LMFDB - Embedded newform 585.2.b.f.181.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [585,2,Mod(181,585)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(585, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("585.181");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 585 = 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	585.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.67124851824$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.2
Root			$-2.30278i$ of defining polynomial
Character	$\chi$	$=$	585.181
Dual form			585.2.b.f.181.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -2.00000 q^{4} +1.00000i q^{5} +3.60555i q^{7} +O(q^{10})$	$q-2.00000i q^{2} -2.00000 q^{4} +1.00000i q^{5} +3.60555i q^{7} +2.00000 q^{10} +3.00000i q^{11} +3.60555i q^{13} +7.21110 q^{14} -4.00000 q^{16} -3.60555 q^{17} +7.21110i q^{19} -2.00000i q^{20} +6.00000 q^{22} +3.60555 q^{23} -1.00000 q^{25} +7.21110 q^{26} -7.21110i q^{28} +7.21110 q^{29} -7.21110i q^{31} +8.00000i q^{32} +7.21110i q^{34} -3.60555 q^{35} +3.60555i q^{37} +14.4222 q^{38} -11.0000i q^{41} -4.00000 q^{43} -6.00000i q^{44} -7.21110i q^{46} -4.00000i q^{47} -6.00000 q^{49} +2.00000i q^{50} -7.21110i q^{52} -10.8167 q^{53} -3.00000 q^{55} -14.4222i q^{58} +12.0000i q^{59} +13.0000 q^{61} -14.4222 q^{62} +8.00000 q^{64} -3.60555 q^{65} +7.21110 q^{68} +7.21110i q^{70} +5.00000i q^{71} -7.21110i q^{73} +7.21110 q^{74} -14.4222i q^{76} -10.8167 q^{77} +13.0000 q^{79} -4.00000i q^{80} -22.0000 q^{82} -6.00000i q^{83} -3.60555i q^{85} +8.00000i q^{86} -3.00000i q^{89} -13.0000 q^{91} -7.21110 q^{92} -8.00000 q^{94} -7.21110 q^{95} +3.60555i q^{97} +12.0000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} + 8 q^{10} - 16 q^{16} + 24 q^{22} - 4 q^{25} - 16 q^{43} - 24 q^{49} - 12 q^{55} + 52 q^{61} + 32 q^{64} + 52 q^{79} - 88 q^{82} - 52 q^{91} - 32 q^{94}+O(q^{100}) $ 4 * q - 8 * q^4 + 8 * q^10 - 16 * q^16 + 24 * q^22 - 4 * q^25 - 16 * q^43 - 24 * q^49 - 12 * q^55 + 52 * q^61 + 32 * q^64 + 52 * q^79 - 88 * q^82 - 52 * q^91 - 32 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/585\mathbb{Z}\right)^\times$.

$n$	$326$	$352$	$496$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 2.00000i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	− 2.00000i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	3.60555i	1.36277i	0.731925	+	0.681385i	$0.238622\pi$
$7$	3.60555i	1.36277i	−0.731925	+	0.681385i	$0.761378\pi$
$8$	0	0
$9$	0	0
$10$	2.00000	0.632456
$11$	3.00000i	0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$	3.00000i	0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$	0	0
$13$	3.60555i	1.00000i
$13$	3.60555i	1.00000i
$14$	7.21110	1.92725
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−3.60555	−0.874475	−0.437237	−	0.899346i	$-0.644043\pi$
$17$	−3.60555	−0.874475	−0.437237	+	0.899346i	$0.644043\pi$
$18$	0	0
$19$	7.21110i	1.65434i	0.561951	+	0.827170i	$0.310051\pi$
$19$	7.21110i	1.65434i	−0.561951	+	0.827170i	$0.689949\pi$
$20$	− 2.00000i	− 0.447214i
$21$	0	0
$22$	6.00000	1.27920
$23$	3.60555	0.751809	0.375905	−	0.926658i	$-0.377332\pi$
$23$	3.60555	0.751809	0.375905	+	0.926658i	$0.377332\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	7.21110	1.41421
$27$	0	0
$28$	− 7.21110i	− 1.36277i
$29$	7.21110	1.33907	0.669534	−	0.742781i	$-0.266494\pi$
$29$	7.21110	1.33907	0.669534	+	0.742781i	$0.266494\pi$
$30$	0	0
$31$	− 7.21110i	− 1.29515i	−0.762001	−	0.647576i	$-0.775783\pi$
$31$	− 7.21110i	− 1.29515i	0.762001	−	0.647576i	$-0.224217\pi$
$32$	8.00000i	1.41421i
$33$	0	0
$34$	7.21110i	1.23669i
$35$	−3.60555	−0.609449
$36$	0	0
$37$	3.60555i	0.592749i	0.955072	+	0.296374i	$0.0957776\pi$
$37$	3.60555i	0.592749i	−0.955072	+	0.296374i	$0.904222\pi$
$38$	14.4222	2.33959
$39$	0	0
$40$	0	0
$41$	− 11.0000i	− 1.71791i	−0.512050	−	0.858956i	$-0.671114\pi$
$41$	− 11.0000i	− 1.71791i	0.512050	−	0.858956i	$-0.328886\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	− 6.00000i	− 0.904534i
$45$	0	0
$46$	− 7.21110i	− 1.06322i
$47$	− 4.00000i	− 0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$	− 4.00000i	− 0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	2.00000i	0.282843i
$51$	0	0
$52$	− 7.21110i	− 1.00000i
$53$	−10.8167	−1.48578	−0.742891	−	0.669413i	$-0.766546\pi$
$53$	−10.8167	−1.48578	−0.742891	+	0.669413i	$0.766546\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	0	0
$57$	0	0
$58$	− 14.4222i	− 1.89373i
$59$	12.0000i	1.56227i	0.624364	+	0.781133i	$0.285358\pi$
$59$	12.0000i	1.56227i	−0.624364	+	0.781133i	$0.714642\pi$
$60$	0	0
$61$	13.0000	1.66448	0.832240	−	0.554416i	$-0.187058\pi$
$61$	13.0000	1.66448	0.832240	+	0.554416i	$0.187058\pi$
$62$	−14.4222	−1.83162
$63$	0	0
$64$	8.00000	1.00000
$65$	−3.60555	−0.447214
