LMFDB - Embedded newform 2064.2.l.h.257.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2064,2,Mod(257,2064)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2064, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2064.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2064 = 2^{4} \cdot 3 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2064.l (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:	$16.4811229772$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 6x^{10} + 29x^{8} + 88x^{6} + 261x^{4} + 486x^{2} + 729 $ x^12 + 6x^10 + 29x^8 + 88x^6 + 261x^4 + 486*x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 129)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			257.6
Root			$-0.583090 - 1.63095i$ of defining polynomial
Character	$\chi$	$=$	2064.257
Dual form			2064.2.l.h.257.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.583090 + 1.63095i) q^{3} +1.38036 q^{5} -0.790787i q^{7} +(-2.32001 - 1.90198i) q^{9} +O(q^{10})$	\(q+(-0.583090 + 1.63095i) q^{3} +1.38036 q^{5} -0.790787i q^{7} +(-2.32001 - 1.90198i) q^{9} +4.15328i q^{11} +3.64002 q^{13} +(-0.804874 + 2.25130i) q^{15} -1.44088i q^{17} +6.14039i q^{19} +(1.28974 + 0.461100i) q^{21} -5.94348i q^{23} -3.09461 q^{25} +(4.45482 - 2.67480i) q^{27} +4.59618 q^{29} +1.15516 q^{31} +(-6.77381 - 2.42174i) q^{33} -1.09157i q^{35} +10.1931i q^{37} +(-2.12246 + 5.93671i) q^{39} +8.82525i q^{41} +(-5.15516 + 4.05269i) q^{43} +(-3.20245 - 2.62542i) q^{45} -5.37060i q^{47} +6.37466 q^{49} +(2.35001 + 0.840166i) q^{51} -5.02128i q^{53} +5.73302i q^{55} +(-10.0147 - 3.58040i) q^{57} +3.80397i q^{59} +4.84348i q^{61} +(-1.50406 + 1.83463i) q^{63} +5.02454 q^{65} +0.515138 q^{67} +(9.69354 + 3.46559i) q^{69} +15.4602 q^{71} +11.8734i q^{73} +(1.80444 - 5.04716i) q^{75} +3.28436 q^{77} -7.60975 q^{79} +(1.76491 + 8.82525i) q^{81} -7.95725i q^{83} -1.98894i q^{85} +(-2.67999 + 7.49615i) q^{87} -0.419798 q^{89} -2.87848i q^{91} +(-0.673563 + 1.88401i) q^{93} +8.47594i q^{95} -11.4995 q^{97} +(7.89948 - 9.63567i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^9	$ 12 q - 12 q^{9} + 12 q^{13} + 8 q^{15} - 4 q^{21} - 16 q^{31} - 32 q^{43} - 24 q^{49} + 12 q^{57} + 8 q^{67} - 56 q^{79} - 44 q^{81} - 48 q^{87} - 4 q^{97} + 40 q^{99}+O(q^{100}) $ 12 * q - 12 * q^9 + 12 * q^13 + 8 * q^15 - 4 * q^21 - 16 * q^31 - 32 * q^43 - 24 * q^49 + 12 * q^57 + 8 * q^67 - 56 * q^79 - 44 * q^81 - 48 * q^87 - 4 * q^97 + 40 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2064\mathbb{Z}\right)^\times$.

$n$	$433$	$517$	$689$	$1807$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	−0.583090	+	1.63095i	−0.336647	+	0.941631i
$3$	−0.583090	+	1.63095i	−0.336647	+	0.941631i
$4$		0			0
$5$	1.38036			0.617315			0.308658	−	0.951173i	$-0.400120\pi$
$5$	1.38036			0.617315			0.308658	+	0.951173i	$0.400120\pi$
$6$		0			0
$7$		−	0.790787i		−	0.298889i	−0.988770	−	0.149445i	$-0.952251\pi$
$7$		−	0.790787i		−	0.298889i	0.988770	−	0.149445i	$-0.0477486\pi$
$8$		0			0
$9$	−2.32001	−	1.90198i	−0.773337	−	0.633995i
$10$		0			0
$11$			4.15328i			1.25226i	0.779718	+	0.626131i	$0.215363\pi$
$11$			4.15328i			1.25226i	−0.779718	+	0.626131i	$0.784637\pi$
$12$		0			0
$13$	3.64002			1.00956			0.504780	−	0.863248i	$-0.331573\pi$
$13$	3.64002			1.00956			0.504780	+	0.863248i	$0.331573\pi$
$14$		0			0
$15$	−0.804874	+	2.25130i	−0.207817	+	0.581283i
$16$		0			0
$17$		−	1.44088i		−	0.349466i	−0.984616	−	0.174733i	$-0.944094\pi$
$17$		−	1.44088i		−	0.349466i	0.984616	−	0.174733i	$-0.0559062\pi$
$18$		0			0
$19$			6.14039i			1.40870i	0.709852	+	0.704351i	$0.248762\pi$
$19$			6.14039i			1.40870i	−0.709852	+	0.704351i	$0.751238\pi$
$20$		0			0
$21$	1.28974	+	0.461100i	0.281443	+	0.100620i
$22$		0			0
$23$		−	5.94348i		−	1.23930i	−0.784877	−	0.619651i	$-0.787274\pi$
$23$		−	5.94348i		−	1.23930i	0.784877	−	0.619651i	$-0.212726\pi$
$24$		0			0
$25$	−3.09461			−0.618922
$26$		0			0
$27$	4.45482	−	2.67480i	0.857331	−	0.514766i
$28$		0			0
$29$	4.59618			0.853489			0.426745	−	0.904372i	$-0.359660\pi$
$29$	4.59618			0.853489			0.426745	+	0.904372i	$0.359660\pi$
$30$		0			0
$31$	1.15516			0.207473			0.103737	−	0.994605i	$-0.466920\pi$
