LMFDB - Embedded newform 4608.2.d.a.2305.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(2305,4608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4608, 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("4608.2305");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.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:	$36.7950652514$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2305.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	4608.2305
Dual form			4608.2.d.a.2305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{5} -4.00000 q^{7} +O(q^{10})$	$q+2.82843i q^{5} -4.00000 q^{7} +1.41421i q^{11} -2.82843i q^{13} +4.00000 q^{17} -7.07107i q^{19} +4.00000 q^{23} -3.00000 q^{25} +8.48528i q^{29} +8.00000 q^{31} -11.3137i q^{35} +2.82843i q^{37} +2.00000 q^{41} -4.24264i q^{43} +9.00000 q^{49} -2.82843i q^{53} -4.00000 q^{55} +4.24264i q^{59} +8.48528i q^{61} +8.00000 q^{65} +4.24264i q^{67} -4.00000 q^{71} +4.00000 q^{73} -5.65685i q^{77} -8.00000 q^{79} +9.89949i q^{83} +11.3137i q^{85} +12.0000 q^{89} +11.3137i q^{91} +20.0000 q^{95} -4.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{7}+O(q^{10}) $ 2 * q - 8 * q^7	$ 2 q - 8 q^{7} + 8 q^{17} + 8 q^{23} - 6 q^{25} + 16 q^{31} + 4 q^{41} + 18 q^{49} - 8 q^{55} + 16 q^{65} - 8 q^{71} + 8 q^{73} - 16 q^{79} + 24 q^{89} + 40 q^{95} - 8 q^{97}+O(q^{100}) $ 2 * q - 8 * q^7 + 8 * q^17 + 8 * q^23 - 6 * q^25 + 16 * q^31 + 4 * q^41 + 18 * q^49 - 8 * q^55 + 16 * q^65 - 8 * q^71 + 8 * q^73 - 16 * q^79 + 24 * q^89 + 40 * q^95 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times$.

$n$	$2053$	$3583$	$4097$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0	0
$4$	0	0
$5$	2.82843i	1.26491i	0.774597	+	0.632456i	$0.217953\pi$
$5$	2.82843i	1.26491i	−0.774597	+	0.632456i	$0.782047\pi$
$6$	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
$9$	0	0
$10$	0	0
$11$	1.41421i	0.426401i	0.977008	+	0.213201i	$0.0683888\pi$
$11$	1.41421i	0.426401i	−0.977008	+	0.213201i	$0.931611\pi$
$12$	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
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	− 7.07107i	− 1.62221i	−0.584898	−	0.811107i	$-0.698865\pi$
$19$	− 7.07107i	− 1.62221i	0.584898	−	0.811107i	$-0.301135\pi$
$20$	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
$25$	−3.00000	−0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.48528i	1.57568i	0.615882	+	0.787839i	$0.288800\pi$
$29$	8.48528i	1.57568i	−0.615882	+	0.787839i	$0.711200\pi$
$30$	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
$35$	− 11.3137i	− 1.91237i
$36$	0	0
$37$	2.82843i	0.464991i	0.972598	+	0.232495i	$0.0746890\pi$
$37$	2.82843i	0.464991i	−0.972598	+	0.232495i	$0.925311\pi$
$38$	0	0
$39$	0	0
$40$	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
$43$	− 4.24264i	− 0.646997i	−0.946229	−	0.323498i	$-0.895141\pi$
$43$	− 4.24264i	− 0.646997i	0.946229	−	0.323498i	$-0.104859\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 2.82843i	− 0.388514i	−0.980951	−	0.194257i	$-0.937770\pi$
$53$	− 2.82843i	− 0.388514i	0.980951	−	0.194257i	$-0.0622296\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.24264i	0.552345i	0.961108	+	0.276172i	$0.0890661\pi$
$59$	4.24264i	0.552345i	−0.961108	+	0.276172i	$0.910934\pi$
$60$	0	0
$61$	8.48528i	1.08643i	0.839594	+	0.543214i	$0.182793\pi$
$61$	8.48528i	1.08643i	−0.839594	+	0.543214i	$0.817207\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	8.00000	0.992278
$66$	0	0
$67$	4.24264i	0.518321i	0.965834	+	0.259161i	$0.0834459\pi$
$67$	4.24264i	0.518321i	−0.965834	+	0.259161i	$0.916554\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.65685i	− 0.644658i
$78$	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
