LMFDB - Embedded newform 2592.2.c.b.2591.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,2,Mod(2591,2592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2592, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("2592.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.c (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:	$20.6972242039$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 12x^{14} + 49x^{12} - 12x^{10} - 600x^{8} + 108x^{6} + 4057x^{4} + 18252x^{2} + 28561 $ x^16 - 12x^14 + 49x^12 - 12x^10 - 600x^8 + 108x^6 + 4057x^4 + 18252*x^2 + 28561
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.9
Root			$0.115299 - 1.50155i$ of defining polynomial
Character	$\chi$	$=$	2592.2591
Dual form			2592.2.c.b.2591.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.30185i q^{5} -3.51498i q^{7} +O(q^{10})$	$q+1.30185i q^{5} -3.51498i q^{7} +1.81949 q^{11} -5.24703 q^{13} +3.03908i q^{17} +3.05816i q^{19} +5.22199 q^{23} +3.30519 q^{25} -3.75134i q^{29} -4.35506i q^{31} +4.57596 q^{35} +1.48502 q^{37} -8.45569i q^{41} +6.44318i q^{43} +12.6983 q^{47} -5.35506 q^{49} -11.2841i q^{53} +2.36869i q^{55} +0.685926 q^{59} -4.15992 q^{61} -6.83083i q^{65} -12.6613i q^{67} -2.50541 q^{71} +12.3052 q^{73} -6.39545i q^{77} +2.08812i q^{79} -17.1671 q^{83} -3.95641 q^{85} -12.5300i q^{89} +18.4432i q^{91} -3.98127 q^{95} -8.79122 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 8 q^{13} - 48 q^{25} + 72 q^{37} - 48 q^{49} - 56 q^{61} + 96 q^{73} - 24 q^{85} - 32 q^{97}+O(q^{100}) $ 16 * q - 8 * q^13 - 48 * q^25 + 72 * q^37 - 48 * q^49 - 56 * q^61 + 96 * q^73 - 24 * q^85 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	1.30185i	0.582204i	0.956692	+	0.291102i	$0.0940219\pi$
$5$	1.30185i	0.582204i	−0.956692	+	0.291102i	$0.905978\pi$
$6$	0	0
$7$	− 3.51498i	− 1.32854i	−0.747494	−	0.664268i	$-0.768743\pi$
$7$	− 3.51498i	− 1.32854i	0.747494	−	0.664268i	$-0.231257\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.81949	0.548596	0.274298	−	0.961645i	$-0.411555\pi$
$11$	1.81949	0.548596	0.274298	+	0.961645i	$0.411555\pi$
$12$	0	0
$13$	−5.24703	−1.45526	−0.727632	−	0.685968i	$-0.759379\pi$
$13$	−5.24703	−1.45526	−0.727632	+	0.685968i	$0.759379\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.03908i	0.737084i	0.929611	+	0.368542i	$0.120143\pi$
$17$	3.03908i	0.737084i	−0.929611	+	0.368542i	$0.879857\pi$
$18$	0	0
$19$	3.05816i	0.701591i	0.936452	+	0.350796i	$0.114089\pi$
$19$	3.05816i	0.701591i	−0.936452	+	0.350796i	$0.885911\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.22199	1.08886	0.544431	−	0.838806i	$-0.316746\pi$
$23$	5.22199	1.08886	0.544431	+	0.838806i	$0.316746\pi$
$24$	0	0
$25$	3.30519	0.661038
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 3.75134i	− 0.696606i	−0.937382	−	0.348303i	$-0.886758\pi$
$29$	− 3.75134i	− 0.696606i	0.937382	−	0.348303i	$-0.113242\pi$
$30$	0	0
$31$	− 4.35506i	− 0.782192i	−0.920350	−	0.391096i	$-0.872096\pi$
$31$	− 4.35506i	− 0.782192i	0.920350	−	0.391096i	$-0.127904\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.57596	0.773479
$36$	0	0
$37$	1.48502	0.244136	0.122068	−	0.992522i	$-0.461047\pi$
$37$	1.48502	0.244136	0.122068	+	0.992522i	$0.461047\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 8.45569i	− 1.32056i	−0.751021	−	0.660279i	$-0.770438\pi$
$41$	− 8.45569i	− 1.32056i	0.751021	−	0.660279i	$-0.229562\pi$
$42$	0	0
$43$	6.44318i	0.982576i	0.870997	+	0.491288i	$0.163474\pi$
$43$	6.44318i	0.982576i	−0.870997	+	0.491288i	$0.836526\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.6983	1.85224	0.926121	−	0.377226i	$-0.123122\pi$
$47$	12.6983	1.85224	0.926121	+	0.377226i	$0.123122\pi$
$48$	0	0
$49$	−5.35506	−0.765009
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 11.2841i	− 1.54999i	−0.631965	−	0.774997i	$-0.717752\pi$
$53$	− 11.2841i	− 1.54999i	0.631965	−	0.774997i	$-0.282248\pi$
$54$	0	0
$55$	2.36869i	0.319395i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.685926	0.0893000	0.0446500	−	0.999003i	$-0.485783\pi$
$59$	0.685926	0.0893000	0.0446500	+	0.999003i	$0.485783\pi$
$60$	0	0
$61$	−4.15992	−0.532623	−0.266311	−	0.963887i	$-0.585805\pi$
$61$	−4.15992	−0.532623	−0.266311	+	0.963887i	$0.585805\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 6.83083i	− 0.847260i
$66$	0	0
$67$	− 12.6613i	− 1.54682i	−0.633907	−	0.773410i	$-0.718550\pi$
$67$	− 12.6613i	− 1.54682i	0.633907	−	0.773410i	$-0.281450\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.50541	−0.297338	−0.148669	−	0.988887i	$-0.547499\pi$
$71$	−2.50541	−0.297338	−0.148669	+	0.988887i	$0.547499\pi$
$72$	0	0
$73$	12.3052	1.44021	0.720107	−	0.693863i	$-0.244093\pi$
