LMFDB - Embedded newform 3920.2.a.br.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3920,2,Mod(1,3920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3920, 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("3920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3920 = 2^{4} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3920.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:	$31.3013575923$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 245)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	3920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{3} +1.00000 q^{5} +2.82843 q^{9} +O(q^{10})$	$q-2.41421 q^{3} +1.00000 q^{5} +2.82843 q^{9} +5.82843 q^{11} +1.58579 q^{13} -2.41421 q^{15} -5.24264 q^{17} -6.00000 q^{19} -4.58579 q^{23} +1.00000 q^{25} +0.414214 q^{27} +2.65685 q^{29} -1.75736 q^{31} -14.0711 q^{33} -6.24264 q^{37} -3.82843 q^{39} +2.24264 q^{41} -2.00000 q^{43} +2.82843 q^{45} -1.24264 q^{47} +12.6569 q^{51} -4.24264 q^{53} +5.82843 q^{55} +14.4853 q^{57} -6.24264 q^{59} +2.82843 q^{61} +1.58579 q^{65} -0.242641 q^{67} +11.0711 q^{69} +8.82843 q^{71} +8.48528 q^{73} -2.41421 q^{75} +15.4853 q^{79} -9.48528 q^{81} -5.24264 q^{85} -6.41421 q^{87} -8.00000 q^{89} +4.24264 q^{93} -6.00000 q^{95} +4.75736 q^{97} +16.4853 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5	$ 2 q - 2 q^{3} + 2 q^{5} + 6 q^{11} + 6 q^{13} - 2 q^{15} - 2 q^{17} - 12 q^{19} - 12 q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{29} - 12 q^{31} - 14 q^{33} - 4 q^{37} - 2 q^{39} - 4 q^{41} - 4 q^{43} + 6 q^{47} + 14 q^{51} + 6 q^{55} + 12 q^{57} - 4 q^{59} + 6 q^{65} + 8 q^{67} + 8 q^{69} + 12 q^{71} - 2 q^{75} + 14 q^{79} - 2 q^{81} - 2 q^{85} - 10 q^{87} - 16 q^{89} - 12 q^{95} + 18 q^{97} + 16 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 + 6 * q^11 + 6 * q^13 - 2 * q^15 - 2 * q^17 - 12 * q^19 - 12 * q^23 + 2 * q^25 - 2 * q^27 - 6 * q^29 - 12 * q^31 - 14 * q^33 - 4 * q^37 - 2 * q^39 - 4 * q^41 - 4 * q^43 + 6 * q^47 + 14 * q^51 + 6 * q^55 + 12 * q^57 - 4 * q^59 + 6 * q^65 + 8 * q^67 + 8 * q^69 + 12 * q^71 - 2 * q^75 + 14 * q^79 - 2 * q^81 - 2 * q^85 - 10 * q^87 - 16 * q^89 - 12 * q^95 + 18 * q^97 + 16 * q^99

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
$3$	−2.41421	−1.39385	−0.696923	−	0.717146i	$-0.745448\pi$
$3$	−2.41421	−1.39385	−0.696923	+	0.717146i	$0.745448\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	5.82843	1.75734	0.878668	−	0.477432i	$-0.158432\pi$
$11$	5.82843	1.75734	0.878668	+	0.477432i	$0.158432\pi$
$12$	0	0
$13$	1.58579	0.439818	0.219909	−	0.975520i	$-0.429424\pi$
$13$	1.58579	0.439818	0.219909	+	0.975520i	$0.429424\pi$
$14$	0	0
$15$	−2.41421	−0.623347
$16$	0	0
$17$	−5.24264	−1.27153	−0.635764	−	0.771884i	$-0.719315\pi$
$17$	−5.24264	−1.27153	−0.635764	+	0.771884i	$0.719315\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.58579	−0.956203	−0.478101	−	0.878305i	$-0.658675\pi$
$23$	−4.58579	−0.956203	−0.478101	+	0.878305i	$0.658675\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0.414214	0.0797154
$28$	0	0
$29$	2.65685	0.493365	0.246683	−	0.969096i	$-0.420659\pi$
$29$	2.65685	0.493365	0.246683	+	0.969096i	$0.420659\pi$
$30$	0	0
$31$	−1.75736	−0.315631	−0.157816	−	0.987469i	$-0.550445\pi$
$31$	−1.75736	−0.315631	−0.157816	+	0.987469i	$0.550445\pi$
$32$	0	0
$33$	−14.0711	−2.44946
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.24264	−1.02628	−0.513142	−	0.858304i	$-0.671519\pi$
$37$	−6.24264	−1.02628	−0.513142	+	0.858304i	$0.671519\pi$
$38$	0	0
$39$	−3.82843	−0.613039
$40$	0	0
$41$	2.24264	0.350242	0.175121	−	0.984547i	$-0.443968\pi$
$41$	2.24264	0.350242	0.175121	+	0.984547i	$0.443968\pi$
$42$	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
$45$	2.82843	0.421637
$46$	0	0
$47$	−1.24264	−0.181258	−0.0906289	−	0.995885i	$-0.528888\pi$
$47$	−1.24264	−0.181258	−0.0906289	+	0.995885i	$0.528888\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	12.6569	1.77231
$52$	0	0
$53$	−4.24264	−0.582772	−0.291386	−	0.956606i	$-0.594116\pi$
$53$	−4.24264	−0.582772	−0.291386	+	0.956606i	$0.594116\pi$
$54$	0	0
$55$	5.82843	0.785905
$56$	0	0
$57$	14.4853	1.91862
$58$	0	0
$59$	−6.24264	−0.812723	−0.406361	−	0.913712i	$-0.633203\pi$
$59$	−6.24264	−0.812723	−0.406361	+	0.913712i	$0.633203\pi$
$60$	0	0
$61$	2.82843	0.362143	0.181071	−	0.983470i	$-0.442043\pi$
$61$	2.82843	0.362143	0.181071	+	0.983470i	$0.442043\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.58579	0.196693
$66$	0	0
$67$	−0.242641	−0.0296433	−0.0148216	−	0.999890i	$-0.504718\pi$
$67$	−0.242641	−0.0296433	−0.0148216	+	0.999890i	$0.504718\pi$
$68$	0	0
