LMFDB - Embedded newform 6040.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$0$
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	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} +1.00000 q^{5} +4.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} +1.00000 q^{5} +4.00000 q^{7} -2.00000 q^{9} +4.00000 q^{11} +2.00000 q^{13} +1.00000 q^{15} +3.00000 q^{17} +6.00000 q^{19} +4.00000 q^{21} +4.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} +8.00000 q^{29} -5.00000 q^{31} +4.00000 q^{33} +4.00000 q^{35} +2.00000 q^{37} +2.00000 q^{39} -2.00000 q^{43} -2.00000 q^{45} -9.00000 q^{47} +9.00000 q^{49} +3.00000 q^{51} -7.00000 q^{53} +4.00000 q^{55} +6.00000 q^{57} -12.0000 q^{59} -1.00000 q^{61} -8.00000 q^{63} +2.00000 q^{65} -9.00000 q^{67} +4.00000 q^{69} +6.00000 q^{71} -2.00000 q^{73} +1.00000 q^{75} +16.0000 q^{77} -6.00000 q^{79} +1.00000 q^{81} +3.00000 q^{83} +3.00000 q^{85} +8.00000 q^{87} -6.00000 q^{89} +8.00000 q^{91} -5.00000 q^{93} +6.00000 q^{95} +5.00000 q^{97} -8.00000 q^{99} +O(q^{100})$

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	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	4.00000	0.676123
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	−7.00000	−0.961524	−0.480762	−	0.876851i	$-0.659640\pi$
$53$	−7.00000	−0.961524	−0.480762	+	0.876851i	$0.659640\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	0	0
$63$	−8.00000	−1.00791
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	−9.00000	−1.09952	−0.549762	−	0.835321i	$-0.685282\pi$
$67$	−9.00000	−1.09952	−0.549762	+	0.835321i	$0.685282\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	16.0000	1.82337
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.00000	0.329293	0.164646	−	0.986353i	$-0.447352\pi$
$83$	3.00000	0.329293	0.164646	+	0.986353i	$0.447352\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	0	0
$87$	8.00000	0.857690
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	0	0
$93$	−5.00000	−0.518476
$94$	0	0
$95$	6.00000	0.615587
$96$	0	0
$97$	5.00000	0.507673	0.253837	−	0.967247i	$-0.418307\pi$
$97$	5.00000	0.507673	0.253837	+	0.967247i	$0.418307\pi$
$98$	0	0
$99$	−8.00000	−0.804030
$100$	0	0
$101$	−11.0000	−1.09454	−0.547270	−	0.836956i	$-0.684333\pi$
$101$	−11.0000	−1.09454	−0.547270	+	0.836956i	$0.684333\pi$
$102$	0	0
$103$	−5.00000	−0.492665	−0.246332	−	0.969185i	$-0.579225\pi$
$103$	−5.00000	−0.492665	−0.246332	+	0.969185i	$0.579225\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	20.0000	1.88144	0.940721	−	0.339182i	$-0.110150\pi$
$113$	20.0000	1.88144	0.940721	+	0.339182i	$0.110150\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	−4.00000	−0.369800
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	19.0000	1.68598	0.842989	−	0.537931i	$-0.180794\pi$
$127$	19.0000	1.68598	0.842989	+	0.537931i	$0.180794\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	0	0
$133$	24.0000	2.08106
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	−15.0000	−1.28154	−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000	−1.28154	−0.640768	+	0.767734i	$0.721384\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	−9.00000	−0.757937
$142$	0	0
$143$	8.00000	0.668994
$144$	0	0
$145$	8.00000	0.664364
$146$	0	0
$147$	9.00000	0.742307
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	−5.00000	−0.401610
$156$	0	0
$157$	−19.0000	−1.51637	−0.758183	−	0.652042i	$-0.773912\pi$
$157$	−19.0000	−1.51637	−0.758183	+	0.652042i	$0.773912\pi$
$158$	0	0
$159$	−7.00000	−0.555136
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	21.0000	1.64485	0.822423	−	0.568876i	$-0.192621\pi$