$73$	12.3052	1.44021	0.720107	+	0.693863i	$0.244093\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 6.39545i	− 0.728829i
$78$	0	0
$79$	2.08812i	0.234932i	0.993077	+	0.117466i	$0.0374771\pi$
$79$	2.08812i	0.234932i	−0.993077	+	0.117466i	$0.962523\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−17.1671	−1.88434	−0.942169	−	0.335139i	$-0.891217\pi$
$83$	−17.1671	−1.88434	−0.942169	+	0.335139i	$0.891217\pi$
$84$	0	0
$85$	−3.95641	−0.429133
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 12.5300i	− 1.32818i	−0.747652	−	0.664091i	$-0.768819\pi$
$89$	− 12.5300i	− 1.32818i	0.747652	−	0.664091i	$-0.231181\pi$
$90$	0	0
$91$	18.4432i	1.93337i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.98127	−0.408469
$96$	0	0
$97$	−8.79122	−0.892613	−0.446307	−	0.894880i	$-0.647261\pi$
$97$	−8.79122	−0.892613	−0.446307	+	0.894880i	$0.647261\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.91635i	− 0.190684i	−0.995445	−	0.0953418i	$-0.969606\pi$
$101$	− 1.91635i	− 0.190684i	0.995445	−	0.0953418i	$-0.0303944\pi$
$102$	0	0
$103$	− 17.8564i	− 1.75944i	−0.475488	−	0.879722i	$-0.657729\pi$
$103$	− 17.8564i	− 1.75944i	0.475488	−	0.879722i	$-0.342271\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.8573	1.82301	0.911503	−	0.411293i	$-0.134923\pi$
$107$	18.8573	1.82301	0.911503	+	0.411293i	$0.134923\pi$
$108$	0	0
$109$	17.1679	1.64439	0.822195	−	0.569206i	$-0.192749\pi$
$109$	17.1679	1.64439	0.822195	+	0.569206i	$0.192749\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.57788i	0.242507i	0.992622	+	0.121253i	$0.0386913\pi$
$113$	2.57788i	0.242507i	−0.992622	+	0.121253i	$0.961309\pi$
$114$	0	0
$115$	6.79824i	0.633939i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	10.6823	0.979243
$120$	0	0
$121$	−7.68947	−0.699043
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.8121i	0.967063i
$126$	0	0
$127$	9.73306i	0.863669i	0.901953	+	0.431835i	$0.142134\pi$
$127$	9.73306i	0.863669i	−0.901953	+	0.431835i	$0.857866\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.0157	1.22456	0.612278	−	0.790642i	$-0.290253\pi$
$131$	14.0157	1.22456	0.612278	+	0.790642i	$0.290253\pi$
$132$	0	0
$133$	10.7494	0.932089
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.88487i	− 0.246471i	−0.992377	−	0.123236i	$-0.960673\pi$
$137$	− 2.88487i	− 0.246471i	0.992377	−	0.123236i	$-0.0393271\pi$
$138$	0	0
$139$	8.17624i	0.693499i	0.937958	+	0.346750i	$0.112715\pi$
$139$	8.17624i	0.693499i	−0.937958	+	0.346750i	$0.887285\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−9.54689	−0.798351
$144$	0	0
$145$	4.88367	0.405567
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 8.03634i	− 0.658363i	−0.944267	−	0.329181i	$-0.893227\pi$
$149$	− 8.03634i	− 0.658363i	0.944267	−	0.329181i	$-0.106773\pi$
$150$	0	0
$151$	− 13.1463i	− 1.06983i	−0.844906	−	0.534915i	$-0.820344\pi$
$151$	− 13.1463i	− 1.06983i	0.844906	−	0.534915i	$-0.179656\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.66963	0.455395
$156$	0	0
$157$	−1.72376	−0.137571	−0.0687853	−	0.997631i	$-0.521912\pi$
$157$	−1.72376	−0.137571	−0.0687853	+	0.997631i	$0.521912\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 18.3552i	− 1.44659i
$162$	0	0
$163$	6.13698i	0.480686i	0.970688	+	0.240343i	$0.0772598\pi$
$163$	6.13698i	0.480686i	−0.970688	+	0.240343i	$0.922740\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.2273	1.56524	0.782619	−	0.622500i	$-0.213883\pi$
$167$	20.2273	1.56524	0.782619	+	0.622500i	$0.213883\pi$
$168$	0	0
$169$	14.5313	1.11779
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 2.59144i	− 0.197024i	−0.995136	−	0.0985119i	$-0.968592\pi$
$173$	− 2.59144i	− 0.197024i	0.995136	−	0.0985119i	$-0.0314082\pi$
$174$	0	0
$175$	− 11.6177i	− 0.878214i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	25.4493	1.90217	0.951086	−	0.308925i	$-0.0999692\pi$
$179$	25.4493	1.90217	0.951086	+	0.308925i	$0.0999692\pi$
$180$	0	0
$181$	−0.811874	−0.0603461	−0.0301730	−	0.999545i	$-0.509606\pi$
$181$	−0.811874	−0.0603461	−0.0301730	+	0.999545i	$0.509606\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.93327i	0.142137i
$186$	0	0
$187$	5.52956i	0.404361i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.4402	1.11721	0.558607	−	0.829432i	$-0.311336\pi$
$191$	15.4402	1.11721	0.558607	+	0.829432i	$0.311336\pi$
$192$	0	0
$193$	−24.8000	−1.78514	−0.892571	−	0.450907i	$-0.851101\pi$
$193$	−24.8000	−1.78514	−0.892571	+	0.450907i	$0.851101\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5.99395i	− 0.427051i	−0.976937	−	0.213526i	$-0.931505\pi$
$197$	− 5.99395i	− 0.427051i	0.976937	−	0.213526i	$-0.0684947\pi$
$198$	0	0
$199$	− 18.7700i	− 1.33057i	−0.746589	−	0.665286i	$-0.768310\pi$
