LMFDB - Embedded newform 6384.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6384,2,Mod(1,6384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6384, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6384.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6384.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:	$50.9764966504$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1596)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} -6.00000 q^{13} +2.00000 q^{15} +2.00000 q^{17} -1.00000 q^{19} +1.00000 q^{21} -6.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} -8.00000 q^{29} +8.00000 q^{31} -2.00000 q^{33} +2.00000 q^{35} -10.0000 q^{37} +6.00000 q^{39} +8.00000 q^{41} +8.00000 q^{43} -2.00000 q^{45} -2.00000 q^{47} +1.00000 q^{49} -2.00000 q^{51} -8.00000 q^{53} -4.00000 q^{55} +1.00000 q^{57} -6.00000 q^{61} -1.00000 q^{63} +12.0000 q^{65} +4.00000 q^{67} +6.00000 q^{69} +6.00000 q^{73} +1.00000 q^{75} -2.00000 q^{77} +1.00000 q^{81} -14.0000 q^{83} -4.00000 q^{85} +8.00000 q^{87} -16.0000 q^{89} +6.00000 q^{91} -8.00000 q^{93} +2.00000 q^{95} -2.00000 q^{97} +2.00000 q^{99} +O(q^{100})$

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	6.00000	0.960769
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	−8.00000	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	12.0000	1.48842
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	0	0
$87$	8.00000	0.857690
$88$	0	0
$89$	−16.0000	−1.69600	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−16.0000	−1.69600	−0.847998	+	0.529999i	$0.822192\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	12.0000	1.11901
$116$	0	0
$117$	−6.00000	−0.554700
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−8.00000	−0.721336
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	2.00000	0.172133
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	2.00000	0.168430
$142$	0	0
$143$	−12.0000	−1.00349
$144$	0	0
$145$	16.0000	1.32873
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	−16.0000	−1.28515
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	8.00000	0.634441
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	4.00000	0.311400
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−8.00000	−0.608229	−0.304114	−	0.952636i	$-0.598361\pi$
$173$	−8.00000	−0.608229	−0.304114	+	0.952636i	$0.598361\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	26.0000	1.93256	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000	1.93256	0.966282	+	0.257485i	$0.0828937\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	20.0000	1.47043
$186$	0	0
$187$	4.00000	0.292509
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	14.0000	1.01300	0.506502	−	0.862239i	$-0.330938\pi$
$191$	14.0000	1.01300	0.506502	+	0.862239i	$0.330938\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	−12.0000	−0.859338
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	8.00000	0.561490
$204$	0	0
$205$	−16.0000	−1.11749
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−16.0000	−1.09119
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	−6.00000	−0.405442
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.0000	1.42306	0.711531	−	0.702655i	$-0.248002\pi$
$239$	22.0000	1.42306	0.711531	+	0.702655i	$0.248002\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.00000	−0.127775
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	14.0000	0.887214
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	0	0
$255$	4.00000	0.250490
$256$	0	0
$257$	28.0000	1.74659	0.873296	−	0.487190i	$-0.161978\pi$
$257$	28.0000	1.74659	0.873296	+	0.487190i	$0.161978\pi$
$258$	0	0
$259$	10.0000	0.621370
$260$	0	0
$261$	−8.00000	−0.495188
$262$	0	0
$263$	−10.0000	−0.616626	−0.308313	−	0.951285i	$-0.599764\pi$
$263$	−10.0000	−0.616626	−0.308313	+	0.951285i	$0.599764\pi$
$264$	0	0
$265$	16.0000	0.982872
$266$	0	0
$267$	16.0000	0.979184
$268$	0	0
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	28.0000	1.70088	0.850439	−	0.526073i	$-0.176336\pi$
$271$	28.0000	1.70088	0.850439	+	0.526073i	$0.176336\pi$
$272$	0	0
$273$	−6.00000	−0.363137
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	−8.00000	−0.472225
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	24.0000	1.40209	0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000	1.40209	0.701047	+	0.713115i	$0.252716\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	36.0000	2.08193
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	−18.0000	−1.03407
$304$	0	0
$305$	12.0000	0.687118
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	22.0000	1.24751	0.623753	−	0.781622i	$-0.285607\pi$
$311$	22.0000	1.24751	0.623753	+	0.781622i	$0.285607\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	0	0
