LMFDB - Embedded newform 1280.2.d.a.641.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,2,Mod(641,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, 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("1280.641");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1280.d (of order $2$, degree $1$, not 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:	$10.2208514587$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			641.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1280.641
Dual form			1280.2.d.a.641.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{5} -4.00000 q^{7} +3.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{5} -4.00000 q^{7} +3.00000 q^{9} +4.00000i q^{11} -2.00000i q^{13} +2.00000 q^{17} -4.00000i q^{19} +4.00000 q^{23} -1.00000 q^{25} -2.00000i q^{29} +8.00000 q^{31} +4.00000i q^{35} -6.00000i q^{37} +6.00000 q^{41} -8.00000i q^{43} -3.00000i q^{45} -4.00000 q^{47} +9.00000 q^{49} -6.00000i q^{53} +4.00000 q^{55} -4.00000i q^{59} -2.00000i q^{61} -12.0000 q^{63} -2.00000 q^{65} -8.00000i q^{67} +6.00000 q^{73} -16.0000i q^{77} +9.00000 q^{81} +16.0000i q^{83} -2.00000i q^{85} +6.00000 q^{89} +8.00000i q^{91} -4.00000 q^{95} -14.0000 q^{97} +12.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q - 8 * q^7 + 6 * q^9	$ 2 q - 8 q^{7} + 6 q^{9} + 4 q^{17} + 8 q^{23} - 2 q^{25} + 16 q^{31} + 12 q^{41} - 8 q^{47} + 18 q^{49} + 8 q^{55} - 24 q^{63} - 4 q^{65} + 12 q^{73} + 18 q^{81} + 12 q^{89} - 8 q^{95} - 28 q^{97}+O(q^{100}) $ 2 * q - 8 * q^7 + 6 * q^9 + 4 * q^17 + 8 * q^23 - 2 * q^25 + 16 * q^31 + 12 * q^41 - 8 * q^47 + 18 * q^49 + 8 * q^55 - 24 * q^63 - 4 * q^65 + 12 * q^73 + 18 * q^81 + 12 * q^89 - 8 * q^95 - 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	4.00000i	1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$	4.00000i	1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	0	0
$21$	0	0
$22$	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
$27$	0	0
$28$	0	0
$29$	− 2.00000i	− 0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$	− 2.00000i	− 0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	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
$35$	4.00000i	0.676123i
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	0	0
$45$	− 3.00000i	− 0.447214i
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 4.00000i	− 0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$	− 4.00000i	− 0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$	0	0
$61$	− 2.00000i	− 0.256074i	−0.991769	−	0.128037i	$-0.959132\pi$
$61$	− 2.00000i	− 0.256074i	0.991769	−	0.128037i	$-0.0408676\pi$
$62$	0	0
$63$	−12.0000	−1.51186
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 16.0000i	− 1.82337i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	16.0000i	1.75623i	0.478451	+	0.878114i	$0.341198\pi$
$83$	16.0000i	1.75623i	−0.478451	+	0.878114i	$0.658802\pi$
$84$	0	0
$85$	− 2.00000i	− 0.216930i
$86$	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
$91$	8.00000i	0.838628i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	12.0000i	1.20605i
$100$	0	0
$101$	− 6.00000i	− 0.597022i	−0.954406	−	0.298511i	$-0.903510\pi$
$101$	− 6.00000i	− 0.597022i	0.954406	−	0.298511i	$-0.0964900\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	14.0000i	1.34096i	0.741929	+	0.670478i	$0.233911\pi$
$109$	14.0000i	1.34096i	−0.741929	+	0.670478i	$0.766089\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	− 4.00000i	− 0.373002i
