0
$243$	5032.00	1.32841
$244$	0	0
$245$	0	0
$246$	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	$-0.688993\pi$
$257$	−4610.00	−1.11893	−0.559463	+	0.828855i	$0.688993\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1258.00	−0.298346
$262$	0	0
$263$	−4848.00	−1.13666	−0.568328	−	0.822802i	$-0.692409\pi$
$263$	−4848.00	−1.13666	−0.568328	+	0.822802i	$0.692409\pi$
$264$	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	$-0.716526\pi$
$269$	−5550.00	−1.25795	−0.628977	+	0.777424i	$0.716526\pi$
$270$	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
$273$	0	0
$274$	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
$287$	0	0
$288$	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	$-0.566164\pi$
$293$	−2070.00	−0.412733	−0.206366	+	0.978475i	$0.566164\pi$
$294$	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
$301$	0	0
$302$	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	$-0.527915\pi$
$313$	−970.000	−0.175168	−0.0875841	+	0.996157i	$0.527915\pi$
$314$	0	0
$315$	0	0
$316$	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
$329$	0	0
$330$	0	0
$331$	−5800.00	−0.963132	−0.481566	−	0.876410i	$-0.659932\pi$
$331$	−5800.00	−0.963132	−0.481566	+	0.876410i	$0.659932\pi$
$332$	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
$343$	0	0
$344$	0	0
$345$	−3840.00	−0.599242
$346$	0	0
$347$	376.000	0.0581693	0.0290846	−	0.999577i	$-0.490741\pi$
$347$	376.000	0.0581693	0.0290846	+	0.999577i	$0.490741\pi$
$348$	0	0
$349$	7586.00	1.16352	0.581761	−	0.813360i	$-0.302364\pi$
$349$	7586.00	1.16352	0.581761	+	0.813360i	$0.302364\pi$
$350$	0	0
$351$	−4000.00	−0.608274
$352$	0	0
$353$	−2530.00	−0.381468	−0.190734	−	0.981642i	$-0.561087\pi$
$353$	−2530.00	−0.381468	−0.190734	+	0.981642i	$0.561087\pi$
$354$	0	0
$355$	4000.00	0.598022
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9680.00	1.42309	0.711547	−	0.702638i	$-0.247995\pi$
$359$	9680.00	1.42309	0.711547	+	0.702638i	$0.247995\pi$
$360$	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
$371$	0	0
$372$	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	$-0.462814\pi$
$379$	1720.00	0.233115	0.116557	+	0.993184i	$0.462814\pi$
$380$	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
$385$	0	0
$386$	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	$-0.410916\pi$
$397$	4370.00	0.552453	0.276227	+	0.961093i	$0.410916\pi$
$398$	0	0
$399$	0	0
$400$	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	$-0.314366\pi$
$409$	9110.00	1.10137	0.550685	+	0.834713i	$0.314366\pi$
$410$	0	0
$411$	−4560.00	−0.547271
$412$	0	0
$413$	0	0
$414$	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
$427$	0	0
$428$	0	0
$429$	16000.0	1.80067
$430$	0	0
$431$	13920.0	1.55569	0.777845	−	0.628456i	$-0.216313\pi$
$431$	13920.0	1.55569	0.777845	+	0.628456i	$0.216313\pi$
$432$	0	0
$433$	−4930.00	−0.547161	−0.273580	−	0.961849i	$-0.588208\pi$
$433$	−4930.00	−0.547161	−0.273580	+	0.961849i	$0.588208\pi$
$434$	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
$441$	0	0
$442$	0	0
$443$	−9288.00	−0.996131	−0.498066	−	0.867139i	$-0.665956\pi$
$443$	−9288.00	−0.996131	−0.498066	+	0.867139i	$0.665956\pi$
$444$	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
$455$	0	0
$456$	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	$-0.689823\pi$
$461$	−11118.0	−1.12325	−0.561624	+	0.827393i	$0.689823\pi$
$462$	0	0
$463$	5792.00	0.581376	0.290688	−	0.956818i	$-0.406116\pi$