$163$	21.0000	1.64485	0.822423	+	0.568876i	$0.192621\pi$
$164$	0	0
$165$	4.00000	0.311400
$166$	0	0
$167$	−3.00000	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−12.0000	−0.917663
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	−12.0000	−0.901975
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	23.0000	1.70958	0.854788	−	0.518977i	$-0.173687\pi$
$181$	23.0000	1.70958	0.854788	+	0.518977i	$0.173687\pi$
$182$	0	0
$183$	−1.00000	−0.0739221
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	12.0000	0.877527
$188$	0	0
$189$	−20.0000	−1.45479
$190$	0	0
$191$	7.00000	0.506502	0.253251	−	0.967401i	$-0.418500\pi$
$191$	7.00000	0.506502	0.253251	+	0.967401i	$0.418500\pi$
$192$	0	0
$193$	15.0000	1.07972	0.539862	−	0.841754i	$-0.318476\pi$
$193$	15.0000	1.07972	0.539862	+	0.841754i	$0.318476\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	−13.0000	−0.926212	−0.463106	−	0.886303i	$-0.653265\pi$
$197$	−13.0000	−0.926212	−0.463106	+	0.886303i	$0.653265\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	−9.00000	−0.634811
$202$	0	0
$203$	32.0000	2.24596
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−8.00000	−0.556038
$208$	0	0
$209$	24.0000	1.66011
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	6.00000	0.411113
$214$	0	0
$215$	−2.00000	−0.136399
$216$	0	0
$217$	−20.0000	−1.35769
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	0	0
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	0	0
$229$	−24.0000	−1.58596	−0.792982	−	0.609245i	$-0.791473\pi$
$229$	−24.0000	−1.58596	−0.792982	+	0.609245i	$0.791473\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−9.00000	−0.587095
$236$	0	0
$237$	−6.00000	−0.389742
$238$	0	0
$239$	9.00000	0.582162	0.291081	−	0.956698i	$-0.405985\pi$
$239$	9.00000	0.582162	0.291081	+	0.956698i	$0.405985\pi$
$240$	0	0
$241$	5.00000	0.322078	0.161039	−	0.986948i	$-0.448515\pi$
$241$	5.00000	0.322078	0.161039	+	0.986948i	$0.448515\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	9.00000	0.574989
$246$	0	0
$247$	12.0000	0.763542
$248$	0	0
$249$	3.00000	0.190117
$250$	0	0
$251$	10.0000	0.631194	0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000	0.631194	0.315597	+	0.948893i	$0.397795\pi$
$252$	0	0
$253$	16.0000	1.00591
$254$	0	0
$255$	3.00000	0.187867
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	−16.0000	−0.990375
$262$	0	0
$263$	−22.0000	−1.35658	−0.678289	−	0.734795i	$-0.737278\pi$
$263$	−22.0000	−1.35658	−0.678289	+	0.734795i	$0.737278\pi$
$264$	0	0
$265$	−7.00000	−0.430007
$266$	0	0
$267$	−6.00000	−0.367194
$268$	0	0
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	18.0000	1.09342	0.546711	−	0.837321i	$-0.315880\pi$
$271$	18.0000	1.09342	0.546711	+	0.837321i	$0.315880\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	−3.00000	−0.180253	−0.0901263	−	0.995930i	$-0.528727\pi$
$277$	−3.00000	−0.180253	−0.0901263	+	0.995930i	$0.528727\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	0	0
$285$	6.00000	0.355409
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	5.00000	0.293105
$292$	0	0
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	0	0
$297$	−20.0000	−1.16052
$298$	0	0
$299$	8.00000	0.462652
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	−11.0000	−0.631933
$304$	0	0
$305$	−1.00000	−0.0572598
$306$	0	0
$307$	−14.0000	−0.799022	−0.399511	−	0.916728i	$-0.630820\pi$
$307$	−14.0000	−0.799022	−0.399511	+	0.916728i	$0.630820\pi$
$308$	0	0
$309$	−5.00000	−0.284440
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	−8.00000	−0.450749
$316$	0	0