$81$	0	0
$82$	0	0
$83$	9.89949i	1.08661i	0.839535	+	0.543305i	$0.182827\pi$
$83$	9.89949i	1.08661i	−0.839535	+	0.543305i	$0.817173\pi$
$84$	0	0
$85$	11.3137i	1.22714i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	11.3137i	1.18600i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	20.0000	2.05196
$96$	0	0
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 2.82843i	− 0.281439i	−0.990050	−	0.140720i	$-0.955058\pi$
$101$	− 2.82843i	− 0.281439i	0.990050	−	0.140720i	$-0.0449416\pi$
$102$	0	0
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 9.89949i	− 0.957020i	−0.878082	−	0.478510i	$-0.841177\pi$
$107$	− 9.89949i	− 0.957020i	0.878082	−	0.478510i	$-0.158823\pi$
$108$	0	0
$109$	14.1421i	1.35457i	0.735720	+	0.677285i	$0.236844\pi$
$109$	14.1421i	1.35457i	−0.735720	+	0.677285i	$0.763156\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	11.3137i	1.05501i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−16.0000	−1.46672
$120$	0	0
$121$	9.00000	0.818182
$122$	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
$131$	7.07107i	0.617802i	0.951094	+	0.308901i	$0.0999612\pi$
$131$	7.07107i	0.617802i	−0.951094	+	0.308901i	$0.900039\pi$
$132$	0	0
$133$	28.2843i	2.45256i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	− 4.24264i	− 0.359856i	−0.983680	−	0.179928i	$-0.942414\pi$
$139$	− 4.24264i	− 0.359856i	0.983680	−	0.179928i	$-0.0575865\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	−24.0000	−1.99309
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.82843i	0.231714i	0.993266	+	0.115857i	$0.0369614\pi$
$149$	2.82843i	0.231714i	−0.993266	+	0.115857i	$0.963039\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	0	0
$154$	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
$161$	−16.0000	−1.26098
$162$	0	0
$163$	− 9.89949i	− 0.775388i	−0.921788	−	0.387694i	$-0.873272\pi$
$163$	− 9.89949i	− 0.775388i	0.921788	−	0.387694i	$-0.126728\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.1421i	1.07521i	0.843198	+	0.537603i	$0.180670\pi$
$173$	14.1421i	1.07521i	−0.843198	+	0.537603i	$0.819330\pi$
$174$	0	0
$175$	12.0000	0.907115
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 1.41421i	− 0.105703i	−0.998602	−	0.0528516i	$-0.983169\pi$
$179$	− 1.41421i	− 0.105703i	0.998602	−	0.0528516i	$-0.0168310\pi$
$180$	0	0
$181$	8.48528i	0.630706i	0.948974	+	0.315353i	$0.102123\pi$
$181$	8.48528i	0.630706i	−0.948974	+	0.315353i	$0.897877\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	5.65685i	0.413670i
$188$	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
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.1421i	1.00759i	0.863825	+	0.503793i	$0.168062\pi$
$197$	14.1421i	1.00759i	−0.863825	+	0.503793i	$0.831938\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 33.9411i	− 2.38220i
$204$	0	0
$205$	5.65685i	0.395092i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	10.0000	0.691714
$210$	0	0
$211$	24.0416i	1.65509i	0.561396	+	0.827547i	$0.310264\pi$
$211$	24.0416i	1.65509i	−0.561396	+	0.827547i	$0.689736\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	12.0000	0.818393
$216$	0	0
$217$	−32.0000	−2.17230
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 11.3137i	− 0.761042i
$222$	0	0
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	21.2132i	1.40797i	0.710215	+	0.703985i	$0.248598\pi$
$227$	21.2132i	1.40797i	−0.710215	+	0.703985i	$0.751402\pi$
$228$	0	0
$229$	− 25.4558i	− 1.68217i	−0.540903	−	0.841085i	$-0.681918\pi$
$229$	− 25.4558i	− 1.68217i	0.540903	−	0.841085i	$-0.318082\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	25.4558i	1.62631i
$246$	0	0
$247$	−20.0000	−1.27257