$199$	− 18.7700i	− 1.33057i	0.746589	−	0.665286i	$-0.231690\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−13.1859	−0.925466
$204$	0	0
$205$	11.0080	0.768834
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.56429i	0.384890i
$210$	0	0
$211$	19.2344i	1.32415i	0.749437	+	0.662075i	$0.230324\pi$
$211$	19.2344i	1.32415i	−0.749437	+	0.662075i	$0.769676\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.38804	−0.572060
$216$	0	0
$217$	−15.3079	−1.03917
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 15.9461i	− 1.07265i
$222$	0	0
$223$	− 5.98637i	− 0.400877i	−0.979706	−	0.200438i	$-0.935763\pi$
$223$	− 5.98637i	− 0.400877i	0.979706	−	0.200438i	$-0.0642367\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.67291	0.509269	0.254634	−	0.967037i	$-0.418045\pi$
$227$	7.67291	0.509269	0.254634	+	0.967037i	$0.418045\pi$
$228$	0	0
$229$	−20.5323	−1.35681	−0.678406	−	0.734687i	$-0.737329\pi$
$229$	−20.5323	−1.35681	−0.678406	+	0.734687i	$0.737329\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	26.0981i	1.70974i	0.518839	+	0.854872i	$0.326364\pi$
$233$	26.0981i	1.70974i	−0.518839	+	0.854872i	$0.673636\pi$
$234$	0	0
$235$	16.5313i	1.07838i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.06629	−0.0689723	−0.0344862	−	0.999405i	$-0.510979\pi$
$239$	−1.06629	−0.0689723	−0.0344862	+	0.999405i	$0.510979\pi$
$240$	0	0
$241$	−11.0007	−0.708620	−0.354310	−	0.935128i	$-0.615284\pi$
$241$	−11.0007	−0.708620	−0.354310	+	0.935128i	$0.615284\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 6.97148i	− 0.445391i
$246$	0	0
$247$	− 16.0463i	− 1.02100i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.66440	0.546892	0.273446	−	0.961887i	$-0.411837\pi$
$251$	8.66440	0.546892	0.273446	+	0.961887i	$0.411837\pi$
$252$	0	0
$253$	9.50134	0.597344
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.2268i	1.13696i	0.822698	+	0.568478i	$0.192468\pi$
$257$	18.2268i	1.13696i	−0.822698	+	0.568478i	$0.807532\pi$
$258$	0	0
$259$	− 5.21982i	− 0.324344i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.25435	0.139009	0.0695045	−	0.997582i	$-0.477858\pi$
$263$	2.25435	0.139009	0.0695045	+	0.997582i	$0.477858\pi$
$264$	0	0
$265$	14.6902	0.902412
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.70436i	0.286830i	0.989663	+	0.143415i	$0.0458083\pi$
$269$	4.70436i	0.286830i	−0.989663	+	0.143415i	$0.954192\pi$
$270$	0	0
$271$	− 6.89999i	− 0.419145i	−0.977793	−	0.209572i	$-0.932793\pi$
$271$	− 6.89999i	− 0.419145i	0.977793	−	0.209572i	$-0.0672072\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.01375	0.362643
$276$	0	0
$277$	−23.9163	−1.43699	−0.718496	−	0.695531i	$-0.755169\pi$
$277$	−23.9163	−1.43699	−0.718496	+	0.695531i	$0.755169\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 13.8507i	− 0.826261i	−0.910672	−	0.413130i	$-0.864435\pi$
$281$	− 13.8507i	− 0.826261i	0.910672	−	0.413130i	$-0.135565\pi$
$282$	0	0
$283$	− 0.101750i	− 0.00604843i	−0.999995	−	0.00302421i	$-0.999037\pi$
$283$	− 0.101750i	− 0.00604843i	0.999995	−	0.00302421i	$-0.000962639\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−29.7216	−1.75441
$288$	0	0
$289$	7.76402	0.456707
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.1168i	1.29208i	0.763304	+	0.646040i	$0.223576\pi$
$293$	22.1168i	1.29208i	−0.763304	+	0.646040i	$0.776424\pi$
$294$	0	0
$295$	0.892972i	0.0519908i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−27.3999	−1.58458
$300$	0	0
$301$	22.6476	1.30539
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 5.41558i	− 0.310095i
$306$	0	0
$307$	11.3850i	0.649777i	0.945752	+	0.324889i	$0.105327\pi$
$307$	11.3850i	0.649777i	−0.945752	+	0.324889i	$0.894673\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−21.9415	−1.24419	−0.622094	−	0.782943i	$-0.713718\pi$
$311$	−21.9415	−1.24419	−0.622094	+	0.782943i	$0.713718\pi$
$312$	0	0
$313$	−8.48677	−0.479700	−0.239850	−	0.970810i	$-0.577098\pi$
$313$	−8.48677	−0.479700	−0.239850	+	0.970810i	$0.577098\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.9048i	1.00564i	0.864393	+	0.502818i	$0.167703\pi$
$317$	17.9048i	1.00564i	−0.864393	+	0.502818i	$0.832297\pi$
$318$	0	0
$319$	− 6.82551i	− 0.382155i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−9.29400	−0.517132
$324$	0	0
$325$	−17.3424	−0.961985
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 44.6344i	− 2.46077i
$330$	0	0
$331$	2.08812i	0.114773i	0.998352	+	0.0573867i	$0.0182768\pi$
$331$	2.08812i	0.114773i	−0.998352	+	0.0573867i	$0.981723\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	16.4830	0.900564
$336$	0	0
$337$	27.6630	1.50690	0.753450	−	0.657505i	$-0.228388\pi$
$337$	27.6630	1.50690	0.753450	+	0.657505i	$0.228388\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 7.92397i	− 0.429107i