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	7.21110	0.874475
$69$	0	0
$70$	7.21110i	0.861892i
$71$	5.00000i	0.593391i	0.954972	+	0.296695i	$0.0958846\pi$
$71$	5.00000i	0.593391i	−0.954972	+	0.296695i	$0.904115\pi$
$72$	0	0
$73$	− 7.21110i	− 0.843996i	−0.906597	−	0.421998i	$-0.861329\pi$
$73$	− 7.21110i	− 0.843996i	0.906597	−	0.421998i	$-0.138671\pi$
$74$	7.21110	0.838274
$75$	0	0
$76$	− 14.4222i	− 1.65434i
$77$	−10.8167	−1.23267
$78$	0	0
$79$	13.0000	1.46261	0.731307	−	0.682048i	$-0.238911\pi$
$79$	13.0000	1.46261	0.731307	+	0.682048i	$0.238911\pi$
$80$	− 4.00000i	− 0.447214i
$81$	0	0
$82$	−22.0000	−2.42949
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	− 3.60555i	− 0.391077i
$86$	8.00000i	0.862662i
$87$	0	0
$88$	0	0
$89$	− 3.00000i	− 0.317999i	−0.987279	−	0.159000i	$-0.949173\pi$
$89$	− 3.00000i	− 0.317999i	0.987279	−	0.159000i	$-0.0508269\pi$
$90$	0	0
$91$	−13.0000	−1.36277
$92$	−7.21110	−0.751809
$93$	0	0
$94$	−8.00000	−0.825137
$95$	−7.21110	−0.739844
$96$	0	0
$97$	3.60555i	0.366088i	0.983105	+	0.183044i	$0.0585951\pi$
$97$	3.60555i	0.366088i	−0.983105	+	0.183044i	$0.941405\pi$
$98$	12.0000i	1.21218i
$99$	0	0
$100$	2.00000	0.200000
$101$	−7.21110	−0.717532	−0.358766	−	0.933428i	$-0.616802\pi$
$101$	−7.21110	−0.717532	−0.358766	+	0.933428i	$0.616802\pi$
$102$	0	0
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	0	0
$105$	0	0
$106$	21.6333i	2.10121i
$107$	−3.60555	−0.348562	−0.174281	−	0.984696i	$-0.555760\pi$
$107$	−3.60555	−0.348562	−0.174281	+	0.984696i	$0.555760\pi$
$108$	0	0
$109$	14.4222i	1.38140i	0.723143	+	0.690698i	$0.242697\pi$
$109$	14.4222i	1.38140i	−0.723143	+	0.690698i	$0.757303\pi$
$110$	6.00000i	0.572078i
$111$	0	0
$112$	− 14.4222i	− 1.36277i
$113$	−7.21110	−0.678363	−0.339182	−	0.940721i	$-0.610150\pi$
$113$	−7.21110	−0.678363	−0.339182	+	0.940721i	$0.610150\pi$
$114$	0	0
$115$	3.60555i	0.336219i
$116$	−14.4222	−1.33907
$117$	0	0
$118$	24.0000	2.20938
$119$	− 13.0000i	− 1.19171i
$120$	0	0
$121$	2.00000	0.181818
$122$	− 26.0000i	− 2.35393i
$123$	0	0
$124$	14.4222i	1.29515i
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	7.21110i	0.632456i
$131$	21.6333	1.89011	0.945055	−	0.326910i	$-0.106007\pi$
$131$	21.6333	1.89011	0.945055	+	0.326910i	$0.106007\pi$
$132$	0	0
$133$	−26.0000	−2.25449
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	0	0
$139$	7.00000	0.593732	0.296866	−	0.954919i	$-0.404058\pi$
$139$	7.00000	0.593732	0.296866	+	0.954919i	$0.404058\pi$
$140$	7.21110	0.609449
$141$	0	0
$142$	10.0000	0.839181
$143$	−10.8167	−0.904534
$144$	0	0
$145$	7.21110i	0.598849i
$146$	−14.4222	−1.19359
$147$	0	0
$148$	− 7.21110i	− 0.592749i
$149$	− 5.00000i	− 0.409616i	−0.978802	−	0.204808i	$-0.934343\pi$
$149$	− 5.00000i	− 0.409616i	0.978802	−	0.204808i	$-0.0656570\pi$
$150$	0	0
$151$	7.21110i	0.586831i	0.955985	+	0.293416i	$0.0947920\pi$
$151$	7.21110i	0.586831i	−0.955985	+	0.293416i	$0.905208\pi$
$152$	0	0
$153$	0	0
$154$	21.6333i	1.74326i
$155$	7.21110	0.579210
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	− 26.0000i	− 2.06845i
$159$	0	0
$160$	−8.00000	−0.632456
$161$	13.0000i	1.02454i
$162$	0	0
$163$	− 10.8167i	− 0.847226i	−0.905843	−	0.423613i	$-0.860762\pi$
$163$	− 10.8167i	− 0.847226i	0.905843	−	0.423613i	$-0.139238\pi$
$164$	22.0000i	1.71791i
$165$	0	0
$166$	−12.0000	−0.931381
$167$	10.0000i	0.773823i	0.922117	+	0.386912i	$0.126458\pi$
$167$	10.0000i	0.773823i	−0.922117	+	0.386912i	$0.873542\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	−7.21110	−0.553066
$171$	0	0
$172$	8.00000	0.609994
$173$	−7.21110	−0.548250	−0.274125	−	0.961694i	$-0.588388\pi$
$173$	−7.21110	−0.548250	−0.274125	+	0.961694i	$0.588388\pi$
$174$	0	0
$175$	− 3.60555i	− 0.272554i
$176$	− 12.0000i	− 0.904534i
$177$	0	0
$178$	−6.00000	−0.449719
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	3.00000	0.222988	0.111494	−	0.993765i	$-0.464436\pi$
$181$	3.00000	0.222988	0.111494	+	0.993765i	$0.464436\pi$
$182$	26.0000i	1.92725i
$183$	0	0
$184$	0	0
$185$	−3.60555	−0.265085
$186$	0	0
$187$	− 10.8167i	− 0.790992i
$188$	8.00000i	0.583460i
$189$	0	0
$190$	14.4222i	1.04630i
$191$	−7.21110	−0.521777	−0.260889	−	0.965369i	$-0.584016\pi$
$191$	−7.21110	−0.521777	−0.260889	+	0.965369i	$0.584016\pi$
$192$	0	0
$193$	3.60555i	0.259533i	0.991545	+	0.129767i	$0.0414228\pi$
$193$	3.60555i	0.259533i	−0.991545	+	0.129767i	$0.958577\pi$
$194$	7.21110	0.517727
$195$	0	0
$196$	12.0000	0.857143
$197$	− 2.00000i	− 0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	− 2.00000i	− 0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	14.4222i	1.01474i
$203$	26.0000i	1.82484i
$204$	0	0
$205$	11.0000	0.768273
$206$	− 20.0000i	− 1.39347i
$207$	0	0
$208$	− 14.4222i	− 1.00000i
$209$	−21.6333	−1.49641
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	21.6333	1.48578
