LMFDB - Embedded newform 2816.2.c.h.1409.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2816,2,Mod(1409,2816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2816, 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("2816.1409");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2816 = 2^{8} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2816.c (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:	$22.4858732092$
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 352)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1409.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2816.1409
Dual form			2816.2.c.h.1409.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +1.00000i q^{5} -6.00000 q^{9} -1.00000i q^{11} +6.00000i q^{13} +3.00000 q^{15} -4.00000 q^{17} -6.00000i q^{19} -3.00000 q^{23} +4.00000 q^{25} +9.00000i q^{27} +4.00000i q^{29} -9.00000 q^{31} -3.00000 q^{33} +7.00000i q^{37} +18.0000 q^{39} +2.00000 q^{41} +6.00000i q^{43} -6.00000i q^{45} +12.0000 q^{47} -7.00000 q^{49} +12.0000i q^{51} +2.00000i q^{53} +1.00000 q^{55} -18.0000 q^{57} +9.00000i q^{59} -8.00000i q^{61} -6.00000 q^{65} +15.0000i q^{67} +9.00000i q^{69} +3.00000 q^{71} +6.00000 q^{73} -12.0000i q^{75} -6.00000 q^{79} +9.00000 q^{81} +6.00000i q^{83} -4.00000i q^{85} +12.0000 q^{87} +5.00000 q^{89} +27.0000i q^{93} +6.00000 q^{95} -3.00000 q^{97} +6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{9} + 6 q^{15} - 8 q^{17} - 6 q^{23} + 8 q^{25} - 18 q^{31} - 6 q^{33} + 36 q^{39} + 4 q^{41} + 24 q^{47} - 14 q^{49} + 2 q^{55} - 36 q^{57} - 12 q^{65} + 6 q^{71} + 12 q^{73} - 12 q^{79} + 18 q^{81}+ \cdots - 6 q^{97}+O(q^{100}) $ 2 * q - 12 * q^9 + 6 * q^15 - 8 * q^17 - 6 * q^23 + 8 * q^25 - 18 * q^31 - 6 * q^33 + 36 * q^39 + 4 * q^41 + 24 * q^47 - 14 * q^49 + 2 * q^55 - 36 * q^57 - 12 * q^65 + 6 * q^71 + 12 * q^73 - 12 * q^79 + 18 * q^81 + 24 * q^87 + 10 * q^89 + 12 * q^95 - 6 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times$.

$n$	$1025$	$1541$	$2047$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 3.00000i	− 1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	− 3.00000i	− 1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$4$	0	0
$5$	1.00000i	0.447214i	0.974679	+	0.223607i	$0.0717831\pi$
$5$	1.00000i	0.447214i	−0.974679	+	0.223607i	$0.928217\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−6.00000	−2.00000
$10$	0	0
$11$	− 1.00000i	− 0.301511i
$11$	− 1.00000i	− 0.301511i
$12$	0	0
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	− 6.00000i	− 1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$	− 6.00000i	− 1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	9.00000i	1.73205i
$28$	0	0
$29$	4.00000i	0.742781i	0.928477	+	0.371391i	$0.121119\pi$
$29$	4.00000i	0.742781i	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.00000i	1.15079i	0.817875	+	0.575396i	$0.195152\pi$
$37$	7.00000i	1.15079i	−0.817875	+	0.575396i	$0.804848\pi$
$38$	0	0
$39$	18.0000	2.88231
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	6.00000i	0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$	6.00000i	0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	0	0
$45$	− 6.00000i	− 0.894427i
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	12.0000i	1.68034i
$52$	0	0
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−18.0000	−2.38416
$58$	0	0
$59$	9.00000i	1.17170i	0.810419	+	0.585850i	$0.199239\pi$
$59$	9.00000i	1.17170i	−0.810419	+	0.585850i	$0.800761\pi$
$60$	0	0
$61$	− 8.00000i	− 1.02430i	−0.858898	−	0.512148i	$-0.828850\pi$
$61$	− 8.00000i	− 1.02430i	0.858898	−	0.512148i	$-0.171150\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.00000	−0.744208
$66$	0	0
$67$	15.0000i	1.83254i	0.400559	+	0.916271i	$0.368816\pi$
$67$	15.0000i	1.83254i	−0.400559	+	0.916271i	$0.631184\pi$
$68$	0	0
$69$	9.00000i	1.08347i
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	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
$75$	− 12.0000i	− 1.38564i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	− 4.00000i	− 0.433861i
$86$	0	0
$87$	12.0000	1.28654
$88$	0	0
$89$	5.00000	0.529999	0.264999	−	0.964249i	$-0.414628\pi$