$69$	11.0711	1.33280
$70$	0	0
$71$	8.82843	1.04774	0.523871	−	0.851798i	$-0.324487\pi$
$71$	8.82843	1.04774	0.523871	+	0.851798i	$0.324487\pi$
$72$	0	0
$73$	8.48528	0.993127	0.496564	−	0.868000i	$-0.334595\pi$
$73$	8.48528	0.993127	0.496564	+	0.868000i	$0.334595\pi$
$74$	0	0
$75$	−2.41421	−0.278769
$76$	0	0
$77$	0	0
$78$	0	0
$79$	15.4853	1.74223	0.871115	−	0.491079i	$-0.163397\pi$
$79$	15.4853	1.74223	0.871115	+	0.491079i	$0.163397\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−5.24264	−0.568644
$86$	0	0
$87$	−6.41421	−0.687676
$88$	0	0
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.24264	0.439941
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	4.75736	0.483037	0.241518	−	0.970396i	$-0.422355\pi$
$97$	4.75736	0.483037	0.241518	+	0.970396i	$0.422355\pi$
$98$	0	0
$99$	16.4853	1.65683
$100$	0	0
$101$	−14.4853	−1.44134	−0.720670	−	0.693279i	$-0.756166\pi$
$101$	−14.4853	−1.44134	−0.720670	+	0.693279i	$0.756166\pi$
$102$	0	0
$103$	−10.7574	−1.05995	−0.529977	−	0.848012i	$-0.677799\pi$
$103$	−10.7574	−1.05995	−0.529977	+	0.848012i	$0.677799\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.4853	−1.40035	−0.700173	−	0.713974i	$-0.746894\pi$
$107$	−14.4853	−1.40035	−0.700173	+	0.713974i	$0.746894\pi$
$108$	0	0
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	0	0
$111$	15.0711	1.43048
$112$	0	0
$113$	1.07107	0.100758	0.0503788	−	0.998730i	$-0.483957\pi$
$113$	1.07107	0.100758	0.0503788	+	0.998730i	$0.483957\pi$
$114$	0	0
$115$	−4.58579	−0.427627
$116$	0	0
$117$	4.48528	0.414664
$118$	0	0
$119$	0	0
$120$	0	0
$121$	22.9706	2.08823
$122$	0	0
$123$	−5.41421	−0.488183
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.242641	0.0215309	0.0107654	−	0.999942i	$-0.496573\pi$
$127$	0.242641	0.0215309	0.0107654	+	0.999942i	$0.496573\pi$
$128$	0	0
$129$	4.82843	0.425119
$130$	0	0
$131$	−3.75736	−0.328282	−0.164141	−	0.986437i	$-0.552485\pi$
$131$	−3.75736	−0.328282	−0.164141	+	0.986437i	$0.552485\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0.414214	0.0356498
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	−16.2426	−1.37768	−0.688841	−	0.724912i	$-0.741880\pi$
$139$	−16.2426	−1.37768	−0.688841	+	0.724912i	$0.741880\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	0	0
$143$	9.24264	0.772908
$144$	0	0
$145$	2.65685	0.220640
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−14.8284	−1.21479	−0.607396	−	0.794399i	$-0.707786\pi$
$149$	−14.8284	−1.21479	−0.607396	+	0.794399i	$0.707786\pi$
$150$	0	0
$151$	−9.48528	−0.771901	−0.385951	−	0.922519i	$-0.626126\pi$
$151$	−9.48528	−0.771901	−0.385951	+	0.922519i	$0.626126\pi$
$152$	0	0
$153$	−14.8284	−1.19881
$154$	0	0
$155$	−1.75736	−0.141154
$156$	0	0
$157$	−20.8284	−1.66229	−0.831145	−	0.556056i	$-0.812314\pi$
$157$	−20.8284	−1.66229	−0.831145	+	0.556056i	$0.812314\pi$
$158$	0	0
$159$	10.2426	0.812294
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1.75736	0.137647	0.0688235	−	0.997629i	$-0.478075\pi$
$163$	1.75736	0.137647	0.0688235	+	0.997629i	$0.478075\pi$
$164$	0	0
$165$	−14.0711	−1.09543
$166$	0	0
$167$	−9.24264	−0.715217	−0.357609	−	0.933872i	$-0.616408\pi$
$167$	−9.24264	−0.715217	−0.357609	+	0.933872i	$0.616408\pi$
$168$	0	0
$169$	−10.4853	−0.806560
$170$	0	0
$171$	−16.9706	−1.29777
$172$	0	0
$173$	1.24264	0.0944762	0.0472381	−	0.998884i	$-0.484958\pi$
$173$	1.24264	0.0944762	0.0472381	+	0.998884i	$0.484958\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	15.0711	1.13281
$178$	0	0
$179$	−2.48528	−0.185759	−0.0928793	−	0.995677i	$-0.529607\pi$
$179$	−2.48528	−0.185759	−0.0928793	+	0.995677i	$0.529607\pi$
$180$	0	0
$181$	6.72792	0.500083	0.250041	−	0.968235i	$-0.419556\pi$
$181$	6.72792	0.500083	0.250041	+	0.968235i	$0.419556\pi$
$182$	0	0
$183$	−6.82843	−0.504772
$184$	0	0
$185$	−6.24264	−0.458968
$186$	0	0
$187$	−30.5563	−2.23450
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.9706	1.44502	0.722510	−	0.691361i	$-0.242989\pi$
$191$	19.9706	1.44502	0.722510	+	0.691361i	$0.242989\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0	0
$195$	−3.82843	−0.274159
$196$	0	0
$197$	10.5858	0.754206	0.377103	−	0.926171i	$-0.376920\pi$
$197$	10.5858	0.754206	0.377103	+	0.926171i	$0.376920\pi$
$198$	0	0
$199$	4.58579	0.325078	0.162539	−	0.986702i	$-0.448032\pi$
$199$	4.58579	0.325078	0.162539	+	0.986702i	$0.448032\pi$
$200$	0	0
$201$	0.585786	0.0413182
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.24264	0.156633
$206$	0	0
$207$	−12.9706	−0.901516
$208$	0	0
$209$	−34.9706	−2.41896
$210$	0	0
$211$	−9.00000	−0.619586	−0.309793	−	0.950804i	$-0.600260\pi$
$211$	−9.00000	−0.619586	−0.309793	+	0.950804i	$0.600260\pi$