$213$	0	0
$214$	7.21110i	0.492941i
$215$	− 4.00000i	− 0.272798i
$216$	0	0
$217$	26.0000	1.76500
$218$	28.8444	1.95359
$219$	0	0
$220$	6.00000	0.404520
$221$	− 13.0000i	− 0.874475i
$222$	0	0
$223$	− 14.4222i	− 0.965782i	−0.875680	−	0.482891i	$-0.839587\pi$
$223$	− 14.4222i	− 0.965782i	0.875680	−	0.482891i	$-0.160413\pi$
$224$	−28.8444	−1.92725
$225$	0	0
$226$	14.4222i	0.959351i
$227$	− 18.0000i	− 1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$	− 18.0000i	− 1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$	0	0
$229$	− 14.4222i	− 0.953046i	−0.879162	−	0.476523i	$-0.841897\pi$
$229$	− 14.4222i	− 0.953046i	0.879162	−	0.476523i	$-0.158103\pi$
$230$	7.21110	0.475486
$231$	0	0
$232$	0	0
$233$	−25.2389	−1.65345	−0.826726	−	0.562604i	$-0.809799\pi$
$233$	−25.2389	−1.65345	−0.826726	+	0.562604i	$0.809799\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	− 24.0000i	− 1.56227i
$237$	0	0
$238$	−26.0000	−1.68533
$239$	− 7.00000i	− 0.452792i	−0.974035	−	0.226396i	$-0.927306\pi$
$239$	− 7.00000i	− 0.452792i	0.974035	−	0.226396i	$-0.0726944\pi$
$240$	0	0
$241$	− 21.6333i	− 1.39352i	−0.717302	−	0.696762i	$-0.754623\pi$
$241$	− 21.6333i	− 1.39352i	0.717302	−	0.696762i	$-0.245377\pi$
$242$	− 4.00000i	− 0.257130i
$243$	0	0
$244$	−26.0000	−1.66448
$245$	− 6.00000i	− 0.383326i
$246$	0	0
$247$	−26.0000	−1.65434
$248$	0	0
$249$	0	0
$250$	−2.00000	−0.126491
$251$	21.6333	1.36548	0.682741	−	0.730660i	$-0.260788\pi$
$251$	21.6333	1.36548	0.682741	+	0.730660i	$0.260788\pi$
$252$	0	0
$253$	10.8167i	0.680037i
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	21.6333	1.34945	0.674724	−	0.738070i	$-0.264263\pi$
$257$	21.6333	1.34945	0.674724	+	0.738070i	$0.264263\pi$
$258$	0	0
$259$	−13.0000	−0.807781
$260$	7.21110	0.447214
$261$	0	0
$262$	− 43.2666i	− 2.67302i
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	− 10.8167i	− 0.664462i
$266$	52.0000i	3.18832i
$267$	0	0
$268$	0	0
$269$	−7.21110	−0.439669	−0.219834	−	0.975537i	$-0.570552\pi$
$269$	−7.21110	−0.439669	−0.219834	+	0.975537i	$0.570552\pi$
$270$	0	0
$271$	− 28.8444i	− 1.75217i	−0.482154	−	0.876087i	$-0.660145\pi$
$271$	− 28.8444i	− 1.75217i	0.482154	−	0.876087i	$-0.339855\pi$
$272$	14.4222	0.874475
$273$	0	0
$274$	24.0000	1.44989
$275$	− 3.00000i	− 0.180907i
$276$	0	0
$277$	4.00000	0.240337	0.120168	−	0.992754i	$-0.461657\pi$
$277$	4.00000	0.240337	0.120168	+	0.992754i	$0.461657\pi$
$278$	− 14.0000i	− 0.839664i
$279$	0	0
$280$	0	0
$281$	30.0000i	1.78965i	0.446417	+	0.894825i	$0.352700\pi$
$281$	30.0000i	1.78965i	−0.446417	+	0.894825i	$0.647300\pi$
$282$	0	0
$283$	26.0000	1.54554	0.772770	−	0.634686i	$-0.218871\pi$
$283$	26.0000	1.54554	0.772770	+	0.634686i	$0.218871\pi$
$284$	− 10.0000i	− 0.593391i
$285$	0	0
$286$	21.6333i	1.27920i
$287$	39.6611	2.34112
$288$	0	0
$289$	−4.00000	−0.235294
$290$	14.4222	0.846901
$291$	0	0
$292$	14.4222i	0.843996i
$293$	− 12.0000i	− 0.701047i	−0.936554	−	0.350524i	$-0.886004\pi$
$293$	− 12.0000i	− 0.701047i	0.936554	−	0.350524i	$-0.113996\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	0	0
$297$	0	0
$298$	−10.0000	−0.579284
$299$	13.0000i	0.751809i
$300$	0	0
$301$	− 14.4222i	− 0.831282i
$302$	14.4222	0.829905
$303$	0	0
$304$	− 28.8444i	− 1.65434i
$305$	13.0000i	0.744378i
$306$	0	0
$307$	18.0278i	1.02890i	0.857521	+	0.514449i	$0.172004\pi$
$307$	18.0278i	1.02890i	−0.857521	+	0.514449i	$0.827996\pi$
$308$	21.6333	1.23267
$309$	0	0
$310$	− 14.4222i	− 0.819126i
$311$	14.4222	0.817808	0.408904	−	0.912577i	$-0.365911\pi$
$311$	14.4222	0.817808	0.408904	+	0.912577i	$0.365911\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	− 4.00000i	− 0.225733i
$315$	0	0
$316$	−26.0000	−1.46261
$317$	− 32.0000i	− 1.79730i	−0.438667	−	0.898650i	$-0.644549\pi$
$317$	− 32.0000i	− 1.79730i	0.438667	−	0.898650i	$-0.355451\pi$
$318$	0	0
$319$	21.6333i	1.21123i
$320$	8.00000i	0.447214i
$321$	0	0
$322$	26.0000	1.44892
$323$	− 26.0000i	− 1.44668i
$324$	0	0
$325$	− 3.60555i	− 0.200000i
$326$	−21.6333	−1.19816
$327$	0	0
$328$	0	0
$329$	14.4222	0.795122
$330$	0	0
$331$	− 28.8444i	− 1.58543i	−0.609591	−	0.792716i	$-0.708666\pi$
$331$	− 28.8444i	− 1.58543i	0.609591	−	0.792716i	$-0.291334\pi$
$332$	12.0000i	0.658586i
$333$	0	0
$334$	20.0000	1.09435
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	26.0000i	1.41421i
$339$	0	0
$340$	7.21110i	0.391077i
$341$	21.6333	1.17151
$342$	0	0
$343$	3.60555i	0.194681i
$344$	0	0
$345$	0	0
$346$	14.4222i	0.775343i
$347$	25.2389	1.35489	0.677446	−	0.735572i	$-0.263087\pi$
$347$	25.2389	1.35489	0.677446	+	0.735572i	$0.263087\pi$
$348$	0	0
$349$	− 7.21110i	− 0.386001i	−0.981199	−	0.193001i	$-0.938178\pi$
$349$	− 7.21110i	− 0.386001i	0.981199	−	0.193001i	$-0.0618220\pi$
$350$	−7.21110	−0.385450
$351$	0	0