$317$	21.0000	1.17948	0.589739	−	0.807594i	$-0.299231\pi$
$317$	21.0000	1.17948	0.589739	+	0.807594i	$0.299231\pi$
$318$	0	0
$319$	32.0000	1.79166
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	18.0000	1.00155
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	−11.0000	−0.608301
$328$	0	0
$329$	−36.0000	−1.98474
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	0	0
$335$	−9.00000	−0.491723
$336$	0	0
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	0	0
$339$	20.0000	1.08625
$340$	0	0
$341$	−20.0000	−1.08306
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	0	0
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	−10.0000	−0.533761
$352$	0	0
$353$	8.00000	0.425797	0.212899	−	0.977074i	$-0.431710\pi$
$353$	8.00000	0.425797	0.212899	+	0.977074i	$0.431710\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	0	0
$357$	12.0000	0.635107
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	−2.00000	−0.104685
$366$	0	0
$367$	−2.00000	−0.104399	−0.0521996	−	0.998637i	$-0.516623\pi$
$367$	−2.00000	−0.104399	−0.0521996	+	0.998637i	$0.516623\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−28.0000	−1.45369
$372$	0	0
$373$	7.00000	0.362446	0.181223	−	0.983442i	$-0.441994\pi$
$373$	7.00000	0.362446	0.181223	+	0.983442i	$0.441994\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	16.0000	0.824042
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	19.0000	0.973399
$382$	0	0
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	16.0000	0.815436
$386$	0	0
$387$	4.00000	0.203331
$388$	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
$393$	−3.00000	−0.151330
$394$	0	0
$395$	−6.00000	−0.301893
$396$	0	0
$397$	−4.00000	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$398$	0	0
$399$	24.0000	1.20150
$400$	0	0
$401$	7.00000	0.349563	0.174782	−	0.984607i	$-0.444078\pi$
$401$	7.00000	0.349563	0.174782	+	0.984607i	$0.444078\pi$
$402$	0	0
$403$	−10.0000	−0.498135
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	0	0
$411$	−15.0000	−0.739895
$412$	0	0
$413$	−48.0000	−2.36193
$414$	0	0
$415$	3.00000	0.147264
$416$	0	0
$417$	−8.00000	−0.391762
$418$	0	0
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	0	0
$421$	5.00000	0.243685	0.121843	−	0.992549i	$-0.461120\pi$
$421$	5.00000	0.243685	0.121843	+	0.992549i	$0.461120\pi$
$422$	0	0
$423$	18.0000	0.875190
$424$	0	0
$425$	3.00000	0.145521
$426$	0	0
$427$	−4.00000	−0.193574
$428$	0	0
$429$	8.00000	0.386244
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	0	0
$435$	8.00000	0.383571
$436$	0	0
$437$	24.0000	1.14808
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	−18.0000	−0.857143
$442$	0	0
$443$	39.0000	1.85295	0.926473	−	0.376361i	$-0.122825\pi$
$443$	39.0000	1.85295	0.926473	+	0.376361i	$0.122825\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	0	0
$447$	6.00000	0.283790
$448$	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
$451$	0	0
$452$	0	0
$453$	−1.00000	−0.0469841
$454$	0	0
$455$	8.00000	0.375046
$456$	0	0
$457$	−13.0000	−0.608114	−0.304057	−	0.952654i	$-0.598341\pi$
$457$	−13.0000	−0.608114	−0.304057	+	0.952654i	$0.598341\pi$
$458$	0	0
$459$	−15.0000	−0.700140
$460$	0	0
$461$	−8.00000	−0.372597	−0.186299	−	0.982493i	$-0.559649\pi$
$461$	−8.00000	−0.372597	−0.186299	+	0.982493i	$0.559649\pi$
$462$	0	0
$463$	−27.0000	−1.25480	−0.627398	−	0.778699i	$-0.715880\pi$
$463$	−27.0000	−1.25480	−0.627398	+	0.778699i	$0.715880\pi$
$464$	0	0
$465$	−5.00000	−0.231869
$466$	0	0
$467$	27.0000	1.24941	0.624705	−	0.780860i	$-0.285219\pi$
$467$	27.0000	1.24941	0.624705	+	0.780860i	$0.285219\pi$
$468$	0	0
$469$	−36.0000	−1.66233