$342$	0	0
$343$	− 5.78192i	− 0.312194i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.88577	0.154916	0.0774582	−	0.996996i	$-0.475320\pi$
$347$	2.88577	0.154916	0.0774582	+	0.996996i	$0.475320\pi$
$348$	0	0
$349$	0.396905	0.0212458	0.0106229	−	0.999944i	$-0.496619\pi$
$349$	0.396905	0.0212458	0.0106229	+	0.999944i	$0.496619\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.25347i	0.492512i	0.969205	+	0.246256i	$0.0792004\pi$
$353$	9.25347i	0.492512i	−0.969205	+	0.246256i	$0.920800\pi$
$354$	0	0
$355$	− 3.26167i	− 0.173111i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−16.8248	−0.887981	−0.443990	−	0.896032i	$-0.646438\pi$
$359$	−16.8248	−0.887981	−0.443990	+	0.896032i	$0.646438\pi$
$360$	0	0
$361$	9.64763	0.507770
$362$	0	0
$363$	0	0
$364$	0	0
$365$	16.0195i	0.838498i
$366$	0	0
$367$	− 0.355062i	− 0.0185341i	−0.999957	−	0.00926703i	$-0.997050\pi$
$367$	− 0.355062i	− 0.0185341i	0.999957	−	0.00926703i	$-0.00294983\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−39.6634	−2.05922
$372$	0	0
$373$	−3.04453	−0.157640	−0.0788200	−	0.996889i	$-0.525115\pi$
$373$	−3.04453	−0.157640	−0.0788200	+	0.996889i	$0.525115\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	19.6834i	1.01375i
$378$	0	0
$379$	− 36.6330i	− 1.88171i	−0.338805	−	0.940857i	$-0.610023\pi$
$379$	− 36.6330i	− 1.88171i	0.338805	−	0.940857i	$-0.389977\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6.97421	0.356365	0.178183	−	0.983997i	$-0.442978\pi$
$383$	6.97421	0.356365	0.178183	+	0.983997i	$0.442978\pi$
$384$	0	0
$385$	8.32590	0.424327
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 30.3779i	− 1.54022i	−0.637911	−	0.770110i	$-0.720201\pi$
$389$	− 30.3779i	− 1.54022i	0.637911	−	0.770110i	$-0.279799\pi$
$390$	0	0
$391$	15.8700i	0.802582i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.71841	−0.136778
$396$	0	0
$397$	−12.6167	−0.633215	−0.316608	−	0.948557i	$-0.602544\pi$
$397$	−12.6167	−0.633215	−0.316608	+	0.948557i	$0.602544\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.2672i	0.562657i	0.959612	+	0.281328i	$0.0907750\pi$
$401$	11.2672i	0.562657i	−0.959612	+	0.281328i	$0.909225\pi$
$402$	0	0
$403$	22.8511i	1.13830i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.70198	0.133932
$408$	0	0
$409$	18.5167	0.915590	0.457795	−	0.889058i	$-0.348639\pi$
$409$	18.5167	0.915590	0.457795	+	0.889058i	$0.348639\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 2.41102i	− 0.118638i
$414$	0	0
$415$	− 22.3490i	− 1.09707i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−23.0897	−1.12800	−0.564002	−	0.825774i	$-0.690739\pi$
$419$	−23.0897	−1.12800	−0.564002	+	0.825774i	$0.690739\pi$
$420$	0	0
$421$	−11.1825	−0.545003	−0.272501	−	0.962155i	$-0.587851\pi$
$421$	−11.1825	−0.545003	−0.272501	+	0.962155i	$0.587851\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	10.0447i	0.487241i
$426$	0	0
$427$	14.6220i	0.707608i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.3110	0.833843	0.416921	−	0.908943i	$-0.363109\pi$
$431$	17.3110	0.833843	0.416921	+	0.908943i	$0.363109\pi$
$432$	0	0
$433$	−22.7194	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$433$	−22.7194	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	15.9697i	0.763935i
$438$	0	0
$439$	14.4987i	0.691983i	0.938238	+	0.345992i	$0.112457\pi$
$439$	14.4987i	0.691983i	−0.938238	+	0.345992i	$0.887543\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.53279	−0.120336	−0.0601682	−	0.998188i	$-0.519164\pi$
$443$	−2.53279	−0.120336	−0.0601682	+	0.998188i	$0.519164\pi$
$444$	0	0
$445$	16.3122	0.773273
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 15.4046i	− 0.726988i	−0.931596	−	0.363494i	$-0.881584\pi$
$449$	− 15.4046i	− 0.726988i	0.931596	−	0.363494i	$-0.118416\pi$
$450$	0	0
$451$	− 15.3850i	− 0.724452i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−24.0102	−1.12562
$456$	0	0
$457$	18.1098	0.847140	0.423570	−	0.905863i	$-0.360777\pi$
$457$	18.1098	0.847140	0.423570	+	0.905863i	$0.360777\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	28.1955i	1.31320i	0.754241	+	0.656598i	$0.228005\pi$
$461$	28.1955i	1.31320i	−0.754241	+	0.656598i	$0.771995\pi$
$462$	0	0
$463$	32.6330i	1.51659i	0.651914	+	0.758293i	$0.273966\pi$
$463$	32.6330i	1.51659i	−0.651914	+	0.758293i	$0.726034\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.6595	1.23366	0.616828	−	0.787098i	$-0.288418\pi$
$467$	26.6595	1.23366	0.616828	+	0.787098i	$0.288418\pi$
$468$	0	0
$469$	−44.5040	−2.05501
$470$	0	0
$471$	0	0
$472$	0	0
$473$	11.7233i	0.539037i
$474$	0	0
$475$	10.1078i	0.463779i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−24.0501	−1.09888	−0.549439	−	0.835534i	$-0.685158\pi$
