LMFDB - Embedded newform 896.2.b.d.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [896,2,Mod(449,896)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(896, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("896.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 896 = 2^{7} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	896.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$7.15459602111$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	896.449
Dual form			896.2.b.d.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{5} +1.00000 q^{7} +3.00000 q^{9} +O(q^{10})$	$q-2.82843i q^{5} +1.00000 q^{7} +3.00000 q^{9} -2.82843i q^{11} -2.82843i q^{13} -2.00000 q^{17} +5.65685i q^{19} -8.00000 q^{23} -3.00000 q^{25} -5.65685i q^{29} +8.00000 q^{31} -2.82843i q^{35} -5.65685i q^{37} -6.00000 q^{41} -2.82843i q^{43} -8.48528i q^{45} +8.00000 q^{47} +1.00000 q^{49} -11.3137i q^{53} -8.00000 q^{55} +11.3137i q^{59} +2.82843i q^{61} +3.00000 q^{63} -8.00000 q^{65} -8.48528i q^{67} -8.00000 q^{71} +6.00000 q^{73} -2.82843i q^{77} -8.00000 q^{79} +9.00000 q^{81} -5.65685i q^{83} +5.65685i q^{85} +6.00000 q^{89} -2.82843i q^{91} +16.0000 q^{95} +14.0000 q^{97} -8.48528i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^7 + 6 * q^9	$ 2 q + 2 q^{7} + 6 q^{9} - 4 q^{17} - 16 q^{23} - 6 q^{25} + 16 q^{31} - 12 q^{41} + 16 q^{47} + 2 q^{49} - 16 q^{55} + 6 q^{63} - 16 q^{65} - 16 q^{71} + 12 q^{73} - 16 q^{79} + 18 q^{81} + 12 q^{89} + 32 q^{95} + 28 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 + 6 * q^9 - 4 * q^17 - 16 * q^23 - 6 * q^25 + 16 * q^31 - 12 * q^41 + 16 * q^47 + 2 * q^49 - 16 * q^55 + 6 * q^63 - 16 * q^65 - 16 * q^71 + 12 * q^73 - 16 * q^79 + 18 * q^81 + 12 * q^89 + 32 * q^95 + 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/896\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$645$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	0	0
$5$	− 2.82843i	− 1.26491i	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	− 2.82843i	− 1.26491i	0.774597	−	0.632456i	$-0.217953\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	− 2.82843i	− 0.852803i	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	− 2.82843i	− 0.852803i	0.904534	−	0.426401i	$-0.140219\pi$
$12$	0	0
$13$	− 2.82843i	− 0.784465i	−0.919866	−	0.392232i	$-0.871703\pi$
$13$	− 2.82843i	− 0.784465i	0.919866	−	0.392232i	$-0.128297\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	5.65685i	1.29777i	0.760886	+	0.648886i	$0.224765\pi$
$19$	5.65685i	1.29777i	−0.760886	+	0.648886i	$0.775235\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 5.65685i	− 1.05045i	−0.850963	−	0.525226i	$-0.823981\pi$
$29$	− 5.65685i	− 1.05045i	0.850963	−	0.525226i	$-0.176019\pi$
$30$	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
$33$	0	0
$34$	0	0
$35$	− 2.82843i	− 0.478091i
$36$	0	0
$37$	− 5.65685i	− 0.929981i	−0.885316	−	0.464991i	$-0.846058\pi$
$37$	− 5.65685i	− 0.929981i	0.885316	−	0.464991i	$-0.153942\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	− 2.82843i	− 0.431331i	−0.976467	−	0.215666i	$-0.930808\pi$
$43$	− 2.82843i	− 0.431331i	0.976467	−	0.215666i	$-0.0691921\pi$
$44$	0	0
$45$	− 8.48528i	− 1.26491i
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 11.3137i	− 1.55406i	−0.629465	−	0.777029i	$-0.716726\pi$
$53$	− 11.3137i	− 1.55406i	0.629465	−	0.777029i	$-0.283274\pi$
$54$	0	0
$55$	−8.00000	−1.07872
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.3137i	1.47292i	0.676481	+	0.736460i	$0.263504\pi$