$116$	0	0
$117$	− 6.00000i	− 0.554700i
$118$	0	0
$119$	−8.00000	−0.733359
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 12.0000i	− 1.04844i	−0.851581	−	0.524222i	$-0.824356\pi$
$131$	− 12.0000i	− 1.04844i	0.851581	−	0.524222i	$-0.175644\pi$
$132$	0	0
$133$	16.0000i	1.38738i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	12.0000i	1.01783i	0.860818	+	0.508913i	$0.169953\pi$
$139$	12.0000i	1.01783i	−0.860818	+	0.508913i	$0.830047\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.00000	0.668994
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.0000i	0.819232i	0.912258	+	0.409616i	$0.134337\pi$
$149$	10.0000i	0.819232i	−0.912258	+	0.409616i	$0.865663\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	− 8.00000i	− 0.642575i
$156$	0	0
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−16.0000	−1.26098
$162$	0	0
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	− 12.0000i	− 0.917663i
$172$	0	0
$173$	14.0000i	1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$	14.0000i	1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 20.0000i	− 1.49487i	−0.664335	−	0.747435i	$-0.731285\pi$
$179$	− 20.0000i	− 1.49487i	0.664335	−	0.747435i	$-0.268715\pi$
$180$	0	0
$181$	10.0000i	0.743294i	0.928374	+	0.371647i	$0.121207\pi$
$181$	10.0000i	0.743294i	−0.928374	+	0.371647i	$0.878793\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	8.00000i	0.585018i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 22.0000i	− 1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$	− 22.0000i	− 1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.00000i	0.561490i
$204$	0	0
$205$	− 6.00000i	− 0.419058i
$206$	0	0
$207$	12.0000	0.834058
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	4.00000i	0.275371i	0.990476	+	0.137686i	$0.0439664\pi$
$211$	4.00000i	0.275371i	−0.990476	+	0.137686i	$0.956034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	−32.0000	−2.17230
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 4.00000i	− 0.269069i
$222$	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
$225$	−3.00000	−0.200000
$226$	0	0
$227$	24.0000i	1.59294i	0.604681	+	0.796468i	$0.293301\pi$
$227$	24.0000i	1.59294i	−0.604681	+	0.796468i	$0.706699\pi$
$228$	0	0
$229$	26.0000i	1.71813i	0.511868	+	0.859064i	$0.328954\pi$
$229$	26.0000i	1.71813i	−0.511868	+	0.859064i	$0.671046\pi$
$230$	0	0
$231$	0	0
$232$	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
$235$	4.00000i	0.260931i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 9.00000i	− 0.574989i
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 12.0000i	− 0.757433i	−0.925513	−	0.378717i	$-0.876365\pi$
$251$	− 12.0000i	− 0.757433i	0.925513	−	0.378717i	$-0.123635\pi$
$252$	0	0
$253$	16.0000i	1.00591i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−30.0000	−1.87135	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000	−1.87135	−0.935674	+	0.352865i	$0.885208\pi$
$258$	0	0
$259$	24.0000i	1.49129i
$260$	0	0
$261$	− 6.00000i	− 0.371391i
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14.0000i	0.853595i	0.904347	+	0.426798i	$0.140358\pi$
$269$	14.0000i	0.853595i	−0.904347	+	0.426798i	$0.859642\pi$
$270$	0	0
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 4.00000i	− 0.241209i
$276$	0	0
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	0	0
$279$	24.0000	1.43684