$31$	1.15516			0.207473			0.103737	+	0.994605i	$0.466920\pi$
$32$		0			0
$33$	−6.77381	−	2.42174i	−1.17917	−	0.421571i
$34$		0			0
$35$		−	1.09157i		−	0.184509i
$36$		0			0
$37$			10.1931i			1.67573i	0.545876	+	0.837866i	$0.316197\pi$
$37$			10.1931i			1.67573i	−0.545876	+	0.837866i	$0.683803\pi$
$38$		0			0
$39$	−2.12246	+	5.93671i	−0.339866	+	0.950634i
$40$		0			0
$41$			8.82525i			1.37827i	0.724632	+	0.689137i	$0.242010\pi$
$41$			8.82525i			1.37827i	−0.724632	+	0.689137i	$0.757990\pi$
$42$		0			0
$43$	−5.15516	+	4.05269i	−0.786155	+	0.618030i
$43$	−5.15516	+	4.05269i	−0.786155	+	0.618030i
$44$		0			0
$45$	−3.20245	−	2.62542i	−0.477393	−	0.391375i
$46$		0			0
$47$		−	5.37060i		−	0.783382i	−0.920097	−	0.391691i	$-0.871890\pi$
$47$		−	5.37060i		−	0.783382i	0.920097	−	0.391691i	$-0.128110\pi$
$48$		0			0
$49$	6.37466			0.910665
$50$		0			0
$51$	2.35001	+	0.840166i	0.329068	+	0.117647i
$52$		0			0
$53$		−	5.02128i		−	0.689726i	−0.938653	−	0.344863i	$-0.887925\pi$
$53$		−	5.02128i		−	0.689726i	0.938653	−	0.344863i	$-0.112075\pi$
$54$		0			0
$55$			5.73302i			0.773041i
$56$		0			0
$57$	−10.0147	−	3.58040i	−1.32648	−	0.474235i
$58$		0			0
$59$			3.80397i			0.495235i	0.968858	+	0.247617i	$0.0796476\pi$
$59$			3.80397i			0.495235i	−0.968858	+	0.247617i	$0.920352\pi$
$60$		0			0
$61$			4.84348i			0.620144i	0.950713	+	0.310072i	$0.100353\pi$
$61$			4.84348i			0.620144i	−0.950713	+	0.310072i	$0.899647\pi$
$62$		0			0
$63$	−1.50406	+	1.83463i	−0.189494	+	0.231142i
$64$		0			0
$65$	5.02454			0.623217
$66$		0			0
$67$	0.515138			0.0629341			0.0314671	−	0.999505i	$-0.489982\pi$
$67$	0.515138			0.0629341			0.0314671	+	0.999505i	$0.489982\pi$
$68$		0			0
$69$	9.69354	+	3.46559i	1.16697	+	0.417208i
$70$		0			0
$71$	15.4602			1.83479			0.917393	−	0.397983i	$-0.130290\pi$
$71$	15.4602			1.83479			0.917393	+	0.397983i	$0.130290\pi$
$72$		0			0
$73$			11.8734i			1.38968i	0.719166	+	0.694839i	$0.244524\pi$
$73$			11.8734i			1.38968i	−0.719166	+	0.694839i	$0.755476\pi$
$74$		0			0
$75$	1.80444	−	5.04716i	0.208358	−	0.582796i
$76$		0			0
$77$	3.28436			0.374288
$78$		0			0
$79$	−7.60975			−0.856163			−0.428082	−	0.903740i	$-0.640810\pi$
$79$	−7.60975			−0.856163			−0.428082	+	0.903740i	$0.640810\pi$
$80$		0			0
$81$	1.76491	+	8.82525i	0.196101	+	0.980584i
$82$		0			0
$83$		−	7.95725i		−	0.873422i	−0.899602	−	0.436711i	$-0.856143\pi$
$83$		−	7.95725i		−	0.873422i	0.899602	−	0.436711i	$-0.143857\pi$
$84$		0			0
$85$		−	1.98894i		−	0.215731i
$86$		0			0
$87$	−2.67999	+	7.49615i	−0.287325	+	0.803672i
$88$		0			0
$89$	−0.419798			−0.0444985			−0.0222492	−	0.999752i	$-0.507083\pi$
$89$	−0.419798			−0.0444985			−0.0222492	+	0.999752i	$0.507083\pi$
$90$		0			0
$91$		−	2.87848i		−	0.301747i
$92$		0			0
$93$	−0.673563	+	1.88401i	−0.0698453	+	0.195363i
$94$		0			0
$95$			8.47594i			0.869613i
$96$		0			0
$97$	−11.4995			−1.16760			−0.583801	−	0.811897i	$-0.698435\pi$
$97$	−11.4995			−1.16760			−0.583801	+	0.811897i	$0.698435\pi$
$98$		0			0
$99$	7.89948	−	9.63567i	0.793928	−	0.968421i
$100$		0			0
$101$			6.16705i			0.613645i	0.951767	+	0.306822i	$0.0992658\pi$
$101$			6.16705i			0.613645i	−0.951767	+	0.306822i	$0.900734\pi$
$102$		0			0
$103$	1.54541			0.152274			0.0761371	−	0.997097i	$-0.475741\pi$
$103$	1.54541			0.152274			0.0761371	+	0.997097i	$0.475741\pi$
$104$		0			0
$105$	1.78030	+	0.636483i	0.173739	+	0.0621144i
$106$		0			0
$107$			16.5589i			1.60081i	0.599458	+	0.800406i	$0.295383\pi$
$107$			16.5589i			1.60081i	−0.599458	+	0.800406i	$0.704617\pi$
$108$		0			0
$109$	−5.57947			−0.534416			−0.267208	−	0.963639i	$-0.586101\pi$
$109$	−5.57947			−0.534416			−0.267208	+	0.963639i	$0.586101\pi$
$110$		0			0
$111$	−16.6244	−	5.94348i	−1.57792	−	0.564131i
$112$		0			0
$113$	−3.11198			−0.292750			−0.146375	−	0.989229i	$-0.546761\pi$
$113$	−3.11198			−0.292750			−0.146375	+	0.989229i	$0.546761\pi$
$114$		0			0
$115$		−	8.20414i		−	0.765040i
$116$		0			0
$117$	−8.44490	−	6.92327i	−0.780731	−	0.640056i
$118$		0			0
$119$	−1.13943			−0.104452
$120$		0			0
$121$	−6.24977			−0.568161
$122$		0			0
$123$	−14.3936	−	5.14592i	−1.29782	−	0.463992i
$124$		0			0
$125$	−11.1735			−0.999385
$126$		0			0