$59$	11.3137i	1.47292i	−0.676481	+	0.736460i	$0.736496\pi$
$60$	0	0
$61$	2.82843i	0.362143i	0.983470	+	0.181071i	$0.0579565\pi$
$61$	2.82843i	0.362143i	−0.983470	+	0.181071i	$0.942043\pi$
$62$	0	0
$63$	3.00000	0.377964
$64$	0	0
$65$	−8.00000	−0.992278
$66$	0	0
$67$	− 8.48528i	− 1.03664i	−0.855186	−	0.518321i	$-0.826557\pi$
$67$	− 8.48528i	− 1.03664i	0.855186	−	0.518321i	$-0.173443\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	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
$75$	0	0
$76$	0	0
$77$	− 2.82843i	− 0.322329i
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	− 5.65685i	− 0.620920i	−0.950586	−	0.310460i	$-0.899517\pi$
$83$	− 5.65685i	− 0.620920i	0.950586	−	0.310460i	$-0.100483\pi$
$84$	0	0
$85$	5.65685i	0.613572i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	− 2.82843i	− 0.296500i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	16.0000	1.64157
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	− 8.48528i	− 0.852803i
$100$	0	0
$101$	8.48528i	0.844317i	0.906522	+	0.422159i	$0.138727\pi$
$101$	8.48528i	0.844317i	−0.906522	+	0.422159i	$0.861273\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.1421i	1.36717i	0.729870	+	0.683586i	$0.239581\pi$
$107$	14.1421i	1.36717i	−0.729870	+	0.683586i	$0.760419\pi$
$108$	0	0
$109$	16.9706i	1.62549i	0.582623	+	0.812743i	$0.302026\pi$
$109$	16.9706i	1.62549i	−0.582623	+	0.812743i	$0.697974\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	22.6274i	2.11002i
$116$	0	0
$117$	− 8.48528i	− 0.784465i
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	3.00000	0.272727
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 5.65685i	− 0.505964i
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 5.65685i	− 0.494242i	−0.968985	−	0.247121i	$-0.920516\pi$
$131$	− 5.65685i	− 0.494242i	0.968985	−	0.247121i	$-0.0794845\pi$
$132$	0	0
$133$	5.65685i	0.490511i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	− 16.9706i	− 1.43942i	−0.694273	−	0.719712i	$-0.744274\pi$
$139$	− 16.9706i	− 1.43942i	0.694273	−	0.719712i	$-0.255726\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	−16.0000	−1.32873
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.3137i	0.926855i	0.886135	+	0.463428i	$0.153381\pi$
$149$	11.3137i	0.926855i	−0.886135	+	0.463428i	$0.846619\pi$
$150$	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
$153$	−6.00000	−0.485071
$154$	0	0
$155$	− 22.6274i	− 1.81748i
$156$	0	0
$157$	8.48528i	0.677199i	0.940931	+	0.338600i	$0.109953\pi$
$157$	8.48528i	0.677199i	−0.940931	+	0.338600i	$0.890047\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	19.7990i	1.55078i	0.631485	+	0.775388i	$0.282446\pi$
$163$	19.7990i	1.55078i	−0.631485	+	0.775388i	$0.717554\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	16.9706i	1.29777i
$172$	0	0
$173$	8.48528i	0.645124i	0.946548	+	0.322562i	$0.104544\pi$
$173$	8.48528i	0.645124i	−0.946548	+	0.322562i	$0.895456\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 8.48528i	− 0.634220i	−0.948389	−	0.317110i	$-0.897288\pi$
$179$	− 8.48528i	− 0.634220i	0.948389	−	0.317110i	$-0.102712\pi$
$180$	0	0
$181$	25.4558i	1.89212i	0.323994	+	0.946059i	$0.394974\pi$
$181$	25.4558i	1.89212i	−0.323994	+	0.946059i	$0.605026\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−16.0000	−1.17634
$186$	0	0
$187$	5.65685i	0.413670i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 5.65685i	− 0.397033i
$204$	0	0
$205$	16.9706i	1.18528i
$206$	0	0
$207$	−24.0000	−1.66812