$463$	5792.00	0.581376	0.290688	+	0.956818i	$0.406116\pi$
$464$	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
$469$	0	0
$470$	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
$483$	0	0
$484$	0	0
$485$	1100.00	0.102986
$486$	0	0
$487$	13264.0	1.23419	0.617094	−	0.786890i	$-0.288310\pi$
$487$	13264.0	1.23419	0.617094	+	0.786890i	$0.288310\pi$
$488$	0	0
$489$	9664.00	0.893704
$490$	0	0
$491$	−4840.00	−0.444860	−0.222430	−	0.974949i	$-0.571399\pi$
$491$	−4840.00	−0.444860	−0.222430	+	0.974949i	$0.571399\pi$
$492$	0	0
$493$	−1020.00	−0.0931815
$494$	0	0
$495$	−14800.0	−1.34386
$496$	0	0
$497$	0	0
$498$	0	0
$499$	19560.0	1.75476	0.877381	−	0.479795i	$-0.159289\pi$
$499$	19560.0	1.75476	0.877381	+	0.479795i	$0.159289\pi$
$500$	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	$-0.175760\pi$
$509$	19554.0	1.70278	0.851391	+	0.524532i	$0.175760\pi$
$510$	0	0
$511$	0	0
$512$	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	$-0.720025\pi$
$521$	−15162.0	−1.27497	−0.637485	+	0.770463i	$0.720025\pi$
$522$	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
$525$	0	0
$526$	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
$539$	0	0
$540$	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	$-0.205768\pi$
$547$	20424.0	1.59647	0.798233	+	0.602348i	$0.205768\pi$
$548$	0	0
$549$	−1110.00	−0.0862908
$550$	0	0
$551$	1360.00	0.105151
$552$	0	0
$553$	0	0
$554$	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
$567$	0	0
$568$	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	$-0.666608\pi$
$571$	−13640.0	−0.999678	−0.499839	+	0.866118i	$0.666608\pi$
$572$	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	$-0.427373\pi$
$577$	6270.00	0.452380	0.226190	+	0.974083i	$0.427373\pi$
$578$	0	0
$579$	40560.0	2.91125
$580$	0	0
$581$	0	0
$582$	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	$-0.560878\pi$
$593$	−5490.00	−0.380181	−0.190090	+	0.981767i	$0.560878\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	23680.0	1.62338
$598$	0	0
$599$	−15440.0	−1.05319	−0.526595	−	0.850116i	$-0.676532\pi$
$599$	−15440.0	−1.05319	−0.526595	+	0.850116i	$0.676532\pi$
$600$	0	0
$601$	−8890.00	−0.603379	−0.301689	−	0.953406i	$-0.597551\pi$
$601$	−8890.00	−0.603379	−0.301689	+	0.953406i	$0.597551\pi$
$602$	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
$609$	0	0
$610$	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
$623$	0	0
$624$	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	$-0.365060\pi$
$631$	13040.0	0.822685	0.411342	+	0.911481i	$0.365060\pi$
$632$	0	0
$633$	−320.000	−0.0200930
$634$	0	0
$635$	−10240.0	−0.639940
$636$	0	0
$637$	0	0
$638$	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
$651$	0	0
$652$	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	$-0.698985\pi$
$659$	−19800.0	−1.17041	−0.585204	+	0.810886i	$0.698985\pi$
$660$	0	0
$661$	−27110.0	−1.59524	−0.797622	−	0.603157i	$-0.793909\pi$
$661$	−27110.0	−1.59524	−0.797622	+	0.603157i	$0.793909\pi$
$662$	0	0
$663$	−12000.0	−0.702928
$664$	0	0
$665$	0	0
$666$	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	$-0.780636\pi$
$677$	−27190.0	−1.54357	−0.771785	+	0.635884i	$0.780636\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−51648.0	−2.90625
$682$	0	0
$683$	−20328.0	−1.13884	−0.569421	−	0.822046i	$-0.692833\pi$
$683$	−20328.0	−1.13884	−0.569421	+	0.822046i	$0.692833\pi$