$212$	0	0
$213$	−21.3137	−1.46039
$214$	0	0
$215$	−2.00000	−0.136399
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−20.4853	−1.38427
$220$	0	0
$221$	−8.31371	−0.559241
$222$	0	0
$223$	−18.2132	−1.21965	−0.609823	−	0.792538i	$-0.708760\pi$
$223$	−18.2132	−1.21965	−0.609823	+	0.792538i	$0.708760\pi$
$224$	0	0
$225$	2.82843	0.188562
$226$	0	0
$227$	9.72792	0.645665	0.322832	−	0.946456i	$-0.395365\pi$
$227$	9.72792	0.645665	0.322832	+	0.946456i	$0.395365\pi$
$228$	0	0
$229$	−30.0416	−1.98521	−0.992603	−	0.121402i	$-0.961261\pi$
$229$	−30.0416	−1.98521	−0.992603	+	0.121402i	$0.961261\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.8284	−0.971443	−0.485721	−	0.874114i	$-0.661443\pi$
$233$	−14.8284	−0.971443	−0.485721	+	0.874114i	$0.661443\pi$
$234$	0	0
$235$	−1.24264	−0.0810609
$236$	0	0
$237$	−37.3848	−2.42840
$238$	0	0
$239$	0.514719	0.0332944	0.0166472	−	0.999861i	$-0.494701\pi$
$239$	0.514719	0.0332944	0.0166472	+	0.999861i	$0.494701\pi$
$240$	0	0
$241$	24.7279	1.59287	0.796433	−	0.604727i	$-0.206718\pi$
$241$	24.7279	1.59287	0.796433	+	0.604727i	$0.206718\pi$
$242$	0	0
$243$	21.6569	1.38929
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−9.51472	−0.605407
$248$	0	0
$249$	0	0
$250$	0	0
$251$	25.2132	1.59144	0.795722	−	0.605663i	$-0.207092\pi$
$251$	25.2132	1.59144	0.795722	+	0.605663i	$0.207092\pi$
$252$	0	0
$253$	−26.7279	−1.68037
$254$	0	0
$255$	12.6569	0.792603
$256$	0	0
$257$	−26.4853	−1.65211	−0.826053	−	0.563592i	$-0.809419\pi$
$257$	−26.4853	−1.65211	−0.826053	+	0.563592i	$0.809419\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	7.51472	0.465149
$262$	0	0
$263$	−28.6274	−1.76524	−0.882621	−	0.470085i	$-0.844223\pi$
$263$	−28.6274	−1.76524	−0.882621	+	0.470085i	$0.844223\pi$
$264$	0	0
$265$	−4.24264	−0.260623
$266$	0	0
$267$	19.3137	1.18198
$268$	0	0
$269$	−7.75736	−0.472975	−0.236487	−	0.971635i	$-0.575996\pi$
$269$	−7.75736	−0.472975	−0.236487	+	0.971635i	$0.575996\pi$
$270$	0	0
$271$	23.3137	1.41621	0.708103	−	0.706109i	$-0.249551\pi$
$271$	23.3137	1.41621	0.708103	+	0.706109i	$0.249551\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.82843	0.351467
$276$	0	0
$277$	−27.2132	−1.63508	−0.817541	−	0.575870i	$-0.804664\pi$
$277$	−27.2132	−1.63508	−0.817541	+	0.575870i	$0.804664\pi$
$278$	0	0
$279$	−4.97056	−0.297580
$280$	0	0
$281$	−20.3137	−1.21181	−0.605907	−	0.795535i	$-0.707190\pi$
$281$	−20.3137	−1.21181	−0.605907	+	0.795535i	$0.707190\pi$
$282$	0	0
$283$	6.55635	0.389735	0.194867	−	0.980830i	$-0.437572\pi$
$283$	6.55635	0.389735	0.194867	+	0.980830i	$0.437572\pi$
$284$	0	0
$285$	14.4853	0.858034
$286$	0	0
$287$	0	0
$288$	0	0
$289$	10.4853	0.616781
$290$	0	0
$291$	−11.4853	−0.673279
$292$	0	0
$293$	0.272078	0.0158950	0.00794748	−	0.999968i	$-0.497470\pi$
$293$	0.272078	0.0158950	0.00794748	+	0.999968i	$0.497470\pi$
$294$	0	0
$295$	−6.24264	−0.363461
$296$	0	0
$297$	2.41421	0.140087
$298$	0	0
$299$	−7.27208	−0.420555
$300$	0	0
$301$	0	0
$302$	0	0
$303$	34.9706	2.00901
$304$	0	0
$305$	2.82843	0.161955
$306$	0	0
$307$	−11.1005	−0.633539	−0.316770	−	0.948503i	$-0.602598\pi$
$307$	−11.1005	−0.633539	−0.316770	+	0.948503i	$0.602598\pi$
$308$	0	0
$309$	25.9706	1.47741
$310$	0	0
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	0	0
$313$	24.2132	1.36861	0.684306	−	0.729195i	$-0.260105\pi$
$313$	24.2132	1.36861	0.684306	+	0.729195i	$0.260105\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.6569	−0.654714	−0.327357	−	0.944901i	$-0.606158\pi$
$317$	−11.6569	−0.654714	−0.327357	+	0.944901i	$0.606158\pi$
$318$	0	0
$319$	15.4853	0.867009
$320$	0	0
$321$	34.9706	1.95187
$322$	0	0
$323$	31.4558	1.75025
$324$	0	0
$325$	1.58579	0.0879636
$326$	0	0
$327$	−12.0711	−0.667532
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−23.4558	−1.28925	−0.644625	−	0.764499i	$-0.722986\pi$
$331$	−23.4558	−1.28925	−0.644625	+	0.764499i	$0.722986\pi$
$332$	0	0
$333$	−17.6569	−0.967590
$334$	0	0
$335$	−0.242641	−0.0132569
$336$	0	0
$337$	13.7574	0.749411	0.374706	−	0.927144i	$-0.377744\pi$
$337$	13.7574	0.749411	0.374706	+	0.927144i	$0.377744\pi$
$338$	0	0
$339$	−2.58579	−0.140441
$340$	0	0
$341$	−10.2426	−0.554670
$342$	0	0
$343$	0	0
$344$	0	0
$345$	11.0711	0.596046
$346$	0	0
$347$	1.07107	0.0574979	0.0287490	−	0.999587i	$-0.490848\pi$
$347$	1.07107	0.0574979	0.0287490	+	0.999587i	$0.490848\pi$
$348$	0	0
$349$	22.9706	1.22959	0.614793	−	0.788688i	$-0.289240\pi$
$349$	22.9706	1.22959	0.614793	+	0.788688i	$0.289240\pi$
$350$	0	0
$351$	0.656854	0.0350603
$352$	0	0
$353$	36.2132	1.92743	0.963717	−	0.266925i	$-0.0860078\pi$
$353$	36.2132	1.92743	0.963717	+	0.266925i	$0.0860078\pi$