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	− 19.7990i	− 1.36302i	−0.731809	−	0.681509i	$-0.761324\pi$
$211$	− 19.7990i	− 1.36302i	0.731809	−	0.681509i	$-0.238676\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.65685i	0.380521i
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	−9.00000	−0.600000
$226$	0	0
$227$	28.2843i	1.87729i	0.344881	+	0.938647i	$0.387919\pi$
$227$	28.2843i	1.87729i	−0.344881	+	0.938647i	$0.612081\pi$
$228$	0	0
$229$	25.4558i	1.68217i	0.540903	+	0.841085i	$0.318082\pi$
$229$	25.4558i	1.68217i	−0.540903	+	0.841085i	$0.681918\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	0	0
$235$	− 22.6274i	− 1.47605i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 2.82843i	− 0.180702i
$246$	0	0
$247$	16.0000	1.01806
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5.65685i	0.357057i	0.983935	+	0.178529i	$0.0571337\pi$
$251$	5.65685i	0.357057i	−0.983935	+	0.178529i	$0.942866\pi$
$252$	0	0
$253$	22.6274i	1.42257i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	− 5.65685i	− 0.351500i
$260$	0	0
$261$	− 16.9706i	− 1.05045i
$262$	0	0
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	−32.0000	−1.96574
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 2.82843i	− 0.172452i	−0.996276	−	0.0862261i	$-0.972519\pi$
$269$	− 2.82843i	− 0.172452i	0.996276	−	0.0862261i	$-0.0274808\pi$
$270$	0	0
$271$	−32.0000	−1.94386	−0.971931	−	0.235267i	$-0.924404\pi$
$271$	−32.0000	−1.94386	−0.971931	+	0.235267i	$0.924404\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.48528i	0.511682i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	24.0000	1.43684
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	22.6274i	1.34506i	0.740070	+	0.672530i	$0.234792\pi$
$283$	22.6274i	1.34506i	−0.740070	+	0.672530i	$0.765208\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.00000	−0.354169
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 14.1421i	− 0.826192i	−0.910687	−	0.413096i	$-0.864447\pi$
$293$	− 14.1421i	− 0.826192i	0.910687	−	0.413096i	$-0.135553\pi$
$294$	0	0
$295$	32.0000	1.86311
$296$	0	0
$297$	0	0
$298$	0	0
$299$	22.6274i	1.30858i
$300$	0	0
$301$	− 2.82843i	− 0.163028i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.00000	0.458079
$306$	0	0
$307$	− 28.2843i	− 1.61427i	−0.590368	−	0.807134i	$-0.701017\pi$
$307$	− 28.2843i	− 1.61427i	0.590368	−	0.807134i	$-0.298983\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	− 8.48528i	− 0.478091i
$316$	0	0
$317$	− 22.6274i	− 1.27088i	−0.772149	−	0.635441i	$-0.780818\pi$
$317$	− 22.6274i	− 1.27088i	0.772149	−	0.635441i	$-0.219182\pi$
$318$	0	0
$319$	−16.0000	−0.895828
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 11.3137i	− 0.629512i
$324$	0	0
$325$	8.48528i	0.470679i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	8.48528i	0.466393i	0.972430	+	0.233197i	$0.0749186\pi$
$331$	8.48528i	0.466393i	−0.972430	+	0.233197i	$0.925081\pi$
$332$	0	0
$333$	− 16.9706i	− 0.929981i
$334$	0	0
$335$	−24.0000	−1.31126
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 22.6274i	− 1.22534i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 25.4558i	− 1.36654i	−0.730165	−	0.683271i	$-0.760557\pi$
$347$	− 25.4558i	− 1.36654i	0.730165	−	0.683271i	$-0.239443\pi$
$348$	0	0
$349$	8.48528i	0.454207i	0.973871	+	0.227103i	$0.0729255\pi$
$349$	8.48528i	0.454207i	−0.973871	+	0.227103i	$0.927074\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	22.6274i	1.20094i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−13.0000	−0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 16.9706i	− 0.888280i