$127$	−10.2947			−0.913509			−0.456755	−	0.889593i	$-0.650988\pi$
$127$	−10.2947			−0.913509			−0.456755	+	0.889593i	$0.650988\pi$
$128$		0			0
$129$	−3.60382	−	10.7709i	−0.317299	−	0.948326i
$130$		0			0
$131$	−21.1454			−1.84749			−0.923743	−	0.383013i	$-0.874887\pi$
$131$	−21.1454			−1.84749			−0.923743	+	0.383013i	$0.874887\pi$
$132$		0			0
$133$	4.85574			0.421046
$134$		0			0
$135$	6.14925	−	3.69218i	0.529244	−	0.317773i
$136$		0			0
$137$	11.9198			1.01838			0.509191	−	0.860654i	$-0.329945\pi$
$137$	11.9198			1.01838			0.509191	+	0.860654i	$0.329945\pi$
$138$		0			0
$139$	14.6850			1.24556			0.622782	−	0.782396i	$-0.286002\pi$
$139$	14.6850			1.24556			0.622782	+	0.782396i	$0.286002\pi$
$140$		0			0
$141$	8.75919	+	3.13154i	0.737657	+	0.263724i
$142$		0			0
$143$			15.1181i			1.26424i
$144$		0			0
$145$	6.34438			0.526872
$146$		0			0
$147$	−3.71700	+	10.3968i	−0.306573	+	0.857510i
$148$		0			0
$149$	−6.08687			−0.498656			−0.249328	−	0.968419i	$-0.580210\pi$
$149$	−6.08687			−0.498656			−0.249328	+	0.968419i	$0.580210\pi$
$150$		0			0
$151$			13.9611i			1.13614i	0.822981	+	0.568069i	$0.192309\pi$
$151$			13.9611i			1.13614i	−0.822981	+	0.568069i	$0.807691\pi$
$152$		0			0
$153$	−2.74054	+	3.34287i	−0.221560	+	0.270255i
$154$		0			0
$155$	1.59454			0.128076
$156$		0			0
$157$			2.85454i			0.227817i	0.993491	+	0.113909i	$0.0363371\pi$
$157$			2.85454i			0.227817i	−0.993491	+	0.113909i	$0.963663\pi$
$158$		0			0
$159$	8.18948	+	2.92786i	0.649468	+	0.232195i
$160$		0			0
$161$	−4.70003			−0.370414
$162$		0			0
$163$		−	3.64533i		−	0.285524i	−0.989757	−	0.142762i	$-0.954402\pi$
$163$		−	3.64533i		−	0.285524i	0.989757	−	0.142762i	$-0.0455984\pi$
$164$		0			0
$165$	−9.35029	−	3.34287i	−0.727919	−	0.260242i
$166$		0			0
$167$		−	12.5750i		−	0.973084i	−0.873657	−	0.486542i	$-0.838258\pi$
$167$		−	12.5750i		−	0.973084i	0.873657	−	0.486542i	$-0.161742\pi$
$168$		0			0
$169$	0.249771			0.0192131
$170$		0			0
$171$	11.6789	−	14.2458i	0.893110	−	1.08940i
$172$		0			0
$173$		−	19.4949i		−	1.48217i	−0.671411	−	0.741085i	$-0.734311\pi$
$173$		−	19.4949i		−	1.48217i	0.671411	−	0.741085i	$-0.265689\pi$
$174$		0			0
$175$			2.44718i			0.184989i
$176$		0			0
$177$	−6.20409	−	2.21806i	−0.466328	−	0.166719i
$178$		0			0
$179$	−3.67949			−0.275018			−0.137509	−	0.990501i	$-0.543910\pi$
$179$	−3.67949			−0.275018			−0.137509	+	0.990501i	$0.543910\pi$
$180$		0			0
$181$	10.8751			0.808341			0.404170	−	0.914684i	$-0.367560\pi$
$181$	10.8751			0.808341			0.404170	+	0.914684i	$0.367560\pi$
$182$		0			0
$183$	−7.89948	−	2.82418i	−0.583947	−	0.208770i
$184$		0			0
$185$			14.0701i			1.03445i
$186$		0			0
$187$	5.98440			0.437623
$188$		0			0
$189$	−2.11520	−	3.52281i	−0.153858	−	0.256247i
$190$		0			0
$191$	4.28023			0.309707			0.154853	−	0.987937i	$-0.450510\pi$
$191$	4.28023			0.309707			0.154853	+	0.987937i	$0.450510\pi$
$192$		0			0
$193$	−14.2342			−1.02460			−0.512299	−	0.858807i	$-0.671206\pi$
$193$	−14.2342			−1.02460			−0.512299	+	0.858807i	$0.671206\pi$
$194$		0			0
$195$	−2.92976	+	8.19478i	−0.209804	+	0.586841i
$196$		0			0
$197$			19.9700i			1.42280i	0.702787	+	0.711401i	$0.251939\pi$
$197$			19.9700i			1.42280i	−0.702787	+	0.711401i	$0.748061\pi$
$198$		0			0
$199$			9.37835i			0.664814i	0.943136	+	0.332407i	$0.107861\pi$
$199$			9.37835i			0.664814i	−0.943136	+	0.332407i	$0.892139\pi$
$200$		0			0
$201$	−0.300372	+	0.840166i	−0.0211866	+	0.0592607i
$202$		0			0
$203$		−	3.63460i		−	0.255099i
$204$		0			0
$205$			12.1820i			0.850829i
$206$		0			0
$207$	−11.3044	+	13.7890i	−0.785711	+	0.958398i
$208$		0			0
$209$	−25.5028			−1.76406
$210$		0			0
$211$		−	8.22808i		−	0.566445i	−0.959054	−	0.283222i	$-0.908597\pi$
$211$		−	8.22808i		−	0.566445i	0.959054	−	0.283222i	$-0.0914034\pi$
$212$		0			0
$213$	−9.01468	+	25.2148i	−0.617676	+	1.72769i
$214$		0			0
$215$	−7.11597	+	5.59417i	−0.485305	+	0.381519i
$216$		0			0
$217$		−	0.913486i		−	0.0620115i
$218$		0			0
$219$	−19.3650	−	6.92327i	−1.30856	−	0.467831i
$220$		0			0
$221$		−	5.24485i		−	0.352807i
$222$		0			0
$223$			4.15145i			0.278002i	0.990292	+	0.139001i	$0.0443891\pi$
$223$			4.15145i			0.278002i	−0.990292	+	0.139001i	$0.955611\pi$
$224$		0			0