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	8.00000i	0.475551i	0.971320	+	0.237775i	$0.0764182\pi$
$283$	8.00000i	0.475551i	−0.971320	+	0.237775i	$0.923582\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−24.0000	−1.41668
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	26.0000i	1.51894i	0.650545	+	0.759468i	$0.274541\pi$
$293$	26.0000i	1.51894i	−0.650545	+	0.759468i	$0.725459\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 8.00000i	− 0.462652i
$300$	0	0
$301$	32.0000i	1.84445i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	8.00000i	0.456584i	0.973593	+	0.228292i	$0.0733141\pi$
$307$	8.00000i	0.456584i	−0.973593	+	0.228292i	$0.926686\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	32.0000	1.81455	0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000	1.81455	0.907277	+	0.420534i	$0.138157\pi$
$312$	0	0
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	0	0
$315$	12.0000i	0.676123i
$316$	0	0
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 8.00000i	− 0.445132i
$324$	0	0
$325$	2.00000i	0.110940i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	16.0000	0.882109
$330$	0	0
$331$	− 12.0000i	− 0.659580i	−0.944054	−	0.329790i	$-0.893022\pi$
$331$	− 12.0000i	− 0.659580i	0.944054	−	0.329790i	$-0.106978\pi$
$332$	0	0
$333$	− 18.0000i	− 0.986394i
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	32.0000i	1.73290i
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 16.0000i	− 0.858925i	−0.903085	−	0.429463i	$-0.858703\pi$
$347$	− 16.0000i	− 0.858925i	0.903085	−	0.429463i	$-0.141297\pi$
$348$	0	0
$349$	30.0000i	1.60586i	0.596071	+	0.802932i	$0.296728\pi$
$349$	30.0000i	1.60586i	−0.596071	+	0.802932i	$0.703272\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 6.00000i	− 0.314054i
$366$	0	0
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	0	0
$369$	18.0000	0.937043
$370$	0	0
$371$	24.0000i	1.24602i
$372$	0	0
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.00000	−0.206010
$378$	0	0
$379$	− 20.0000i	− 1.02733i	−0.857991	−	0.513665i	$-0.828287\pi$
$379$	− 20.0000i	− 1.02733i	0.857991	−	0.513665i	$-0.171713\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	0	0
$385$	−16.0000	−0.815436
$386$	0	0
$387$	− 24.0000i	− 1.21999i
$388$	0	0
$389$	− 6.00000i	− 0.304212i	−0.988364	−	0.152106i	$-0.951394\pi$
$389$	− 6.00000i	− 0.304212i	0.988364	−	0.152106i	$-0.0486055\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	− 16.0000i	− 0.797017i
$404$	0	0
$405$	− 9.00000i	− 0.447214i
$406$	0	0
$407$	24.0000	1.18964
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	16.0000i	0.787309i
$414$	0	0
$415$	16.0000	0.785409
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 36.0000i	− 1.75872i	−0.476162	−	0.879358i	$-0.657972\pi$
$419$	− 36.0000i	− 1.75872i	0.476162	−	0.879358i	$-0.342028\pi$
$420$	0	0
$421$	− 6.00000i	− 0.292422i	−0.989253	−	0.146211i	$-0.953292\pi$
$421$	− 6.00000i	− 0.292422i	0.989253	−	0.146211i	$-0.0467079\pi$
$422$	0	0
$423$	−12.0000	−0.583460
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	8.00000i	0.387147i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	40.0000	1.92673	0.963366	−	0.268190i	$-0.0864254\pi$
$431$	40.0000	1.92673	0.963366	+	0.268190i	$0.0864254\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 16.0000i	− 0.765384i
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	27.0000	1.28571
$442$	0	0
$443$	24.0000i	1.14027i	0.821549	+	0.570137i	$0.193110\pi$