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.24264i	0.267793i	0.990995	+	0.133897i	$0.0427490\pi$
$251$	4.24264i	0.267793i	−0.990995	+	0.133897i	$0.957251\pi$
$252$	0	0
$253$	5.65685i	0.355643i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	− 11.3137i	− 0.703000i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	28.0000	1.72655	0.863277	−	0.504730i	$-0.168408\pi$
$263$	28.0000	1.72655	0.863277	+	0.504730i	$0.168408\pi$
$264$	0	0
$265$	8.00000	0.491436
$266$	0	0
$267$	0	0
$268$	0	0
$269$	19.7990i	1.20717i	0.797300	+	0.603583i	$0.206261\pi$
$269$	19.7990i	1.20717i	−0.797300	+	0.603583i	$0.793739\pi$
$270$	0	0
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 4.24264i	− 0.255841i
$276$	0	0
$277$	− 8.48528i	− 0.509831i	−0.966963	−	0.254916i	$-0.917952\pi$
$277$	− 8.48528i	− 0.509831i	0.966963	−	0.254916i	$-0.0820477\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−20.0000	−1.19310	−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000	−1.19310	−0.596550	+	0.802576i	$0.703462\pi$
$282$	0	0
$283$	− 24.0416i	− 1.42913i	−0.699571	−	0.714563i	$-0.746625\pi$
$283$	− 24.0416i	− 1.42913i	0.699571	−	0.714563i	$-0.253375\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.00000	−0.472225
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 8.48528i	− 0.495715i	−0.968796	−	0.247858i	$-0.920273\pi$
$293$	− 8.48528i	− 0.495715i	0.968796	−	0.247858i	$-0.0797265\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 11.3137i	− 0.654289i
$300$	0	0
$301$	16.9706i	0.978167i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−24.0000	−1.37424
$306$	0	0
$307$	− 7.07107i	− 0.403567i	−0.979430	−	0.201784i	$-0.935326\pi$
$307$	− 7.07107i	− 0.403567i	0.979430	−	0.201784i	$-0.0646738\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.82843i	− 0.158860i	−0.996840	−	0.0794301i	$-0.974690\pi$
$317$	− 2.82843i	− 0.158860i	0.996840	−	0.0794301i	$-0.0253101\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 28.2843i	− 1.57378i
$324$	0	0
$325$	8.48528i	0.470679i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 12.7279i	− 0.699590i	−0.936826	−	0.349795i	$-0.886251\pi$
$331$	− 12.7279i	− 0.699590i	0.936826	−	0.349795i	$-0.113749\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−12.0000	−0.655630
$336$	0	0
$337$	30.0000	1.63420	0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000	1.63420	0.817102	+	0.576493i	$0.195579\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	11.3137i	0.612672i
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.5563i	0.835109i	0.908652	+	0.417554i	$0.137113\pi$
$347$	15.5563i	0.835109i	−0.908652	+	0.417554i	$0.862887\pi$
$348$	0	0
$349$	− 19.7990i	− 1.05982i	−0.848055	−	0.529908i	$-0.822227\pi$
$349$	− 19.7990i	− 1.05982i	0.848055	−	0.529908i	$-0.177773\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	− 11.3137i	− 0.600469i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−36.0000	−1.90001	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	−36.0000	−1.90001	−0.950004	+	0.312239i	$0.898921\pi$
$360$	0	0
$361$	−31.0000	−1.63158
$362$	0	0
$363$	0	0
$364$	0	0
$365$	11.3137i	0.592187i
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	11.3137i	0.587378i
$372$	0	0
$373$	36.7696i	1.90386i	0.306323	+	0.951928i	$0.400901\pi$
$373$	36.7696i	1.90386i	−0.306323	+	0.951928i	$0.599099\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.0000	1.23606
$378$	0	0
$379$	29.6985i	1.52551i	0.646688	+	0.762754i	$0.276153\pi$
$379$	29.6985i	1.52551i	−0.646688	+	0.762754i	$0.723847\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	0	0
$385$	16.0000	0.815436
$386$	0	0
$387$	0	0
$388$	0	0
$389$	19.7990i	1.00385i	0.864912	+	0.501924i	$0.167374\pi$