$470$	0	0
$471$	−19.0000	−0.875474
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	6.00000	0.275299
$476$	0	0
$477$	14.0000	0.641016
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	16.0000	0.728025
$484$	0	0
$485$	5.00000	0.227038
$486$	0	0
$487$	23.0000	1.04223	0.521115	−	0.853487i	$-0.325516\pi$
$487$	23.0000	1.04223	0.521115	+	0.853487i	$0.325516\pi$
$488$	0	0
$489$	21.0000	0.949653
$490$	0	0
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	0	0
$493$	24.0000	1.08091
$494$	0	0
$495$	−8.00000	−0.359573
$496$	0	0
$497$	24.0000	1.07655
$498$	0	0
$499$	3.00000	0.134298	0.0671492	−	0.997743i	$-0.478610\pi$
$499$	3.00000	0.134298	0.0671492	+	0.997743i	$0.478610\pi$
$500$	0	0
$501$	−3.00000	−0.134030
$502$	0	0
$503$	39.0000	1.73892	0.869462	−	0.494000i	$-0.164466\pi$
$503$	39.0000	1.73892	0.869462	+	0.494000i	$0.164466\pi$
$504$	0	0
$505$	−11.0000	−0.489494
$506$	0	0
$507$	−9.00000	−0.399704
$508$	0	0
$509$	43.0000	1.90594	0.952971	−	0.303062i	$-0.0980090\pi$
$509$	43.0000	1.90594	0.952971	+	0.303062i	$0.0980090\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	0	0
$513$	−30.0000	−1.32453
$514$	0	0
$515$	−5.00000	−0.220326
$516$	0	0
$517$	−36.0000	−1.58328
$518$	0	0
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	11.0000	0.481919	0.240959	−	0.970535i	$-0.422538\pi$
$521$	11.0000	0.481919	0.240959	+	0.970535i	$0.422538\pi$
$522$	0	0
$523$	−1.00000	−0.0437269	−0.0218635	−	0.999761i	$-0.506960\pi$
$523$	−1.00000	−0.0437269	−0.0218635	+	0.999761i	$0.506960\pi$
$524$	0	0
$525$	4.00000	0.174574
$526$	0	0
$527$	−15.0000	−0.653410
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	24.0000	1.04151
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	−4.00000	−0.172613
$538$	0	0
$539$	36.0000	1.55063
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	0	0
$543$	23.0000	0.987024
$544$	0	0
$545$	−11.0000	−0.471188
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	48.0000	2.04487
$552$	0	0
$553$	−24.0000	−1.02058
$554$	0	0
$555$	2.00000	0.0848953
$556$	0	0
$557$	−21.0000	−0.889799	−0.444899	−	0.895581i	$-0.646761\pi$
$557$	−21.0000	−0.889799	−0.444899	+	0.895581i	$0.646761\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	12.0000	0.506640
$562$	0	0
$563$	−30.0000	−1.26435	−0.632175	−	0.774826i	$-0.717837\pi$
$563$	−30.0000	−1.26435	−0.632175	+	0.774826i	$0.717837\pi$
$564$	0	0
$565$	20.0000	0.841406
$566$	0	0
$567$	4.00000	0.167984
$568$	0	0
$569$	−9.00000	−0.377300	−0.188650	−	0.982044i	$-0.560411\pi$
$569$	−9.00000	−0.377300	−0.188650	+	0.982044i	$0.560411\pi$
$570$	0	0
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	0	0
$573$	7.00000	0.292429
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	−17.0000	−0.707719	−0.353860	−	0.935299i	$-0.615131\pi$
$577$	−17.0000	−0.707719	−0.353860	+	0.935299i	$0.615131\pi$
$578$	0	0
$579$	15.0000	0.623379
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	−28.0000	−1.15964
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	−30.0000	−1.23613
$590$	0	0
$591$	−13.0000	−0.534749
$592$	0	0
$593$	−36.0000	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	0	0
$595$	12.0000	0.491952
$596$	0	0
$597$	−8.00000	−0.327418
$598$	0	0
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	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
$603$	18.0000	0.733017
$604$	0	0
$605$	5.00000	0.203279
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	0	0
$609$	32.0000	1.29671
$610$	0	0
$611$	−18.0000	−0.728202
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	−40.0000	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$620$	0	0