$443$	24.0000i	1.14027i	−0.821549	+	0.570137i	$0.806890\pi$
$444$	0	0
$445$	− 6.00000i	− 0.284427i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	24.0000i	1.13012i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	8.00000	0.375046
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 18.0000i	− 0.838344i	−0.907907	−	0.419172i	$-0.862320\pi$
$461$	− 18.0000i	− 0.838344i	0.907907	−	0.419172i	$-0.137680\pi$
$462$	0	0
$463$	−12.0000	−0.557687	−0.278844	−	0.960337i	$-0.589951\pi$
$463$	−12.0000	−0.557687	−0.278844	+	0.960337i	$0.589951\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 8.00000i	− 0.370196i	−0.982720	−	0.185098i	$-0.940740\pi$
$467$	− 8.00000i	− 0.370196i	0.982720	−	0.185098i	$-0.0592602\pi$
$468$	0	0
$469$	32.0000i	1.47762i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	32.0000	1.47136
$474$	0	0
$475$	4.00000i	0.183533i
$476$	0	0
$477$	− 18.0000i	− 0.824163i
$478$	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
$481$	−12.0000	−0.547153
$482$	0	0
$483$	0	0
$484$	0	0
$485$	14.0000i	0.635707i
$486$	0	0
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	36.0000i	1.62466i	0.583200	+	0.812329i	$0.301800\pi$
$491$	36.0000i	1.62466i	−0.583200	+	0.812329i	$0.698200\pi$
$492$	0	0
$493$	− 4.00000i	− 0.180151i
$494$	0	0
$495$	12.0000	0.539360
$496$	0	0
$497$	0	0
$498$	0	0
$499$	28.0000i	1.25345i	0.779240	+	0.626726i	$0.215605\pi$
$499$	28.0000i	1.25345i	−0.779240	+	0.626726i	$0.784395\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 2.00000i	− 0.0886484i	−0.999017	−	0.0443242i	$-0.985887\pi$
$509$	− 2.00000i	− 0.0886484i	0.999017	−	0.0443242i	$-0.0141135\pi$
$510$	0	0
$511$	−24.0000	−1.06170
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 4.00000i	− 0.176261i
$516$	0	0
$517$	− 16.0000i	− 0.703679i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	− 8.00000i	− 0.349816i	−0.984585	−	0.174908i	$-0.944037\pi$
$523$	− 8.00000i	− 0.349816i	0.984585	−	0.174908i	$-0.0559627\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16.0000	0.696971
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	− 12.0000i	− 0.520756i
$532$	0	0
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	36.0000i	1.55063i
$540$	0	0
$541$	− 2.00000i	− 0.0859867i	−0.999075	−	0.0429934i	$-0.986311\pi$
$541$	− 2.00000i	− 0.0859867i	0.999075	−	0.0429934i	$-0.0136894\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	14.0000	0.599694
$546$	0	0
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	0	0
$549$	− 6.00000i	− 0.256074i
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.0000i	0.593199i	0.955002	+	0.296600i	$0.0958526\pi$
$557$	14.0000i	0.593199i	−0.955002	+	0.296600i	$0.904147\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 16.0000i	− 0.674320i	−0.941447	−	0.337160i	$-0.890534\pi$
$563$	− 16.0000i	− 0.674320i	0.941447	−	0.337160i	$-0.109466\pi$
$564$	0	0
$565$	− 18.0000i	− 0.757266i
$566$	0	0
$567$	−36.0000	−1.51186
$568$	0	0
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	0	0
$571$	4.00000i	0.167395i	0.996491	+	0.0836974i	$0.0266729\pi$
$571$	4.00000i	0.167395i	−0.996491	+	0.0836974i	$0.973327\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.00000	−0.166812
$576$	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
$581$	− 64.0000i	− 2.65517i
$582$	0	0
$583$	24.0000	0.993978
$584$	0	0
$585$	−6.00000	−0.248069
$586$	0	0
$587$	− 48.0000i	− 1.98117i	−0.136892	−	0.990586i	$-0.543711\pi$
$587$	− 48.0000i	− 1.98117i	0.136892	−	0.990586i	$-0.456289\pi$