$389$	19.7990i	1.00385i	−0.864912	+	0.501924i	$0.832626\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 22.6274i	− 1.13851i
$396$	0	0
$397$	2.82843i	0.141955i	0.997478	+	0.0709773i	$0.0226118\pi$
$397$	2.82843i	0.141955i	−0.997478	+	0.0709773i	$0.977388\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	36.0000	1.79775	0.898877	−	0.438201i	$-0.144384\pi$
$401$	36.0000	1.79775	0.898877	+	0.438201i	$0.144384\pi$
$402$	0	0
$403$	− 22.6274i	− 1.12715i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.00000	−0.198273
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 16.9706i	− 0.835067i
$414$	0	0
$415$	−28.0000	−1.37447
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 26.8701i	− 1.31269i	−0.754462	−	0.656344i	$-0.772102\pi$
$419$	− 26.8701i	− 1.31269i	0.754462	−	0.656344i	$-0.227898\pi$
$420$	0	0
$421$	8.48528i	0.413547i	0.978389	+	0.206774i	$0.0662964\pi$
$421$	8.48528i	0.413547i	−0.978389	+	0.206774i	$0.933704\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−12.0000	−0.582086
$426$	0	0
$427$	− 33.9411i	− 1.64253i
$428$	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
$433$	−20.0000	−0.961139	−0.480569	−	0.876957i	$-0.659570\pi$
$433$	−20.0000	−0.961139	−0.480569	+	0.876957i	$0.659570\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 28.2843i	− 1.35302i
$438$	0	0
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	35.3553i	1.67978i	0.542754	+	0.839891i	$0.317381\pi$
$443$	35.3553i	1.67978i	−0.542754	+	0.839891i	$0.682619\pi$
$444$	0	0
$445$	33.9411i	1.60896i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	2.82843i	0.133185i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−32.0000	−1.50018
$456$	0	0
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 14.1421i	− 0.658665i	−0.944214	−	0.329332i	$-0.893176\pi$
$461$	− 14.1421i	− 0.658665i	0.944214	−	0.329332i	$-0.106824\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.3848i	0.850746i	0.905018	+	0.425373i	$0.139857\pi$
$467$	18.3848i	0.850746i	−0.905018	+	0.425373i	$0.860143\pi$
$468$	0	0
$469$	− 16.9706i	− 0.783628i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.00000	0.275880
$474$	0	0
$475$	21.2132i	0.973329i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 11.3137i	− 0.513729i
$486$	0	0
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15.5563i	0.702048i	0.936366	+	0.351024i	$0.114166\pi$
$491$	15.5563i	0.702048i	−0.936366	+	0.351024i	$0.885834\pi$
$492$	0	0
$493$	33.9411i	1.52863i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16.0000	0.717698
$498$	0	0
$499$	12.7279i	0.569780i	0.958560	+	0.284890i	$0.0919571\pi$
$499$	12.7279i	0.569780i	−0.958560	+	0.284890i	$0.908043\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	20.0000	0.891756	0.445878	−	0.895094i	$-0.352892\pi$
$503$	20.0000	0.891756	0.445878	+	0.895094i	$0.352892\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.82843i	0.125368i	0.998033	+	0.0626839i	$0.0199660\pi$
$509$	2.82843i	0.125368i	−0.998033	+	0.0626839i	$0.980034\pi$
$510$	0	0
$511$	−16.0000	−0.707798
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 33.9411i	− 1.49562i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	0	0
$523$	9.89949i	0.432875i	0.976297	+	0.216437i	$0.0694437\pi$
$523$	9.89949i	0.432875i	−0.976297	+	0.216437i	$0.930556\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	32.0000	1.39394
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 5.65685i	− 0.245026i
$534$	0	0
$535$	28.0000	1.21055
$536$	0	0
$537$	0	0
$538$	0	0
$539$	12.7279i	0.548230i
$540$	0	0
$541$	8.48528i	0.364811i	0.983223	+	0.182405i	$0.0583883\pi$
$541$	8.48528i	0.364811i	−0.983223	+	0.182405i	$0.941612\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−40.0000	−1.71341