$684$	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
$693$	0	0
$694$	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
$707$	0	0
$708$	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
$721$	0	0
$722$	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	$-0.621200\pi$
$733$	−14750.0	−0.743252	−0.371626	+	0.928383i	$0.621200\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−31040.0	−1.55139
$738$	0	0
$739$	−2360.00	−0.117475	−0.0587375	−	0.998273i	$-0.518707\pi$
$739$	−2360.00	−0.117475	−0.0587375	+	0.998273i	$0.518707\pi$
$740$	0	0
$741$	16000.0	0.793218
$742$	0	0
$743$	32208.0	1.59031	0.795153	−	0.606409i	$-0.207391\pi$
$743$	32208.0	1.59031	0.795153	+	0.606409i	$0.207391\pi$
$744$	0	0
$745$	−20100.0	−0.988466
$746$	0	0
$747$	−20424.0	−1.00037
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−36640.0	−1.78031	−0.890155	−	0.455658i	$-0.849404\pi$
$751$	−36640.0	−1.78031	−0.890155	+	0.455658i	$0.849404\pi$
$752$	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	$-0.474819\pi$
$761$	3318.00	0.158052	0.0790259	+	0.996873i	$0.474819\pi$
$762$	0	0
$763$	0	0
$764$	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	$-0.586950\pi$
$769$	−11506.0	−0.539554	−0.269777	+	0.962923i	$0.586950\pi$
$770$	0	0
$771$	36880.0	1.72270
$772$	0	0
$773$	−22230.0	−1.03436	−0.517178	−	0.855878i	$-0.673018\pi$
$773$	−22230.0	−1.03436	−0.517178	+	0.855878i	$0.673018\pi$
$774$	0	0
$775$	8000.00	0.370798
$776$	0	0
$777$	0	0
$778$	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
$791$	0	0
$792$	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	$-0.449066\pi$
$797$	7170.00	0.318663	0.159332	+	0.987225i	$0.449066\pi$
$798$	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
$805$	0	0
$806$	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
$819$	0	0
$820$	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	$-0.567041\pi$
$823$	−9872.00	−0.418124	−0.209062	+	0.977902i	$0.567041\pi$
$824$	0	0
$825$	−8000.00	−0.337605
$826$	0	0
$827$	−5704.00	−0.239840	−0.119920	−	0.992784i	$-0.538264\pi$
$827$	−5704.00	−0.239840	−0.119920	+	0.992784i	$0.538264\pi$
$828$	0	0
$829$	−27230.0	−1.14082	−0.570408	−	0.821361i	$-0.693215\pi$
$829$	−27230.0	−1.14082	−0.570408	+	0.821361i	$0.693215\pi$
$830$	0	0
$831$	−8240.00	−0.343974
$832$	0	0
$833$	0	0
$834$	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
$847$	0	0
$848$	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	$-0.421985\pi$
$853$	12090.0	0.485292	0.242646	+	0.970115i	$0.421985\pi$
$854$	0	0
$855$	−14800.0	−0.591988
$856$	0	0
$857$	470.000	0.0187338	0.00936692	−	0.999956i	$-0.497018\pi$
$857$	470.000	0.0187338	0.00936692	+	0.999956i	$0.497018\pi$
$858$	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
$861$	0	0
$862$	0	0
$863$	−22592.0	−0.891125	−0.445562	−	0.895251i	$-0.646996\pi$
$863$	−22592.0	−0.891125	−0.445562	+	0.895251i	$0.646996\pi$
$864$	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
$875$	0	0
$876$	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	$-0.294478\pi$
$881$	31470.0	1.20346	0.601732	+	0.798698i	$0.294478\pi$
$882$	0	0
$883$	−3352.00	−0.127751	−0.0638753	−	0.997958i	$-0.520346\pi$
$883$	−3352.00	−0.127751	−0.0638753	+	0.997958i	$0.520346\pi$
$884$	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
$889$	0	0
$890$	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
$903$	0	0
$904$	0	0
$905$	4420.00	0.162349
$906$	0	0