$479$	−24.0501	−1.09888	−0.549439	+	0.835534i	$0.685158\pi$
$480$	0	0
$481$	−7.79196	−0.355283
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 11.4448i	− 0.519683i
$486$	0	0
$487$	17.4449i	0.790505i	0.918573	+	0.395252i	$0.129343\pi$
$487$	17.4449i	0.790505i	−0.918573	+	0.395252i	$0.870657\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	13.3170	0.600987	0.300494	−	0.953784i	$-0.402849\pi$
$491$	13.3170	0.600987	0.300494	+	0.953784i	$0.402849\pi$
$492$	0	0
$493$	11.4006	0.513457
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.80647i	0.395024i
$498$	0	0
$499$	33.3894i	1.49472i	0.664421	+	0.747358i	$0.268678\pi$
$499$	33.3894i	1.49472i	−0.664421	+	0.747358i	$0.731322\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−24.2884	−1.08297	−0.541483	−	0.840712i	$-0.682137\pi$
$503$	−24.2884	−1.08297	−0.541483	+	0.840712i	$0.682137\pi$
$504$	0	0
$505$	2.49479	0.111017
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 23.5729i	− 1.04485i	−0.852685	−	0.522425i	$-0.825027\pi$
$509$	− 23.5729i	− 1.04485i	0.852685	−	0.522425i	$-0.174973\pi$
$510$	0	0
$511$	− 43.2525i	− 1.91338i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	23.2463	1.02436
$516$	0	0
$517$	23.1044	1.01613
$518$	0	0
$519$	0	0
$520$	0	0
$521$	32.5603i	1.42649i	0.700913	+	0.713246i	$0.252776\pi$
$521$	32.5603i	1.42649i	−0.700913	+	0.713246i	$0.747224\pi$
$522$	0	0
$523$	5.17449i	0.226265i	0.993580	+	0.113132i	$0.0360884\pi$
$523$	5.17449i	0.226265i	−0.993580	+	0.113132i	$0.963912\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13.2354	0.576542
$528$	0	0
$529$	4.26922	0.185618
$530$	0	0
$531$	0	0
$532$	0	0
$533$	44.3673i	1.92176i
$534$	0	0
$535$	24.5494i	1.06136i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−9.74346	−0.419680
$540$	0	0
$541$	15.6620	0.673362	0.336681	−	0.941619i	$-0.390696\pi$
$541$	15.6620	0.673362	0.336681	+	0.941619i	$0.390696\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	22.3500i	0.957371i
$546$	0	0
$547$	− 28.4295i	− 1.21556i	−0.794106	−	0.607780i	$-0.792060\pi$
$547$	− 28.4295i	− 1.21556i	0.794106	−	0.607780i	$-0.207940\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	11.4722	0.488733
$552$	0	0
$553$	7.33969	0.312115
$554$	0	0
$555$	0	0
$556$	0	0
$557$	45.9337i	1.94627i	0.230225	+	0.973137i	$0.426054\pi$
$557$	45.9337i	1.94627i	−0.230225	+	0.973137i	$0.573946\pi$
$558$	0	0
$559$	− 33.8075i	− 1.42991i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−35.6042	−1.50054	−0.750269	−	0.661133i	$-0.770076\pi$
$563$	−35.6042	−1.50054	−0.750269	+	0.661133i	$0.770076\pi$
$564$	0	0
$565$	−3.35601	−0.141188
$566$	0	0
$567$	0	0
$568$	0	0
$569$	34.5548i	1.44861i	0.689478	+	0.724307i	$0.257840\pi$
$569$	34.5548i	1.44861i	−0.689478	+	0.724307i	$0.742160\pi$
$570$	0	0
$571$	− 2.90755i	− 0.121677i	−0.998148	−	0.0608386i	$-0.980623\pi$
$571$	− 2.90755i	− 0.121677i	0.998148	−	0.0608386i	$-0.0193775\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	17.2597	0.719779
$576$	0	0
$577$	−28.1251	−1.17086	−0.585431	−	0.810722i	$-0.699075\pi$
$577$	−28.1251	−1.17086	−0.585431	+	0.810722i	$0.699075\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	60.3421i	2.50341i
$582$	0	0
$583$	− 20.5313i	− 0.850319i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−28.5082	−1.17666	−0.588330	−	0.808621i	$-0.700214\pi$
$587$	−28.5082	−1.17666	−0.588330	+	0.808621i	$0.700214\pi$
$588$	0	0
$589$	13.3185	0.548779
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 8.02682i	− 0.329622i	−0.986325	−	0.164811i	$-0.947299\pi$
$593$	− 8.02682i	− 0.329622i	0.986325	−	0.164811i	$-0.0527014\pi$
$594$	0	0
$595$	13.9067i	0.570119i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	17.2070	0.703060	0.351530	−	0.936177i	$-0.385662\pi$
$599$	17.2070	0.703060	0.351530	+	0.936177i	$0.385662\pi$
$600$	0	0
$601$	−3.45681	−0.141006	−0.0705032	−	0.997512i	$-0.522460\pi$
$601$	−3.45681	−0.141006	−0.0705032	+	0.997512i	$0.522460\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 10.0105i	− 0.406986i
$606$	0	0
$607$	9.07274i	0.368251i	0.982903	+	0.184126i	$0.0589453\pi$
$607$	9.07274i	0.368251i	−0.982903	+	0.184126i	$0.941055\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−66.6285	−2.69550
$612$	0	0
$613$	13.1044	0.529283	0.264642	−	0.964347i	$-0.414746\pi$
$613$	13.1044	0.529283	0.264642	+	0.964347i	$0.414746\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 3.72358i	− 0.149906i	−0.997187	−	0.0749528i	$-0.976119\pi$
$617$	− 3.72358i	− 0.149906i	0.997187	−	0.0749528i	$-0.0238806\pi$
$618$	0	0
$619$	11.4687i	0.460966i	0.973076	+	0.230483i	$0.0740306\pi$
$619$	11.4687i	0.460966i	−0.973076	+	0.230483i	$0.925969\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−44.0428	−1.76454