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	0	0
$369$	−18.0000	−0.937043
$370$	0	0
$371$	− 11.3137i	− 0.587378i
$372$	0	0
$373$	33.9411i	1.75740i	0.477370	+	0.878702i	$0.341590\pi$
$373$	33.9411i	1.75740i	−0.477370	+	0.878702i	$0.658410\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−16.0000	−0.824042
$378$	0	0
$379$	8.48528i	0.435860i	0.975964	+	0.217930i	$0.0699304\pi$
$379$	8.48528i	0.435860i	−0.975964	+	0.217930i	$0.930070\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	0	0
$387$	− 8.48528i	− 0.431331i
$388$	0	0
$389$	16.9706i	0.860442i	0.902724	+	0.430221i	$0.141564\pi$
$389$	16.9706i	0.860442i	−0.902724	+	0.430221i	$0.858436\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	0	0
$394$	0	0
$395$	22.6274i	1.13851i
$396$	0	0
$397$	− 25.4558i	− 1.27759i	−0.769376	−	0.638796i	$-0.779433\pi$
$397$	− 25.4558i	− 1.27759i	0.769376	−	0.638796i	$-0.220567\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	− 22.6274i	− 1.12715i
$404$	0	0
$405$	− 25.4558i	− 1.26491i
$406$	0	0
$407$	−16.0000	−0.793091
$408$	0	0
$409$	−38.0000	−1.87898	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000	−1.87898	−0.939490	+	0.342578i	$0.888700\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	11.3137i	0.556711i
$414$	0	0
$415$	−16.0000	−0.785409
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	− 22.6274i	− 1.10279i	−0.834243	−	0.551396i	$-0.814095\pi$
$421$	− 22.6274i	− 1.10279i	0.834243	−	0.551396i	$-0.185905\pi$
$422$	0	0
$423$	24.0000	1.16692
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	2.82843i	0.136877i
$428$	0	0
$429$	0	0
$430$	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
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 45.2548i	− 2.16483i
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	2.82843i	0.134383i	0.997740	+	0.0671913i	$0.0214038\pi$
$443$	2.82843i	0.134383i	−0.997740	+	0.0671913i	$0.978596\pi$
$444$	0	0
$445$	− 16.9706i	− 0.804482i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	16.9706i	0.799113i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−8.00000	−0.375046
$456$	0	0
$457$	34.0000	1.59045	0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000	1.59045	0.795226	+	0.606313i	$0.207352\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.48528i	0.395199i	0.980283	+	0.197599i	$0.0633145\pi$
$461$	8.48528i	0.395199i	−0.980283	+	0.197599i	$0.936685\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.65685i	0.261768i	0.991398	+	0.130884i	$0.0417815\pi$
$467$	5.65685i	0.261768i	−0.991398	+	0.130884i	$0.958218\pi$
$468$	0	0
$469$	− 8.48528i	− 0.391814i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	− 16.9706i	− 0.778663i
$476$	0	0
$477$	− 33.9411i	− 1.55406i
$478$	0	0
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 39.5980i	− 1.79805i
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 31.1127i	− 1.40410i	−0.712129	−	0.702048i	$-0.752269\pi$
$491$	− 31.1127i	− 1.40410i	0.712129	−	0.702048i	$-0.247731\pi$
$492$	0	0
$493$	11.3137i	0.509544i
$494$	0	0
$495$	−24.0000	−1.07872
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	25.4558i	1.13956i	0.821797	+	0.569780i	$0.192972\pi$
$499$	25.4558i	1.13956i	−0.821797	+	0.569780i	$0.807028\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−40.0000	−1.78351	−0.891756	−	0.452517i	$-0.850526\pi$
$503$	−40.0000	−1.78351	−0.891756	+	0.452517i	$0.850526\pi$
$504$	0	0
$505$	24.0000	1.06799