$588$	0	0
$589$	− 32.0000i	− 1.31854i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	8.00000i	0.327968i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	− 24.0000i	− 0.977356i
$604$	0	0
$605$	5.00000i	0.203279i
$606$	0	0
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.00000i	0.323645i
$612$	0	0
$613$	42.0000i	1.69636i	0.529705	+	0.848182i	$0.322303\pi$
$613$	42.0000i	1.69636i	−0.529705	+	0.848182i	$0.677697\pi$
$614$	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
$619$	− 4.00000i	− 0.160774i	−0.996764	−	0.0803868i	$-0.974384\pi$
$619$	− 4.00000i	− 0.160774i	0.996764	−	0.0803868i	$-0.0256155\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−24.0000	−0.961540
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 12.0000i	− 0.478471i
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 12.0000i	− 0.476205i
$636$	0	0
$637$	− 18.0000i	− 0.713186i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	48.0000i	1.89294i	0.322799	+	0.946468i	$0.395376\pi$
$643$	48.0000i	1.89294i	−0.322799	+	0.946468i	$0.604624\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 34.0000i	− 1.33052i	−0.746611	−	0.665261i	$-0.768320\pi$
$653$	− 34.0000i	− 1.33052i	0.746611	−	0.665261i	$-0.231680\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	0	0
$657$	18.0000	0.702247
$658$	0	0
$659$	12.0000i	0.467454i	0.972302	+	0.233727i	$0.0750921\pi$
$659$	12.0000i	0.467454i	−0.972302	+	0.233727i	$0.924908\pi$
$660$	0	0
$661$	42.0000i	1.63361i	0.576913	+	0.816805i	$0.304257\pi$
$661$	42.0000i	1.63361i	−0.576913	+	0.816805i	$0.695743\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	16.0000	0.620453
$666$	0	0
$667$	− 8.00000i	− 0.309761i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	18.0000	0.693849	0.346925	−	0.937893i	$-0.387226\pi$
$673$	18.0000	0.693849	0.346925	+	0.937893i	$0.387226\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 22.0000i	− 0.845529i	−0.906240	−	0.422764i	$-0.861060\pi$
$677$	− 22.0000i	− 0.845529i	0.906240	−	0.422764i	$-0.138940\pi$
$678$	0	0
$679$	56.0000	2.14908
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24.0000i	0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$	24.0000i	0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$	0	0
$685$	10.0000i	0.382080i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	− 44.0000i	− 1.67384i	−0.547326	−	0.836919i	$-0.684354\pi$
$691$	− 44.0000i	− 1.67384i	0.547326	−	0.836919i	$-0.315646\pi$
$692$	0	0
$693$	− 48.0000i	− 1.82337i
$694$	0	0
$695$	12.0000	0.455186
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 34.0000i	− 1.28416i	−0.766637	−	0.642081i	$-0.778071\pi$
$701$	− 34.0000i	− 1.28416i	0.766637	−	0.642081i	$-0.221929\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	0	0
$705$	0	0
$706$	0	0
$707$	24.0000i	0.902613i
$708$	0	0
$709$	− 38.0000i	− 1.42712i	−0.700594	−	0.713560i	$-0.747082\pi$
$709$	− 38.0000i	− 1.42712i	0.700594	−	0.713560i	$-0.252918\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	32.0000	1.19841
$714$	0	0
$715$	− 8.00000i	− 0.299183i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	32.0000	1.19340	0.596699	−	0.802465i	$-0.296479\pi$
$719$	32.0000	1.19340	0.596699	+	0.802465i	$0.296479\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.00000i	0.0742781i
$726$	0	0
$727$	44.0000	1.63187	0.815935	−	0.578144i	$-0.196223\pi$