$225$	7.17953	+	5.88590i	0.478635	+	0.392393i
$226$		0			0
$227$	21.4635			1.42458			0.712290	−	0.701885i	$-0.247658\pi$
$227$	21.4635			1.42458			0.712290	+	0.701885i	$0.247658\pi$
$228$		0			0
$229$	15.8439			1.04700			0.523498	−	0.852027i	$-0.324627\pi$
$229$	15.8439			1.04700			0.523498	+	0.852027i	$0.324627\pi$
$230$		0			0
$231$	−1.91508	+	5.35664i	−0.126003	+	0.352441i
$232$		0			0
$233$	−6.05364			−0.396587			−0.198294	−	0.980143i	$-0.563540\pi$
$233$	−6.05364			−0.396587			−0.198294	+	0.980143i	$0.563540\pi$
$234$		0			0
$235$		−	7.41335i		−	0.483594i
$236$		0			0
$237$	4.43717	−	12.4111i	0.288225	−	0.806190i
$238$		0			0
$239$			10.0968i			0.653106i	0.945179	+	0.326553i	$0.105887\pi$
$239$			10.0968i			0.653106i	−0.945179	+	0.326553i	$0.894113\pi$
$240$		0			0
$241$		−	18.8046i		−	1.21131i	−0.795727	−	0.605655i	$-0.792911\pi$
$241$		−	18.8046i		−	1.21131i	0.795727	−	0.605655i	$-0.207089\pi$
$242$		0			0
$243$	−15.4227	−	2.26744i	−0.989365	−	0.145456i
$244$		0			0
$245$	8.79931			0.562168
$246$		0			0
$247$			22.3512i			1.42217i
$248$		0			0
$249$	12.9779	+	4.63980i	0.822441	+	0.294035i
$250$		0			0
$251$		−	8.43231i		−	0.532243i	−0.963940	−	0.266121i	$-0.914258\pi$
$251$		−	8.43231i		−	0.532243i	0.963940	−	0.266121i	$-0.0857422\pi$
$252$		0			0
$253$	24.6850			1.55193
$254$		0			0
$255$	3.24386	+	1.15973i	0.203139	+	0.0726251i
$256$		0			0
$257$	20.0564			1.25108			0.625541	−	0.780191i	$-0.284878\pi$
$257$	20.0564			1.25108			0.625541	+	0.780191i	$0.284878\pi$
$258$		0			0
$259$	8.06055			0.500858
$260$		0			0
$261$	−10.6632	−	8.74187i	−0.660035	−	0.541108i
$262$		0			0
$263$	9.75780			0.601692			0.300846	−	0.953673i	$-0.402731\pi$
$263$	9.75780			0.601692			0.300846	+	0.953673i	$0.402731\pi$
$264$		0			0
$265$		−	6.93117i		−	0.425779i
$266$		0			0
$267$	0.244780	−	0.684670i	0.0149803	−	0.0419011i
$268$		0			0
$269$		−	10.2225i		−	0.623278i	−0.950201	−	0.311639i	$-0.899122\pi$
$269$		−	10.2225i		−	0.623278i	0.950201	−	0.311639i	$-0.100878\pi$
$270$		0			0
$271$	1.73463			0.105371			0.0526857	−	0.998611i	$-0.483222\pi$
$271$	1.73463			0.105371			0.0526857	+	0.998611i	$0.483222\pi$
$272$		0			0
$273$	4.69467	+	1.67841i	0.284134	+	0.101582i
$274$		0			0
$275$		−	12.8528i		−	0.775053i
$276$		0			0
$277$			26.5026i			1.59239i	0.605042	+	0.796194i	$0.293156\pi$
$277$			26.5026i			1.59239i	−0.605042	+	0.796194i	$0.706844\pi$
$278$		0			0
$279$	−2.67999	−	2.19710i	−0.160447	−	0.131537i
$280$		0			0
$281$		−	4.79771i		−	0.286208i	−0.989708	−	0.143104i	$-0.954292\pi$
$281$		−	4.79771i		−	0.286208i	0.989708	−	0.143104i	$-0.0457083\pi$
$282$		0			0
$283$	12.1249			0.720750			0.360375	−	0.932808i	$-0.382649\pi$
$283$	12.1249			0.720750			0.360375	+	0.932808i	$0.382649\pi$
$284$		0			0
$285$	−13.8239	−	4.94224i	−0.818854	−	0.292753i
$286$		0			0
$287$	6.97889			0.411951
$288$		0			0
$289$	14.9239			0.877874
$290$		0			0
$291$	6.70527	−	18.7552i	0.393070	−	1.09945i
$292$		0			0
$293$		−	17.8741i		−	1.04421i	−0.852880	−	0.522107i	$-0.825146\pi$
$293$		−	17.8741i		−	1.04421i	0.852880	−	0.522107i	$-0.174854\pi$
$294$		0			0
$295$			5.25084i			0.305716i
$296$		0			0
$297$	11.1092	+	18.5021i	0.644622	+	1.07360i
$298$		0			0
$299$		−	21.6344i		−	1.25115i
$300$		0			0
$301$	3.20482	+	4.07663i	0.184722	+	0.234973i
$302$		0			0
$303$	−10.0582	−	3.59595i	−0.577827	−	0.206582i
$304$		0			0
$305$			6.68574i			0.382824i
$306$		0			0
$307$	12.1854			0.695460			0.347730	−	0.937595i	$-0.386953\pi$
$307$	12.1854			0.695460			0.347730	+	0.937595i	$0.386953\pi$
$308$		0			0
$309$	−0.901116	+	2.52050i	−0.0512627	+	0.143386i
$310$		0			0
$311$			4.15328i			0.235511i	0.993043	+	0.117756i	$0.0375699\pi$
$311$			4.15328i			0.235511i	−0.993043	+	0.117756i	$0.962430\pi$
$312$		0			0
$313$		−	5.61032i		−	0.317114i	−0.987350	−	0.158557i	$-0.949316\pi$
$313$		−	5.61032i		−	0.317114i	0.987350	−	0.158557i	$-0.0506842\pi$
$314$		0			0
$315$	−2.07615	+	2.53245i	−0.116978	+	0.142688i
$316$		0			0
$317$		−	20.7122i		−	1.16331i	−0.813434	−	0.581657i	$-0.802405\pi$
$317$		−	20.7122i		−	1.16331i	0.813434	−	0.581657i	$-0.197595\pi$
$318$		0			0
$319$			19.0892i			1.06879i
$320$		0			0
$321$	−27.0068	−	9.65535i	−1.50737	−	0.538909i