$352$	−24.0000	−1.27920
$353$	24.0000i	1.27739i	0.769460	+	0.638696i	$0.220526\pi$
$353$	24.0000i	1.27739i	−0.769460	+	0.638696i	$0.779474\pi$
$354$	0	0
$355$	−5.00000	−0.265372
$356$	6.00000i	0.317999i
$357$	0	0
$358$	0	0
$359$	4.00000i	0.211112i	0.994413	+	0.105556i	$0.0336622\pi$
$359$	4.00000i	0.211112i	−0.994413	+	0.105556i	$0.966338\pi$
$360$	0	0
$361$	−33.0000	−1.73684
$362$	− 6.00000i	− 0.315353i
$363$	0	0
$364$	26.0000	1.36277
$365$	7.21110	0.377446
$366$	0	0
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	−14.4222	−0.751809
$369$	0	0
$370$	7.21110i	0.374887i
$371$	− 39.0000i	− 2.02478i
$372$	0	0
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	−21.6333	−1.11863
$375$	0	0
$376$	0	0
$377$	26.0000i	1.33907i
$378$	0	0
$379$	21.6333i	1.11123i	0.831440	+	0.555614i	$0.187517\pi$
$379$	21.6333i	1.11123i	−0.831440	+	0.555614i	$0.812483\pi$
$380$	14.4222	0.739844
$381$	0	0
$382$	14.4222i	0.737904i
$383$	− 8.00000i	− 0.408781i	−0.978889	−	0.204390i	$-0.934479\pi$
$383$	− 8.00000i	− 0.408781i	0.978889	−	0.204390i	$-0.0655212\pi$
$384$	0	0
$385$	− 10.8167i	− 0.551268i
$386$	7.21110	0.367035
$387$	0	0
$388$	− 7.21110i	− 0.366088i
$389$	−28.8444	−1.46247	−0.731235	−	0.682126i	$-0.761056\pi$
$389$	−28.8444	−1.46247	−0.731235	+	0.682126i	$0.761056\pi$
$390$	0	0
$391$	−13.0000	−0.657438
$392$	0	0
$393$	0	0
$394$	−4.00000	−0.201517
$395$	13.0000i	0.654101i
$396$	0	0
$397$	18.0278i	0.904787i	0.891818	+	0.452394i	$0.149430\pi$
$397$	18.0278i	0.904787i	−0.891818	+	0.452394i	$0.850570\pi$
$398$	0	0
$399$	0	0
$400$	4.00000	0.200000
$401$	− 10.0000i	− 0.499376i	−0.968326	−	0.249688i	$-0.919672\pi$
$401$	− 10.0000i	− 0.499376i	0.968326	−	0.249688i	$-0.0803281\pi$
$402$	0	0
$403$	26.0000	1.29515
$404$	14.4222	0.717532
$405$	0	0
$406$	52.0000	2.58072
$407$	−10.8167	−0.536162
$408$	0	0
$409$	− 21.6333i	− 1.06970i	−0.844948	−	0.534849i	$-0.820368\pi$
$409$	− 21.6333i	− 1.06970i	0.844948	−	0.534849i	$-0.179632\pi$
$410$	− 22.0000i	− 1.08650i
$411$	0	0
$412$	−20.0000	−0.985329
$413$	−43.2666	−2.12901
$414$	0	0
$415$	6.00000	0.294528
$416$	−28.8444	−1.41421
$417$	0	0
$418$	43.2666i	2.11624i
$419$	21.6333	1.05686	0.528428	−	0.848978i	$-0.322782\pi$
$419$	21.6333	1.05686	0.528428	+	0.848978i	$0.322782\pi$
$420$	0	0
$421$	7.21110i	0.351448i	0.984440	+	0.175724i	$0.0562266\pi$
$421$	7.21110i	0.351448i	−0.984440	+	0.175724i	$0.943773\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.60555	0.174895
$426$	0	0
$427$	46.8722i	2.26830i
$428$	7.21110	0.348562
$429$	0	0
$430$	−8.00000	−0.385794
$431$	− 24.0000i	− 1.15604i	−0.816023	−	0.578020i	$-0.803826\pi$
$431$	− 24.0000i	− 1.15604i	0.816023	−	0.578020i	$-0.196174\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	− 52.0000i	− 2.49608i
$435$	0	0
$436$	− 28.8444i	− 1.38140i
$437$	26.0000i	1.24375i
$438$	0	0
$439$	25.0000	1.19318	0.596592	−	0.802544i	$-0.296521\pi$
$439$	25.0000	1.19318	0.596592	+	0.802544i	$0.296521\pi$
$440$	0	0
$441$	0	0
$442$	−26.0000	−1.23669
$443$	10.8167	0.513915	0.256957	−	0.966423i	$-0.417280\pi$
$443$	10.8167	0.513915	0.256957	+	0.966423i	$0.417280\pi$
$444$	0	0
$445$	3.00000	0.142214
$446$	−28.8444	−1.36582
$447$	0	0
$448$	28.8444i	1.36277i
$449$	27.0000i	1.27421i	0.770778	+	0.637104i	$0.219868\pi$
$449$	27.0000i	1.27421i	−0.770778	+	0.637104i	$0.780132\pi$
$450$	0	0
$451$	33.0000	1.55391
$452$	14.4222	0.678363
$453$	0	0
$454$	−36.0000	−1.68956
$455$	− 13.0000i	− 0.609449i
$456$	0	0
$457$	18.0278i	0.843303i	0.906758	+	0.421651i	$0.138549\pi$
$457$	18.0278i	0.843303i	−0.906758	+	0.421651i	$0.861451\pi$
$458$	−28.8444	−1.34781
$459$	0	0
$460$	− 7.21110i	− 0.336219i
$461$	5.00000i	0.232873i	0.993198	+	0.116437i	$0.0371472\pi$
$461$	5.00000i	0.232873i	−0.993198	+	0.116437i	$0.962853\pi$
$462$	0	0
$463$	18.0278i	0.837821i	0.908028	+	0.418910i	$0.137588\pi$
$463$	18.0278i	0.837821i	−0.908028	+	0.418910i	$0.862412\pi$
$464$	−28.8444	−1.33907
$465$	0	0
$466$	50.4777i	2.33834i
$467$	32.4500	1.50161	0.750803	−	0.660527i	$-0.229667\pi$
$467$	32.4500	1.50161	0.750803	+	0.660527i	$0.229667\pi$
$468$	0	0
$469$	0	0
$470$	− 8.00000i	− 0.369012i
$471$	0	0
$472$	0	0
$473$	− 12.0000i	− 0.551761i
$474$	0	0
$475$	− 7.21110i	− 0.330868i
$476$	26.0000i	1.19171i
$477$	0	0
$478$	−14.0000	−0.640345
$479$	− 3.00000i	− 0.137073i	−0.997649	−	0.0685367i	$-0.978167\pi$
$479$	− 3.00000i	− 0.137073i	0.997649	−	0.0685367i	$-0.0218330\pi$
$480$	0	0
$481$	−13.0000	−0.592749
$482$	−43.2666	−1.97074
$483$	0	0
$484$	−4.00000	−0.181818
$485$	−3.60555	−0.163720
$486$	0	0
$487$	− 10.8167i	− 0.490149i	−0.969504	−	0.245075i	$-0.921188\pi$
$487$	− 10.8167i	− 0.490149i	0.969504	−	0.245075i	$-0.0788125\pi$
$488$	0	0
$489$	0	0
$490$	−12.0000	−0.542105
$491$	7.21110	0.325433	0.162716	−	0.986673i	$-0.447974\pi$