$907$	16216.0	0.593653	0.296827	−	0.954931i	$-0.404072\pi$
$907$	16216.0	0.593653	0.296827	+	0.954931i	$0.404072\pi$
$908$	0	0
$909$	40626.0	1.48238
$910$	0	0
$911$	49440.0	1.79805	0.899023	−	0.437901i	$-0.144278\pi$
$911$	49440.0	1.79805	0.899023	+	0.437901i	$0.144278\pi$
$912$	0	0
$913$	22080.0	0.800374
$914$	0	0
$915$	2400.00	0.0867121
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−16080.0	−0.577182	−0.288591	−	0.957452i	$-0.593187\pi$
$919$	−16080.0	−0.577182	−0.288591	+	0.957452i	$0.593187\pi$
$920$	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	$-0.435999\pi$
$929$	11310.0	0.399428	0.199714	+	0.979854i	$0.435999\pi$
$930$	0	0
$931$	0	0
$932$	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	$-0.644341\pi$
$937$	−25130.0	−0.876159	−0.438080	+	0.898936i	$0.644341\pi$
$938$	0	0
$939$	7760.00	0.269689
$940$	0	0
$941$	22322.0	0.773301	0.386651	−	0.922226i	$-0.373632\pi$
$941$	22322.0	0.773301	0.386651	+	0.922226i	$0.373632\pi$
$942$	0	0
$943$	−19680.0	−0.679607
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36456.0	1.25096	0.625481	−	0.780239i	$-0.284903\pi$
$947$	36456.0	1.25096	0.625481	+	0.780239i	$0.284903\pi$
$948$	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
$959$	0	0
$960$	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	$-0.304206\pi$
$967$	34704.0	1.15409	0.577045	+	0.816712i	$0.304206\pi$
$968$	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
$973$	0	0
$974$	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
$987$	0	0
$988$	0	0
$989$	7296.00	0.234580
$990$	0	0
$991$	−47680.0	−1.52836	−0.764180	−	0.645003i	$-0.776856\pi$
$991$	−47680.0	−1.52836	−0.764180	+	0.645003i	$0.776856\pi$
$992$	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	$-0.282896\pi$
$997$	39690.0	1.26078	0.630389	+	0.776280i	$0.282896\pi$
$998$	0	0
$999$	−24800.0	−0.785423

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.4.a.c.1.1		1
4.3	odd	2		1568.4.a.o.1.1		1
7.6	odd	2		32.4.a.c.1.1	yes	1
21.20	even	2		288.4.a.i.1.1		1
28.27	even	2		32.4.a.a.1.1	✓	1
35.13	even	4		800.4.c.a.449.2		2
35.27	even	4		800.4.c.a.449.1		2
35.34	odd	2		800.4.a.a.1.1		1
56.13	odd	2		64.4.a.a.1.1		1
56.27	even	2		64.4.a.e.1.1		1
84.83	odd	2		288.4.a.h.1.1		1
112.13	odd	4		256.4.b.c.129.2		2
112.27	even	4		256.4.b.e.129.2		2
112.69	odd	4		256.4.b.c.129.1		2
112.83	even	4		256.4.b.e.129.1		2
140.27	odd	4		800.4.c.b.449.2		2
140.83	odd	4		800.4.c.b.449.1		2
140.139	even	2		800.4.a.k.1.1		1
168.83	odd	2		576.4.a.g.1.1		1
168.125	even	2		576.4.a.h.1.1		1
280.69	odd	2		1600.4.a.bw.1.1		1
280.139	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	28.27	even	2
32.4.a.c.1.1	yes	1	7.6	odd	2
64.4.a.a.1.1		1	56.13	odd	2
64.4.a.e.1.1		1	56.27	even	2
256.4.b.c.129.1		2	112.69	odd	4
256.4.b.c.129.2		2	112.13	odd	4
256.4.b.e.129.1		2	112.83	even	4
256.4.b.e.129.2		2	112.27	even	4
288.4.a.h.1.1		1	84.83	odd	2
288.4.a.i.1.1		1	21.20	even	2
576.4.a.g.1.1		1	168.83	odd	2
576.4.a.h.1.1		1	168.125	even	2
800.4.a.a.1.1		1	35.34	odd	2
800.4.a.k.1.1		1	140.139	even	2
800.4.c.a.449.1		2	35.27	even	4
800.4.c.a.449.2		2	35.13	even	4
800.4.c.b.449.1		2	140.83	odd	4
800.4.c.b.449.2		2	140.27	odd	4
1568.4.a.c.1.1		1	1.1	even	1	trivial
1568.4.a.o.1.1		1	4.3	odd	2
1600.4.a.e.1.1		1	280.139	even	2
1600.4.a.bw.1.1		1	280.69	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1568.a (trivial)

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