$89$	5.00000	0.529999	0.264999	+	0.964249i	$0.414628\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	27.0000i	2.79977i
$94$	0	0
$95$	6.00000	0.615587
$96$	0	0
$97$	−3.00000	−0.304604	−0.152302	−	0.988334i	$-0.548669\pi$
$97$	−3.00000	−0.304604	−0.152302	+	0.988334i	$0.548669\pi$
$98$	0	0
$99$	6.00000i	0.603023i
$100$	0	0
$101$	16.0000i	1.59206i	0.605257	+	0.796030i	$0.293070\pi$
$101$	16.0000i	1.59206i	−0.605257	+	0.796030i	$0.706930\pi$
$102$	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
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	6.00000i	0.574696i	0.957826	+	0.287348i	$0.0927736\pi$
$109$	6.00000i	0.574696i	−0.957826	+	0.287348i	$0.907226\pi$
$110$	0	0
$111$	21.0000	1.99323
$112$	0	0
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	0	0
$115$	− 3.00000i	− 0.279751i
$116$	0	0
$117$	− 36.0000i	− 3.32820i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	− 6.00000i	− 0.541002i
$124$	0	0
$125$	9.00000i	0.804984i
$126$	0	0
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	0	0
$129$	18.0000	1.58481
$130$	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
$133$	0	0
$134$	0	0
$135$	−9.00000	−0.774597
$136$	0	0
$137$	−5.00000	−0.427179	−0.213589	−	0.976924i	$-0.568515\pi$
$137$	−5.00000	−0.427179	−0.213589	+	0.976924i	$0.568515\pi$
$138$	0	0
$139$	− 12.0000i	− 1.01783i	−0.860818	−	0.508913i	$-0.830047\pi$
$139$	− 12.0000i	− 1.01783i	0.860818	−	0.508913i	$-0.169953\pi$
$140$	0	0
$141$	− 36.0000i	− 3.03175i
$142$	0	0
$143$	6.00000	0.501745
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	21.0000i	1.73205i
$148$	0	0
$149$	− 16.0000i	− 1.31077i	−0.755295	−	0.655386i	$-0.772506\pi$
$149$	− 16.0000i	− 1.31077i	0.755295	−	0.655386i	$-0.227494\pi$
$150$	0	0
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	0	0
$153$	24.0000	1.94029
$154$	0	0
$155$	− 9.00000i	− 0.722897i
$156$	0	0
$157$	− 13.0000i	− 1.03751i	−0.854922	−	0.518756i	$-0.826395\pi$
$157$	− 13.0000i	− 1.03751i	0.854922	−	0.518756i	$-0.173605\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	0	0
$165$	− 3.00000i	− 0.233550i
$166$	0	0
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	36.0000i	2.75299i
$172$	0	0
$173$	− 10.0000i	− 0.760286i	−0.924928	−	0.380143i	$-0.875875\pi$
$173$	− 10.0000i	− 0.760286i	0.924928	−	0.380143i	$-0.124125\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	27.0000	2.02944
$178$	0	0
$179$	15.0000i	1.12115i	0.828103	+	0.560576i	$0.189420\pi$
$179$	15.0000i	1.12115i	−0.828103	+	0.560576i	$0.810580\pi$
$180$	0	0
$181$	3.00000i	0.222988i	0.993765	+	0.111494i	$0.0355636\pi$
$181$	3.00000i	0.222988i	−0.993765	+	0.111494i	$0.964436\pi$
$182$	0	0
$183$	−24.0000	−1.77413
$184$	0	0
$185$	−7.00000	−0.514650
$186$	0	0
$187$	4.00000i	0.292509i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	21.0000	1.51951	0.759753	−	0.650211i	$-0.225320\pi$
$191$	21.0000	1.51951	0.759753	+	0.650211i	$0.225320\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	0	0
$195$	18.0000i	1.28901i
$196$	0	0
$197$	22.0000i	1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$	22.0000i	1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	45.0000	3.17406
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.00000i	0.139686i
$206$	0	0
$207$	18.0000	1.25109
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	− 12.0000i	− 0.826114i	−0.910705	−	0.413057i	$-0.864461\pi$
$211$	− 12.0000i	− 0.826114i	0.910705	−	0.413057i	$-0.135539\pi$
$212$	0	0
$213$	− 9.00000i	− 0.616670i
$214$	0	0
$215$	−6.00000	−0.409197
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 18.0000i	− 1.21633i
$220$	0	0
$221$	− 24.0000i	− 1.61441i
$222$	0	0
$223$	−21.0000	−1.40626	−0.703132	−	0.711059i	$-0.748216\pi$
$223$	−21.0000	−1.40626	−0.703132	+	0.711059i	$0.748216\pi$
$224$	0	0
$225$	−24.0000	−1.60000
$226$	0	0
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	15.0000i	0.991228i	0.868543	+	0.495614i	$0.165057\pi$
$229$	15.0000i	0.991228i	−0.868543	+	0.495614i	$0.834943\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.00000	0.262049	0.131024	−	0.991379i	$-0.458173\pi$
$233$	4.00000	0.262049	0.131024	+	0.991379i	$0.458173\pi$