$491$	7.21110	0.325433	0.162716	+	0.986673i	$0.447974\pi$
$492$	0	0
$493$	−26.0000	−1.17098
$494$	52.0000i	2.33959i
$495$	0	0
$496$	28.8444i	1.29515i
$497$	−18.0278	−0.808655
$498$	0	0
$499$	28.8444i	1.29125i	0.763653	+	0.645627i	$0.223404\pi$
$499$	28.8444i	1.29125i	−0.763653	+	0.645627i	$0.776596\pi$
$500$	2.00000i	0.0894427i
$501$	0	0
$502$	− 43.2666i	− 1.93108i
$503$	−14.4222	−0.643054	−0.321527	−	0.946900i	$-0.604196\pi$
$503$	−14.4222	−0.643054	−0.321527	+	0.946900i	$0.604196\pi$
$504$	0	0
$505$	− 7.21110i	− 0.320890i
$506$	21.6333	0.961718
$507$	0	0
$508$	0	0
$509$	15.0000i	0.664863i	0.943127	+	0.332432i	$0.107869\pi$
$509$	15.0000i	0.664863i	−0.943127	+	0.332432i	$0.892131\pi$
$510$	0	0
$511$	26.0000	1.15017
$512$	− 32.0000i	− 1.41421i
$513$	0	0
$514$	− 43.2666i	− 1.90841i
$515$	10.0000i	0.440653i
$516$	0	0
$517$	12.0000	0.527759
$518$	26.0000i	1.14237i
$519$	0	0
$520$	0	0
$521$	−28.8444	−1.26370	−0.631848	−	0.775092i	$-0.717703\pi$
$521$	−28.8444	−1.26370	−0.631848	+	0.775092i	$0.717703\pi$
$522$	0	0
$523$	−34.0000	−1.48672	−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000	−1.48672	−0.743358	+	0.668894i	$0.766768\pi$
$524$	−43.2666	−1.89011
$525$	0	0
$526$	0	0
$527$	26.0000i	1.13258i
$528$	0	0
$529$	−10.0000	−0.434783
$530$	−21.6333	−0.939691
$531$	0	0
$532$	52.0000	2.25449
$533$	39.6611	1.71791
$534$	0	0
$535$	− 3.60555i	− 0.155882i
$536$	0	0
$537$	0	0
$538$	14.4222i	0.621785i
$539$	− 18.0000i	− 0.775315i
$540$	0	0
$541$	− 7.21110i	− 0.310030i	−0.987912	−	0.155015i	$-0.950457\pi$
$541$	− 7.21110i	− 0.310030i	0.987912	−	0.155015i	$-0.0495425\pi$
$542$	−57.6888	−2.47795
$543$	0	0
$544$	− 28.8444i	− 1.23669i
$545$	−14.4222	−0.617779
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	− 24.0000i	− 1.02523i
$549$	0	0
$550$	−6.00000	−0.255841
$551$	52.0000i	2.21527i
$552$	0	0
$553$	46.8722i	1.99321i
$554$	− 8.00000i	− 0.339887i
$555$	0	0
$556$	−14.0000	−0.593732
$557$	− 42.0000i	− 1.77960i	−0.456354	−	0.889799i	$-0.650845\pi$
$557$	− 42.0000i	− 1.77960i	0.456354	−	0.889799i	$-0.349155\pi$
$558$	0	0
$559$	− 14.4222i	− 0.609994i
$560$	14.4222	0.609449
$561$	0	0
$562$	60.0000	2.53095
$563$	32.4500	1.36760	0.683801	−	0.729668i	$-0.260325\pi$
$563$	32.4500	1.36760	0.683801	+	0.729668i	$0.260325\pi$
$564$	0	0
$565$	− 7.21110i	− 0.303373i
$566$	− 52.0000i	− 2.18572i
$567$	0	0
$568$	0	0
$569$	43.2666	1.81383	0.906915	−	0.421313i	$-0.138431\pi$
$569$	43.2666	1.81383	0.906915	+	0.421313i	$0.138431\pi$
$570$	0	0
$571$	−13.0000	−0.544033	−0.272017	−	0.962293i	$-0.587691\pi$
$571$	−13.0000	−0.544033	−0.272017	+	0.962293i	$0.587691\pi$
$572$	21.6333	0.904534
$573$	0	0
$574$	− 79.3221i	− 3.31084i
$575$	−3.60555	−0.150362
$576$	0	0
$577$	3.60555i	0.150101i	0.997180	+	0.0750505i	$0.0239118\pi$
$577$	3.60555i	0.150101i	−0.997180	+	0.0750505i	$0.976088\pi$
$578$	8.00000i	0.332756i
$579$	0	0
$580$	− 14.4222i	− 0.598849i
$581$	21.6333	0.897501
$582$	0	0
$583$	− 32.4500i	− 1.34394i
$584$	0	0
$585$	0	0
$586$	−24.0000	−0.991431
$587$	28.0000i	1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$	28.0000i	1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$	0	0
$589$	52.0000	2.14262
$590$	24.0000i	0.988064i
$591$	0	0
$592$	− 14.4222i	− 0.592749i
$593$	− 4.00000i	− 0.164260i	−0.996622	−	0.0821302i	$-0.973828\pi$
$593$	− 4.00000i	− 0.164260i	0.996622	−	0.0821302i	$-0.0261723\pi$
$594$	0	0
$595$	13.0000	0.532948
$596$	10.0000i	0.409616i
$597$	0	0
$598$	26.0000	1.06322
$599$	−7.21110	−0.294638	−0.147319	−	0.989089i	$-0.547064\pi$
$599$	−7.21110	−0.294638	−0.147319	+	0.989089i	$0.547064\pi$
$600$	0	0
$601$	−13.0000	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$602$	−28.8444	−1.17561
$603$	0	0
$604$	− 14.4222i	− 0.586831i
$605$	2.00000i	0.0813116i
$606$	0	0
$607$	−20.0000	−0.811775	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000	−0.811775	−0.405887	+	0.913923i	$0.633038\pi$
$608$	−57.6888	−2.33959
$609$	0	0
$610$	26.0000	1.05271
$611$	14.4222	0.583460
$612$	0	0
$613$	3.60555i	0.145627i	0.997346	+	0.0728134i	$0.0231978\pi$
$613$	3.60555i	0.145627i	−0.997346	+	0.0728134i	$0.976802\pi$
$614$	36.0555	1.45508
$615$	0	0
$616$	0	0
$617$	− 34.0000i	− 1.36879i	−0.729112	−	0.684394i	$-0.760067\pi$
$617$	− 34.0000i	− 1.36879i	0.729112	−	0.684394i	$-0.239933\pi$
$618$	0	0
$619$	− 14.4222i	− 0.579677i	−0.957076	−	0.289839i	$-0.906398\pi$
$619$	− 14.4222i	− 0.579677i	0.957076	−	0.289839i	$-0.0936017\pi$
$620$	−14.4222	−0.579210
$621$	0	0
$622$	− 28.8444i	− 1.15656i
$623$	10.8167	0.433360
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 52.0000i	− 2.07834i
$627$	0	0
$628$	−4.00000	−0.159617
$629$	− 13.0000i	− 0.518344i
$630$	0	0
$631$	28.8444i	1.14828i	0.818758	+	0.574139i	$0.194663\pi$
$631$	28.8444i	1.14828i	−0.818758	+	0.574139i	$0.805337\pi$
$632$	0	0