$546$	0	0
$547$	26.8701i	1.14888i	0.818546	+	0.574440i	$0.194780\pi$
$547$	26.8701i	1.14888i	−0.818546	+	0.574440i	$0.805220\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	60.0000	2.55609
$552$	0	0
$553$	32.0000	1.36078
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 31.1127i	− 1.31829i	−0.752017	−	0.659144i	$-0.770919\pi$
$557$	− 31.1127i	− 1.31829i	0.752017	−	0.659144i	$-0.229081\pi$
$558$	0	0
$559$	−12.0000	−0.507546
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7.07107i	0.298010i	0.988836	+	0.149005i	$0.0476070\pi$
$563$	7.07107i	0.298010i	−0.988836	+	0.149005i	$0.952393\pi$
$564$	0	0
$565$	− 39.5980i	− 1.66590i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	32.5269i	1.36121i	0.732651	+	0.680604i	$0.238283\pi$
$571$	32.5269i	1.36121i	−0.732651	+	0.680604i	$0.761717\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12.0000	−0.500435
$576$	0	0
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 39.5980i	− 1.64280i
$582$	0	0
$583$	4.00000	0.165663
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.41421i	0.0583708i	0.999574	+	0.0291854i	$0.00929133\pi$
$587$	1.41421i	0.0583708i	−0.999574	+	0.0291854i	$0.990709\pi$
$588$	0	0
$589$	− 56.5685i	− 2.33087i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	− 45.2548i	− 1.85527i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	4.00000	0.163163	0.0815817	−	0.996667i	$-0.474003\pi$
$601$	4.00000	0.163163	0.0815817	+	0.996667i	$0.474003\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	25.4558i	1.03493i
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	8.48528i	0.342717i	0.985209	+	0.171359i	$0.0548157\pi$
$613$	8.48528i	0.342717i	−0.985209	+	0.171359i	$0.945184\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	28.0000	1.12724	0.563619	−	0.826035i	$-0.309409\pi$
$617$	28.0000	1.12724	0.563619	+	0.826035i	$0.309409\pi$
$618$	0	0
$619$	− 1.41421i	− 0.0568420i	−0.999596	−	0.0284210i	$-0.990952\pi$
$619$	− 1.41421i	− 0.0568420i	0.999596	−	0.0284210i	$-0.00904791\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−48.0000	−1.92308
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.3137i	0.451107i
$630$	0	0
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	22.6274i	0.897942i
$636$	0	0
$637$	− 25.4558i	− 1.00860i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4.00000	0.157991	0.0789953	−	0.996875i	$-0.474829\pi$
$641$	4.00000	0.157991	0.0789953	+	0.996875i	$0.474829\pi$
$642$	0	0
$643$	4.24264i	0.167313i	0.996495	+	0.0836567i	$0.0266599\pi$
$643$	4.24264i	0.167313i	−0.996495	+	0.0836567i	$0.973340\pi$
$644$	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
$649$	−6.00000	−0.235521
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 48.0833i	− 1.88164i	−0.338902	−	0.940822i	$-0.610055\pi$
$653$	− 48.0833i	− 1.88164i	0.338902	−	0.940822i	$-0.389945\pi$
$654$	0	0
$655$	−20.0000	−0.781465
$656$	0	0
$657$	0	0
$658$	0	0
$659$	32.5269i	1.26707i	0.773715	+	0.633534i	$0.218396\pi$
$659$	32.5269i	1.26707i	−0.773715	+	0.633534i	$0.781604\pi$
$660$	0	0
$661$	31.1127i	1.21014i	0.796171	+	0.605072i	$0.206856\pi$
$661$	31.1127i	1.21014i	−0.796171	+	0.605072i	$0.793144\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−80.0000	−3.10227
$666$	0	0
$667$	33.9411i	1.31421i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	−20.0000	−0.770943	−0.385472	−	0.922720i	$-0.625961\pi$
$673$	−20.0000	−0.770943	−0.385472	+	0.922720i	$0.625961\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.4264i	1.63058i	0.579053	+	0.815290i	$0.303422\pi$
$677$	42.4264i	1.63058i	−0.579053	+	0.815290i	$0.696578\pi$
$678$	0	0
$679$	16.0000	0.614024