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 2.82843i	− 0.125368i	−0.998033	−	0.0626839i	$-0.980034\pi$
$509$	− 2.82843i	− 0.125368i	0.998033	−	0.0626839i	$-0.0199660\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 22.6274i	− 0.995153i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	0	0
$523$	− 11.3137i	− 0.494714i	−0.968924	−	0.247357i	$-0.920438\pi$
$523$	− 11.3137i	− 0.494714i	0.968924	−	0.247357i	$-0.0795620\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−16.0000	−0.696971
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	33.9411i	1.47292i
$532$	0	0
$533$	16.9706i	0.735077i
$534$	0	0
$535$	40.0000	1.72935
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 2.82843i	− 0.121829i
$540$	0	0
$541$	− 11.3137i	− 0.486414i	−0.969974	−	0.243207i	$-0.921801\pi$
$541$	− 11.3137i	− 0.486414i	0.969974	−	0.243207i	$-0.0781995\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	48.0000	2.05609
$546$	0	0
$547$	− 2.82843i	− 0.120935i	−0.998170	−	0.0604674i	$-0.980741\pi$
$547$	− 2.82843i	− 0.120935i	0.998170	−	0.0604674i	$-0.0192591\pi$
$548$	0	0
$549$	8.48528i	0.362143i
$550$	0	0
$551$	32.0000	1.36325
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 45.2548i	− 1.91751i	−0.284236	−	0.958754i	$-0.591740\pi$
$557$	− 45.2548i	− 1.91751i	0.284236	−	0.958754i	$-0.408260\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11.3137i	0.476816i	0.971165	+	0.238408i	$0.0766255\pi$
$563$	11.3137i	0.476816i	−0.971165	+	0.238408i	$0.923374\pi$
$564$	0	0
$565$	16.9706i	0.713957i
$566$	0	0
$567$	9.00000	0.377964
$568$	0	0
$569$	2.00000	0.0838444	0.0419222	−	0.999121i	$-0.486652\pi$
$569$	2.00000	0.0838444	0.0419222	+	0.999121i	$0.486652\pi$
$570$	0	0
$571$	19.7990i	0.828562i	0.910149	+	0.414281i	$0.135967\pi$
$571$	19.7990i	0.828562i	−0.910149	+	0.414281i	$0.864033\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	24.0000	1.00087
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 5.65685i	− 0.234686i
$582$	0	0
$583$	−32.0000	−1.32530
$584$	0	0
$585$	−24.0000	−0.992278
$586$	0	0
$587$	− 5.65685i	− 0.233483i	−0.993162	−	0.116742i	$-0.962755\pi$
$587$	− 5.65685i	− 0.233483i	0.993162	−	0.116742i	$-0.0372450\pi$
$588$	0	0
$589$	45.2548i	1.86469i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	5.65685i	0.231908i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	− 25.4558i	− 1.03664i
$604$	0	0
$605$	− 8.48528i	− 0.344976i
$606$	0	0
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 22.6274i	− 0.915407i
$612$	0	0
$613$	28.2843i	1.14239i	0.820814	+	0.571195i	$0.193520\pi$
$613$	28.2843i	1.14239i	−0.820814	+	0.571195i	$0.806480\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−26.0000	−1.04672	−0.523360	−	0.852111i	$-0.675322\pi$
$617$	−26.0000	−1.04672	−0.523360	+	0.852111i	$0.675322\pi$
$618$	0	0
$619$	− 11.3137i	− 0.454736i	−0.973809	−	0.227368i	$-0.926988\pi$
$619$	− 11.3137i	− 0.454736i	0.973809	−	0.227368i	$-0.0730121\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.00000	0.240385
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.3137i	0.451107i
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 22.6274i	− 0.897942i
$636$	0	0
$637$	− 2.82843i	− 0.112066i
$638$	0	0
$639$	−24.0000	−0.949425
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	− 33.9411i	− 1.33851i	−0.743034	−	0.669254i	$-0.766614\pi$
$643$	− 33.9411i	− 1.33851i	0.743034	−	0.669254i	$-0.233386\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16.0000	−0.629025	−0.314512	−	0.949253i	$-0.601841\pi$