$234$	0	0
$235$	12.0000i	0.782794i
$236$	0	0
$237$	18.0000i	1.16923i
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 7.00000i	− 0.447214i
$246$	0	0
$247$	36.0000	2.29063
$248$	0	0
$249$	18.0000	1.14070
$250$	0	0
$251$	21.0000i	1.32551i	0.748837	+	0.662754i	$0.230613\pi$
$251$	21.0000i	1.32551i	−0.748837	+	0.662754i	$0.769387\pi$
$252$	0	0
$253$	3.00000i	0.188608i
$254$	0	0
$255$	−12.0000	−0.751469
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 24.0000i	− 1.48556i
$262$	0	0
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	− 15.0000i	− 0.917985i
$268$	0	0
$269$	2.00000i	0.121942i	0.998140	+	0.0609711i	$0.0194197\pi$
$269$	2.00000i	0.121942i	−0.998140	+	0.0609711i	$0.980580\pi$
$270$	0	0
$271$	−6.00000	−0.364474	−0.182237	−	0.983255i	$-0.558334\pi$
$271$	−6.00000	−0.364474	−0.182237	+	0.983255i	$0.558334\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 4.00000i	− 0.241209i
$276$	0	0
$277$	− 6.00000i	− 0.360505i	−0.983620	−	0.180253i	$-0.942309\pi$
$277$	− 6.00000i	− 0.360505i	0.983620	−	0.180253i	$-0.0576915\pi$
$278$	0	0
$279$	54.0000	3.23290
$280$	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
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	− 18.0000i	− 1.06623i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	9.00000i	0.527589i
$292$	0	0
$293$	− 22.0000i	− 1.28525i	−0.766179	−	0.642627i	$-0.777845\pi$
$293$	− 22.0000i	− 1.28525i	0.766179	−	0.642627i	$-0.222155\pi$
$294$	0	0
$295$	−9.00000	−0.524000
$296$	0	0
$297$	9.00000	0.522233
$298$	0	0
$299$	− 18.0000i	− 1.04097i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	48.0000	2.75753
$304$	0	0
$305$	8.00000	0.458079
$306$	0	0
$307$	− 6.00000i	− 0.342438i	−0.985233	−	0.171219i	$-0.945229\pi$
$307$	− 6.00000i	− 0.342438i	0.985233	−	0.171219i	$-0.0547706\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.0000i	1.29181i	0.763418	+	0.645904i	$0.223520\pi$
$317$	23.0000i	1.29181i	−0.763418	+	0.645904i	$0.776480\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	36.0000	2.00932
$322$	0	0
$323$	24.0000i	1.33540i
$324$	0	0
$325$	24.0000i	1.33128i
$326$	0	0
$327$	18.0000	0.995402
$328$	0	0
$329$	0	0
$330$	0	0
$331$	15.0000i	0.824475i	0.911077	+	0.412237i	$0.135253\pi$
$331$	15.0000i	0.824475i	−0.911077	+	0.412237i	$0.864747\pi$
$332$	0	0
$333$	− 42.0000i	− 2.30159i
$334$	0	0
$335$	−15.0000	−0.819538
$336$	0	0
$337$	24.0000	1.30736	0.653682	−	0.756770i	$-0.273224\pi$
$337$	24.0000	1.30736	0.653682	+	0.756770i	$0.273224\pi$
$338$	0	0
$339$	− 3.00000i	− 0.162938i
$340$	0	0
$341$	9.00000i	0.487377i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−9.00000	−0.484544
$346$	0	0
$347$	6.00000i	0.322097i	0.986947	+	0.161048i	$0.0514875\pi$
$347$	6.00000i	0.322097i	−0.986947	+	0.161048i	$0.948512\pi$
$348$	0	0
$349$	28.0000i	1.49881i	0.662114	+	0.749403i	$0.269659\pi$
$349$	28.0000i	1.49881i	−0.662114	+	0.749403i	$0.730341\pi$
$350$	0	0
$351$	−54.0000	−2.88231
$352$	0	0
$353$	−5.00000	−0.266123	−0.133062	−	0.991108i	$-0.542481\pi$
$353$	−5.00000	−0.266123	−0.133062	+	0.991108i	$0.542481\pi$
$354$	0	0
$355$	3.00000i	0.159223i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	0	0
$363$	3.00000i	0.157459i
$364$	0	0
$365$	6.00000i	0.314054i
$366$	0	0
$367$	15.0000	0.782994	0.391497	−	0.920179i	$-0.371957\pi$
$367$	15.0000	0.782994	0.391497	+	0.920179i	$0.371957\pi$
$368$	0	0
$369$	−12.0000	−0.624695
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 4.00000i	− 0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$	− 4.00000i	− 0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$	0	0
$375$	27.0000	1.39427
$376$	0	0
$377$	−24.0000	−1.23606
$378$	0	0
$379$	− 21.0000i	− 1.07870i	−0.842082	−	0.539349i	$-0.818670\pi$
$379$	− 21.0000i	− 1.07870i	0.842082	−	0.539349i	$-0.181330\pi$
$380$	0	0
$381$	18.0000i	0.922168i
$382$	0	0
$383$	−21.0000	−1.07305	−0.536525	−	0.843884i	$-0.680263\pi$
$383$	−21.0000	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 36.0000i	− 1.82998i
$388$	0	0
$389$	5.00000i	0.253510i	0.991934	+	0.126755i	$0.0404562\pi$
$389$	5.00000i	0.253510i	−0.991934	+	0.126755i	$0.959544\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	−36.0000	−1.81596