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 29.6985i	− 1.13638i	−0.822897	−	0.568190i	$-0.807644\pi$
$683$	− 29.6985i	− 1.13638i	0.822897	−	0.568190i	$-0.192356\pi$
$684$	0	0
$685$	− 50.9117i	− 1.94524i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−8.00000	−0.304776
$690$	0	0
$691$	46.6690i	1.77537i	0.460447	+	0.887687i	$0.347689\pi$
$691$	46.6690i	1.77537i	−0.460447	+	0.887687i	$0.652311\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	12.0000	0.455186
$696$	0	0
$697$	8.00000	0.303022
$698$	0	0
$699$	0	0
$700$	0	0
$701$	48.0833i	1.81608i	0.418884	+	0.908040i	$0.362421\pi$
$701$	48.0833i	1.81608i	−0.418884	+	0.908040i	$0.637579\pi$
$702$	0	0
$703$	20.0000	0.754314
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11.3137i	0.425496i
$708$	0	0
$709$	− 8.48528i	− 0.318671i	−0.987224	−	0.159336i	$-0.949065\pi$
$709$	− 8.48528i	− 0.318671i	0.987224	−	0.159336i	$-0.0509352\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	32.0000	1.19841
$714$	0	0
$715$	11.3137i	0.423109i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	48.0000	1.78761
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 25.4558i	− 0.945406i
$726$	0	0
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 16.9706i	− 0.627679i
$732$	0	0
$733$	− 19.7990i	− 0.731292i	−0.930754	−	0.365646i	$-0.880848\pi$
$733$	− 19.7990i	− 0.731292i	0.930754	−	0.365646i	$-0.119152\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.00000	−0.221013
$738$	0	0
$739$	24.0416i	0.884386i	0.896920	+	0.442193i	$0.145799\pi$
$739$	24.0416i	0.884386i	−0.896920	+	0.442193i	$0.854201\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4.00000	0.146746	0.0733729	−	0.997305i	$-0.476624\pi$
$743$	4.00000	0.146746	0.0733729	+	0.997305i	$0.476624\pi$
$744$	0	0
$745$	−8.00000	−0.293097
$746$	0	0
$747$	0	0
$748$	0	0
$749$	39.5980i	1.44688i
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 11.3137i	− 0.411748i
$756$	0	0
$757$	− 25.4558i	− 0.925208i	−0.886565	−	0.462604i	$-0.846915\pi$
$757$	− 25.4558i	− 0.925208i	0.886565	−	0.462604i	$-0.153085\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	− 56.5685i	− 2.04792i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.0000	0.433295
$768$	0	0
$769$	44.0000	1.58668	0.793340	−	0.608778i	$-0.208340\pi$
$769$	44.0000	1.58668	0.793340	+	0.608778i	$0.208340\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.48528i	0.305194i	0.988288	+	0.152597i	$0.0487637\pi$
$773$	8.48528i	0.305194i	−0.988288	+	0.152597i	$0.951236\pi$
$774$	0	0
$775$	−24.0000	−0.862105
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 14.1421i	− 0.506695i
$780$	0	0
$781$	− 5.65685i	− 0.202418i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−24.0000	−0.856597
$786$	0	0
$787$	46.6690i	1.66357i	0.555097	+	0.831786i	$0.312681\pi$
$787$	46.6690i	1.66357i	−0.555097	+	0.831786i	$0.687319\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	56.0000	1.99113
$792$	0	0
$793$	24.0000	0.852265
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 19.7990i	− 0.701316i	−0.936504	−	0.350658i	$-0.885958\pi$
$797$	− 19.7990i	− 0.701316i	0.936504	−	0.350658i	$-0.114042\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5.65685i	0.199626i
$804$	0	0
$805$	− 45.2548i	− 1.59502i
$806$	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
$811$	− 24.0416i	− 0.844216i	−0.906546	−	0.422108i	$-0.861290\pi$
$811$	− 24.0416i	− 0.844216i	0.906546	−	0.422108i	$-0.138710\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	28.0000	0.980797
$816$	0	0
$817$	−30.0000	−1.04957
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8.48528i	0.296138i	0.988977	+	0.148069i	$0.0473058\pi$
$821$	8.48528i	0.296138i	−0.988977	+	0.148069i	$0.952694\pi$