$727$	44.0000	1.63187	0.815935	+	0.578144i	$0.196223\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	− 16.0000i	− 0.591781i
$732$	0	0
$733$	30.0000i	1.10808i	0.832492	+	0.554038i	$0.186914\pi$
$733$	30.0000i	1.10808i	−0.832492	+	0.554038i	$0.813086\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	32.0000	1.17874
$738$	0	0
$739$	− 36.0000i	− 1.32428i	−0.749380	−	0.662141i	$-0.769648\pi$
$739$	− 36.0000i	− 1.32428i	0.749380	−	0.662141i	$-0.230352\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	0	0
$747$	48.0000i	1.75623i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	16.0000i	0.582300i
$756$	0	0
$757$	10.0000i	0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$	10.0000i	0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	− 56.0000i	− 2.02734i
$764$	0	0
$765$	− 6.00000i	− 0.216930i
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	42.0000i	1.51064i	0.655359	+	0.755318i	$0.272517\pi$
$773$	42.0000i	1.51064i	−0.655359	+	0.755318i	$0.727483\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 24.0000i	− 0.859889i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	− 8.00000i	− 0.285169i	−0.989783	−	0.142585i	$-0.954459\pi$
$787$	− 8.00000i	− 0.285169i	0.989783	−	0.142585i	$-0.0455413\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−72.0000	−2.56003
$792$	0	0
$793$	−4.00000	−0.142044
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 18.0000i	− 0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$	− 18.0000i	− 0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	18.0000	0.635999
$802$	0	0
$803$	24.0000i	0.846942i
$804$	0	0
$805$	16.0000i	0.563926i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	0	0
$811$	− 28.0000i	− 0.983213i	−0.870817	−	0.491606i	$-0.836410\pi$
$811$	− 28.0000i	− 0.983213i	0.870817	−	0.491606i	$-0.163590\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−16.0000	−0.560456
$816$	0	0
$817$	−32.0000	−1.11954
$818$	0	0
$819$	24.0000i	0.838628i
$820$	0	0
$821$	− 22.0000i	− 0.767805i	−0.923374	−	0.383903i	$-0.874580\pi$
$821$	− 22.0000i	− 0.767805i	0.923374	−	0.383903i	$-0.125420\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	48.0000i	1.66912i	0.550914	+	0.834562i	$0.314279\pi$
$827$	48.0000i	1.66912i	−0.550914	+	0.834562i	$0.685721\pi$
$828$	0	0
$829$	− 2.00000i	− 0.0694629i	−0.999397	−	0.0347314i	$-0.988942\pi$
$829$	− 2.00000i	− 0.0694629i	0.999397	−	0.0347314i	$-0.0110576\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	− 12.0000i	− 0.415277i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 9.00000i	− 0.309609i
$846$	0	0
$847$	20.0000	0.687208
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 24.0000i	− 0.822709i
$852$	0	0
$853$	− 38.0000i	− 1.30110i	−0.759465	−	0.650548i	$-0.774539\pi$
$853$	− 38.0000i	− 1.30110i	0.759465	−	0.650548i	$-0.225461\pi$
$854$	0	0
$855$	−12.0000	−0.410391
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	− 4.00000i	− 0.136478i	−0.997669	−	0.0682391i	$-0.978262\pi$
$859$	− 4.00000i	− 0.136478i	0.997669	−	0.0682391i	$-0.0217381\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	−42.0000	−1.42148
$874$	0	0
$875$	− 4.00000i	− 0.135225i
$876$	0	0
$877$	46.0000i	1.55331i	0.629926	+	0.776655i	$0.283085\pi$
$877$	46.0000i	1.55331i	−0.629926	+	0.776655i	$0.716915\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	16.0000i	0.538443i	0.963078	+	0.269221i	$0.0867663\pi$
$883$	16.0000i	0.538443i	−0.963078	+	0.269221i	$0.913234\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−20.0000	−0.671534	−0.335767	−	0.941945i	$-0.608996\pi$