$394$	0	0
$395$	− 6.00000i	− 0.301893i
$396$	0	0
$397$	34.0000i	1.70641i	0.521575	+	0.853206i	$0.325345\pi$
$397$	34.0000i	1.70641i	−0.521575	+	0.853206i	$0.674655\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	0	0
$403$	− 54.0000i	− 2.68993i
$404$	0	0
$405$	9.00000i	0.447214i
$406$	0	0
$407$	7.00000	0.346977
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	15.0000i	0.739895i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	−36.0000	−1.76293
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	6.00000i	0.292422i	0.989253	+	0.146211i	$0.0467079\pi$
$421$	6.00000i	0.292422i	−0.989253	+	0.146211i	$0.953292\pi$
$422$	0	0
$423$	−72.0000	−3.50076
$424$	0	0
$425$	−16.0000	−0.776114
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 18.0000i	− 0.869048i
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	0	0
$435$	12.0000i	0.575356i
$436$	0	0
$437$	18.0000i	0.861057i
$438$	0	0
$439$	−36.0000	−1.71819	−0.859093	−	0.511819i	$-0.828972\pi$
$439$	−36.0000	−1.71819	−0.859093	+	0.511819i	$0.828972\pi$
$440$	0	0
$441$	42.0000	2.00000
$442$	0	0
$443$	9.00000i	0.427603i	0.976877	+	0.213801i	$0.0685846\pi$
$443$	9.00000i	0.427603i	−0.976877	+	0.213801i	$0.931415\pi$
$444$	0	0
$445$	5.00000i	0.237023i
$446$	0	0
$447$	−48.0000	−2.27032
$448$	0	0
$449$	−17.0000	−0.802280	−0.401140	−	0.916017i	$-0.631386\pi$
$449$	−17.0000	−0.802280	−0.401140	+	0.916017i	$0.631386\pi$
$450$	0	0
$451$	− 2.00000i	− 0.0941763i
$452$	0	0
$453$	− 18.0000i	− 0.845714i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.0000	1.12267	0.561336	−	0.827588i	$-0.310287\pi$
$457$	24.0000	1.12267	0.561336	+	0.827588i	$0.310287\pi$
$458$	0	0
$459$	− 36.0000i	− 1.68034i
$460$	0	0
$461$	40.0000i	1.86299i	0.363760	+	0.931493i	$0.381493\pi$
$461$	40.0000i	1.86299i	−0.363760	+	0.931493i	$0.618507\pi$
$462$	0	0
$463$	−15.0000	−0.697109	−0.348555	−	0.937288i	$-0.613327\pi$
$463$	−15.0000	−0.697109	−0.348555	+	0.937288i	$0.613327\pi$
$464$	0	0
$465$	−27.0000	−1.25210
$466$	0	0
$467$	15.0000i	0.694117i	0.937843	+	0.347059i	$0.112820\pi$
$467$	15.0000i	0.694117i	−0.937843	+	0.347059i	$0.887180\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−39.0000	−1.79703
$472$	0	0
$473$	6.00000	0.275880
$474$	0	0
$475$	− 24.0000i	− 1.10120i
$476$	0	0
$477$	− 12.0000i	− 0.549442i
$478$	0	0
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	−42.0000	−1.91504
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 3.00000i	− 0.136223i
$486$	0	0
$487$	−3.00000	−0.135943	−0.0679715	−	0.997687i	$-0.521653\pi$
$487$	−3.00000	−0.135943	−0.0679715	+	0.997687i	$0.521653\pi$
$488$	0	0
$489$	−36.0000	−1.62798
$490$	0	0
$491$	− 24.0000i	− 1.08310i	−0.840667	−	0.541552i	$-0.817837\pi$
$491$	− 24.0000i	− 1.08310i	0.840667	−	0.541552i	$-0.182163\pi$
$492$	0	0
$493$	− 16.0000i	− 0.720604i
$494$	0	0
$495$	−6.00000	−0.269680
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 12.0000i	− 0.537194i	−0.963253	−	0.268597i	$-0.913440\pi$
$499$	− 12.0000i	− 0.537194i	0.963253	−	0.268597i	$-0.0865599\pi$
$500$	0	0
$501$	54.0000i	2.41254i
$502$	0	0
$503$	−42.0000	−1.87269	−0.936344	−	0.351085i	$-0.885813\pi$
$503$	−42.0000	−1.87269	−0.936344	+	0.351085i	$0.885813\pi$
$504$	0	0
$505$	−16.0000	−0.711991
$506$	0	0
$507$	69.0000i	3.06440i
$508$	0	0
$509$	1.00000i	0.0443242i	0.999754	+	0.0221621i	$0.00705500\pi$
$509$	1.00000i	0.0443242i	−0.999754	+	0.0221621i	$0.992945\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	54.0000	2.38416
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 12.0000i	− 0.527759i
$518$	0	0
$519$	−30.0000	−1.31685
$520$	0	0
$521$	31.0000	1.35813	0.679067	−	0.734076i	$-0.262384\pi$
$521$	31.0000	1.35813	0.679067	+	0.734076i	$0.262384\pi$
$522$	0	0
$523$	− 30.0000i	− 1.31181i	−0.754844	−	0.655904i	$-0.772288\pi$
$523$	− 30.0000i	− 1.31181i	0.754844	−	0.655904i	$-0.227712\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	36.0000	1.56818
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	− 54.0000i	− 2.34340i
$532$	0	0
$533$	12.0000i	0.519778i
$534$	0	0
$535$	−12.0000	−0.518805
$536$	0	0
$537$	45.0000	1.94189