$822$	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
$827$	− 21.2132i	− 0.737655i	−0.929498	−	0.368828i	$-0.879759\pi$
$827$	− 21.2132i	− 0.737655i	0.929498	−	0.368828i	$-0.120241\pi$
$828$	0	0
$829$	− 25.4558i	− 0.884118i	−0.896986	−	0.442059i	$-0.854248\pi$
$829$	− 25.4558i	− 0.884118i	0.896986	−	0.442059i	$-0.145752\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	36.0000	1.24733
$834$	0	0
$835$	− 33.9411i	− 1.17458i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	20.0000	0.690477	0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000	0.690477	0.345238	+	0.938515i	$0.387798\pi$
$840$	0	0
$841$	−43.0000	−1.48276
$842$	0	0
$843$	0	0
$844$	0	0
$845$	14.1421i	0.486504i
$846$	0	0
$847$	−36.0000	−1.23697
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11.3137i	0.387829i
$852$	0	0
$853$	− 42.4264i	− 1.45265i	−0.687350	−	0.726326i	$-0.741226\pi$
$853$	− 42.4264i	− 1.45265i	0.687350	−	0.726326i	$-0.258774\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−34.0000	−1.16142	−0.580709	−	0.814111i	$-0.697225\pi$
$857$	−34.0000	−1.16142	−0.580709	+	0.814111i	$0.697225\pi$
$858$	0	0
$859$	− 38.1838i	− 1.30281i	−0.758729	−	0.651407i	$-0.774179\pi$
$859$	− 38.1838i	− 1.30281i	0.758729	−	0.651407i	$-0.225821\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	−40.0000	−1.36004
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 11.3137i	− 0.383791i
$870$	0	0
$871$	12.0000	0.406604
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 22.6274i	− 0.764946i
$876$	0	0
$877$	− 8.48528i	− 0.286528i	−0.989685	−	0.143264i	$-0.954240\pi$
$877$	− 8.48528i	− 0.286528i	0.989685	−	0.143264i	$-0.0457597\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	34.0000	1.14549	0.572745	−	0.819734i	$-0.305879\pi$
$881$	34.0000	1.14549	0.572745	+	0.819734i	$0.305879\pi$
$882$	0	0
$883$	− 41.0122i	− 1.38017i	−0.723728	−	0.690085i	$-0.757573\pi$
$883$	− 41.0122i	− 1.38017i	0.723728	−	0.690085i	$-0.242427\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28.0000	0.940148	0.470074	−	0.882627i	$-0.344227\pi$
$887$	28.0000	0.940148	0.470074	+	0.882627i	$0.344227\pi$
$888$	0	0
$889$	−32.0000	−1.07325
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	4.00000	0.133705
$896$	0	0
$897$	0	0
$898$	0	0
$899$	67.8823i	2.26400i
$900$	0	0
$901$	− 11.3137i	− 0.376914i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−24.0000	−0.797787
$906$	0	0
$907$	7.07107i	0.234791i	0.993085	+	0.117395i	$0.0374545\pi$
$907$	7.07107i	0.234791i	−0.993085	+	0.117395i	$0.962545\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	−14.0000	−0.463332
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 28.2843i	− 0.934029i
$918$	0	0
$919$	44.0000	1.45143	0.725713	−	0.687998i	$-0.241510\pi$
$919$	44.0000	1.45143	0.725713	+	0.687998i	$0.241510\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	11.3137i	0.372395i
$924$	0	0
$925$	− 8.48528i	− 0.278994i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−60.0000	−1.96854	−0.984268	−	0.176682i	$-0.943464\pi$
$929$	−60.0000	−1.96854	−0.984268	+	0.176682i	$0.943464\pi$
$930$	0	0
$931$	− 63.6396i	− 2.08570i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−16.0000	−0.523256
$936$	0	0
$937$	20.0000	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 36.7696i	− 1.19865i	−0.800505	−	0.599327i	$-0.795435\pi$
$941$	− 36.7696i	− 1.19865i	0.800505	−	0.599327i	$-0.204565\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 60.8112i	− 1.97610i	−0.154140	−	0.988049i	$-0.549261\pi$
$947$	− 60.8112i	− 1.97610i	0.154140	−	0.988049i	$-0.450739\pi$
$948$	0	0
$949$	− 11.3137i	− 0.367259i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	0	0