$621$	−20.0000	−0.802572
$622$	0	0
$623$	−24.0000	−0.961540
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	24.0000	0.958468
$628$	0	0
$629$	6.00000	0.239236
$630$	0	0
$631$	−36.0000	−1.43314	−0.716569	−	0.697517i	$-0.754288\pi$
$631$	−36.0000	−1.43314	−0.716569	+	0.697517i	$0.754288\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	19.0000	0.753992
$636$	0	0
$637$	18.0000	0.713186
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	−11.0000	−0.434474	−0.217237	−	0.976119i	$-0.569704\pi$
$641$	−11.0000	−0.434474	−0.217237	+	0.976119i	$0.569704\pi$
$642$	0	0
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	0	0
$645$	−2.00000	−0.0787499
$646$	0	0
$647$	27.0000	1.06148	0.530740	−	0.847535i	$-0.321914\pi$
$647$	27.0000	1.06148	0.530740	+	0.847535i	$0.321914\pi$
$648$	0	0
$649$	−48.0000	−1.88416
$650$	0	0
$651$	−20.0000	−0.783862
$652$	0	0
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	−3.00000	−0.117220
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	−40.0000	−1.55818	−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000	−1.55818	−0.779089	+	0.626913i	$0.784318\pi$
$660$	0	0
$661$	29.0000	1.12797	0.563985	−	0.825785i	$-0.309268\pi$
$661$	29.0000	1.12797	0.563985	+	0.825785i	$0.309268\pi$
$662$	0	0
$663$	6.00000	0.233021
$664$	0	0
$665$	24.0000	0.930680
$666$	0	0
$667$	32.0000	1.23904
$668$	0	0
$669$	4.00000	0.154649
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	−27.0000	−1.04077	−0.520387	−	0.853931i	$-0.674212\pi$
$673$	−27.0000	−1.04077	−0.520387	+	0.853931i	$0.674212\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	20.0000	0.767530
$680$	0	0
$681$	24.0000	0.919682
$682$	0	0
$683$	−11.0000	−0.420903	−0.210452	−	0.977604i	$-0.567493\pi$
$683$	−11.0000	−0.420903	−0.210452	+	0.977604i	$0.567493\pi$
$684$	0	0
$685$	−15.0000	−0.573121
$686$	0	0
$687$	−24.0000	−0.915657
$688$	0	0
$689$	−14.0000	−0.533358
$690$	0	0
$691$	−15.0000	−0.570627	−0.285313	−	0.958434i	$-0.592098\pi$
$691$	−15.0000	−0.570627	−0.285313	+	0.958434i	$0.592098\pi$
$692$	0	0
$693$	−32.0000	−1.21558
$694$	0	0
$695$	−8.00000	−0.303457
$696$	0	0
$697$	0	0
$698$	0	0
$699$	6.00000	0.226941
$700$	0	0
$701$	44.0000	1.66186	0.830929	−	0.556379i	$-0.187810\pi$
$701$	44.0000	1.66186	0.830929	+	0.556379i	$0.187810\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	0	0
$705$	−9.00000	−0.338960
$706$	0	0
$707$	−44.0000	−1.65479
$708$	0	0
$709$	8.00000	0.300446	0.150223	−	0.988652i	$-0.452001\pi$
$709$	8.00000	0.300446	0.150223	+	0.988652i	$0.452001\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	0	0
$713$	−20.0000	−0.749006
$714$	0	0
$715$	8.00000	0.299183
$716$	0	0
$717$	9.00000	0.336111
$718$	0	0
$719$	52.0000	1.93927	0.969636	−	0.244551i	$-0.0786406\pi$
$719$	52.0000	1.93927	0.969636	+	0.244551i	$0.0786406\pi$
$720$	0	0
$721$	−20.0000	−0.744839
$722$	0	0
$723$	5.00000	0.185952
$724$	0	0
$725$	8.00000	0.297113
$726$	0	0
$727$	37.0000	1.37225	0.686127	−	0.727482i	$-0.259309\pi$
$727$	37.0000	1.37225	0.686127	+	0.727482i	$0.259309\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−6.00000	−0.221918
$732$	0	0
$733$	−15.0000	−0.554038	−0.277019	−	0.960864i	$-0.589346\pi$
$733$	−15.0000	−0.554038	−0.277019	+	0.960864i	$0.589346\pi$
$734$	0	0
$735$	9.00000	0.331970
$736$	0	0
$737$	−36.0000	−1.32608
$738$	0	0
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	0	0
$743$	−28.0000	−1.02722	−0.513610	−	0.858024i	$-0.671692\pi$
$743$	−28.0000	−1.02722	−0.513610	+	0.858024i	$0.671692\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	0	0
$747$	−6.00000	−0.219529