$354$	0	0
$355$	8.82843	0.468564
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.3137	−0.597115	−0.298557	−	0.954392i	$-0.596505\pi$
$359$	−11.3137	−0.597115	−0.298557	+	0.954392i	$0.596505\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−55.4558	−2.91068
$364$	0	0
$365$	8.48528	0.444140
$366$	0	0
$367$	−29.8701	−1.55920	−0.779602	−	0.626275i	$-0.784579\pi$
$367$	−29.8701	−1.55920	−0.779602	+	0.626275i	$0.784579\pi$
$368$	0	0
$369$	6.34315	0.330211
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0.485281	0.0251269	0.0125635	−	0.999921i	$-0.496001\pi$
$373$	0.485281	0.0251269	0.0125635	+	0.999921i	$0.496001\pi$
$374$	0	0
$375$	−2.41421	−0.124669
$376$	0	0
$377$	4.21320	0.216991
$378$	0	0
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	0	0
$381$	−0.585786	−0.0300107
$382$	0	0
$383$	12.4853	0.637968	0.318984	−	0.947760i	$-0.396658\pi$
$383$	12.4853	0.637968	0.318984	+	0.947760i	$0.396658\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.65685	−0.287554
$388$	0	0
$389$	−35.1421	−1.78178	−0.890889	−	0.454222i	$-0.849917\pi$
$389$	−35.1421	−1.78178	−0.890889	+	0.454222i	$0.849917\pi$
$390$	0	0
$391$	24.0416	1.21584
$392$	0	0
$393$	9.07107	0.457575
$394$	0	0
$395$	15.4853	0.779149
$396$	0	0
$397$	4.41421	0.221543	0.110772	−	0.993846i	$-0.464668\pi$
$397$	4.41421	0.221543	0.110772	+	0.993846i	$0.464668\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.17157	0.308194	0.154097	−	0.988056i	$-0.450753\pi$
$401$	6.17157	0.308194	0.154097	+	0.988056i	$0.450753\pi$
$402$	0	0
$403$	−2.78680	−0.138820
$404$	0	0
$405$	−9.48528	−0.471327
$406$	0	0
$407$	−36.3848	−1.80353
$408$	0	0
$409$	−14.4853	−0.716251	−0.358126	−	0.933673i	$-0.616584\pi$
$409$	−14.4853	−0.716251	−0.358126	+	0.933673i	$0.616584\pi$
$410$	0	0
$411$	−28.9706	−1.42901
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	39.2132	1.92028
$418$	0	0
$419$	6.72792	0.328681	0.164340	−	0.986404i	$-0.447450\pi$
$419$	6.72792	0.328681	0.164340	+	0.986404i	$0.447450\pi$
$420$	0	0
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	0	0
$423$	−3.51472	−0.170891
$424$	0	0
$425$	−5.24264	−0.254305
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−22.3137	−1.07732
$430$	0	0
$431$	−10.7990	−0.520169	−0.260085	−	0.965586i	$-0.583750\pi$
$431$	−10.7990	−0.520169	−0.260085	+	0.965586i	$0.583750\pi$
$432$	0	0
$433$	−22.9706	−1.10389	−0.551947	−	0.833879i	$-0.686115\pi$
$433$	−22.9706	−1.10389	−0.551947	+	0.833879i	$0.686115\pi$
$434$	0	0
$435$	−6.41421	−0.307538
$436$	0	0
$437$	27.5147	1.31621
$438$	0	0
$439$	−6.38478	−0.304729	−0.152364	−	0.988324i	$-0.548689\pi$
$439$	−6.38478	−0.304729	−0.152364	+	0.988324i	$0.548689\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−20.8284	−0.989588	−0.494794	−	0.869010i	$-0.664757\pi$
$443$	−20.8284	−0.989588	−0.494794	+	0.869010i	$0.664757\pi$
$444$	0	0
$445$	−8.00000	−0.379236
$446$	0	0
$447$	35.7990	1.69323
$448$	0	0
$449$	−29.8284	−1.40769	−0.703845	−	0.710353i	$-0.748535\pi$
$449$	−29.8284	−1.40769	−0.703845	+	0.710353i	$0.748535\pi$
$450$	0	0
$451$	13.0711	0.615493
$452$	0	0
$453$	22.8995	1.07591
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−20.2426	−0.946911	−0.473455	−	0.880818i	$-0.656993\pi$
$457$	−20.2426	−0.946911	−0.473455	+	0.880818i	$0.656993\pi$
$458$	0	0
$459$	−2.17157	−0.101360
$460$	0	0
$461$	36.9706	1.72189	0.860945	−	0.508697i	$-0.169873\pi$
$461$	36.9706	1.72189	0.860945	+	0.508697i	$0.169873\pi$
$462$	0	0
$463$	−29.4558	−1.36893	−0.684465	−	0.729046i	$-0.739964\pi$
$463$	−29.4558	−1.36893	−0.684465	+	0.729046i	$0.739964\pi$
$464$	0	0
$465$	4.24264	0.196748
$466$	0	0
$467$	−19.7279	−0.912899	−0.456450	−	0.889749i	$-0.650879\pi$
$467$	−19.7279	−0.912899	−0.456450	+	0.889749i	$0.650879\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	50.2843	2.31698
$472$	0	0
$473$	−11.6569	−0.535983
$474$	0	0
$475$	−6.00000	−0.275299
$476$	0	0
$477$	−12.0000	−0.549442
$478$	0	0
$479$	−20.2426	−0.924910	−0.462455	−	0.886643i	$-0.653031\pi$
$479$	−20.2426	−0.924910	−0.462455	+	0.886643i	$0.653031\pi$
$480$	0	0
$481$	−9.89949	−0.451378
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.75736	0.216021
$486$	0	0
$487$	31.6985	1.43640	0.718198	−	0.695839i	$-0.244967\pi$
$487$	31.6985	1.43640	0.718198	+	0.695839i	$0.244967\pi$
$488$	0	0
$489$	−4.24264	−0.191859
$490$	0	0
$491$	−19.2843	−0.870287	−0.435143	−	0.900361i	$-0.643302\pi$
$491$	−19.2843	−0.870287	−0.435143	+	0.900361i	$0.643302\pi$
$492$	0	0
$493$	−13.9289	−0.627328
$494$	0	0
$495$	16.4853	0.740958
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−3.00000	−0.134298	−0.0671492	−	0.997743i	$-0.521390\pi$