$538$	0	0
$539$	7.00000i	0.301511i
$540$	0	0
$541$	6.00000i	0.257960i	0.991647	+	0.128980i	$0.0411703\pi$
$541$	6.00000i	0.257960i	−0.991647	+	0.128980i	$0.958830\pi$
$542$	0	0
$543$	9.00000	0.386227
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	− 36.0000i	− 1.53925i	−0.638497	−	0.769624i	$-0.720443\pi$
$547$	− 36.0000i	− 1.53925i	0.638497	−	0.769624i	$-0.279557\pi$
$548$	0	0
$549$	48.0000i	2.04859i
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	0	0
$554$	0	0
$555$	21.0000i	0.891400i
$556$	0	0
$557$	16.0000i	0.677942i	0.940797	+	0.338971i	$0.110079\pi$
$557$	16.0000i	0.677942i	−0.940797	+	0.338971i	$0.889921\pi$
$558$	0	0
$559$	−36.0000	−1.52264
$560$	0	0
$561$	12.0000	0.506640
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	1.00000i	0.0420703i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−44.0000	−1.84458	−0.922288	−	0.386503i	$-0.873683\pi$
$569$	−44.0000	−1.84458	−0.922288	+	0.386503i	$0.873683\pi$
$570$	0	0
$571$	− 12.0000i	− 0.502184i	−0.967963	−	0.251092i	$-0.919210\pi$
$571$	− 12.0000i	− 0.502184i	0.967963	−	0.251092i	$-0.0807897\pi$
$572$	0	0
$573$	− 63.0000i	− 2.63186i
$574$	0	0
$575$	−12.0000	−0.500435
$576$	0	0
$577$	29.0000	1.20729	0.603643	−	0.797255i	$-0.293715\pi$
$577$	29.0000	1.20729	0.603643	+	0.797255i	$0.293715\pi$
$578$	0	0
$579$	− 12.0000i	− 0.498703i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2.00000	0.0828315
$584$	0	0
$585$	36.0000	1.48842
$586$	0	0
$587$	36.0000i	1.48588i	0.669359	+	0.742940i	$0.266569\pi$
$587$	36.0000i	1.48588i	−0.669359	+	0.742940i	$0.733431\pi$
$588$	0	0
$589$	54.0000i	2.22503i
$590$	0	0
$591$	66.0000	2.71488
$592$	0	0
$593$	20.0000	0.821302	0.410651	−	0.911793i	$-0.365302\pi$
$593$	20.0000	0.821302	0.410651	+	0.911793i	$0.365302\pi$
$594$	0	0
$595$	0	0
$596$	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
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	0	0
$603$	− 90.0000i	− 3.66508i
$604$	0	0
$605$	− 1.00000i	− 0.0406558i
$606$	0	0
$607$	−18.0000	−0.730597	−0.365299	−	0.930890i	$-0.619033\pi$
$607$	−18.0000	−0.730597	−0.365299	+	0.930890i	$0.619033\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	72.0000i	2.91281i
$612$	0	0
$613$	− 16.0000i	− 0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$	− 16.0000i	− 0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	0	0
$615$	6.00000	0.241943
$616$	0	0
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	0	0
$619$	15.0000i	0.602901i	0.953482	+	0.301450i	$0.0974708\pi$
$619$	15.0000i	0.602901i	−0.953482	+	0.301450i	$0.902529\pi$
$620$	0	0
$621$	− 27.0000i	− 1.08347i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	18.0000i	0.718851i
$628$	0	0
$629$	− 28.0000i	− 1.11643i
$630$	0	0
$631$	−15.0000	−0.597141	−0.298570	−	0.954388i	$-0.596510\pi$
$631$	−15.0000	−0.597141	−0.298570	+	0.954388i	$0.596510\pi$
$632$	0	0
$633$	−36.0000	−1.43087
$634$	0	0
$635$	− 6.00000i	− 0.238103i
$636$	0	0
$637$	− 42.0000i	− 1.66410i
$638$	0	0
$639$	−18.0000	−0.712069
$640$	0	0
$641$	−37.0000	−1.46141	−0.730706	−	0.682692i	$-0.760809\pi$
$641$	−37.0000	−1.46141	−0.730706	+	0.682692i	$0.760809\pi$
$642$	0	0
$643$	− 21.0000i	− 0.828159i	−0.910241	−	0.414080i	$-0.864104\pi$
$643$	− 21.0000i	− 0.828159i	0.910241	−	0.414080i	$-0.135896\pi$
$644$	0	0
$645$	18.0000i	0.708749i
$646$	0	0
$647$	3.00000	0.117942	0.0589711	−	0.998260i	$-0.481218\pi$
$647$	3.00000	0.117942	0.0589711	+	0.998260i	$0.481218\pi$
$648$	0	0
$649$	9.00000	0.353281
$650$	0	0
$651$	0	0
$652$	0	0
$653$	17.0000i	0.665261i	0.943057	+	0.332631i	$0.107936\pi$
$653$	17.0000i	0.665261i	−0.943057	+	0.332631i	$0.892064\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	0	0
$657$	−36.0000	−1.40449
$658$	0	0
$659$	18.0000i	0.701180i	0.936529	+	0.350590i	$0.114019\pi$
$659$	18.0000i	0.701180i	−0.936529	+	0.350590i	$0.885981\pi$
$660$	0	0
$661$	− 23.0000i	− 0.894596i	−0.894385	−	0.447298i	$-0.852386\pi$
$661$	− 23.0000i	− 0.894596i	0.894385	−	0.447298i	$-0.147614\pi$
$662$	0	0
$663$	−72.0000	−2.79625
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 12.0000i	− 0.464642i
$668$	0	0
$669$	63.0000i	2.43572i