$647$	−16.0000	−0.629025	−0.314512	+	0.949253i	$0.601841\pi$
$648$	0	0
$649$	32.0000	1.25611
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.65685i	0.221370i	0.993856	+	0.110685i	$0.0353044\pi$
$653$	5.65685i	0.221370i	−0.993856	+	0.110685i	$0.964696\pi$
$654$	0	0
$655$	−16.0000	−0.625172
$656$	0	0
$657$	18.0000	0.702247
$658$	0	0
$659$	8.48528i	0.330540i	0.986248	+	0.165270i	$0.0528495\pi$
$659$	8.48528i	0.330540i	−0.986248	+	0.165270i	$0.947151\pi$
$660$	0	0
$661$	− 2.82843i	− 0.110013i	−0.998486	−	0.0550065i	$-0.982482\pi$
$661$	− 2.82843i	− 0.110013i	0.998486	−	0.0550065i	$-0.0175180\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	16.0000	0.620453
$666$	0	0
$667$	45.2548i	1.75227i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.82843i	0.108705i	0.998522	+	0.0543526i	$0.0173095\pi$
$677$	2.82843i	0.108705i	−0.998522	+	0.0543526i	$0.982690\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	0	0
$681$	0	0
$682$	0	0
$683$	36.7696i	1.40695i	0.710721	+	0.703474i	$0.248369\pi$
$683$	36.7696i	1.40695i	−0.710721	+	0.703474i	$0.751631\pi$
$684$	0	0
$685$	16.9706i	0.648412i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−32.0000	−1.21910
$690$	0	0
$691$	− 16.9706i	− 0.645591i	−0.946469	−	0.322795i	$-0.895377\pi$
$691$	− 16.9706i	− 0.645591i	0.946469	−	0.322795i	$-0.104623\pi$
$692$	0	0
$693$	− 8.48528i	− 0.322329i
$694$	0	0
$695$	−48.0000	−1.82074
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	0	0
$700$	0	0
$701$	5.65685i	0.213656i	0.994277	+	0.106828i	$0.0340695\pi$
$701$	5.65685i	0.213656i	−0.994277	+	0.106828i	$0.965931\pi$
$702$	0	0
$703$	32.0000	1.20690
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.48528i	0.319122i
$708$	0	0
$709$	16.9706i	0.637343i	0.947865	+	0.318671i	$0.103237\pi$
$709$	16.9706i	0.637343i	−0.947865	+	0.318671i	$0.896763\pi$
$710$	0	0
$711$	−24.0000	−0.900070
$712$	0	0
$713$	−64.0000	−2.39682
$714$	0	0
$715$	22.6274i	0.846217i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	16.9706i	0.630271i
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	5.65685i	0.209226i
$732$	0	0
$733$	14.1421i	0.522352i	0.965291	+	0.261176i	$0.0841102\pi$
$733$	14.1421i	0.522352i	−0.965291	+	0.261176i	$0.915890\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24.0000	−0.884051
$738$	0	0
$739$	− 25.4558i	− 0.936408i	−0.883620	−	0.468204i	$-0.844901\pi$
$739$	− 25.4558i	− 0.936408i	0.883620	−	0.468204i	$-0.155099\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	32.0000	1.17239
$746$	0	0
$747$	− 16.9706i	− 0.620920i
$748$	0	0
$749$	14.1421i	0.516742i
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	22.6274i	0.823496i
$756$	0	0
$757$	− 39.5980i	− 1.43921i	−0.694382	−	0.719607i	$-0.744322\pi$
$757$	− 39.5980i	− 1.43921i	0.694382	−	0.719607i	$-0.255678\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	0	0
$763$	16.9706i	0.614376i
$764$	0	0
$765$	16.9706i	0.613572i
$766$	0	0
$767$	32.0000	1.15545
$768$	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
$771$	0	0
$772$	0	0
$773$	19.7990i	0.712120i	0.934463	+	0.356060i	$0.115880\pi$
$773$	19.7990i	0.712120i	−0.934463	+	0.356060i	$0.884120\pi$
$774$	0	0
$775$	−24.0000	−0.862105
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 33.9411i	− 1.21607i
$780$	0	0
$781$	22.6274i	0.809673i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	24.0000	0.856597
$786$	0	0
$787$	− 28.2843i	− 1.00823i	−0.863638	−	0.504113i	$-0.831820\pi$
$787$	− 28.2843i	− 1.00823i	0.863638	−	0.504113i	$-0.168180\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	8.00000	0.284088