$748$	0	0
$749$	−16.0000	−0.584627
$750$	0	0
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	0	0
$753$	10.0000	0.364420
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	−36.0000	−1.30844	−0.654221	−	0.756303i	$-0.727003\pi$
$757$	−36.0000	−1.30844	−0.654221	+	0.756303i	$0.727003\pi$
$758$	0	0
$759$	16.0000	0.580763
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	−44.0000	−1.59291
$764$	0	0
$765$	−6.00000	−0.216930
$766$	0	0
$767$	−24.0000	−0.866590
$768$	0	0
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	0	0
$773$	−16.0000	−0.575480	−0.287740	−	0.957709i	$-0.592904\pi$
$773$	−16.0000	−0.575480	−0.287740	+	0.957709i	$0.592904\pi$
$774$	0	0
$775$	−5.00000	−0.179605
$776$	0	0
$777$	8.00000	0.286998
$778$	0	0
$779$	0	0
$780$	0	0
$781$	24.0000	0.858788
$782$	0	0
$783$	−40.0000	−1.42948
$784$	0	0
$785$	−19.0000	−0.678139
$786$	0	0
$787$	44.0000	1.56843	0.784215	−	0.620489i	$-0.213066\pi$
$787$	44.0000	1.56843	0.784215	+	0.620489i	$0.213066\pi$
$788$	0	0
$789$	−22.0000	−0.783221
$790$	0	0
$791$	80.0000	2.84447
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	0	0
$795$	−7.00000	−0.248264
$796$	0	0
$797$	−22.0000	−0.779280	−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000	−0.779280	−0.389640	+	0.920967i	$0.627401\pi$
$798$	0	0
$799$	−27.0000	−0.955191
$800$	0	0
$801$	12.0000	0.423999
$802$	0	0
$803$	−8.00000	−0.282314
$804$	0	0
$805$	16.0000	0.563926
$806$	0	0
$807$	−30.0000	−1.05605
$808$	0	0
$809$	16.0000	0.562530	0.281265	−	0.959630i	$-0.409246\pi$
$809$	16.0000	0.562530	0.281265	+	0.959630i	$0.409246\pi$
$810$	0	0
$811$	33.0000	1.15879	0.579393	−	0.815048i	$-0.303290\pi$
$811$	33.0000	1.15879	0.579393	+	0.815048i	$0.303290\pi$
$812$	0	0
$813$	18.0000	0.631288
$814$	0	0
$815$	21.0000	0.735598
$816$	0	0
$817$	−12.0000	−0.419827
$818$	0	0
$819$	−16.0000	−0.559085
$820$	0	0
$821$	23.0000	0.802706	0.401353	−	0.915924i	$-0.368540\pi$
$821$	23.0000	0.802706	0.401353	+	0.915924i	$0.368540\pi$
$822$	0	0
$823$	5.00000	0.174289	0.0871445	−	0.996196i	$-0.472226\pi$
$823$	5.00000	0.174289	0.0871445	+	0.996196i	$0.472226\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	−6.00000	−0.208640	−0.104320	−	0.994544i	$-0.533267\pi$
$827$	−6.00000	−0.208640	−0.104320	+	0.994544i	$0.533267\pi$
$828$	0	0
$829$	4.00000	0.138926	0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000	0.138926	0.0694629	+	0.997585i	$0.477871\pi$
$830$	0	0
$831$	−3.00000	−0.104069
$832$	0	0
$833$	27.0000	0.935495
$834$	0	0
$835$	−3.00000	−0.103819
$836$	0	0
$837$	25.0000	0.864126
$838$	0	0
$839$	17.0000	0.586905	0.293453	−	0.955974i	$-0.405196\pi$
$839$	17.0000	0.586905	0.293453	+	0.955974i	$0.405196\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	20.0000	0.687208
$848$	0	0
$849$	−28.0000	−0.960958
$850$	0	0
$851$	8.00000	0.274236
$852$	0	0
$853$	28.0000	0.958702	0.479351	−	0.877623i	$-0.340872\pi$
$853$	28.0000	0.958702	0.479351	+	0.877623i	$0.340872\pi$
$854$	0	0
$855$	−12.0000	−0.410391
$856$	0	0
$857$	−34.0000	−1.16142	−0.580709	−	0.814111i	$-0.697225\pi$
$857$	−34.0000	−1.16142	−0.580709	+	0.814111i	$0.697225\pi$
$858$	0	0
$859$	1.00000	0.0341196	0.0170598	−	0.999854i	$-0.494569\pi$
$859$	1.00000	0.0341196	0.0170598	+	0.999854i	$0.494569\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	0	0
$865$	−2.00000	−0.0680020
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	−18.0000	−0.609907
$872$	0	0
$873$	−10.0000	−0.338449
$874$	0	0
$875$	4.00000	0.135225
$876$	0	0
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	0	0
$879$	−2.00000	−0.0674583
$880$	0	0