$624$	0	0
$625$	2.45026	0.0980103
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.51310i	0.179949i
$630$	0	0
$631$	− 10.1436i	− 0.403810i	−0.979405	−	0.201905i	$-0.935287\pi$
$631$	− 10.1436i	− 0.403810i	0.979405	−	0.201905i	$-0.0647132\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−12.6710	−0.502832
$636$	0	0
$637$	28.0982	1.11329
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 11.5075i	− 0.454521i	−0.973834	−	0.227260i	$-0.927023\pi$
$641$	− 11.5075i	− 0.454521i	0.973834	−	0.227260i	$-0.0729768\pi$
$642$	0	0
$643$	28.0000i	1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$	28.0000i	1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.5438	1.08286	0.541430	−	0.840746i	$-0.317883\pi$
$647$	27.5438	1.08286	0.541430	+	0.840746i	$0.317883\pi$
$648$	0	0
$649$	1.24803	0.0489896
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 45.9560i	− 1.79840i	−0.437541	−	0.899199i	$-0.644150\pi$
$653$	− 45.9560i	− 1.79840i	0.437541	−	0.899199i	$-0.355850\pi$
$654$	0	0
$655$	18.2463i	0.712942i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	33.3332	1.29848	0.649238	−	0.760585i	$-0.275088\pi$
$659$	33.3332	1.29848	0.649238	+	0.760585i	$0.275088\pi$
$660$	0	0
$661$	−11.7026	−0.455177	−0.227588	−	0.973757i	$-0.573084\pi$
$661$	−11.7026	−0.455177	−0.227588	+	0.973757i	$0.573084\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	13.9941i	0.542666i
$666$	0	0
$667$	− 19.5895i	− 0.758507i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.56891	−0.292194
$672$	0	0
$673$	−49.3252	−1.90135	−0.950674	−	0.310193i	$-0.899606\pi$
$673$	−49.3252	−1.90135	−0.950674	+	0.310193i	$0.899606\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16.5827i	0.637326i	0.947868	+	0.318663i	$0.103234\pi$
$677$	16.5827i	0.637326i	−0.947868	+	0.318663i	$0.896766\pi$
$678$	0	0
$679$	30.9009i	1.18587i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−7.39972	−0.283142	−0.141571	−	0.989928i	$-0.545215\pi$
$683$	−7.39972	−0.283142	−0.141571	+	0.989928i	$0.545215\pi$
$684$	0	0
$685$	3.75566	0.143496
$686$	0	0
$687$	0	0
$688$	0	0
$689$	59.2081i	2.25565i
$690$	0	0
$691$	12.2855i	0.467364i	0.972313	+	0.233682i	$0.0750775\pi$
$691$	12.2855i	0.467364i	−0.972313	+	0.233682i	$0.924923\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10.6442	−0.403758
$696$	0	0
$697$	25.6975	0.973362
$698$	0	0
$699$	0	0
$700$	0	0
$701$	23.5927i	0.891084i	0.895261	+	0.445542i	$0.146989\pi$
$701$	23.5927i	0.891084i	−0.895261	+	0.445542i	$0.853011\pi$
$702$	0	0
$703$	4.54145i	0.171284i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.73591	−0.253330
$708$	0	0
$709$	3.46770	0.130232	0.0651160	−	0.997878i	$-0.479258\pi$
$709$	3.46770	0.130232	0.0651160	+	0.997878i	$0.479258\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 22.7421i	− 0.851699i
$714$	0	0
$715$	− 12.4286i	− 0.464803i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−24.7874	−0.924413	−0.462206	−	0.886772i	$-0.652942\pi$
$719$	−24.7874	−0.924413	−0.462206	+	0.886772i	$0.652942\pi$
$720$	0	0
$721$	−62.7649	−2.33749
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 12.3989i	− 0.460483i
$726$	0	0
$727$	10.6194i	0.393852i	0.980418	+	0.196926i	$0.0630959\pi$
$727$	10.6194i	0.393852i	−0.980418	+	0.196926i	$0.936904\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−19.5813	−0.724241
$732$	0	0
$733$	16.8119	0.620961	0.310480	−	0.950580i	$-0.399510\pi$
$733$	16.8119	0.620961	0.310480	+	0.950580i	$0.399510\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 23.0370i	− 0.848578i
$738$	0	0
$739$	− 46.0507i	− 1.69400i	−0.531591	−	0.847001i	$-0.678406\pi$
$739$	− 46.0507i	− 1.69400i	0.531591	−	0.847001i	$-0.321594\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−20.5183	−0.752744	−0.376372	−	0.926469i	$-0.622828\pi$
$743$	−20.5183	−0.752744	−0.376372	+	0.926469i	$0.622828\pi$
$744$	0	0
$745$	10.4621	0.383301
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 66.2830i	− 2.42193i
$750$	0	0
$751$	− 37.0410i	− 1.35165i	−0.737064	−	0.675823i	$-0.763789\pi$
$751$	− 37.0410i	− 1.35165i	0.737064	−	0.675823i	$-0.236211\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	17.1145	0.622859
$756$	0	0
$757$	−16.2740	−0.591487	−0.295744	−	0.955267i	$-0.595567\pi$
$757$	−16.2740	−0.591487	−0.295744	+	0.955267i	$0.595567\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 15.9597i	− 0.578538i	−0.957248	−	0.289269i	$-0.906588\pi$
$761$	− 15.9597i	− 0.578538i	0.957248	−	0.289269i	$-0.0934122\pi$
$762$	0	0
$763$	− 60.3449i	− 2.18463i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.59907	−0.129955
$768$	0	0
$769$	−4.02995	−0.145324	−0.0726619	−	0.997357i	$-0.523149\pi$