$633$	0	0
$634$	−64.0000	−2.54176
$635$	0	0
$636$	0	0
$637$	− 21.6333i	− 0.857143i
$638$	43.2666	1.71294
$639$	0	0
$640$	0	0
$641$	−43.2666	−1.70893	−0.854464	−	0.519510i	$-0.826114\pi$
$641$	−43.2666	−1.70893	−0.854464	+	0.519510i	$0.826114\pi$
$642$	0	0
$643$	25.2389i	0.995323i	0.867371	+	0.497662i	$0.165808\pi$
$643$	25.2389i	0.995323i	−0.867371	+	0.497662i	$0.834192\pi$
$644$	− 26.0000i	− 1.02454i
$645$	0	0
$646$	−52.0000	−2.04591
$647$	−18.0278	−0.708744	−0.354372	−	0.935104i	$-0.615305\pi$
$647$	−18.0278	−0.708744	−0.354372	+	0.935104i	$0.615305\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	−7.21110	−0.282843
$651$	0	0
$652$	21.6333i	0.847226i
$653$	−7.21110	−0.282192	−0.141096	−	0.989996i	$-0.545063\pi$
$653$	−7.21110	−0.282192	−0.141096	+	0.989996i	$0.545063\pi$
$654$	0	0
$655$	21.6333i	0.845283i
$656$	44.0000i	1.71791i
$657$	0	0
$658$	− 28.8444i	− 1.12447i
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	7.21110i	0.280479i	0.990118	+	0.140240i	$0.0447873\pi$
$661$	7.21110i	0.280479i	−0.990118	+	0.140240i	$0.955213\pi$
$662$	−57.6888	−2.24214
$663$	0	0
$664$	0	0
$665$	− 26.0000i	− 1.00824i
$666$	0	0
$667$	26.0000	1.00672
$668$	− 20.0000i	− 0.773823i
$669$	0	0
$670$	0	0
$671$	39.0000i	1.50558i
$672$	0	0
$673$	40.0000	1.54189	0.770943	−	0.636904i	$-0.219785\pi$
$673$	40.0000	1.54189	0.770943	+	0.636904i	$0.219785\pi$
$674$	0	0
$675$	0	0
$676$	26.0000	1.00000
$677$	−10.8167	−0.415718	−0.207859	−	0.978159i	$-0.566649\pi$
$677$	−10.8167	−0.415718	−0.207859	+	0.978159i	$0.566649\pi$
$678$	0	0
$679$	−13.0000	−0.498894
$680$	0	0
$681$	0	0
$682$	− 43.2666i	− 1.65676i
$683$	38.0000i	1.45403i	0.686622	+	0.727015i	$0.259093\pi$
$683$	38.0000i	1.45403i	−0.686622	+	0.727015i	$0.740907\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	7.21110	0.275321
$687$	0	0
$688$	16.0000	0.609994
$689$	− 39.0000i	− 1.48578i
$690$	0	0
$691$	− 14.4222i	− 0.548647i	−0.961638	−	0.274323i	$-0.911546\pi$
$691$	− 14.4222i	− 0.548647i	0.961638	−	0.274323i	$-0.0884538\pi$
$692$	14.4222	0.548250
$693$	0	0
$694$	− 50.4777i	− 1.91611i
$695$	7.00000i	0.265525i
$696$	0	0
$697$	39.6611i	1.50227i
$698$	−14.4222	−0.545889
$699$	0	0
$700$	7.21110i	0.272554i
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	−26.0000	−0.980609
$704$	24.0000i	0.904534i
$705$	0	0
$706$	48.0000	1.80650
$707$	− 26.0000i	− 0.977831i
$708$	0	0
$709$	− 28.8444i	− 1.08327i	−0.840612	−	0.541637i	$-0.817805\pi$
$709$	− 28.8444i	− 1.08327i	0.840612	−	0.541637i	$-0.182195\pi$
$710$	10.0000i	0.375293i
$711$	0	0
$712$	0	0
$713$	− 26.0000i	− 0.973708i
$714$	0	0
$715$	− 10.8167i	− 0.404520i
$716$	0	0
$717$	0	0
$718$	8.00000	0.298557
$719$	21.6333	0.806786	0.403393	−	0.915027i	$-0.367831\pi$
$719$	21.6333	0.806786	0.403393	+	0.915027i	$0.367831\pi$
$720$	0	0
$721$	36.0555i	1.34278i
$722$	66.0000i	2.45627i
$723$	0	0
$724$	−6.00000	−0.222988
$725$	−7.21110	−0.267814
$726$	0	0
$727$	26.0000	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$728$	0	0
$729$	0	0
$730$	− 14.4222i	− 0.533790i
$731$	14.4222	0.533425
$732$	0	0
$733$	− 32.4500i	− 1.19857i	−0.800537	−	0.599283i	$-0.795452\pi$
$733$	− 32.4500i	− 1.19857i	0.800537	−	0.599283i	$-0.204548\pi$
$734$	36.0000i	1.32878i
$735$	0	0
$736$	28.8444i	1.06322i
$737$	0	0
$738$	0	0
$739$	21.6333i	0.795794i	0.917430	+	0.397897i	$0.130260\pi$
$739$	21.6333i	0.795794i	−0.917430	+	0.397897i	$0.869740\pi$
$740$	7.21110	0.265085
$741$	0	0
$742$	−78.0000	−2.86347
$743$	− 2.00000i	− 0.0733729i	−0.999327	−	0.0366864i	$-0.988320\pi$
$743$	− 2.00000i	− 0.0733729i	0.999327	−	0.0366864i	$-0.0116803\pi$
$744$	0	0
$745$	5.00000	0.183186
$746$	52.0000i	1.90386i
$747$	0	0
$748$	21.6333i	0.790992i
$749$	− 13.0000i	− 0.475010i
$750$	0	0
$751$	25.0000	0.912263	0.456131	−	0.889912i	$-0.349235\pi$
$751$	25.0000	0.912263	0.456131	+	0.889912i	$0.349235\pi$
$752$	16.0000i	0.583460i
$753$	0	0
$754$	52.0000	1.89373
$755$	−7.21110	−0.262439
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	43.2666	1.57151
$759$	0	0
$760$	0	0
$761$	− 38.0000i	− 1.37750i	−0.724999	−	0.688749i	$-0.758160\pi$
$761$	− 38.0000i	− 1.37750i	0.724999	−	0.688749i	$-0.241840\pi$
$762$	0	0
$763$	−52.0000	−1.88253
$764$	14.4222	0.521777
$765$	0	0
$766$	−16.0000	−0.578103
$767$	−43.2666	−1.56227
$768$	0	0
$769$	− 7.21110i	− 0.260039i	−0.991511	−	0.130020i	$-0.958496\pi$
$769$	− 7.21110i	− 0.260039i	0.991511	−	0.130020i	$-0.0415040\pi$
$770$	−21.6333	−0.779610
$771$	0	0
$772$	− 7.21110i	− 0.259533i
$773$	18.0000i	0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$	18.0000i	0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$	0	0
$775$	7.21110i	0.259030i
$776$	0	0
$777$	0	0
$778$	57.6888i	2.06824i
$779$	79.3221	2.84201
$780$	0	0
$781$	−15.0000	−0.536742
$782$	26.0000i	0.929758i
$783$	0	0