$499$	−3.00000	−0.134298	−0.0671492	+	0.997743i	$0.521390\pi$
$500$	0	0
$501$	22.3137	0.996903
$502$	0	0
$503$	32.7574	1.46058	0.730289	−	0.683138i	$-0.239385\pi$
$503$	32.7574	1.46058	0.730289	+	0.683138i	$0.239385\pi$
$504$	0	0
$505$	−14.4853	−0.644587
$506$	0	0
$507$	25.3137	1.12422
$508$	0	0
$509$	17.2132	0.762962	0.381481	−	0.924377i	$-0.375414\pi$
$509$	17.2132	0.762962	0.381481	+	0.924377i	$0.375414\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2.48528	−0.109728
$514$	0	0
$515$	−10.7574	−0.474026
$516$	0	0
$517$	−7.24264	−0.318531
$518$	0	0
$519$	−3.00000	−0.131685
$520$	0	0
$521$	−18.9706	−0.831115	−0.415558	−	0.909567i	$-0.636414\pi$
$521$	−18.9706	−0.831115	−0.415558	+	0.909567i	$0.636414\pi$
$522$	0	0
$523$	−15.5147	−0.678411	−0.339206	−	0.940712i	$-0.610158\pi$
$523$	−15.5147	−0.678411	−0.339206	+	0.940712i	$0.610158\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.21320	0.401333
$528$	0	0
$529$	−1.97056	−0.0856766
$530$	0	0
$531$	−17.6569	−0.766242
$532$	0	0
$533$	3.55635	0.154043
$534$	0	0
$535$	−14.4853	−0.626253
$536$	0	0
$537$	6.00000	0.258919
$538$	0	0
$539$	0	0
$540$	0	0
$541$	11.9706	0.514655	0.257327	−	0.966324i	$-0.417158\pi$
$541$	11.9706	0.514655	0.257327	+	0.966324i	$0.417158\pi$
$542$	0	0
$543$	−16.2426	−0.697038
$544$	0	0
$545$	5.00000	0.214176
$546$	0	0
$547$	7.51472	0.321306	0.160653	−	0.987011i	$-0.448640\pi$
$547$	7.51472	0.321306	0.160653	+	0.987011i	$0.448640\pi$
$548$	0	0
$549$	8.00000	0.341432
$550$	0	0
$551$	−15.9411	−0.679115
$552$	0	0
$553$	0	0
$554$	0	0
$555$	15.0711	0.639731
$556$	0	0
$557$	31.7990	1.34737	0.673683	−	0.739020i	$-0.264711\pi$
$557$	31.7990	1.34737	0.673683	+	0.739020i	$0.264711\pi$
$558$	0	0
$559$	−3.17157	−0.134143
$560$	0	0
$561$	73.7696	3.11455
$562$	0	0
$563$	−35.9411	−1.51474	−0.757369	−	0.652987i	$-0.773516\pi$
$563$	−35.9411	−1.51474	−0.757369	+	0.652987i	$0.773516\pi$
$564$	0	0
$565$	1.07107	0.0450602
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.14214	−0.0898030	−0.0449015	−	0.998991i	$-0.514297\pi$
$569$	−2.14214	−0.0898030	−0.0449015	+	0.998991i	$0.514297\pi$
$570$	0	0
$571$	34.4853	1.44316	0.721582	−	0.692329i	$-0.243415\pi$
$571$	34.4853	1.44316	0.721582	+	0.692329i	$0.243415\pi$
$572$	0	0
$573$	−48.2132	−2.01414
$574$	0	0
$575$	−4.58579	−0.191241
$576$	0	0
$577$	−9.72792	−0.404979	−0.202489	−	0.979284i	$-0.564903\pi$
$577$	−9.72792	−0.404979	−0.202489	+	0.979284i	$0.564903\pi$
$578$	0	0
$579$	38.6274	1.60530
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−24.7279	−1.02413
$584$	0	0
$585$	4.48528	0.185444
$586$	0	0
$587$	−13.4558	−0.555382	−0.277691	−	0.960670i	$-0.589569\pi$
$587$	−13.4558	−0.555382	−0.277691	+	0.960670i	$0.589569\pi$
$588$	0	0
$589$	10.5442	0.434464
$590$	0	0
$591$	−25.5563	−1.05125
$592$	0	0
$593$	10.7574	0.441752	0.220876	−	0.975302i	$-0.429108\pi$
$593$	10.7574	0.441752	0.220876	+	0.975302i	$0.429108\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−11.0711	−0.453109
$598$	0	0
$599$	12.1716	0.497317	0.248658	−	0.968591i	$-0.420010\pi$
$599$	12.1716	0.497317	0.248658	+	0.968591i	$0.420010\pi$
$600$	0	0
$601$	22.9706	0.936989	0.468494	−	0.883466i	$-0.344797\pi$
$601$	22.9706	0.936989	0.468494	+	0.883466i	$0.344797\pi$
$602$	0	0
$603$	−0.686292	−0.0279480
$604$	0	0
$605$	22.9706	0.933886
$606$	0	0
$607$	24.8995	1.01064	0.505320	−	0.862932i	$-0.331375\pi$
$607$	24.8995	1.01064	0.505320	+	0.862932i	$0.331375\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.97056	−0.0797204
$612$	0	0
$613$	43.9411	1.77477	0.887383	−	0.461034i	$-0.152521\pi$
$613$	43.9411	1.77477	0.887383	+	0.461034i	$0.152521\pi$
$614$	0	0
$615$	−5.41421	−0.218322
$616$	0	0
$617$	7.41421	0.298485	0.149242	−	0.988801i	$-0.452316\pi$
$617$	7.41421	0.298485	0.149242	+	0.988801i	$0.452316\pi$
$618$	0	0
$619$	−31.0711	−1.24885	−0.624426	−	0.781084i	$-0.714667\pi$
$619$	−31.0711	−1.24885	−0.624426	+	0.781084i	$0.714667\pi$
$620$	0	0
$621$	−1.89949	−0.0762241
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	84.4264	3.37167
$628$	0	0
$629$	32.7279	1.30495
$630$	0	0
$631$	−8.45584	−0.336622	−0.168311	−	0.985734i	$-0.553831\pi$
$631$	−8.45584	−0.336622	−0.168311	+	0.985734i	$0.553831\pi$
$632$	0	0
$633$	21.7279	0.863607
$634$	0	0
$635$	0.242641	0.00962890
$636$	0	0
$637$	0	0
$638$	0	0
$639$	24.9706	0.987820
$640$	0	0
$641$	−0.686292	−0.0271069	−0.0135534	−	0.999908i	$-0.504314\pi$
$641$	−0.686292	−0.0271069	−0.0135534	+	0.999908i	$0.504314\pi$
$642$	0	0
$643$	−27.7279	−1.09348	−0.546741	−	0.837302i	$-0.684132\pi$
$643$	−27.7279	−1.09348	−0.546741	+	0.837302i	$0.684132\pi$
$644$	0	0
$645$	4.82843	0.190119