$769$	−4.02995	−0.145324	−0.0726619	+	0.997357i	$0.523149\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 7.21587i	− 0.259537i	−0.991544	−	0.129768i	$-0.958577\pi$
$773$	− 7.21587i	− 0.259537i	0.991544	−	0.129768i	$-0.0414234\pi$
$774$	0	0
$775$	− 14.3943i	− 0.517059i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	25.8589	0.926491
$780$	0	0
$781$	−4.55856	−0.163118
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 2.24407i	− 0.0800942i
$786$	0	0
$787$	− 47.0969i	− 1.67882i	−0.543497	−	0.839411i	$-0.682900\pi$
$787$	− 47.0969i	− 1.67882i	0.543497	−	0.839411i	$-0.317100\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	9.06119	0.322179
$792$	0	0
$793$	21.8272	0.775106
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 19.2072i	− 0.680354i	−0.940361	−	0.340177i	$-0.889513\pi$
$797$	− 19.2072i	− 0.680354i	0.940361	−	0.340177i	$-0.110487\pi$
$798$	0	0
$799$	38.5912i	1.36526i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	22.3891	0.790095
$804$	0	0
$805$	23.8957	0.842211
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 36.8312i	− 1.29492i	−0.762101	−	0.647458i	$-0.775832\pi$
$809$	− 36.8312i	− 1.29492i	0.762101	−	0.647458i	$-0.224168\pi$
$810$	0	0
$811$	− 37.2208i	− 1.30700i	−0.756927	−	0.653499i	$-0.773300\pi$
$811$	− 37.2208i	− 1.30700i	0.756927	−	0.653499i	$-0.226700\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−7.98942	−0.279857
$816$	0	0
$817$	−19.7043	−0.689366
$818$	0	0
$819$	0	0
$820$	0	0
$821$	13.9895i	0.488237i	0.969745	+	0.244118i	$0.0784985\pi$
$821$	13.9895i	0.488237i	−0.969745	+	0.244118i	$0.921501\pi$
$822$	0	0
$823$	8.49258i	0.296033i	0.988985	+	0.148016i	$0.0472888\pi$
$823$	8.49258i	0.296033i	−0.988985	+	0.148016i	$0.952711\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−51.1755	−1.77955	−0.889774	−	0.456402i	$-0.849138\pi$
$827$	−51.1755	−1.77955	−0.889774	+	0.456402i	$0.849138\pi$
$828$	0	0
$829$	−37.9762	−1.31897	−0.659484	−	0.751718i	$-0.729225\pi$
$829$	−37.9762	−1.31897	−0.659484	+	0.751718i	$0.729225\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 16.2744i	− 0.563876i
$834$	0	0
$835$	26.3329i	0.911288i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	9.79613	0.338200	0.169100	−	0.985599i	$-0.445914\pi$
$839$	9.79613	0.338200	0.169100	+	0.985599i	$0.445914\pi$
$840$	0	0
$841$	14.9275	0.514740
$842$	0	0
$843$	0	0
$844$	0	0
$845$	18.9175i	0.650783i
$846$	0	0
$847$	27.0283i	0.928704i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7.75478	0.265831
$852$	0	0
$853$	−50.6264	−1.73342	−0.866708	−	0.498816i	$-0.833768\pi$
$853$	−50.6264	−1.73342	−0.866708	+	0.498816i	$0.833768\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	23.3633i	0.798074i	0.916935	+	0.399037i	$0.130656\pi$
$857$	23.3633i	0.798074i	−0.916935	+	0.399037i	$0.869344\pi$
$858$	0	0
$859$	− 10.4150i	− 0.355354i	−0.984089	−	0.177677i	$-0.943142\pi$
$859$	− 10.4150i	− 0.355354i	0.984089	−	0.177677i	$-0.0568583\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	15.7458	0.535992	0.267996	−	0.963420i	$-0.413639\pi$
$863$	15.7458	0.535992	0.267996	+	0.963420i	$0.413639\pi$
$864$	0	0
$865$	3.37366	0.114708
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.79930i	0.128882i
$870$	0	0
$871$	66.4340i	2.25103i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	38.0043	1.28478
$876$	0	0
$877$	25.4612	0.859765	0.429883	−	0.902885i	$-0.358555\pi$
$877$	25.4612	0.859765	0.429883	+	0.902885i	$0.358555\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	20.8884i	0.703747i	0.936048	+	0.351873i	$0.114455\pi$
$881$	20.8884i	0.703747i	−0.936048	+	0.351873i	$0.885545\pi$
$882$	0	0
$883$	15.3039i	0.515018i	0.966276	+	0.257509i	$0.0829017\pi$
$883$	15.3039i	0.515018i	−0.966276	+	0.257509i	$0.917098\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−27.2289	−0.914258	−0.457129	−	0.889400i	$-0.651122\pi$
$887$	−27.2289	−0.914258	−0.457129	+	0.889400i	$0.651122\pi$
$888$	0	0
$889$	34.2115	1.14742
$890$	0	0
$891$	0	0
$892$	0	0
$893$	38.8336i	1.29952i
$894$	0	0
$895$	33.1312i	1.10745i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−16.3373	−0.544880
$900$	0	0
$901$	34.2933	1.14248
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1.05694i	− 0.0351337i
$906$	0	0
$907$	− 15.7607i	− 0.523326i	−0.965159	−	0.261663i	$-0.915729\pi$
$907$	− 15.7607i	− 0.523326i	0.965159	−	0.261663i	$-0.0842710\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−27.3074	−0.904734	−0.452367	−	0.891832i	$-0.649420\pi$
$911$	−27.3074	−0.904734	−0.452367	+	0.891832i	$0.649420\pi$
$912$	0	0
$913$	−31.2353	−1.03374
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 49.2648i	− 1.62687i
$918$	0	0
$919$	39.7723i	1.31197i	0.754775	+	0.655984i	$0.227746\pi$