$887$	−20.0000	−0.671534	−0.335767	+	0.941945i	$0.608996\pi$
$888$	0	0
$889$	−48.0000	−1.60987
$890$	0	0
$891$	36.0000i	1.20605i
$892$	0	0
$893$	16.0000i	0.535420i
$894$	0	0
$895$	−20.0000	−0.668526
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 16.0000i	− 0.533630i
$900$	0	0
$901$	− 12.0000i	− 0.399778i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10.0000	0.332411
$906$	0	0
$907$	48.0000i	1.59381i	0.604102	+	0.796907i	$0.293532\pi$
$907$	48.0000i	1.59381i	−0.604102	+	0.796907i	$0.706468\pi$
$908$	0	0
$909$	− 18.0000i	− 0.597022i
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	−64.0000	−2.11809
$914$	0	0
$915$	0	0
$916$	0	0
$917$	48.0000i	1.58510i
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	6.00000i	0.197279i
$926$	0	0
$927$	12.0000	0.394132
$928$	0	0
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	− 36.0000i	− 1.17985i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	46.0000i	1.49956i	0.661689	+	0.749779i	$0.269840\pi$
$941$	46.0000i	1.49956i	−0.661689	+	0.749779i	$0.730160\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.00000i	0.259965i	0.991516	+	0.129983i	$0.0414921\pi$
$947$	8.00000i	0.259965i	−0.991516	+	0.129983i	$0.958508\pi$
$948$	0	0
$949$	− 12.0000i	− 0.389536i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	8.00000i	0.258874i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	40.0000	1.29167
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	0	0
$964$	0	0
$965$	14.0000i	0.450676i
$966$	0	0
$967$	−36.0000	−1.15768	−0.578841	−	0.815440i	$-0.696495\pi$
$967$	−36.0000	−1.15768	−0.578841	+	0.815440i	$0.696495\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	20.0000i	0.641831i	0.947108	+	0.320915i	$0.103990\pi$
$971$	20.0000i	0.641831i	−0.947108	+	0.320915i	$0.896010\pi$
$972$	0	0
$973$	− 48.0000i	− 1.53881i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.00000	0.0639857	0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000	0.0639857	0.0319928	+	0.999488i	$0.489815\pi$
$978$	0	0
$979$	24.0000i	0.767043i
$980$	0	0
$981$	42.0000i	1.34096i
$982$	0	0
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	0	0
$985$	−22.0000	−0.700978
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 32.0000i	− 1.01754i
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 8.00000i	− 0.253617i
$996$	0	0
$997$	42.0000i	1.33015i	0.746775	+	0.665077i	$0.231601\pi$
$997$	42.0000i	1.33015i	−0.746775	+	0.665077i	$0.768399\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.2.d.a.641.1		2
4.3	odd	2		1280.2.d.j.641.1		2
8.3	odd	2		1280.2.d.j.641.2		2
8.5	even	2	inner	1280.2.d.a.641.2		2
16.3	odd	4		320.2.a.c.1.1		1
16.5	even	4		80.2.a.a.1.1		1
16.11	odd	4		40.2.a.a.1.1	✓	1
16.13	even	4		320.2.a.d.1.1		1
48.5	odd	4		720.2.a.e.1.1		1
48.11	even	4		360.2.a.a.1.1		1
48.29	odd	4		2880.2.a.bg.1.1		1
48.35	even	4		2880.2.a.t.1.1		1
80.3	even	4		1600.2.c.k.449.2		2
80.13	odd	4		1600.2.c.m.449.1		2
80.19	odd	4		1600.2.a.o.1.1		1
80.27	even	4		200.2.c.b.49.1		2
80.29	even	4		1600.2.a.k.1.1		1
80.37	odd	4		400.2.c.d.49.2		2
80.43	even	4		200.2.c.b.49.2		2
80.53	odd	4		400.2.c.d.49.1		2
80.59	odd	4		200.2.a.c.1.1		1
80.67	even	4		1600.2.c.k.449.1		2
80.69	even	4		400.2.a.e.1.1		1
80.77	odd	4		1600.2.c.m.449.2		2
112.11	odd	12		1960.2.q.h.961.1		2
112.27	even	4		1960.2.a.g.1.1		1
112.59	even	12		1960.2.q.i.961.1		2