$784$	24.0000	0.857143
$785$	2.00000i	0.0713831i
$786$	0	0
$787$	− 28.8444i	− 1.02819i	−0.857733	−	0.514096i	$-0.828127\pi$
$787$	− 28.8444i	− 1.02819i	0.857733	−	0.514096i	$-0.171873\pi$
$788$	4.00000i	0.142494i
$789$	0	0
$790$	26.0000	0.925038
$791$	− 26.0000i	− 0.924454i
$792$	0	0
$793$	46.8722i	1.66448i
$794$	36.0555	1.27956
$795$	0	0
$796$	0	0
$797$	−39.6611	−1.40487	−0.702433	−	0.711749i	$-0.747903\pi$
$797$	−39.6611	−1.40487	−0.702433	+	0.711749i	$0.747903\pi$
$798$	0	0
$799$	14.4222i	0.510221i
$800$	− 8.00000i	− 0.282843i
$801$	0	0
$802$	−20.0000	−0.706225
$803$	21.6333	0.763423
$804$	0	0
$805$	−13.0000	−0.458190
$806$	− 52.0000i	− 1.83162i
$807$	0	0
$808$	0	0
$809$	−21.6333	−0.760587	−0.380293	−	0.924866i	$-0.624177\pi$
$809$	−21.6333	−0.760587	−0.380293	+	0.924866i	$0.624177\pi$
$810$	0	0
$811$	− 28.8444i	− 1.01286i	−0.862280	−	0.506432i	$-0.830964\pi$
$811$	− 28.8444i	− 1.01286i	0.862280	−	0.506432i	$-0.169036\pi$
$812$	− 52.0000i	− 1.82484i
$813$	0	0
$814$	21.6333i	0.758247i
$815$	10.8167	0.378891
$816$	0	0
$817$	− 28.8444i	− 1.00914i
$818$	−43.2666	−1.51278
$819$	0	0
$820$	−22.0000	−0.768273
$821$	− 15.0000i	− 0.523504i	−0.965135	−	0.261752i	$-0.915700\pi$
$821$	− 15.0000i	− 0.523504i	0.965135	−	0.261752i	$-0.0843002\pi$
$822$	0	0
$823$	30.0000	1.04573	0.522867	−	0.852414i	$-0.324862\pi$
$823$	30.0000	1.04573	0.522867	+	0.852414i	$0.324862\pi$
$824$	0	0
$825$	0	0
$826$	86.5332i	3.01088i
$827$	− 4.00000i	− 0.139094i	−0.997579	−	0.0695468i	$-0.977845\pi$
$827$	− 4.00000i	− 0.139094i	0.997579	−	0.0695468i	$-0.0221553\pi$
$828$	0	0
$829$	−26.0000	−0.903017	−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000	−0.903017	−0.451509	+	0.892267i	$0.649114\pi$
$830$	− 12.0000i	− 0.416526i
$831$	0	0
$832$	28.8444i	1.00000i
$833$	21.6333	0.749550
$834$	0	0
$835$	−10.0000	−0.346064
$836$	43.2666	1.49641
$837$	0	0
$838$	− 43.2666i	− 1.49462i
$839$	51.0000i	1.76072i	0.474310	+	0.880358i	$0.342698\pi$
$839$	51.0000i	1.76072i	−0.474310	+	0.880358i	$0.657302\pi$
$840$	0	0
$841$	23.0000	0.793103
$842$	14.4222	0.497022
$843$	0	0
$844$	0	0
$845$	− 13.0000i	− 0.447214i
$846$	0	0
$847$	7.21110i	0.247776i
$848$	43.2666	1.48578
$849$	0	0
$850$	− 7.21110i	− 0.247339i
$851$	13.0000i	0.445634i
$852$	0	0
$853$	54.0833i	1.85178i	0.377798	+	0.925888i	$0.376681\pi$
$853$	54.0833i	1.85178i	−0.377798	+	0.925888i	$0.623319\pi$
$854$	93.7443	3.20787
$855$	0	0
$856$	0	0
$857$	18.0278	0.615816	0.307908	−	0.951416i	$-0.400371\pi$
$857$	18.0278	0.615816	0.307908	+	0.951416i	$0.400371\pi$
$858$	0	0
$859$	−13.0000	−0.443554	−0.221777	−	0.975097i	$-0.571186\pi$
$859$	−13.0000	−0.443554	−0.221777	+	0.975097i	$0.571186\pi$
$860$	8.00000i	0.272798i
$861$	0	0
$862$	−48.0000	−1.63489
$863$	6.00000i	0.204242i	0.994772	+	0.102121i	$0.0325630\pi$
$863$	6.00000i	0.204242i	−0.994772	+	0.102121i	$0.967437\pi$
$864$	0	0
$865$	− 7.21110i	− 0.245185i
$866$	− 52.0000i	− 1.76703i
$867$	0	0
$868$	−52.0000	−1.76500
$869$	39.0000i	1.32298i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	52.0000	1.75893
$875$	3.60555	0.121890
$876$	0	0
$877$	50.4777i	1.70451i	0.523125	+	0.852256i	$0.324766\pi$
$877$	50.4777i	1.70451i	−0.523125	+	0.852256i	$0.675234\pi$
$878$	− 50.0000i	− 1.68742i
$879$	0	0
$880$	12.0000	0.404520
$881$	43.2666	1.45769	0.728845	−	0.684679i	$-0.240058\pi$
$881$	43.2666	1.45769	0.728845	+	0.684679i	$0.240058\pi$
$882$	0	0
$883$	−26.0000	−0.874970	−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000	−0.874970	−0.437485	+	0.899226i	$0.644131\pi$
$884$	26.0000i	0.874475i
$885$	0	0
$886$	− 21.6333i	− 0.726785i
$887$	−25.2389	−0.847438	−0.423719	−	0.905794i	$-0.639276\pi$
$887$	−25.2389	−0.847438	−0.423719	+	0.905794i	$0.639276\pi$
$888$	0	0
$889$	0	0
$890$	− 6.00000i	− 0.201120i
$891$	0	0
$892$	28.8444i	0.965782i
$893$	28.8444	0.965241
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	54.0000	1.80200
$899$	− 52.0000i	− 1.73430i
$900$	0	0
$901$	39.0000	1.29928
$902$	− 66.0000i	− 2.19756i
$903$	0	0
$904$	0	0
$905$	3.00000i	0.0997234i
$906$	0	0
$907$	52.0000	1.72663	0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000	1.72663	0.863316	+	0.504664i	$0.168384\pi$
$908$	36.0000i	1.19470i
$909$	0	0
$910$	−26.0000	−0.861892
$911$	−36.0555	−1.19457	−0.597286	−	0.802028i	$-0.703754\pi$
$911$	−36.0555	−1.19457	−0.597286	+	0.802028i	$0.703754\pi$
$912$	0	0
$913$	18.0000	0.595713
$914$	36.0555	1.19261
$915$	0	0
$916$	28.8444i	0.953046i
$917$	78.0000i	2.57579i
$918$	0	0
$919$	13.0000	0.428830	0.214415	−	0.976743i	$-0.431215\pi$
$919$	13.0000	0.428830	0.214415	+	0.976743i	$0.431215\pi$
$920$	0	0
$921$	0	0
$922$	10.0000	0.329332
$923$	−18.0278	−0.593391
$924$	0	0
$925$	− 3.60555i	− 0.118550i
$926$	36.0555	1.18486
$927$	0	0
$928$	57.6888i	1.89373i