$322$		0			0
$323$	8.84759			0.492293
$324$		0			0
$325$	−11.2645			−0.624839
$326$		0			0
$327$	3.25333	−	9.09985i	0.179910	−	0.503223i
$328$		0			0
$329$	−4.24700			−0.234145
$330$		0			0
$331$			7.62320i			0.419009i	0.977808	+	0.209505i	$0.0671851\pi$
$331$			7.62320i			0.419009i	−0.977808	+	0.209505i	$0.932815\pi$
$332$		0			0
$333$	19.3871	−	23.6481i	1.06241	−	1.29591i
$334$		0			0
$335$	0.711075			0.0388502
$336$		0			0
$337$	19.8898			1.08347			0.541733	−	0.840551i	$-0.317768\pi$
$337$	19.8898			1.08347			0.541733	+	0.840551i	$0.317768\pi$
$338$		0			0
$339$	1.81456	−	5.07548i	0.0985535	−	0.275662i
$340$		0			0
$341$			4.79771i			0.259811i
$342$		0			0
$343$		−	10.5765i		−	0.571077i
$344$		0			0
$345$	13.3806	+	4.78375i	0.720385	+	0.257549i
$346$		0			0
$347$	−27.8502			−1.49508			−0.747538	−	0.664219i	$-0.768764\pi$
$347$	−27.8502			−1.49508			−0.747538	+	0.664219i	$0.768764\pi$
$348$		0			0
$349$		−	23.3395i		−	1.24933i	−0.780892	−	0.624666i	$-0.785235\pi$
$349$		−	23.3395i		−	1.24933i	0.780892	−	0.624666i	$-0.214765\pi$
$350$		0			0
$351$	16.2157	−	9.73634i	0.865528	−	0.519687i
$352$		0			0
$353$		−	27.2828i		−	1.45212i	−0.687633	−	0.726058i	$-0.741350\pi$
$353$		−	27.2828i		−	1.45212i	0.687633	−	0.726058i	$-0.258650\pi$
$354$		0			0
$355$	21.3406			1.13264
$356$		0			0
$357$	0.664392	−	1.85836i	0.0351633	−	0.0983549i
$358$		0			0
$359$			0.627085i			0.0330963i	0.999863	+	0.0165481i	$0.00526768\pi$
$359$			0.627085i			0.0330963i	−0.999863	+	0.0165481i	$0.994732\pi$
$360$		0			0
$361$	−18.7044			−0.984440
$362$		0			0
$363$	3.64418	−	10.1931i	0.191270	−	0.534998i
$364$		0			0
$365$			16.3896i			0.857869i
$366$		0			0
$367$	16.2909			0.850380			0.425190	−	0.905104i	$-0.360207\pi$
$367$	16.2909			0.850380			0.425190	+	0.905104i	$0.360207\pi$
$368$		0			0
$369$	16.7855	−	20.4747i	0.873818	−	1.06587i
$370$		0			0
$371$	−3.97077			−0.206152
$372$		0			0
$373$		−	7.53605i		−	0.390202i	−0.980783	−	0.195101i	$-0.937497\pi$
$373$		−	7.53605i		−	0.390202i	0.980783	−	0.195101i	$-0.0625035\pi$
$374$		0			0
$375$	6.51514	−	18.2234i	0.336440	−	0.941052i
$376$		0			0
$377$	16.7302			0.861650
$378$		0			0
$379$	29.0440			1.49189			0.745946	−	0.666006i	$-0.231998\pi$
$379$	29.0440			1.49189			0.745946	+	0.666006i	$0.231998\pi$
$380$		0			0
$381$	6.00275	−	16.7902i	0.307530	−	0.860188i
$382$		0			0
$383$	−8.98467			−0.459095			−0.229548	−	0.973297i	$-0.573725\pi$
$383$	−8.98467			−0.459095			−0.229548	+	0.973297i	$0.573725\pi$
$384$		0			0
$385$	4.53360			0.231054
$386$		0			0
$387$	19.6682	+	0.402746i	0.999790	+	0.0204727i
$388$		0			0
$389$	−27.6542			−1.40212			−0.701062	−	0.713101i	$-0.747290\pi$
$389$	−27.6542			−1.40212			−0.701062	+	0.713101i	$0.747290\pi$
$390$		0			0
$391$	−8.56387			−0.433094
$392$		0			0
$393$	12.3297	−	34.4872i	0.621951	−	1.73965i
$394$		0			0
$395$	−10.5042			−0.528523
$396$		0			0
$397$	−4.28383			−0.214999			−0.107500	−	0.994205i	$-0.534284\pi$
$397$	−4.28383			−0.214999			−0.107500	+	0.994205i	$0.534284\pi$
$398$		0			0
$399$	−2.83133	+	7.91948i	−0.141744	+	0.396470i
$400$		0			0
$401$			13.1043i			0.654397i	0.944956	+	0.327198i	$0.106105\pi$
$401$			13.1043i			0.654397i	−0.944956	+	0.327198i	$0.893895\pi$
$402$		0			0
$403$	4.20482			0.209457
$404$		0			0
$405$	2.43621	+	12.1820i	0.121056	+	0.605329i
$406$		0			0
$407$	−42.3348			−2.09846
$408$		0			0
$409$			0.913486i			0.0451690i	0.999745	+	0.0225845i	$0.00718948\pi$
$409$			0.913486i			0.0451690i	−0.999745	+	0.0225845i	$0.992811\pi$
$410$		0			0
$411$	−6.95035	+	19.4407i	−0.342835	+	0.958939i
$412$		0			0
$413$	3.00813			0.148020
$414$		0			0
$415$		−	10.9839i		−	0.539177i
$416$		0			0
$417$	−8.56267	+	23.9505i	−0.419316	+	1.17286i
$418$		0			0
$419$	19.6345			0.959208			0.479604	−	0.877485i	$-0.340780\pi$
$419$	19.6345			0.959208			0.479604	+	0.877485i	$0.340780\pi$
$420$		0			0
$421$		−	1.89018i		−	0.0921217i	−0.998939	−	0.0460609i	$-0.985333\pi$
$421$		−	1.89018i		−	0.0921217i	0.998939	−	0.0460609i	$-0.0146668\pi$
$422$		0			0
$423$	−10.2148	+	12.4599i	−0.496660	+	0.605819i
$424$		0			0
$425$			4.45898i			0.216292i
$426$		0			0
$427$	3.83016			0.185354
$428$		0			0