112.69	odd	4		3920.2.a.s.1.1		1
112.75	even	12		1960.2.q.i.361.1		2
112.107	odd	12		1960.2.q.h.361.1		2
144.11	even	12		3240.2.q.x.1081.1		2
144.43	odd	12		3240.2.q.k.1081.1		2
144.59	even	12		3240.2.q.x.2161.1		2
144.139	odd	12		3240.2.q.k.2161.1		2
176.21	odd	4		9680.2.a.q.1.1		1
176.43	even	4		4840.2.a.f.1.1		1
208.155	odd	4		6760.2.a.i.1.1		1
240.53	even	4		3600.2.f.t.2449.1		2
240.59	even	4		1800.2.a.v.1.1		1
240.107	odd	4		1800.2.f.a.649.1		2
240.149	odd	4		3600.2.a.h.1.1		1
240.197	even	4		3600.2.f.t.2449.2		2
240.203	odd	4		1800.2.f.a.649.2		2
560.139	even	4		9800.2.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.2.a.a.1.1	✓	1	16.11	odd	4
80.2.a.a.1.1		1	16.5	even	4
200.2.a.c.1.1		1	80.59	odd	4
200.2.c.b.49.1		2	80.27	even	4
200.2.c.b.49.2		2	80.43	even	4
320.2.a.c.1.1		1	16.3	odd	4
320.2.a.d.1.1		1	16.13	even	4
360.2.a.a.1.1		1	48.11	even	4
400.2.a.e.1.1		1	80.69	even	4
400.2.c.d.49.1		2	80.53	odd	4
400.2.c.d.49.2		2	80.37	odd	4
720.2.a.e.1.1		1	48.5	odd	4
1280.2.d.a.641.1		2	1.1	even	1	trivial
1280.2.d.a.641.2		2	8.5	even	2	inner
1280.2.d.j.641.1		2	4.3	odd	2
1280.2.d.j.641.2		2	8.3	odd	2
1600.2.a.k.1.1		1	80.29	even	4
1600.2.a.o.1.1		1	80.19	odd	4
1600.2.c.k.449.1		2	80.67	even	4
1600.2.c.k.449.2		2	80.3	even	4
1600.2.c.m.449.1		2	80.13	odd	4
1600.2.c.m.449.2		2	80.77	odd	4
1800.2.a.v.1.1		1	240.59	even	4
1800.2.f.a.649.1		2	240.107	odd	4
1800.2.f.a.649.2		2	240.203	odd	4
1960.2.a.g.1.1		1	112.27	even	4
1960.2.q.h.361.1		2	112.107	odd	12
1960.2.q.h.961.1		2	112.11	odd	12
1960.2.q.i.361.1		2	112.75	even	12
1960.2.q.i.961.1		2	112.59	even	12
2880.2.a.t.1.1		1	48.35	even	4
2880.2.a.bg.1.1		1	48.29	odd	4
3240.2.q.k.1081.1		2	144.43	odd	12
3240.2.q.k.2161.1		2	144.139	odd	12
3240.2.q.x.1081.1		2	144.11	even	12
3240.2.q.x.2161.1		2	144.59	even	12
3600.2.a.h.1.1		1	240.149	odd	4
3600.2.f.t.2449.1		2	240.53	even	4
3600.2.f.t.2449.2		2	240.197	even	4
3920.2.a.s.1.1		1	112.69	odd	4
4840.2.a.f.1.1		1	176.43	even	4
6760.2.a.i.1.1		1	208.155	odd	4
9680.2.a.q.1.1		1	176.21	odd	4
9800.2.a.x.1.1		1	560.139	even	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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1280.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{5} -4.00000 q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{5} -4.00000 q^{7} +3.00000 q^{9} +4.00000i q^{11} -2.00000i q^{13} +2.00000 q^{17} -4.00000i q^{19} +4.00000 q^{23} -1.00000 q^{25} -2.00000i q^{29} +8.00000 q^{31} +4.00000i q^{35} -6.00000i q^{37} +6.00000 q^{41} -8.00000i q^{43} -3.00000i q^{45} -4.00000 q^{47} +9.00000 q^{49} -6.00000i q^{53} +4.00000 q^{55} -4.00000i q^{59} -2.00000i q^{61} -12.0000 q^{63} -2.00000 q^{65} -8.00000i q^{67} +6.00000 q^{73} -16.0000i q^{77} +9.00000 q^{81} +16.0000i q^{83} -2.00000i q^{85} +6.00000 q^{89} +8.00000i q^{91} -4.00000 q^{95} -14.0000 q^{97} +12.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q - 8 * q^7 + 6 * q^9	\( 2 q - 8 q^{7} + 6 q^{9} + 4 q^{17} + 8 q^{23} - 2 q^{25} + 16 q^{31} + 12 q^{41} - 8 q^{47} + 18 q^{49} + 8 q^{55} - 24 q^{63} - 4 q^{65} + 12 q^{73} + 18 q^{81} + 12 q^{89} - 8 q^{95} - 28 q^{97}+O(q^{100}) \) 2 * q - 8 * q^7 + 6 * q^9 + 4 * q^17 + 8 * q^23 - 2 * q^25 + 16 * q^31 + 12 * q^41 - 8 * q^47 + 18 * q^49 + 8 * q^55 - 24 * q^63 - 4 * q^65 + 12 * q^73 + 18 * q^81 + 12 * q^89 - 8 * q^95 - 28 * q^97

Introduction

L-functions

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