$929$	− 5.00000i	− 0.164045i	−0.996630	−	0.0820223i	$-0.973862\pi$
$929$	− 5.00000i	− 0.164045i	0.996630	−	0.0820223i	$-0.0261379\pi$
$930$	0	0
$931$	− 43.2666i	− 1.41801i
$932$	50.4777	1.65345
$933$	0	0
$934$	− 64.8999i	− 2.12359i
$935$	10.8167	0.353742
$936$	0	0
$937$	−52.0000	−1.69877	−0.849383	−	0.527777i	$-0.823026\pi$
$937$	−52.0000	−1.69877	−0.849383	+	0.527777i	$0.823026\pi$
$938$	0	0
$939$	0	0
$940$	−8.00000	−0.260931
$941$	59.0000i	1.92335i	0.274201	+	0.961673i	$0.411587\pi$
$941$	59.0000i	1.92335i	−0.274201	+	0.961673i	$0.588413\pi$
$942$	0	0
$943$	− 39.6611i	− 1.29154i
$944$	− 48.0000i	− 1.56227i
$945$	0	0
$946$	−24.0000	−0.780307
$947$	42.0000i	1.36482i	0.730971	+	0.682408i	$0.239067\pi$
$947$	42.0000i	1.36482i	−0.730971	+	0.682408i	$0.760933\pi$
$948$	0	0
$949$	26.0000	0.843996
$950$	−14.4222	−0.467918
$951$	0	0
$952$	0	0
$953$	10.8167	0.350386	0.175193	−	0.984534i	$-0.443945\pi$
$953$	10.8167	0.350386	0.175193	+	0.984534i	$0.443945\pi$
$954$	0	0
$955$	− 7.21110i	− 0.233346i
$956$	14.0000i	0.452792i
$957$	0	0
$958$	−6.00000	−0.193851
$959$	−43.2666	−1.39715
$960$	0	0
$961$	−21.0000	−0.677419
$962$	26.0000i	0.838274i
$963$	0	0
$964$	43.2666i	1.39352i
$965$	−3.60555	−0.116067
$966$	0	0
$967$	14.4222i	0.463787i	0.972741	+	0.231893i	$0.0744921\pi$
$967$	14.4222i	0.463787i	−0.972741	+	0.231893i	$0.925508\pi$
$968$	0	0
$969$	0	0
$970$	7.21110i	0.231535i
$971$	−21.6333	−0.694246	−0.347123	−	0.937820i	$-0.612841\pi$
$971$	−21.6333	−0.694246	−0.347123	+	0.937820i	$0.612841\pi$
$972$	0	0
$973$	25.2389i	0.809121i
$974$	−21.6333	−0.693176
$975$	0	0
$976$	−52.0000	−1.66448
$977$	− 2.00000i	− 0.0639857i	−0.999488	−	0.0319928i	$-0.989815\pi$
$977$	− 2.00000i	− 0.0639857i	0.999488	−	0.0319928i	$-0.0101854\pi$
$978$	0	0
$979$	9.00000	0.287641
$980$	12.0000i	0.383326i
$981$	0	0
$982$	− 14.4222i	− 0.460231i
$983$	− 4.00000i	− 0.127580i	−0.997963	−	0.0637901i	$-0.979681\pi$
$983$	− 4.00000i	− 0.127580i	0.997963	−	0.0637901i	$-0.0203188\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	52.0000i	1.65602i
$987$	0	0
$988$	52.0000	1.65434
$989$	−14.4222	−0.458599
$990$	0	0
$991$	−5.00000	−0.158830	−0.0794151	−	0.996842i	$-0.525305\pi$
$991$	−5.00000	−0.158830	−0.0794151	+	0.996842i	$0.525305\pi$
$992$	57.6888	1.83162
$993$	0	0
$994$	36.0555i	1.14361i
$995$	0	0
$996$	0	0
$997$	−6.00000	−0.190022	−0.0950110	−	0.995476i	$-0.530289\pi$
$997$	−6.00000	−0.190022	−0.0950110	+	0.995476i	$0.530289\pi$
$998$	57.6888	1.82611
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	585.2.b.f.181.2	yes	4
3.2	odd	2	inner	585.2.b.f.181.4	yes	4
13.5	odd	4		7605.2.a.w.1.1		2
13.8	odd	4		7605.2.a.bl.1.2		2
13.12	even	2	inner	585.2.b.f.181.3	yes	4
39.5	even	4		7605.2.a.bl.1.1		2
39.8	even	4		7605.2.a.w.1.2		2
39.38	odd	2	inner	585.2.b.f.181.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.b.f.181.1	✓	4	39.38	odd	2	inner
585.2.b.f.181.2	yes	4	1.1	even	1	trivial
585.2.b.f.181.3	yes	4	13.12	even	2	inner
585.2.b.f.181.4	yes	4	3.2	odd	2	inner
7605.2.a.w.1.1		2	13.5	odd	4
7605.2.a.w.1.2		2	39.8	even	4
7605.2.a.bl.1.1		2	39.5	even	4
7605.2.a.bl.1.2		2	13.8	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:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	585.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -2.00000 q^{4} +1.00000i q^{5} +3.60555i q^{7} +O(q^{10})\)	\(q-2.00000i q^{2} -2.00000 q^{4} +1.00000i q^{5} +3.60555i q^{7} +2.00000 q^{10} +3.00000i q^{11} +3.60555i q^{13} +7.21110 q^{14} -4.00000 q^{16} -3.60555 q^{17} +7.21110i q^{19} -2.00000i q^{20} +6.00000 q^{22} +3.60555 q^{23} -1.00000 q^{25} +7.21110 q^{26} -7.21110i q^{28} +7.21110 q^{29} -7.21110i q^{31} +8.00000i q^{32} +7.21110i q^{34} -3.60555 q^{35} +3.60555i q^{37} +14.4222 q^{38} -11.0000i q^{41} -4.00000 q^{43} -6.00000i q^{44} -7.21110i q^{46} -4.00000i q^{47} -6.00000 q^{49} +2.00000i q^{50} -7.21110i q^{52} -10.8167 q^{53} -3.00000 q^{55} -14.4222i q^{58} +12.0000i q^{59} +13.0000 q^{61} -14.4222 q^{62} +8.00000 q^{64} -3.60555 q^{65} +7.21110 q^{68} +7.21110i q^{70} +5.00000i q^{71} -7.21110i q^{73} +7.21110 q^{74} -14.4222i q^{76} -10.8167 q^{77} +13.0000 q^{79} -4.00000i q^{80} -22.0000 q^{82} -6.00000i q^{83} -3.60555i q^{85} +8.00000i q^{86} -3.00000i q^{89} -13.0000 q^{91} -7.21110 q^{92} -8.00000 q^{94} -7.21110 q^{95} +3.60555i q^{97} +12.0000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} + 8 q^{10} - 16 q^{16} + 24 q^{22} - 4 q^{25} - 16 q^{43} - 24 q^{49} - 12 q^{55} + 52 q^{61} + 32 q^{64} + 52 q^{79} - 88 q^{82} - 52 q^{91} - 32 q^{94}+O(q^{100}) \) 4 * q - 8 * q^4 + 8 * q^10 - 16 * q^16 + 24 * q^22 - 4 * q^25 - 16 * q^43 - 24 * q^49 - 12 * q^55 + 52 * q^61 + 32 * q^64 + 52 * q^79 - 88 * q^82 - 52 * q^91 - 32 * q^94

Introduction

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