$646$	0	0
$647$	28.4853	1.11987	0.559936	−	0.828536i	$-0.310826\pi$
$647$	28.4853	1.11987	0.559936	+	0.828536i	$0.310826\pi$
$648$	0	0
$649$	−36.3848	−1.42823
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.02944	0.0402850	0.0201425	−	0.999797i	$-0.493588\pi$
$653$	1.02944	0.0402850	0.0201425	+	0.999797i	$0.493588\pi$
$654$	0	0
$655$	−3.75736	−0.146812
$656$	0	0
$657$	24.0000	0.936329
$658$	0	0
$659$	13.9706	0.544216	0.272108	−	0.962267i	$-0.412279\pi$
$659$	13.9706	0.544216	0.272108	+	0.962267i	$0.412279\pi$
$660$	0	0
$661$	−37.4558	−1.45686	−0.728432	−	0.685118i	$-0.759750\pi$
$661$	−37.4558	−1.45686	−0.728432	+	0.685118i	$0.759750\pi$
$662$	0	0
$663$	20.0711	0.779496
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12.1838	−0.471757
$668$	0	0
$669$	43.9706	1.70000
$670$	0	0
$671$	16.4853	0.636407
$672$	0	0
$673$	20.4853	0.789650	0.394825	−	0.918756i	$-0.370805\pi$
$673$	20.4853	0.789650	0.394825	+	0.918756i	$0.370805\pi$
$674$	0	0
$675$	0.414214	0.0159431
$676$	0	0
$677$	1.78680	0.0686722	0.0343361	−	0.999410i	$-0.489068\pi$
$677$	1.78680	0.0686722	0.0343361	+	0.999410i	$0.489068\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−23.4853	−0.899958
$682$	0	0
$683$	7.79899	0.298420	0.149210	−	0.988806i	$-0.452327\pi$
$683$	7.79899	0.298420	0.149210	+	0.988806i	$0.452327\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	72.5269	2.76707
$688$	0	0
$689$	−6.72792	−0.256313
$690$	0	0
$691$	−9.17157	−0.348903	−0.174452	−	0.984666i	$-0.555815\pi$
$691$	−9.17157	−0.348903	−0.174452	+	0.984666i	$0.555815\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−16.2426	−0.616118
$696$	0	0
$697$	−11.7574	−0.445342
$698$	0	0
$699$	35.7990	1.35404
$700$	0	0
$701$	−4.45584	−0.168295	−0.0841475	−	0.996453i	$-0.526817\pi$
$701$	−4.45584	−0.168295	−0.0841475	+	0.996453i	$0.526817\pi$
$702$	0	0
$703$	37.4558	1.41267
$704$	0	0
$705$	3.00000	0.112987
$706$	0	0
$707$	0	0
$708$	0	0
$709$	17.0000	0.638448	0.319224	−	0.947679i	$-0.396578\pi$
$709$	17.0000	0.638448	0.319224	+	0.947679i	$0.396578\pi$
$710$	0	0
$711$	43.7990	1.64259
$712$	0	0
$713$	8.05887	0.301807
$714$	0	0
$715$	9.24264	0.345655
$716$	0	0
$717$	−1.24264	−0.0464073
$718$	0	0
$719$	−17.2132	−0.641944	−0.320972	−	0.947089i	$-0.604010\pi$
$719$	−17.2132	−0.641944	−0.320972	+	0.947089i	$0.604010\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−59.6985	−2.22021
$724$	0	0
$725$	2.65685	0.0986731
$726$	0	0
$727$	29.3137	1.08719	0.543593	−	0.839349i	$-0.317064\pi$
$727$	29.3137	1.08719	0.543593	+	0.839349i	$0.317064\pi$
$728$	0	0
$729$	−23.8284	−0.882534
$730$	0	0
$731$	10.4853	0.387812
$732$	0	0
$733$	−44.6985	−1.65098	−0.825488	−	0.564420i	$-0.809100\pi$
$733$	−44.6985	−1.65098	−0.825488	+	0.564420i	$0.809100\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.41421	−0.0520932
$738$	0	0
$739$	3.97056	0.146060	0.0730298	−	0.997330i	$-0.476733\pi$
$739$	3.97056	0.146060	0.0730298	+	0.997330i	$0.476733\pi$
$740$	0	0
$741$	22.9706	0.843845
$742$	0	0
$743$	−36.7279	−1.34742	−0.673708	−	0.738997i	$-0.735300\pi$
$743$	−36.7279	−1.34742	−0.673708	+	0.738997i	$0.735300\pi$
$744$	0	0
$745$	−14.8284	−0.543272
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−12.5147	−0.456669	−0.228334	−	0.973583i	$-0.573328\pi$
$751$	−12.5147	−0.456669	−0.228334	+	0.973583i	$0.573328\pi$
$752$	0	0
$753$	−60.8701	−2.21823
$754$	0	0
$755$	−9.48528	−0.345205
$756$	0	0
$757$	−16.4853	−0.599168	−0.299584	−	0.954070i	$-0.596848\pi$
$757$	−16.4853	−0.599168	−0.299584	+	0.954070i	$0.596848\pi$
$758$	0	0
$759$	64.5269	2.34218
$760$	0	0
$761$	12.7279	0.461387	0.230693	−	0.973026i	$-0.425901\pi$
$761$	12.7279	0.461387	0.230693	+	0.973026i	$0.425901\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−14.8284	−0.536123
$766$	0	0
$767$	−9.89949	−0.357450
$768$	0	0
$769$	−3.17157	−0.114370	−0.0571849	−	0.998364i	$-0.518212\pi$
$769$	−3.17157	−0.114370	−0.0571849	+	0.998364i	$0.518212\pi$
$770$	0	0
$771$	63.9411	2.30278
$772$	0	0
$773$	36.2132	1.30250	0.651249	−	0.758864i	$-0.274245\pi$
$773$	36.2132	1.30250	0.651249	+	0.758864i	$0.274245\pi$
$774$	0	0
$775$	−1.75736	−0.0631262
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−13.4558	−0.482106
$780$	0	0
$781$	51.4558	1.84123
$782$	0	0
$783$	1.10051	0.0393288
$784$	0	0
$785$	−20.8284	−0.743398
$786$	0	0
$787$	−45.7279	−1.63002	−0.815012	−	0.579444i	$-0.803270\pi$
$787$	−45.7279	−1.63002	−0.815012	+	0.579444i	$0.803270\pi$
$788$	0	0
$789$	69.1127	2.46048
$790$	0	0
$791$	0	0
$792$	0	0
$793$	4.48528	0.159277
$794$	0	0
$795$	10.2426	0.363269
$796$	0	0
$797$	55.1838	1.95471	0.977355	−	0.211608i	$-0.0678700\pi$