$429$	−24.6568	−	8.81519i	−1.19044	−	0.425601i
$430$		0			0
$431$		−	38.4169i		−	1.85048i	−0.379384	−	0.925239i	$-0.623864\pi$
$431$		−	38.4169i		−	1.85048i	0.379384	−	0.925239i	$-0.376136\pi$
$432$		0			0
$433$		−	23.8456i		−	1.14595i	−0.819574	−	0.572973i	$-0.805790\pi$
$433$		−	23.8456i		−	1.14595i	0.819574	−	0.572973i	$-0.194210\pi$
$434$		0			0
$435$	−3.69935	+	10.3474i	−0.177370	+	0.496119i
$436$		0			0
$437$	36.4953			1.74581
$438$		0			0
$439$	1.71526			0.0818647			0.0409323	−	0.999162i	$-0.486967\pi$
$439$	1.71526			0.0818647			0.0409323	+	0.999162i	$0.486967\pi$
$440$		0			0
$441$	−14.7893	−	12.1245i	−0.704251	−	0.577357i
$442$		0			0
$443$		−	30.7374i		−	1.46038i	−0.683244	−	0.730190i	$-0.739432\pi$
$443$		−	30.7374i		−	1.46038i	0.683244	−	0.730190i	$-0.260568\pi$
$444$		0			0
$445$	−0.579471			−0.0274696
$446$		0			0
$447$	3.54920	−	9.92740i	0.167871	−	0.469550i
$448$		0			0
$449$	34.7969			1.64217			0.821084	−	0.570808i	$-0.193370\pi$
$449$	34.7969			1.64217			0.821084	+	0.570808i	$0.193370\pi$
$450$		0			0
$451$	−36.6538			−1.72596
$452$		0			0
$453$	−22.7699	−	8.14058i	−1.06982	−	0.382478i
$454$		0			0
$455$		−	3.97334i		−	0.186273i
$456$		0			0
$457$		−	4.33736i		−	0.202893i	−0.994841	−	0.101446i	$-0.967653\pi$
$457$		−	4.33736i		−	0.202893i	0.994841	−	0.101446i	$-0.0323471\pi$
$458$		0			0
$459$	−3.85408	−	6.41889i	−0.179893	−	0.299608i
$460$		0			0
$461$			16.0297i			0.746576i	0.927715	+	0.373288i	$0.121770\pi$
$461$			16.0297i			0.746576i	−0.927715	+	0.373288i	$0.878230\pi$
$462$		0			0
$463$		−	9.09369i		−	0.422619i	−0.977419	−	0.211310i	$-0.932227\pi$
$463$		−	9.09369i		−	0.422619i	0.977419	−	0.211310i	$-0.0677729\pi$
$464$		0			0
$465$	−0.929759	+	2.60061i	−0.0431165	+	0.120601i
$466$		0			0
$467$	8.32602			0.385282			0.192641	−	0.981269i	$-0.438295\pi$
$467$	8.32602			0.385282			0.192641	+	0.981269i	$0.438295\pi$
$468$		0			0
$469$		−	0.407364i		−	0.0188103i
$470$		0			0
$471$	−4.65562	−	1.66445i	−0.214520	−	0.0766940i
$472$		0			0
$473$	−16.8320	−	21.4109i	−0.773935	−	0.984472i
$474$		0			0
$475$		−	19.0021i		−	0.871876i
$476$		0			0
$477$	−9.55041	+	11.6494i	−0.437283	+	0.533391i
$478$		0			0
$479$			39.2307i			1.79250i	0.443552	+	0.896249i	$0.353718\pi$
$479$			39.2307i			1.79250i	−0.443552	+	0.896249i	$0.646282\pi$
$480$		0			0
$481$			37.1030i			1.69175i
$482$		0			0
$483$	2.74054	−	7.66552i	0.124699	−	0.348793i
$484$		0			0
$485$	−15.8735			−0.720778
$486$		0			0
$487$	19.0984			0.865431			0.432715	−	0.901531i	$-0.357556\pi$
$487$	19.0984			0.865431			0.432715	+	0.901531i	$0.357556\pi$
$488$		0			0
$489$	5.94536	+	2.12555i	0.268858	+	0.0961209i
$490$		0			0
$491$	−4.49233			−0.202736			−0.101368	−	0.994849i	$-0.532322\pi$
$491$	−4.49233			−0.202736			−0.101368	+	0.994849i	$0.532322\pi$
$492$		0			0
$493$		−	6.62257i		−	0.298265i
$494$		0			0
$495$	10.9041	−	13.3007i	0.490104	−	0.597821i
$496$		0			0
$497$		−	12.2257i		−	0.548398i
$498$		0			0
$499$		−	41.7606i		−	1.86946i	−0.355356	−	0.934731i	$-0.615640\pi$
$499$		−	41.7606i		−	1.86946i	0.355356	−	0.934731i	$-0.384360\pi$
$500$		0			0
$501$	20.5093	+	7.33237i	0.916286	+	0.327586i
$502$		0			0
$503$	3.35290			0.149499			0.0747493	−	0.997202i	$-0.476184\pi$
$503$	3.35290			0.149499			0.0747493	+	0.997202i	$0.476184\pi$
$504$		0			0
$505$			8.51275i			0.378812i
$506$		0			0
$507$	−0.145639	+	0.407364i	−0.00646805	+	0.0180917i
$508$		0			0
$509$			30.9784i			1.37309i	0.727087	+	0.686546i	$0.240874\pi$
$509$			30.9784i			1.37309i	−0.727087	+	0.686546i	$0.759126\pi$
$510$		0			0
$511$	9.38934			0.415360
$512$		0			0
$513$	16.4243	+	27.3543i	0.725151	+	1.20772i
$514$		0			0
$515$	2.13323			0.0940012
$516$		0			0
$517$	22.3056			0.981000
$518$		0			0
$519$	31.7953	+	11.3673i	1.39566	+	0.498969i
$520$		0			0
$521$	−13.6912			−0.599822			−0.299911	−	0.953967i	$-0.596957\pi$
$521$	−13.6912			−0.599822			−0.299911	+	0.953967i	$0.596957\pi$
$522$		0			0
$523$		−	22.4499i		−	0.981666i	−0.871254	−	0.490833i	$-0.836692\pi$
$523$		−	22.4499i		−	0.981666i	0.871254	−	0.490833i	$-0.163308\pi$
$524$		0			0
$525$	−3.99123	−	1.42692i	−0.174191	−	0.0622761i
$526$		0			0
$527$		−	1.66445i		−	0.0725048i
$528$		0			0
$529$	−12.3250			−0.535870