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	0	0
$675$	36.0000i	1.38564i
$676$	0	0
$677$	− 8.00000i	− 0.307465i	−0.988113	−	0.153732i	$-0.950871\pi$
$677$	− 8.00000i	− 0.307465i	0.988113	−	0.153732i	$-0.0491294\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−36.0000	−1.37952
$682$	0	0
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	0	0
$685$	− 5.00000i	− 0.191040i
$686$	0	0
$687$	45.0000	1.71686
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	− 33.0000i	− 1.25538i	−0.778464	−	0.627690i	$-0.784001\pi$
$691$	− 33.0000i	− 1.25538i	0.778464	−	0.627690i	$-0.215999\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	12.0000	0.455186
$696$	0	0
$697$	−8.00000	−0.303022
$698$	0	0
$699$	− 12.0000i	− 0.453882i
$700$	0	0
$701$	40.0000i	1.51078i	0.655276	+	0.755390i	$0.272552\pi$
$701$	40.0000i	1.51078i	−0.655276	+	0.755390i	$0.727448\pi$
$702$	0	0
$703$	42.0000	1.58406
$704$	0	0
$705$	36.0000	1.35584
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 33.0000i	− 1.23934i	−0.784862	−	0.619671i	$-0.787266\pi$
$709$	− 33.0000i	− 1.23934i	0.784862	−	0.619671i	$-0.212734\pi$
$710$	0	0
$711$	36.0000	1.35011
$712$	0	0
$713$	27.0000	1.01116
$714$	0	0
$715$	6.00000i	0.224387i
$716$	0	0
$717$	18.0000i	0.672222i
$718$	0	0
$719$	27.0000	1.00693	0.503465	−	0.864016i	$-0.332058\pi$
$719$	27.0000	1.00693	0.503465	+	0.864016i	$0.332058\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	36.0000i	1.33885i
$724$	0	0
$725$	16.0000i	0.594225i
$726$	0	0
$727$	−3.00000	−0.111264	−0.0556319	−	0.998451i	$-0.517717\pi$
$727$	−3.00000	−0.111264	−0.0556319	+	0.998451i	$0.517717\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	− 24.0000i	− 0.887672i
$732$	0	0
$733$	− 48.0000i	− 1.77292i	−0.462805	−	0.886460i	$-0.653157\pi$
$733$	− 48.0000i	− 1.77292i	0.462805	−	0.886460i	$-0.346843\pi$
$734$	0	0
$735$	−21.0000	−0.774597
$736$	0	0
$737$	15.0000	0.552532
$738$	0	0
$739$	6.00000i	0.220714i	0.993892	+	0.110357i	$0.0351994\pi$
$739$	6.00000i	0.220714i	−0.993892	+	0.110357i	$0.964801\pi$
$740$	0	0
$741$	− 108.000i	− 3.96748i
$742$	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
$745$	16.0000	0.586195
$746$	0	0
$747$	− 36.0000i	− 1.31717i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−3.00000	−0.109472	−0.0547358	−	0.998501i	$-0.517432\pi$
$751$	−3.00000	−0.109472	−0.0547358	+	0.998501i	$0.517432\pi$
$752$	0	0
$753$	63.0000	2.29585
$754$	0	0
$755$	6.00000i	0.218362i
$756$	0	0
$757$	6.00000i	0.218074i	0.994038	+	0.109037i	$0.0347767\pi$
$757$	6.00000i	0.218074i	−0.994038	+	0.109037i	$0.965223\pi$
$758$	0	0
$759$	9.00000	0.326679
$760$	0	0
$761$	26.0000	0.942499	0.471250	−	0.882000i	$-0.343803\pi$
$761$	26.0000	0.942499	0.471250	+	0.882000i	$0.343803\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	24.0000i	0.867722i
$766$	0	0
$767$	−54.0000	−1.94983
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	− 42.0000i	− 1.51259i
$772$	0	0
$773$	2.00000i	0.0719350i	0.999353	+	0.0359675i	$0.0114513\pi$
$773$	2.00000i	0.0719350i	−0.999353	+	0.0359675i	$0.988549\pi$
$774$	0	0
$775$	−36.0000	−1.29316
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 12.0000i	− 0.429945i
$780$	0	0
$781$	− 3.00000i	− 0.107348i
$782$	0	0
$783$	−36.0000	−1.28654
$784$	0	0
$785$	13.0000	0.463990
$786$	0	0
$787$	36.0000i	1.28326i	0.767014	+	0.641631i	$0.221742\pi$
$787$	36.0000i	1.28326i	−0.767014	+	0.641631i	$0.778258\pi$
$788$	0	0
$789$	72.0000i	2.56327i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	48.0000	1.70453
$794$	0	0
$795$	6.00000i	0.212798i
$796$	0	0
$797$	− 49.0000i	− 1.73567i	−0.496853	−	0.867835i	$-0.665511\pi$
$797$	− 49.0000i	− 1.73567i	0.496853	−	0.867835i	$-0.334489\pi$
$798$	0	0
$799$	−48.0000	−1.69812
$800$	0	0
$801$	−30.0000	−1.06000
$802$	0	0
$803$	− 6.00000i	− 0.211735i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.00000	0.211210
$808$	0	0
$809$	−46.0000	−1.61727	−0.808637	−	0.588308i	$-0.799794\pi$
$809$	−46.0000	−1.61727	−0.808637	+	0.588308i	$0.799794\pi$
$810$	0	0
$811$	30.0000i	1.05344i	0.850038	+	0.526721i	$0.176579\pi$
$811$	30.0000i	1.05344i	−0.850038	+	0.526721i	$0.823421\pi$
$812$	0	0
$813$	18.0000i	0.631288i