$315$	2.00000	0.112687
$316$	0	0
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	0	0
$319$	−16.0000	−0.895828
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	−2.00000	−0.111283
$324$	0	0
$325$	6.00000	0.332820
$326$	0	0
$327$	6.00000	0.331801
$328$	0	0
$329$	2.00000	0.110264
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	−10.0000	−0.547997
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−34.0000	−1.85210	−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−34.0000	−1.85210	−0.926049	+	0.377403i	$0.876817\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	16.0000	0.866449
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−12.0000	−0.646058
$346$	0	0
$347$	−14.0000	−0.751559	−0.375780	−	0.926709i	$-0.622625\pi$
$347$	−14.0000	−0.751559	−0.375780	+	0.926709i	$0.622625\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.00000	0.105851
$358$	0	0
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	−12.0000	−0.628109
$366$	0	0
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	0	0
$369$	8.00000	0.416463
$370$	0	0
$371$	8.00000	0.415339
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	0	0
$377$	48.0000	2.47213
$378$	0	0
$379$	−36.0000	−1.84920	−0.924598	−	0.380945i	$-0.875599\pi$
$379$	−36.0000	−1.84920	−0.924598	+	0.380945i	$0.875599\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	0	0
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	0	0
$387$	8.00000	0.406663
$388$	0	0
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	−6.00000	−0.302660
$394$	0	0
$395$	0	0
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	−48.0000	−2.39105
$404$	0	0
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	−20.0000	−0.991363
$408$	0	0
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	0	0
$413$	0	0
$414$	0	0
$415$	28.0000	1.37447
$416$	0	0
$417$	4.00000	0.195881
$418$	0	0
$419$	−10.0000	−0.488532	−0.244266	−	0.969708i	$-0.578547\pi$
$419$	−10.0000	−0.488532	−0.244266	+	0.969708i	$0.578547\pi$
$420$	0	0
$421$	−14.0000	−0.682318	−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000	−0.682318	−0.341159	+	0.940006i	$0.610819\pi$
$422$	0	0
$423$	−2.00000	−0.0972433
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	6.00000	0.290360
$428$	0	0
$429$	12.0000	0.579365
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	−16.0000	−0.767141
$436$	0	0
$437$	6.00000	0.287019
$438$	0	0
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	0	0
$445$	32.0000	1.51695
$446$	0	0
$447$	2.00000	0.0945968
$448$	0	0
$449$	24.0000	1.13263	0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000	1.13263	0.566315	+	0.824189i	$0.308369\pi$
$450$	0	0
$451$	16.0000	0.753411
$452$	0	0
$453$	8.00000	0.375873
$454$	0	0
$455$	−12.0000	−0.562569
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	0	0
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	26.0000	1.21094	0.605470	−	0.795868i	$-0.292985\pi$
$461$	26.0000	1.21094	0.605470	+	0.795868i	$0.292985\pi$
$462$	0	0
$463$	28.0000	1.30127	0.650635	−	0.759390i	$-0.274503\pi$
$463$	28.0000	1.30127	0.650635	+	0.759390i	$0.274503\pi$
$464$	0	0
$465$	16.0000	0.741982
$466$	0	0
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	16.0000	0.735681
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	−8.00000	−0.366295
$478$	0	0
$479$	2.00000	0.0913823	0.0456912	−	0.998956i	$-0.485451\pi$
$479$	2.00000	0.0913823	0.0456912	+	0.998956i	$0.485451\pi$
$480$	0	0
$481$	60.0000	2.73576
$482$	0	0
$483$	−6.00000	−0.273009
$484$	0	0
$485$	4.00000	0.181631
$486$	0	0
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	18.0000	0.812329	0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000	0.812329	0.406164	+	0.913800i	$0.366866\pi$
$492$	0	0
$493$	−16.0000	−0.720604
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	0	0
$498$	0	0
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	0	0
$503$	2.00000	0.0891756	0.0445878	−	0.999005i	$-0.485803\pi$
$503$	2.00000	0.0891756	0.0445878	+	0.999005i	$0.485803\pi$
$504$	0	0
$505$	−36.0000	−1.60198
$506$	0	0
$507$	−23.0000	−1.02147
$508$	0	0
$509$	−24.0000	−1.06378	−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000	−1.06378	−0.531891	+	0.846813i	$0.678518\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−16.0000	−0.705044
$516$	0	0
$517$	−4.00000	−0.175920
$518$	0	0
$519$	8.00000	0.351161
$520$	0	0
$521$	8.00000	0.350486	0.175243	−	0.984525i	$-0.443929\pi$
$521$	8.00000	0.350486	0.175243	+	0.984525i	$0.443929\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	16.0000	0.696971
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−48.0000	−2.07911
$534$	0	0
$535$	−24.0000	−1.03761
$536$	0	0
$537$	−4.00000	−0.172613
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	−26.0000	−1.11577