$955$	− 22.6274i	− 0.732206i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	72.0000	2.32500
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 11.3137i	− 0.364201i
$966$	0	0
$967$	−20.0000	−0.643157	−0.321578	−	0.946883i	$-0.604213\pi$
$967$	−20.0000	−0.643157	−0.321578	+	0.946883i	$0.604213\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	15.5563i	0.499227i	0.968346	+	0.249614i	$0.0803036\pi$
$971$	15.5563i	0.499227i	−0.968346	+	0.249614i	$0.919696\pi$
$972$	0	0
$973$	16.9706i	0.544051i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	0	0
$979$	16.9706i	0.542382i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	0	0
$985$	−40.0000	−1.27451
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 16.9706i	− 0.539633i
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 11.3137i	− 0.358669i
$996$	0	0
$997$	14.1421i	0.447886i	0.974602	+	0.223943i	$0.0718929\pi$
$997$	14.1421i	0.447886i	−0.974602	+	0.223943i	$0.928107\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.d.a.2305.2		2
3.2	odd	2		512.2.b.a.257.2		2
4.3	odd	2		4608.2.d.b.2305.2		2
8.3	odd	2		4608.2.d.b.2305.1		2
8.5	even	2	inner	4608.2.d.a.2305.1		2
12.11	even	2		512.2.b.b.257.1		2
16.3	odd	4		4608.2.a.f.1.2		2
16.5	even	4		4608.2.a.m.1.1		2
16.11	odd	4		4608.2.a.f.1.1		2
16.13	even	4		4608.2.a.m.1.2		2
24.5	odd	2		512.2.b.a.257.1		2
24.11	even	2		512.2.b.b.257.2		2
48.5	odd	4		512.2.a.d.1.2	yes	2
48.11	even	4		512.2.a.c.1.1	✓	2
48.29	odd	4		512.2.a.d.1.1	yes	2
48.35	even	4		512.2.a.c.1.2	yes	2
96.5	odd	8		1024.2.e.c.257.1		2
96.11	even	8		1024.2.e.b.257.1		2
96.29	odd	8		1024.2.e.c.769.1		2
96.35	even	8		1024.2.e.e.769.1		2
96.53	odd	8		1024.2.e.d.257.1		2
96.59	even	8		1024.2.e.e.257.1		2
96.77	odd	8		1024.2.e.d.769.1		2
96.83	even	8		1024.2.e.b.769.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.a.c.1.1	✓	2	48.11	even	4
512.2.a.c.1.2	yes	2	48.35	even	4
512.2.a.d.1.1	yes	2	48.29	odd	4
512.2.a.d.1.2	yes	2	48.5	odd	4
512.2.b.a.257.1		2	24.5	odd	2
512.2.b.a.257.2		2	3.2	odd	2
512.2.b.b.257.1		2	12.11	even	2
512.2.b.b.257.2		2	24.11	even	2
1024.2.e.b.257.1		2	96.11	even	8
1024.2.e.b.769.1		2	96.83	even	8
1024.2.e.c.257.1		2	96.5	odd	8
1024.2.e.c.769.1		2	96.29	odd	8
1024.2.e.d.257.1		2	96.53	odd	8
1024.2.e.d.769.1		2	96.77	odd	8
1024.2.e.e.257.1		2	96.59	even	8
1024.2.e.e.769.1		2	96.35	even	8
4608.2.a.f.1.1		2	16.11	odd	4
4608.2.a.f.1.2		2	16.3	odd	4
4608.2.a.m.1.1		2	16.5	even	4
4608.2.a.m.1.2		2	16.13	even	4
4608.2.d.a.2305.1		2	8.5	even	2	inner
4608.2.d.a.2305.2		2	1.1	even	1	trivial
4608.2.d.b.2305.1		2	8.3	odd	2
4608.2.d.b.2305.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{5} -4.00000 q^{7} +O(q^{10})\)	\(q+2.82843i q^{5} -4.00000 q^{7} +1.41421i q^{11} -2.82843i q^{13} +4.00000 q^{17} -7.07107i q^{19} +4.00000 q^{23} -3.00000 q^{25} +8.48528i q^{29} +8.00000 q^{31} -11.3137i q^{35} +2.82843i q^{37} +2.00000 q^{41} -4.24264i q^{43} +9.00000 q^{49} -2.82843i q^{53} -4.00000 q^{55} +4.24264i q^{59} +8.48528i q^{61} +8.00000 q^{65} +4.24264i q^{67} -4.00000 q^{71} +4.00000 q^{73} -5.65685i q^{77} -8.00000 q^{79} +9.89949i q^{83} +11.3137i q^{85} +12.0000 q^{89} +11.3137i q^{91} +20.0000 q^{95} -4.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{7}+O(q^{10}) \) 2 * q - 8 * q^7	\( 2 q - 8 q^{7} + 8 q^{17} + 8 q^{23} - 6 q^{25} + 16 q^{31} + 4 q^{41} + 18 q^{49} - 8 q^{55} + 16 q^{65} - 8 q^{71} + 8 q^{73} - 16 q^{79} + 24 q^{89} + 40 q^{95} - 8 q^{97}+O(q^{100}) \) 2 * q - 8 * q^7 + 8 * q^17 + 8 * q^23 - 6 * q^25 + 16 * q^31 + 4 * q^41 + 18 * q^49 - 8 * q^55 + 16 * q^65 - 8 * q^71 + 8 * q^73 - 16 * q^79 + 24 * q^89 + 40 * q^95 - 8 * q^97

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