$919$	39.7723i	1.31197i	−0.754775	+	0.655984i	$0.772254\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	13.1460	0.432705
$924$	0	0
$925$	4.90829	0.161383
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 34.8487i	− 1.14335i	−0.820481	−	0.571674i	$-0.806294\pi$
$929$	− 34.8487i	− 1.14335i	0.820481	−	0.571674i	$-0.193706\pi$
$930$	0	0
$931$	− 16.3767i	− 0.536723i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−7.19864	−0.235421
$936$	0	0
$937$	34.4177	1.12438	0.562188	−	0.827010i	$-0.309960\pi$
$937$	34.4177	1.12438	0.562188	+	0.827010i	$0.309960\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.21959i	0.0397575i	0.999802	+	0.0198788i	$0.00632802\pi$
$941$	1.21959i	0.0397575i	−0.999802	+	0.0198788i	$0.993672\pi$
$942$	0	0
$943$	− 44.1556i	− 1.43790i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−11.5249	−0.374508	−0.187254	−	0.982312i	$-0.559959\pi$
$947$	−11.5249	−0.374508	−0.187254	+	0.982312i	$0.559959\pi$
$948$	0	0
$949$	−64.5657	−2.09589
$950$	0	0
$951$	0	0
$952$	0	0
$953$	50.5559i	1.63767i	0.574031	+	0.818834i	$0.305379\pi$
$953$	50.5559i	1.63767i	−0.574031	+	0.818834i	$0.694621\pi$
$954$	0	0
$955$	20.1008i	0.650447i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.1403	−0.327446
$960$	0	0
$961$	12.0334	0.388175
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 32.2858i	− 1.03932i
$966$	0	0
$967$	− 12.4644i	− 0.400827i	−0.979711	−	0.200414i	$-0.935771\pi$
$967$	− 12.4644i	− 0.400827i	0.979711	−	0.200414i	$-0.0642286\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	18.9467	0.608029	0.304015	−	0.952667i	$-0.401673\pi$
$971$	18.9467	0.608029	0.304015	+	0.952667i	$0.401673\pi$
$972$	0	0
$973$	28.7393	0.921339
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 5.18431i	− 0.165861i	−0.996555	−	0.0829304i	$-0.973572\pi$
$977$	− 5.18431i	− 0.165861i	0.996555	−	0.0829304i	$-0.0264279\pi$
$978$	0	0
$979$	− 22.7982i	− 0.728635i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25.5419	−0.814660	−0.407330	−	0.913281i	$-0.633540\pi$
$983$	−25.5419	−0.814660	−0.407330	+	0.913281i	$0.633540\pi$
$984$	0	0
$985$	7.80321	0.248631
$986$	0	0
$987$	0	0
$988$	0	0
$989$	33.6462i	1.06989i
$990$	0	0
$991$	39.9173i	1.26801i	0.773327	+	0.634007i	$0.218591\pi$
$991$	39.9173i	1.26801i	−0.773327	+	0.634007i	$0.781409\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	24.4357	0.774664
$996$	0	0
$997$	−4.34276	−0.137537	−0.0687684	−	0.997633i	$-0.521907\pi$
$997$	−4.34276	−0.137537	−0.0687684	+	0.997633i	$0.521907\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.c.b.2591.9	yes	16
3.2	odd	2	inner	2592.2.c.b.2591.7	✓	16
4.3	odd	2	inner	2592.2.c.b.2591.10	yes	16
8.3	odd	2		5184.2.c.l.5183.8		16
8.5	even	2		5184.2.c.l.5183.7		16
9.2	odd	6		2592.2.s.j.863.5		16
9.4	even	3		2592.2.s.i.1727.5		16
9.5	odd	6		2592.2.s.i.1727.4		16
9.7	even	3		2592.2.s.j.863.4		16
12.11	even	2	inner	2592.2.c.b.2591.8	yes	16
24.5	odd	2		5184.2.c.l.5183.9		16
24.11	even	2		5184.2.c.l.5183.10		16
36.7	odd	6		2592.2.s.i.863.4		16
36.11	even	6		2592.2.s.i.863.5		16
36.23	even	6		2592.2.s.j.1727.4		16
36.31	odd	6		2592.2.s.j.1727.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2592.2.c.b.2591.7	✓	16	3.2	odd	2	inner
2592.2.c.b.2591.8	yes	16	12.11	even	2	inner
2592.2.c.b.2591.9	yes	16	1.1	even	1	trivial
2592.2.c.b.2591.10	yes	16	4.3	odd	2	inner
2592.2.s.i.863.4		16	36.7	odd	6
2592.2.s.i.863.5		16	36.11	even	6
2592.2.s.i.1727.4		16	9.5	odd	6
2592.2.s.i.1727.5		16	9.4	even	3
2592.2.s.j.863.4		16	9.7	even	3
2592.2.s.j.863.5		16	9.2	odd	6
2592.2.s.j.1727.4		16	36.23	even	6
2592.2.s.j.1727.5		16	36.31	odd	6
5184.2.c.l.5183.7		16	8.5	even	2
5184.2.c.l.5183.8		16	8.3	odd	2
5184.2.c.l.5183.9		16	24.5	odd	2
5184.2.c.l.5183.10		16	24.11	even	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:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.30185i q^{5} -3.51498i q^{7} +O(q^{10})\)	\(q+1.30185i q^{5} -3.51498i q^{7} +1.81949 q^{11} -5.24703 q^{13} +3.03908i q^{17} +3.05816i q^{19} +5.22199 q^{23} +3.30519 q^{25} -3.75134i q^{29} -4.35506i q^{31} +4.57596 q^{35} +1.48502 q^{37} -8.45569i q^{41} +6.44318i q^{43} +12.6983 q^{47} -5.35506 q^{49} -11.2841i q^{53} +2.36869i q^{55} +0.685926 q^{59} -4.15992 q^{61} -6.83083i q^{65} -12.6613i q^{67} -2.50541 q^{71} +12.3052 q^{73} -6.39545i q^{77} +2.08812i q^{79} -17.1671 q^{83} -3.95641 q^{85} -12.5300i q^{89} +18.4432i q^{91} -3.98127 q^{95} -8.79122 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 8 q^{13} - 48 q^{25} + 72 q^{37} - 48 q^{49} - 56 q^{61} + 96 q^{73} - 24 q^{85} - 32 q^{97}+O(q^{100}) \) 16 * q - 8 * q^13 - 48 * q^25 + 72 * q^37 - 48 * q^49 - 56 * q^61 + 96 * q^73 - 24 * q^85 - 32 * q^97

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