$544$	0	0
$545$	12.0000	0.514024
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	−6.00000	−0.256074
$550$	0	0
$551$	8.00000	0.340811
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−20.0000	−0.848953
$556$	0	0
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	0	0
$559$	−48.0000	−2.03018
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	0	0
$565$	24.0000	1.00969
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−32.0000	−1.34151	−0.670755	−	0.741679i	$-0.734030\pi$
$569$	−32.0000	−1.34151	−0.670755	+	0.741679i	$0.734030\pi$
$570$	0	0
$571$	24.0000	1.00437	0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000	1.00437	0.502184	+	0.864761i	$0.332530\pi$
$572$	0	0
$573$	−14.0000	−0.584858
$574$	0	0
$575$	6.00000	0.250217
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	0	0
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	14.0000	0.580818
$582$	0	0
$583$	−16.0000	−0.662652
$584$	0	0
$585$	12.0000	0.496139
$586$	0	0
$587$	−38.0000	−1.56843	−0.784214	−	0.620491i	$-0.786934\pi$
$587$	−38.0000	−1.56843	−0.784214	+	0.620491i	$0.786934\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	−26.0000	−1.06769	−0.533846	−	0.845582i	$-0.679254\pi$
$593$	−26.0000	−1.06769	−0.533846	+	0.845582i	$0.679254\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	0	0
$597$	12.0000	0.491127
$598$	0	0
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	0	0
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	14.0000	0.569181
$606$	0	0
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	34.0000	1.37325	0.686624	−	0.727013i	$-0.259092\pi$
$613$	34.0000	1.37325	0.686624	+	0.727013i	$0.259092\pi$
$614$	0	0
$615$	16.0000	0.645182
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	6.00000	0.240772
$622$	0	0
$623$	16.0000	0.641026
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	2.00000	0.0798723
$628$	0	0
$629$	−20.0000	−0.797452
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	32.0000	1.26988
$636$	0	0
$637$	−6.00000	−0.237729
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	0	0
$645$	16.0000	0.629999
$646$	0	0
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	8.00000	0.313545
$652$	0	0
$653$	−42.0000	−1.64359	−0.821794	−	0.569785i	$-0.807026\pi$
$653$	−42.0000	−1.64359	−0.821794	+	0.569785i	$0.807026\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	−32.0000	−1.24654	−0.623272	−	0.782006i	$-0.714197\pi$
$659$	−32.0000	−1.24654	−0.623272	+	0.782006i	$0.714197\pi$
$660$	0	0
$661$	46.0000	1.78919	0.894596	−	0.446875i	$-0.147463\pi$
$661$	46.0000	1.78919	0.894596	+	0.446875i	$0.147463\pi$
$662$	0	0
$663$	12.0000	0.466041
$664$	0	0
$665$	−2.00000	−0.0775567
$666$	0	0
$667$	48.0000	1.85857
$668$	0	0
$669$	−8.00000	−0.309298
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−28.0000	−1.07613	−0.538064	−	0.842904i	$-0.680844\pi$
$677$	−28.0000	−1.07613	−0.538064	+	0.842904i	$0.680844\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	0	0
$687$	2.00000	0.0763048
$688$	0	0
$689$	48.0000	1.82865
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	0	0
$693$	−2.00000	−0.0759737
$694$	0	0
$695$	8.00000	0.303457
$696$	0	0
$697$	16.0000	0.606043
$698$	0	0
$699$	−18.0000	−0.680823
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	10.0000	0.377157
$704$	0	0
$705$	−4.00000	−0.150649
$706$	0	0
$707$	−18.0000	−0.676960
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−48.0000	−1.79761
$714$	0	0
$715$	24.0000	0.897549
$716$	0	0
$717$	−22.0000	−0.821605
$718$	0	0
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	−22.0000	−0.818189
$724$	0	0
$725$	8.00000	0.297113
$726$	0	0
$727$	4.00000	0.148352	0.0741759	−	0.997245i	$-0.476367\pi$
$727$	4.00000	0.148352	0.0741759	+	0.997245i	$0.476367\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.0000	0.591781
$732$	0	0
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	0	0
$735$	2.00000	0.0737711
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	0	0
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	0	0
$745$	4.00000	0.146549
$746$	0	0
$747$	−14.0000	−0.512233
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	48.0000	1.75154	0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000	1.75154	0.875772	+	0.482724i	$0.160353\pi$
$752$	0	0
$753$	−2.00000	−0.0728841
$754$	0	0
$755$	16.0000	0.582300
$756$	0	0
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	0	0
$759$	12.0000	0.435572
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	6.00000	0.217215
$764$	0	0
$765$	−4.00000	−0.144620
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	−28.0000	−1.00840
$772$	0	0
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	−10.0000	−0.358748
$778$	0	0
$779$	−8.00000	−0.286630
$780$	0	0
$781$	0	0