$814$	0	0
$815$	12.0000	0.420342
$816$	0	0
$817$	36.0000	1.25948
$818$	0	0
$819$	0	0
$820$	0	0
$821$	22.0000i	0.767805i	0.923374	+	0.383903i	$0.125420\pi$
$821$	22.0000i	0.767805i	−0.923374	+	0.383903i	$0.874580\pi$
$822$	0	0
$823$	33.0000	1.15031	0.575154	−	0.818045i	$-0.304942\pi$
$823$	33.0000	1.15031	0.575154	+	0.818045i	$0.304942\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	− 18.0000i	− 0.625921i	−0.949766	−	0.312961i	$-0.898679\pi$
$827$	− 18.0000i	− 0.625921i	0.949766	−	0.312961i	$-0.101321\pi$
$828$	0	0
$829$	− 21.0000i	− 0.729360i	−0.931133	−	0.364680i	$-0.881178\pi$
$829$	− 21.0000i	− 0.729360i	0.931133	−	0.364680i	$-0.118822\pi$
$830$	0	0
$831$	−18.0000	−0.624413
$832$	0	0
$833$	28.0000	0.970143
$834$	0	0
$835$	− 18.0000i	− 0.622916i
$836$	0	0
$837$	− 81.0000i	− 2.79977i
$838$	0	0
$839$	33.0000	1.13929	0.569643	−	0.821892i	$-0.307081\pi$
$839$	33.0000	1.13929	0.569643	+	0.821892i	$0.307081\pi$
$840$	0	0
$841$	13.0000	0.448276
$842$	0	0
$843$	30.0000i	1.03325i
$844$	0	0
$845$	− 23.0000i	− 0.791224i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 21.0000i	− 0.719871i
$852$	0	0
$853$	46.0000i	1.57501i	0.616308	+	0.787505i	$0.288628\pi$
$853$	46.0000i	1.57501i	−0.616308	+	0.787505i	$0.711372\pi$
$854$	0	0
$855$	−36.0000	−1.23117
$856$	0	0
$857$	32.0000	1.09310	0.546550	−	0.837427i	$-0.315941\pi$
$857$	32.0000	1.09310	0.546550	+	0.837427i	$0.315941\pi$
$858$	0	0
$859$	− 27.0000i	− 0.921228i	−0.887601	−	0.460614i	$-0.847629\pi$
$859$	− 27.0000i	− 0.921228i	0.887601	−	0.460614i	$-0.152371\pi$
$860$	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
$865$	10.0000	0.340010
$866$	0	0
$867$	3.00000i	0.101885i
$868$	0	0
$869$	6.00000i	0.203536i
$870$	0	0
$871$	−90.0000	−3.04953
$872$	0	0
$873$	18.0000	0.609208
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 14.0000i	− 0.472746i	−0.971662	−	0.236373i	$-0.924041\pi$
$877$	− 14.0000i	− 0.472746i	0.971662	−	0.236373i	$-0.0759588\pi$
$878$	0	0
$879$	−66.0000	−2.22612
$880$	0	0
$881$	29.0000	0.977035	0.488517	−	0.872554i	$-0.337538\pi$
$881$	29.0000	0.977035	0.488517	+	0.872554i	$0.337538\pi$
$882$	0	0
$883$	12.0000i	0.403832i	0.979403	+	0.201916i	$0.0647168\pi$
$883$	12.0000i	0.403832i	−0.979403	+	0.201916i	$0.935283\pi$
$884$	0	0
$885$	27.0000i	0.907595i
$886$	0	0
$887$	−6.00000	−0.201460	−0.100730	−	0.994914i	$-0.532118\pi$
$887$	−6.00000	−0.201460	−0.100730	+	0.994914i	$0.532118\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 9.00000i	− 0.301511i
$892$	0	0
$893$	− 72.0000i	− 2.40939i
$894$	0	0
$895$	−15.0000	−0.501395
$896$	0	0
$897$	−54.0000	−1.80301
$898$	0	0
$899$	− 36.0000i	− 1.20067i
$900$	0	0
$901$	− 8.00000i	− 0.266519i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.00000	−0.0997234
$906$	0	0
$907$	− 12.0000i	− 0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$	− 12.0000i	− 0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$	0	0
$909$	− 96.0000i	− 3.18412i
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	6.00000	0.198571
$914$	0	0
$915$	− 24.0000i	− 0.793416i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−30.0000	−0.989609	−0.494804	−	0.869004i	$-0.664760\pi$
$919$	−30.0000	−0.989609	−0.494804	+	0.869004i	$0.664760\pi$
$920$	0	0
$921$	−18.0000	−0.593120
$922$	0	0
$923$	18.0000i	0.592477i
$924$	0	0
$925$	28.0000i	0.920634i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−22.0000	−0.721797	−0.360898	−	0.932605i	$-0.617530\pi$
$929$	−22.0000	−0.721797	−0.360898	+	0.932605i	$0.617530\pi$
$930$	0	0
$931$	42.0000i	1.37649i
$932$	0	0
$933$	− 72.0000i	− 2.35717i
$934$	0	0
$935$	−4.00000	−0.130814
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	− 3.00000i	− 0.0979013i
$940$	0	0
$941$	10.0000i	0.325991i	0.986627	+	0.162995i	$0.0521156\pi$
$941$	10.0000i	0.325991i	−0.986627	+	0.162995i	$0.947884\pi$
$942$	0	0
$943$	−6.00000	−0.195387
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.0000i	0.487435i	0.969846	+	0.243717i	$0.0783669\pi$
$947$	15.0000i	0.487435i	−0.969846	+	0.243717i	$0.921633\pi$
$948$	0	0
$949$	36.0000i	1.16861i
$950$	0	0
$951$	69.0000	2.23748
$952$	0	0