$797$	55.1838	1.95471	0.977355	+	0.211608i	$0.0678700\pi$
$798$	0	0
$799$	6.51472	0.230474
$800$	0	0
$801$	−22.6274	−0.799500
$802$	0	0
$803$	49.4558	1.74526
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18.7279	0.659254
$808$	0	0
$809$	30.5980	1.07577	0.537884	−	0.843019i	$-0.319224\pi$
$809$	30.5980	1.07577	0.537884	+	0.843019i	$0.319224\pi$
$810$	0	0
$811$	23.3553	0.820117	0.410058	−	0.912059i	$-0.365508\pi$
$811$	23.3553	0.820117	0.410058	+	0.912059i	$0.365508\pi$
$812$	0	0
$813$	−56.2843	−1.97398
$814$	0	0
$815$	1.75736	0.0615576
$816$	0	0
$817$	12.0000	0.419827
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.51472	−0.227365	−0.113683	−	0.993517i	$-0.536265\pi$
$821$	−6.51472	−0.227365	−0.113683	+	0.993517i	$0.536265\pi$
$822$	0	0
$823$	8.72792	0.304236	0.152118	−	0.988362i	$-0.451391\pi$
$823$	8.72792	0.304236	0.152118	+	0.988362i	$0.451391\pi$
$824$	0	0
$825$	−14.0711	−0.489892
$826$	0	0
$827$	6.04163	0.210088	0.105044	−	0.994468i	$-0.466502\pi$
$827$	6.04163	0.210088	0.105044	+	0.994468i	$0.466502\pi$
$828$	0	0
$829$	54.0416	1.87694	0.938472	−	0.345356i	$-0.112242\pi$
$829$	54.0416	1.87694	0.938472	+	0.345356i	$0.112242\pi$
$830$	0	0
$831$	65.6985	2.27906
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−9.24264	−0.319855
$836$	0	0
$837$	−0.727922	−0.0251607
$838$	0	0
$839$	48.7279	1.68227	0.841137	−	0.540822i	$-0.181887\pi$
$839$	48.7279	1.68227	0.841137	+	0.540822i	$0.181887\pi$
$840$	0	0
$841$	−21.9411	−0.756591
$842$	0	0
$843$	49.0416	1.68908
$844$	0	0
$845$	−10.4853	−0.360705
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−15.8284	−0.543230
$850$	0	0
$851$	28.6274	0.981335
$852$	0	0
$853$	−22.9706	−0.786497	−0.393249	−	0.919432i	$-0.628649\pi$
$853$	−22.9706	−0.786497	−0.393249	+	0.919432i	$0.628649\pi$
$854$	0	0
$855$	−16.9706	−0.580381
$856$	0	0
$857$	−5.51472	−0.188379	−0.0941896	−	0.995554i	$-0.530026\pi$
$857$	−5.51472	−0.188379	−0.0941896	+	0.995554i	$0.530026\pi$
$858$	0	0
$859$	−48.7696	−1.66400	−0.831998	−	0.554779i	$-0.812803\pi$
$859$	−48.7696	−1.66400	−0.831998	+	0.554779i	$0.812803\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.3848	0.625825	0.312913	−	0.949782i	$-0.398695\pi$
$863$	18.3848	0.625825	0.312913	+	0.949782i	$0.398695\pi$
$864$	0	0
$865$	1.24264	0.0422511
$866$	0	0
$867$	−25.3137	−0.859699
$868$	0	0
$869$	90.2548	3.06169
$870$	0	0
$871$	−0.384776	−0.0130376
$872$	0	0
$873$	13.4558	0.455411
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46.9706	1.58608	0.793042	−	0.609167i	$-0.208496\pi$
$877$	46.9706	1.58608	0.793042	+	0.609167i	$0.208496\pi$
$878$	0	0
$879$	−0.656854	−0.0221551
$880$	0	0
$881$	−19.0294	−0.641118	−0.320559	−	0.947229i	$-0.603871\pi$
$881$	−19.0294	−0.641118	−0.320559	+	0.947229i	$0.603871\pi$
$882$	0	0
$883$	−48.4853	−1.63166	−0.815830	−	0.578292i	$-0.803719\pi$
$883$	−48.4853	−1.63166	−0.815830	+	0.578292i	$0.803719\pi$
$884$	0	0
$885$	15.0711	0.506608
$886$	0	0
$887$	30.9706	1.03989	0.519945	−	0.854200i	$-0.325952\pi$
$887$	30.9706	1.03989	0.519945	+	0.854200i	$0.325952\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−55.2843	−1.85209
$892$	0	0
$893$	7.45584	0.249500
$894$	0	0
$895$	−2.48528	−0.0830738
$896$	0	0
$897$	17.5563	0.586189
$898$	0	0
$899$	−4.66905	−0.155721
$900$	0	0
$901$	22.2426	0.741010
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.72792	0.223644
$906$	0	0
$907$	−32.1838	−1.06864	−0.534322	−	0.845281i	$-0.679433\pi$
$907$	−32.1838	−1.06864	−0.534322	+	0.845281i	$0.679433\pi$
$908$	0	0
$909$	−40.9706	−1.35891
$910$	0	0
$911$	5.65685	0.187420	0.0937100	−	0.995600i	$-0.470127\pi$
$911$	5.65685	0.187420	0.0937100	+	0.995600i	$0.470127\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−6.82843	−0.225741
$916$	0	0
$917$	0	0
$918$	0	0
$919$	38.4558	1.26854	0.634271	−	0.773111i	$-0.281301\pi$
$919$	38.4558	1.26854	0.634271	+	0.773111i	$0.281301\pi$
$920$	0	0
$921$	26.7990	0.883057
$922$	0	0
$923$	14.0000	0.460816
$924$	0	0
$925$	−6.24264	−0.205257
$926$	0	0
$927$	−30.4264	−0.999334
$928$	0	0
$929$	54.7279	1.79556	0.897782	−	0.440439i	$-0.145177\pi$
$929$	54.7279	1.79556	0.897782	+	0.440439i	$0.145177\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−24.1421	−0.790378
$934$	0	0
$935$	−30.5563	−0.999299
$936$	0	0
$937$	−17.4437	−0.569859	−0.284930	−	0.958548i	$-0.591970\pi$
$937$	−17.4437	−0.569859	−0.284930	+	0.958548i	$0.591970\pi$
$938$	0	0
$939$	−58.4558	−1.90763
$940$	0	0
$941$	−4.97056	−0.162036	−0.0810179	−	0.996713i	$-0.525817\pi$
$941$	−4.97056	−0.162036	−0.0810179	+	0.996713i	$0.525817\pi$
$942$	0	0
$943$	−10.2843	−0.334902
$944$	0	0
$945$	0	0
$946$	0	0