$530$		0			0
$531$	7.23509	−	8.82525i	0.313976	−	0.382983i
$532$		0			0
$533$			32.1241i			1.39145i
$534$		0			0
$535$			22.8573i			0.988206i
$536$		0			0
$537$	2.14547	−	6.00107i	0.0925840	−	0.258965i
$538$		0			0
$539$			26.4758i			1.14039i
$540$		0			0
$541$	24.2947			1.04451			0.522256	−	0.852789i	$-0.325091\pi$
$541$	24.2947			1.04451			0.522256	+	0.852789i	$0.325091\pi$
$542$		0			0
$543$	−6.34117	+	17.7368i	−0.272126	+	0.761159i
$544$		0			0
$545$	−7.70167			−0.329903
$546$		0			0
$547$	31.9726			1.36705			0.683525	−	0.729927i	$-0.260446\pi$
$547$	31.9726			1.36705			0.683525	+	0.729927i	$0.260446\pi$
$548$		0			0
$549$	9.21222	−	11.2369i	0.393168	−	0.479580i
$550$		0			0
$551$			28.2223i			1.20231i
$552$		0			0
$553$			6.01769i			0.255898i
$554$		0			0
$555$	−22.9477	−	8.20414i	−0.974075	−	0.348246i
$556$		0			0
$557$		−	15.6473i		−	0.662998i	−0.943456	−	0.331499i	$-0.892446\pi$
$557$		−	15.6473i		−	0.662998i	0.943456	−	0.331499i	$-0.107554\pi$
$558$		0			0
$559$	−18.7649	+	14.7519i	−0.793671	+	0.623939i
$560$		0			0
$561$	−3.48945	+	9.76028i	−0.147325	+	0.412079i
$562$		0			0
$563$			13.0764i			0.551103i	0.961286	+	0.275551i	$0.0888605\pi$
$563$			13.0764i			0.551103i	−0.961286	+	0.275551i	$0.911140\pi$
$564$		0			0
$565$	−4.29564			−0.180719
$566$		0			0
$567$	6.97889	−	1.39567i	0.293086	−	0.0586125i
$568$		0			0
$569$		−	3.28528i		−	0.137726i	−0.997626	−	0.0688631i	$-0.978063\pi$
$569$		−	3.28528i		−	0.137726i	0.997626	−	0.0688631i	$-0.0219372\pi$
$570$		0			0
$571$			9.40229i			0.393474i	0.980456	+	0.196737i	$0.0630345\pi$
$571$			9.40229i			0.393474i	−0.980456	+	0.196737i	$0.936966\pi$
$572$		0			0
$573$	−2.49576	+	6.98085i	−0.104262	+	0.291629i
$574$		0			0
$575$			18.3928i			0.767031i
$576$		0			0
$577$		−	11.6759i		−	0.486074i	−0.970017	−	0.243037i	$-0.921856\pi$
$577$		−	11.6759i		−	0.486074i	0.970017	−	0.243037i	$-0.0781436\pi$
$578$		0			0
$579$	8.29981	−	23.2153i	0.344928	−	0.964793i
$580$		0			0
$581$	−6.29249			−0.261057
$582$		0			0
$583$	20.8548			0.863718
$584$		0			0
$585$	−11.6570	−	9.55660i	−0.481957	−	0.395117i
$586$		0			0
$587$	9.96549			0.411320			0.205660	−	0.978624i	$-0.434066\pi$
$587$	9.96549			0.411320			0.205660	+	0.978624i	$0.434066\pi$
$588$		0			0
$589$			7.09314i			0.292268i
$590$		0			0
$591$	−32.5701	−	11.6443i	−1.33975	−	0.478982i
$592$		0			0
$593$	−7.66844			−0.314905			−0.157453	−	0.987527i	$-0.550328\pi$
$593$	−7.66844			−0.314905			−0.157453	+	0.987527i	$0.550328\pi$
$594$		0			0
$595$	−1.57283			−0.0644796
$596$		0			0
$597$	−15.2956	−	5.46842i	−0.626009	−	0.223808i
$598$		0			0
$599$		−	31.9548i		−	1.30564i	−0.757515	−	0.652818i	$-0.773587\pi$
$599$		−	31.9548i		−	1.30564i	0.757515	−	0.652818i	$-0.226413\pi$
$600$		0			0
$601$		−	21.1530i		−	0.862849i	−0.902149	−	0.431424i	$-0.858011\pi$
$601$		−	21.1530i		−	0.862849i	0.902149	−	0.431424i	$-0.141989\pi$
$602$		0			0
$603$	−1.19513	−	0.979785i	−0.0486693	−	0.0398999i
$604$		0			0
$605$	−8.62693			−0.350734
$606$		0			0
$607$		−	9.30353i		−	0.377619i	−0.982014	−	0.188809i	$-0.939537\pi$
$607$		−	9.30353i		−	0.377619i	0.982014	−	0.188809i	$-0.0604628\pi$
$608$		0			0
$609$	5.92786	+	2.11930i	0.240209	+	0.0858783i
$610$		0			0
$611$		−	19.5491i		−	0.790872i
$612$		0			0
$613$	20.8898			0.843731			0.421865	−	0.906658i	$-0.361375\pi$
$613$	20.8898			0.843731			0.421865	+	0.906658i	$0.361375\pi$
$614$		0			0
$615$	−19.8683	−	7.10321i	−0.801167	−	0.286429i
$616$		0			0
$617$			40.0920i			1.61404i	0.590523	+	0.807021i	$0.298921\pi$
$617$			40.0920i			1.61404i	−0.590523	+	0.807021i	$0.701079\pi$
$618$		0			0
$619$	−28.7181			−1.15428			−0.577139	−	0.816646i	$-0.695831\pi$
$619$	−28.7181			−1.15428			−0.577139	+	0.816646i	$0.695831\pi$
$620$		0			0
$621$	−15.8976	−	26.4772i	−0.637950	−	1.06249i
$622$		0			0
$623$			0.331970i			0.0133001i
$624$		0			0
$625$	0.0496535			0.00198614
$626$		0			0
$627$	14.8704	−	41.5938i	0.593867	−	1.66110i
$628$		0			0
$629$	14.6871			0.585611
$630$		0			0
$631$		−	23.0432i		−	0.917335i	−0.888608	−	0.458667i	$-0.848327\pi$
$631$		−	23.0432i		−	0.917335i	0.888608	−	0.458667i	$-0.151673\pi$
$632$		0			0
$633$	13.4196	+	4.79771i	0.533382	+	0.190692i