$782$	0	0
$783$	8.00000	0.285897
$784$	0	0
$785$	28.0000	0.999363
$786$	0	0
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	0	0
$789$	10.0000	0.356009
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	36.0000	1.27840
$794$	0	0
$795$	−16.0000	−0.567462
$796$	0	0
$797$	24.0000	0.850124	0.425062	−	0.905164i	$-0.360252\pi$
$797$	24.0000	0.850124	0.425062	+	0.905164i	$0.360252\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	0	0
$801$	−16.0000	−0.565332
$802$	0	0
$803$	12.0000	0.423471
$804$	0	0
$805$	−12.0000	−0.422944
$806$	0	0
$807$	−24.0000	−0.844840
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	−28.0000	−0.982003
$814$	0	0
$815$	8.00000	0.280228
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	6.00000	0.209657
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	0	0
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	0	0
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	0	0
$831$	26.0000	0.901930
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	−16.0000	−0.553703
$836$	0	0
$837$	−8.00000	−0.276520
$838$	0	0
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	−8.00000	−0.275535
$844$	0	0
$845$	−46.0000	−1.58245
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	−16.0000	−0.549119
$850$	0	0
$851$	60.0000	2.05677
$852$	0	0
$853$	−42.0000	−1.43805	−0.719026	−	0.694983i	$-0.755412\pi$
$853$	−42.0000	−1.43805	−0.719026	+	0.694983i	$0.755412\pi$
$854$	0	0
$855$	2.00000	0.0683986
$856$	0	0
$857$	−32.0000	−1.09310	−0.546550	−	0.837427i	$-0.684059\pi$
$857$	−32.0000	−1.09310	−0.546550	+	0.837427i	$0.684059\pi$
$858$	0	0
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	0	0
$861$	8.00000	0.272639
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	16.0000	0.544016
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	−12.0000	−0.405674
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	0	0
$879$	−24.0000	−0.809500
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	2.00000	0.0669274
$894$	0	0
$895$	−8.00000	−0.267411
$896$	0	0
$897$	−36.0000	−1.20201
$898$	0	0
$899$	−64.0000	−2.13452
$900$	0	0
$901$	−16.0000	−0.533037
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	−52.0000	−1.72854
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	0	0
$909$	18.0000	0.597022
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	−28.0000	−0.926665
$914$	0	0
$915$	−12.0000	−0.396708
$916$	0	0
$917$	−6.00000	−0.198137
$918$	0	0
$919$	48.0000	1.58337	0.791687	−	0.610927i	$-0.209203\pi$
$919$	48.0000	1.58337	0.791687	+	0.610927i	$0.209203\pi$
$920$	0	0
$921$	28.0000	0.922631
$922$	0	0
$923$	0	0
$924$	0	0
$925$	10.0000	0.328798
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	46.0000	1.50921	0.754606	−	0.656179i	$-0.227828\pi$
$929$	46.0000	1.50921	0.754606	+	0.656179i	$0.227828\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	−22.0000	−0.720248
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	0	0
$939$	−22.0000	−0.717943
$940$	0	0
$941$	16.0000	0.521585	0.260793	−	0.965395i	$-0.416016\pi$
$941$	16.0000	0.521585	0.260793	+	0.965395i	$0.416016\pi$
$942$	0	0
$943$	−48.0000	−1.56310
$944$	0	0
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	50.0000	1.62478	0.812391	−	0.583113i	$-0.198166\pi$
$947$	50.0000	1.62478	0.812391	+	0.583113i	$0.198166\pi$
$948$	0	0
$949$	−36.0000	−1.16861
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	0	0
$955$	−28.0000	−0.906059
$956$	0	0
$957$	16.0000	0.517207
$958$	0	0
$959$	−6.00000	−0.193750
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	12.0000	0.386695
$964$	0	0
$965$	−4.00000	−0.128765
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	0	0
$969$	2.00000	0.0642493
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	0	0
$973$	4.00000	0.128234
$974$	0	0
$975$	−6.00000	−0.192154
$976$	0	0
$977$	−28.0000	−0.895799	−0.447900	−	0.894084i	$-0.647828\pi$
$977$	−28.0000	−0.895799	−0.447900	+	0.894084i	$0.647828\pi$
$978$	0	0
$979$	−32.0000	−1.02272
$980$	0	0
$981$	−6.00000	−0.191565
$982$	0	0
$983$	32.0000	1.02064	0.510321	−	0.859984i	$-0.329527\pi$
$983$	32.0000	1.02064	0.510321	+	0.859984i	$0.329527\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	0	0
$987$	−2.00000	−0.0636607
$988$	0	0
$989$	−48.0000	−1.52631
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	0	0
$995$	24.0000	0.760851
$996$	0	0
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.d.1.1		1
4.3	odd	2		1596.2.a.d.1.1	✓	1
12.11	even	2		4788.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1596.2.a.d.1.1	✓	1	4.3	odd	2
4788.2.a.f.1.1		1	12.11	even	2
6384.2.a.d.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6384.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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