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 2.82843i	− 0.100188i	−0.998745	−	0.0500940i	$-0.984048\pi$
$797$	− 2.82843i	− 0.100188i	0.998745	−	0.0500940i	$-0.0159521\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	18.0000	0.635999
$802$	0	0
$803$	− 16.9706i	− 0.598878i
$804$	0	0
$805$	22.6274i	0.797512i
$806$	0	0
$807$	0	0
$808$	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
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	56.0000	1.96159
$816$	0	0
$817$	16.0000	0.559769
$818$	0	0
$819$	− 8.48528i	− 0.296500i
$820$	0	0
$821$	45.2548i	1.57940i	0.613490	+	0.789702i	$0.289765\pi$
$821$	45.2548i	1.57940i	−0.613490	+	0.789702i	$0.710235\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.7696i	1.27860i	0.768956	+	0.639301i	$0.220776\pi$
$827$	36.7696i	1.27860i	−0.768956	+	0.639301i	$0.779224\pi$
$828$	0	0
$829$	48.0833i	1.67000i	0.550249	+	0.835000i	$0.314533\pi$
$829$	48.0833i	1.67000i	−0.550249	+	0.835000i	$0.685467\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−32.0000	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$840$	0	0
$841$	−3.00000	−0.103448
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 14.1421i	− 0.486504i
$846$	0	0
$847$	3.00000	0.103081
$848$	0	0
$849$	0	0
$850$	0	0
$851$	45.2548i	1.55132i
$852$	0	0
$853$	25.4558i	0.871592i	0.900046	+	0.435796i	$0.143533\pi$
$853$	25.4558i	0.871592i	−0.900046	+	0.435796i	$0.856467\pi$
$854$	0	0
$855$	48.0000	1.64157
$856$	0	0
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	0	0
$859$	39.5980i	1.35107i	0.737330	+	0.675533i	$0.236086\pi$
$859$	39.5980i	1.35107i	−0.737330	+	0.675533i	$0.763914\pi$
$860$	0	0
$861$	0	0
$862$	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
$865$	24.0000	0.816024
$866$	0	0
$867$	0	0
$868$	0	0
$869$	22.6274i	0.767583i
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	42.0000	1.42148
$874$	0	0
$875$	− 5.65685i	− 0.191237i
$876$	0	0
$877$	− 16.9706i	− 0.573055i	−0.958072	−	0.286528i	$-0.907499\pi$
$877$	− 16.9706i	− 0.573055i	0.958072	−	0.286528i	$-0.0925010\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	25.4558i	0.856657i	0.903623	+	0.428329i	$0.140897\pi$
$883$	25.4558i	0.856657i	−0.903623	+	0.428329i	$0.859103\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−32.0000	−1.07445	−0.537227	−	0.843437i	$-0.680528\pi$
$887$	−32.0000	−1.07445	−0.537227	+	0.843437i	$0.680528\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	− 25.4558i	− 0.852803i
$892$	0	0
$893$	45.2548i	1.51440i
$894$	0	0
$895$	−24.0000	−0.802232
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 45.2548i	− 1.50933i
$900$	0	0
$901$	22.6274i	0.753829i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	72.0000	2.39336
$906$	0	0
$907$	− 19.7990i	− 0.657415i	−0.944432	−	0.328707i	$-0.893387\pi$
$907$	− 19.7990i	− 0.657415i	0.944432	−	0.328707i	$-0.106613\pi$
$908$	0	0
$909$	25.4558i	0.844317i
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 5.65685i	− 0.186806i
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	22.6274i	0.744791i
$924$	0	0
$925$	16.9706i	0.557989i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	5.65685i	0.185396i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	16.0000	0.523256
$936$	0	0
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.82843i	0.0922041i	0.998937	+	0.0461020i	$0.0146799\pi$
$941$	2.82843i	0.0922041i	−0.998937	+	0.0461020i	$0.985320\pi$
$942$	0	0
$943$	48.0000	1.56310