$947$	43.7574	1.42192	0.710962	−	0.703231i	$-0.248260\pi$
$947$	43.7574	1.42192	0.710962	+	0.703231i	$0.248260\pi$
$948$	0	0
$949$	13.4558	0.436795
$950$	0	0
$951$	28.1421	0.912571
$952$	0	0
$953$	−29.0122	−0.939797	−0.469899	−	0.882720i	$-0.655710\pi$
$953$	−29.0122	−0.939797	−0.469899	+	0.882720i	$0.655710\pi$
$954$	0	0
$955$	19.9706	0.646232
$956$	0	0
$957$	−37.3848	−1.20848
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.9117	−0.900377
$962$	0	0
$963$	−40.9706	−1.32026
$964$	0	0
$965$	−16.0000	−0.515058
$966$	0	0
$967$	−24.4264	−0.785500	−0.392750	−	0.919645i	$-0.628476\pi$
$967$	−24.4264	−0.785500	−0.392750	+	0.919645i	$0.628476\pi$
$968$	0	0
$969$	−75.9411	−2.43958
$970$	0	0
$971$	42.7279	1.37120	0.685602	−	0.727976i	$-0.259539\pi$
$971$	42.7279	1.37120	0.685602	+	0.727976i	$0.259539\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−3.82843	−0.122608
$976$	0	0
$977$	−30.7696	−0.984405	−0.492203	−	0.870481i	$-0.663808\pi$
$977$	−30.7696	−0.984405	−0.492203	+	0.870481i	$0.663808\pi$
$978$	0	0
$979$	−46.6274	−1.49022
$980$	0	0
$981$	14.1421	0.451524
$982$	0	0
$983$	42.2132	1.34639	0.673196	−	0.739464i	$-0.264921\pi$
$983$	42.2132	1.34639	0.673196	+	0.739464i	$0.264921\pi$
$984$	0	0
$985$	10.5858	0.337291
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.17157	0.291639
$990$	0	0
$991$	47.9411	1.52290	0.761450	−	0.648224i	$-0.224488\pi$
$991$	47.9411	1.52290	0.761450	+	0.648224i	$0.224488\pi$
$992$	0	0
$993$	56.6274	1.79702
$994$	0	0
$995$	4.58579	0.145379
$996$	0	0
$997$	3.72792	0.118064	0.0590322	−	0.998256i	$-0.481199\pi$
$997$	3.72792	0.118064	0.0590322	+	0.998256i	$0.481199\pi$
$998$	0	0
$999$	−2.58579	−0.0818107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3920.2.a.br.1.1		2
4.3	odd	2		245.2.a.f.1.2	yes	2
7.6	odd	2		3920.2.a.bw.1.2		2
12.11	even	2		2205.2.a.t.1.1		2
20.3	even	4		1225.2.b.j.99.2		4
20.7	even	4		1225.2.b.j.99.3		4
20.19	odd	2		1225.2.a.p.1.1		2
28.3	even	6		245.2.e.g.226.1		4
28.11	odd	6		245.2.e.f.226.1		4
28.19	even	6		245.2.e.g.116.1		4
28.23	odd	6		245.2.e.f.116.1		4
28.27	even	2		245.2.a.e.1.2	✓	2
84.83	odd	2		2205.2.a.v.1.1		2
140.27	odd	4		1225.2.b.i.99.4		4
140.83	odd	4		1225.2.b.i.99.1		4
140.139	even	2		1225.2.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.a.e.1.2	✓	2	28.27	even	2
245.2.a.f.1.2	yes	2	4.3	odd	2
245.2.e.f.116.1		4	28.23	odd	6
245.2.e.f.226.1		4	28.11	odd	6
245.2.e.g.116.1		4	28.19	even	6
245.2.e.g.226.1		4	28.3	even	6
1225.2.a.p.1.1		2	20.19	odd	2
1225.2.a.r.1.1		2	140.139	even	2
1225.2.b.i.99.1		4	140.83	odd	4
1225.2.b.i.99.4		4	140.27	odd	4
1225.2.b.j.99.2		4	20.3	even	4
1225.2.b.j.99.3		4	20.7	even	4
2205.2.a.t.1.1		2	12.11	even	2
2205.2.a.v.1.1		2	84.83	odd	2
3920.2.a.br.1.1		2	1.1	even	1	trivial
3920.2.a.bw.1.2		2	7.6	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:	\( N \)	\(=\)	\( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3920.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{3} +1.00000 q^{5} +2.82843 q^{9} +O(q^{10})\)	\(q-2.41421 q^{3} +1.00000 q^{5} +2.82843 q^{9} +5.82843 q^{11} +1.58579 q^{13} -2.41421 q^{15} -5.24264 q^{17} -6.00000 q^{19} -4.58579 q^{23} +1.00000 q^{25} +0.414214 q^{27} +2.65685 q^{29} -1.75736 q^{31} -14.0711 q^{33} -6.24264 q^{37} -3.82843 q^{39} +2.24264 q^{41} -2.00000 q^{43} +2.82843 q^{45} -1.24264 q^{47} +12.6569 q^{51} -4.24264 q^{53} +5.82843 q^{55} +14.4853 q^{57} -6.24264 q^{59} +2.82843 q^{61} +1.58579 q^{65} -0.242641 q^{67} +11.0711 q^{69} +8.82843 q^{71} +8.48528 q^{73} -2.41421 q^{75} +15.4853 q^{79} -9.48528 q^{81} -5.24264 q^{85} -6.41421 q^{87} -8.00000 q^{89} +4.24264 q^{93} -6.00000 q^{95} +4.75736 q^{97} +16.4853 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5	\( 2 q - 2 q^{3} + 2 q^{5} + 6 q^{11} + 6 q^{13} - 2 q^{15} - 2 q^{17} - 12 q^{19} - 12 q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{29} - 12 q^{31} - 14 q^{33} - 4 q^{37} - 2 q^{39} - 4 q^{41} - 4 q^{43} + 6 q^{47} + 14 q^{51} + 6 q^{55} + 12 q^{57} - 4 q^{59} + 6 q^{65} + 8 q^{67} + 8 q^{69} + 12 q^{71} - 2 q^{75} + 14 q^{79} - 2 q^{81} - 2 q^{85} - 10 q^{87} - 16 q^{89} - 12 q^{95} + 18 q^{97} + 16 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 6 * q^11 + 6 * q^13 - 2 * q^15 - 2 * q^17 - 12 * q^19 - 12 * q^23 + 2 * q^25 - 2 * q^27 - 6 * q^29 - 12 * q^31 - 14 * q^33 - 4 * q^37 - 2 * q^39 - 4 * q^41 - 4 * q^43 + 6 * q^47 + 14 * q^51 + 6 * q^55 + 12 * q^57 - 4 * q^59 + 6 * q^65 + 8 * q^67 + 8 * q^69 + 12 * q^71 - 2 * q^75 + 14 * q^79 - 2 * q^81 - 2 * q^85 - 10 * q^87 - 16 * q^89 - 12 * q^95 + 18 * q^97 + 16 * q^99

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