$881$	−16.0000	−0.539054	−0.269527	−	0.962993i	$-0.586867\pi$
$881$	−16.0000	−0.539054	−0.269527	+	0.962993i	$0.586867\pi$
$882$	0	0
$883$	34.0000	1.14419	0.572096	−	0.820187i	$-0.306131\pi$
$883$	34.0000	1.14419	0.572096	+	0.820187i	$0.306131\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	0	0
$887$	2.00000	0.0671534	0.0335767	−	0.999436i	$-0.489310\pi$
$887$	2.00000	0.0671534	0.0335767	+	0.999436i	$0.489310\pi$
$888$	0	0
$889$	76.0000	2.54896
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	−54.0000	−1.80704
$894$	0	0
$895$	−4.00000	−0.133705
$896$	0	0
$897$	8.00000	0.267112
$898$	0	0
$899$	−40.0000	−1.33407
$900$	0	0
$901$	−21.0000	−0.699611
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	23.0000	0.764546
$906$	0	0
$907$	14.0000	0.464862	0.232431	−	0.972613i	$-0.425332\pi$
$907$	14.0000	0.464862	0.232431	+	0.972613i	$0.425332\pi$
$908$	0	0
$909$	22.0000	0.729694
$910$	0	0
$911$	−27.0000	−0.894550	−0.447275	−	0.894397i	$-0.647605\pi$
$911$	−27.0000	−0.894550	−0.447275	+	0.894397i	$0.647605\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	−1.00000	−0.0330590
$916$	0	0
$917$	−12.0000	−0.396275
$918$	0	0
$919$	−26.0000	−0.857661	−0.428830	−	0.903385i	$-0.641074\pi$
$919$	−26.0000	−0.857661	−0.428830	+	0.903385i	$0.641074\pi$
$920$	0	0
$921$	−14.0000	−0.461316
$922$	0	0
$923$	12.0000	0.394985
$924$	0	0
$925$	2.00000	0.0657596
$926$	0	0
$927$	10.0000	0.328443
$928$	0	0
$929$	48.0000	1.57483	0.787414	−	0.616424i	$-0.211419\pi$
$929$	48.0000	1.57483	0.787414	+	0.616424i	$0.211419\pi$
$930$	0	0
$931$	54.0000	1.76978
$932$	0	0
$933$	0	0
$934$	0	0
$935$	12.0000	0.392442
$936$	0	0
$937$	−7.00000	−0.228680	−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000	−0.228680	−0.114340	+	0.993442i	$0.536475\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−20.0000	−0.650600
$946$	0	0
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	21.0000	0.680972
$952$	0	0
$953$	−7.00000	−0.226752	−0.113376	−	0.993552i	$-0.536167\pi$
$953$	−7.00000	−0.226752	−0.113376	+	0.993552i	$0.536167\pi$
$954$	0	0
$955$	7.00000	0.226515
$956$	0	0
$957$	32.0000	1.03441
$958$	0	0
$959$	−60.0000	−1.93750
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	8.00000	0.257796
$964$	0	0
$965$	15.0000	0.482867
$966$	0	0
$967$	−28.0000	−0.900419	−0.450210	−	0.892923i	$-0.648651\pi$
$967$	−28.0000	−0.900419	−0.450210	+	0.892923i	$0.648651\pi$
$968$	0	0
$969$	18.0000	0.578243
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−32.0000	−1.02587
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	−52.0000	−1.66363	−0.831814	−	0.555055i	$-0.812697\pi$
$977$	−52.0000	−1.66363	−0.831814	+	0.555055i	$0.812697\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	0	0
$981$	22.0000	0.702406
$982$	0	0
$983$	−60.0000	−1.91370	−0.956851	−	0.290578i	$-0.906153\pi$
$983$	−60.0000	−1.91370	−0.956851	+	0.290578i	$0.906153\pi$
$984$	0	0
$985$	−13.0000	−0.414214
$986$	0	0
$987$	−36.0000	−1.14589
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	27.0000	0.857683	0.428842	−	0.903380i	$-0.358922\pi$
$991$	27.0000	0.857683	0.428842	+	0.903380i	$0.358922\pi$
$992$	0	0
$993$	20.0000	0.634681
$994$	0	0
$995$	−8.00000	−0.253617
$996$	0	0
$997$	−40.0000	−1.26681	−0.633406	−	0.773819i	$-0.718344\pi$
$997$	−40.0000	−1.26681	−0.633406	+	0.773819i	$0.718344\pi$
$998$	0	0
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.i.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.i.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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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