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 25.4558i	− 0.827204i	−0.910458	−	0.413602i	$-0.864271\pi$
$947$	− 25.4558i	− 0.827204i	0.910458	−	0.413602i	$-0.135729\pi$
$948$	0	0
$949$	− 16.9706i	− 0.550888i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10.0000	0.323932	0.161966	−	0.986796i	$-0.448217\pi$
$953$	10.0000	0.323932	0.161966	+	0.986796i	$0.448217\pi$
$954$	0	0
$955$	− 67.8823i	− 2.19662i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.00000	−0.193750
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	42.4264i	1.36717i
$964$	0	0
$965$	5.65685i	0.182101i
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 11.3137i	− 0.363074i	−0.983384	−	0.181537i	$-0.941893\pi$
$971$	− 11.3137i	− 0.363074i	0.983384	−	0.181537i	$-0.0581072\pi$
$972$	0	0
$973$	− 16.9706i	− 0.544051i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.0000	−0.319928	−0.159964	−	0.987123i	$-0.551138\pi$
$977$	−10.0000	−0.319928	−0.159964	+	0.987123i	$0.551138\pi$
$978$	0	0
$979$	− 16.9706i	− 0.542382i
$980$	0	0
$981$	50.9117i	1.62549i
$982$	0	0
$983$	−48.0000	−1.53096	−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000	−1.53096	−0.765481	+	0.643458i	$0.777499\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	22.6274i	0.719510i
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 25.4558i	− 0.806195i	−0.915157	−	0.403097i	$-0.867934\pi$
$997$	− 25.4558i	− 0.806195i	0.915157	−	0.403097i	$-0.132066\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.b.d.449.1	yes	2
4.3	odd	2		896.2.b.b.449.1	✓	2
8.3	odd	2		896.2.b.b.449.2	yes	2
8.5	even	2	inner	896.2.b.d.449.2	yes	2
16.3	odd	4		1792.2.a.o.1.1		2
16.5	even	4		1792.2.a.m.1.2		2
16.11	odd	4		1792.2.a.o.1.2		2
16.13	even	4		1792.2.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.b.b.449.1	✓	2	4.3	odd	2
896.2.b.b.449.2	yes	2	8.3	odd	2
896.2.b.d.449.1	yes	2	1.1	even	1	trivial
896.2.b.d.449.2	yes	2	8.5	even	2	inner
1792.2.a.m.1.1		2	16.13	even	4
1792.2.a.m.1.2		2	16.5	even	4
1792.2.a.o.1.1		2	16.3	odd	4
1792.2.a.o.1.2		2	16.11	odd	4

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:	\( N \)	\(=\)	\( 896 = 2^{7} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	896.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{5} +1.00000 q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-2.82843i q^{5} +1.00000 q^{7} +3.00000 q^{9} -2.82843i q^{11} -2.82843i q^{13} -2.00000 q^{17} +5.65685i q^{19} -8.00000 q^{23} -3.00000 q^{25} -5.65685i q^{29} +8.00000 q^{31} -2.82843i q^{35} -5.65685i q^{37} -6.00000 q^{41} -2.82843i q^{43} -8.48528i q^{45} +8.00000 q^{47} +1.00000 q^{49} -11.3137i q^{53} -8.00000 q^{55} +11.3137i q^{59} +2.82843i q^{61} +3.00000 q^{63} -8.00000 q^{65} -8.48528i q^{67} -8.00000 q^{71} +6.00000 q^{73} -2.82843i q^{77} -8.00000 q^{79} +9.00000 q^{81} -5.65685i q^{83} +5.65685i q^{85} +6.00000 q^{89} -2.82843i q^{91} +16.0000 q^{95} +14.0000 q^{97} -8.48528i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^7 + 6 * q^9	\( 2 q + 2 q^{7} + 6 q^{9} - 4 q^{17} - 16 q^{23} - 6 q^{25} + 16 q^{31} - 12 q^{41} + 16 q^{47} + 2 q^{49} - 16 q^{55} + 6 q^{63} - 16 q^{65} - 16 q^{71} + 12 q^{73} - 16 q^{79} + 18 q^{81} + 12 q^{89} + 32 q^{95} + 28 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 + 6 * q^9 - 4 * q^17 - 16 * q^23 - 6 * q^25 + 16 * q^31 - 12 * q^41 + 16 * q^47 + 2 * q^49 - 16 * q^55 + 6 * q^63 - 16 * q^65 - 16 * q^71 + 12 * q^73 - 16 * q^79 + 18 * q^81 + 12 * q^89 + 32 * q^95 + 28 * q^97

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