$953$	40.0000	1.29573	0.647864	−	0.761756i	$-0.275663\pi$
$953$	40.0000	1.29573	0.647864	+	0.761756i	$0.275663\pi$
$954$	0	0
$955$	21.0000i	0.679544i
$956$	0	0
$957$	− 12.0000i	− 0.387905i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	50.0000	1.61290
$962$	0	0
$963$	− 72.0000i	− 2.32017i
$964$	0	0
$965$	4.00000i	0.128765i
$966$	0	0
$967$	−12.0000	−0.385894	−0.192947	−	0.981209i	$-0.561805\pi$
$967$	−12.0000	−0.385894	−0.192947	+	0.981209i	$0.561805\pi$
$968$	0	0
$969$	72.0000	2.31297
$970$	0	0
$971$	15.0000i	0.481373i	0.970603	+	0.240686i	$0.0773725\pi$
$971$	15.0000i	0.481373i	−0.970603	+	0.240686i	$0.922627\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	72.0000	2.30585
$976$	0	0
$977$	−23.0000	−0.735835	−0.367918	−	0.929858i	$-0.619929\pi$
$977$	−23.0000	−0.735835	−0.367918	+	0.929858i	$0.619929\pi$
$978$	0	0
$979$	− 5.00000i	− 0.159801i
$980$	0	0
$981$	− 36.0000i	− 1.14939i
$982$	0	0
$983$	45.0000	1.43528	0.717639	−	0.696416i	$-0.245223\pi$
$983$	45.0000	1.43528	0.717639	+	0.696416i	$0.245223\pi$
$984$	0	0
$985$	−22.0000	−0.700978
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 18.0000i	− 0.572367i
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	45.0000	1.42803
$994$	0	0
$995$	0	0
$996$	0	0
$997$	8.00000i	0.253363i	0.991943	+	0.126681i	$0.0404325\pi$
$997$	8.00000i	0.253363i	−0.991943	+	0.126681i	$0.959567\pi$
$998$	0	0
$999$	−63.0000	−1.99323

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2816.2.c.h.1409.1		2
4.3	odd	2		2816.2.c.g.1409.2		2
8.3	odd	2		2816.2.c.g.1409.1		2
8.5	even	2	inner	2816.2.c.h.1409.2		2
16.3	odd	4		352.2.a.a.1.1	✓	1
16.5	even	4		704.2.a.a.1.1		1
16.11	odd	4		704.2.a.k.1.1		1
16.13	even	4		352.2.a.f.1.1	yes	1
48.5	odd	4		6336.2.a.bs.1.1		1
48.11	even	4		6336.2.a.bt.1.1		1
48.29	odd	4		3168.2.a.i.1.1		1
48.35	even	4		3168.2.a.h.1.1		1
80.19	odd	4		8800.2.a.bb.1.1		1
80.29	even	4		8800.2.a.a.1.1		1
176.21	odd	4		7744.2.a.a.1.1		1
176.43	even	4		7744.2.a.bj.1.1		1
176.109	odd	4		3872.2.a.m.1.1		1
176.131	even	4		3872.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.a.a.1.1	✓	1	16.3	odd	4
352.2.a.f.1.1	yes	1	16.13	even	4
704.2.a.a.1.1		1	16.5	even	4
704.2.a.k.1.1		1	16.11	odd	4
2816.2.c.g.1409.1		2	8.3	odd	2
2816.2.c.g.1409.2		2	4.3	odd	2
2816.2.c.h.1409.1		2	1.1	even	1	trivial
2816.2.c.h.1409.2		2	8.5	even	2	inner
3168.2.a.h.1.1		1	48.35	even	4
3168.2.a.i.1.1		1	48.29	odd	4
3872.2.a.a.1.1		1	176.131	even	4
3872.2.a.m.1.1		1	176.109	odd	4
6336.2.a.bs.1.1		1	48.5	odd	4
6336.2.a.bt.1.1		1	48.11	even	4
7744.2.a.a.1.1		1	176.21	odd	4
7744.2.a.bj.1.1		1	176.43	even	4
8800.2.a.a.1.1		1	80.29	even	4
8800.2.a.bb.1.1		1	80.19	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:	\( N \)	\(=\)	\( 2816 = 2^{8} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2816.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +1.00000i q^{5} -6.00000 q^{9} -1.00000i q^{11} +6.00000i q^{13} +3.00000 q^{15} -4.00000 q^{17} -6.00000i q^{19} -3.00000 q^{23} +4.00000 q^{25} +9.00000i q^{27} +4.00000i q^{29} -9.00000 q^{31} -3.00000 q^{33} +7.00000i q^{37} +18.0000 q^{39} +2.00000 q^{41} +6.00000i q^{43} -6.00000i q^{45} +12.0000 q^{47} -7.00000 q^{49} +12.0000i q^{51} +2.00000i q^{53} +1.00000 q^{55} -18.0000 q^{57} +9.00000i q^{59} -8.00000i q^{61} -6.00000 q^{65} +15.0000i q^{67} +9.00000i q^{69} +3.00000 q^{71} +6.00000 q^{73} -12.0000i q^{75} -6.00000 q^{79} +9.00000 q^{81} +6.00000i q^{83} -4.00000i q^{85} +12.0000 q^{87} +5.00000 q^{89} +27.0000i q^{93} +6.00000 q^{95} -3.00000 q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{9} + 6 q^{15} - 8 q^{17} - 6 q^{23} + 8 q^{25} - 18 q^{31} - 6 q^{33} + 36 q^{39} + 4 q^{41} + 24 q^{47} - 14 q^{49} + 2 q^{55} - 36 q^{57} - 12 q^{65} + 6 q^{71} + 12 q^{73} - 12 q^{79} + 18 q^{81}+ \cdots - 6 q^{97}+O(q^{100}) \) 2 * q - 12 * q^9 + 6 * q^15 - 8 * q^17 - 6 * q^23 + 8 * q^25 - 18 * q^31 - 6 * q^33 + 36 * q^39 + 4 * q^41 + 24 * q^47 - 14 * q^49 + 2 * q^55 - 36 * q^57 - 12 * q^65 + 6 * q^71 + 12 * q^73 - 12 * q^79 + 18 * q^81 + 24 * q^87 + 10 * q^89 + 12 * q^95 - 6 * q^97

Introduction

L-functions

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