$634$		0			0
$635$	−14.2104			−0.563923
$636$		0			0
$637$	23.2039			0.919372
$638$		0			0
$639$	−35.8678	−	29.4050i	−1.41891	−	1.16324i
$640$		0			0
$641$	−11.9349			−0.471400			−0.235700	−	0.971826i	$-0.575738\pi$
$641$	−11.9349			−0.471400			−0.235700	+	0.971826i	$0.575738\pi$
$642$		0			0
$643$	23.3288			0.919997			0.459999	−	0.887920i	$-0.347850\pi$
$643$	23.3288			0.919997			0.459999	+	0.887920i	$0.347850\pi$
$644$		0			0
$645$	−4.97457	−	14.8677i	−0.195873	−	0.585416i
$646$		0			0
$647$	24.7629			0.973529			0.486765	−	0.873533i	$-0.338177\pi$
$647$	24.7629			0.973529			0.486765	+	0.873533i	$0.338177\pi$
$648$		0			0
$649$	−15.7990			−0.620164
$650$		0			0
$651$	1.48985	+	0.532645i	0.0583919	+	0.0208760i
$652$		0			0
$653$	−26.0682			−1.02013			−0.510064	−	0.860136i	$-0.670378\pi$
$653$	−26.0682			−1.02013			−0.510064	+	0.860136i	$0.670378\pi$
$654$		0			0
$655$	−29.1883			−1.14048
$656$		0			0
$657$	22.5830	−	27.5465i	0.881048	−	1.07469i
$658$		0			0
$659$			10.5545i			0.411144i	0.978642	+	0.205572i	$0.0659055\pi$
$659$			10.5545i			0.411144i	−0.978642	+	0.205572i	$0.934094\pi$
$660$		0			0
$661$	−48.3094			−1.87902			−0.939509	−	0.342524i	$-0.888718\pi$
$661$	−48.3094			−1.87902			−0.939509	+	0.342524i	$0.888718\pi$
$662$		0			0
$663$	8.55411	+	3.05822i	0.332214	+	0.118772i
$664$		0			0
$665$	6.70266			0.259918
$666$		0			0
$667$		−	27.3173i		−	1.05773i
$668$		0			0
$669$	−6.77082	−	2.42067i	−0.261775	−	0.0935885i
$670$		0			0
$671$	−20.1163			−0.776583
$672$		0			0
$673$		−	20.1763i		−	0.777740i	−0.921293	−	0.388870i	$-0.872866\pi$
$673$		−	20.1763i		−	0.777740i	0.921293	−	0.388870i	$-0.127134\pi$
$674$		0			0
$675$	−13.7859	+	8.27746i	−0.530621	+	0.318600i
$676$		0			0
$677$	−34.1094			−1.31093			−0.655466	−	0.755225i	$-0.727528\pi$
$677$	−34.1094			−1.31093			−0.655466	+	0.755225i	$0.727528\pi$
$678$		0			0
$679$			9.09369i			0.348984i
$680$		0			0
$681$	−12.5151	+	35.0059i	−0.479581	+	1.34143i
$682$		0			0
$683$			13.5793i			0.519599i	0.965663	+	0.259800i	$0.0836565\pi$
$683$			13.5793i			0.519599i	−0.965663	+	0.259800i	$0.916344\pi$
$684$		0			0
$685$	16.4537			0.628662
$686$		0			0
$687$	−9.23843	+	25.8407i	−0.352468	+	0.985884i
$688$		0			0
$689$		−	18.2776i		−	0.696321i
$690$		0			0
$691$			43.0220i			1.63663i	0.574768	+	0.818317i	$0.305092\pi$
$691$			43.0220i			1.63663i	−0.574768	+	0.818317i	$0.694908\pi$
$692$		0			0
$693$	−7.61976	−	6.24681i	−0.289451	−	0.237297i
$694$		0			0
$695$	20.2705			0.768905
$696$		0			0
$697$	12.7162			0.481659
$698$		0			0
$699$	3.52982	−	9.87320i	0.133510	−	0.373439i
$700$		0			0
$701$		−	22.2615i		−	0.840806i	−0.907338	−	0.420403i	$-0.861889\pi$
$701$		−	22.2615i		−	0.840806i	0.907338	−	0.420403i	$-0.138111\pi$
$702$		0			0
$703$	−62.5895			−2.36061
$704$		0			0
$705$	12.0908	+	4.32265i	0.455367	+	0.162801i
$706$		0			0
$707$	4.87682			0.183412
$708$		0			0
$709$	−8.11021			−0.304585			−0.152293	−	0.988335i	$-0.548666\pi$
$709$	−8.11021			−0.304585			−0.152293	+	0.988335i	$0.548666\pi$
$710$		0			0
$711$	17.6547	+	14.4736i	0.662103	+	0.542803i
$712$		0			0
$713$		−	6.86568i		−	0.257122i
$714$		0			0
$715$			20.8683i			0.780432i
$716$		0			0
$717$	−16.4673	−	5.88733i	−0.614984	−	0.219866i
$718$		0			0
$719$		−	2.88177i		−	0.107472i	−0.998555	−	0.0537359i	$-0.982887\pi$
$719$		−	2.88177i		−	0.107472i	0.998555	−	0.0537359i	$-0.0171129\pi$
$720$		0			0
$721$		−	1.22209i		−	0.0455131i
$722$		0			0
$723$	30.6694	+	10.9648i	1.14061	+	0.407784i
$724$		0			0
$725$	−14.2234			−0.528243
$726$		0			0
$727$		−	23.1180i		−	0.857399i	−0.903447	−	0.428700i	$-0.858972\pi$
$727$		−	23.1180i		−	0.857399i	0.903447	−	0.428700i	$-0.141028\pi$
$728$		0			0
$729$	12.6909	−	23.8315i	0.470033	−	0.882649i
$730$		0			0
$731$	5.83946	+	7.42799i	0.215980	+	0.274734i
$732$		0			0
$733$		−	32.3583i		−	1.19518i	−0.801801	−	0.597591i	$-0.796125\pi$
$733$		−	32.3583i		−	1.19518i	0.801801	−	0.597591i	$-0.203875\pi$
$734$		0			0
$735$	−5.13079	+	14.3513i	−0.189252	+	0.529354i
$736$		0			0
$737$			2.13951i			0.0788100i
$738$		0			0
$739$		−	7.67408i		−	0.282295i	−0.989989	−	0.141148i	$-0.954921\pi$
$739$		−	7.67408i		−	0.282295i	0.989989	−	0.141148i	$-0.0450793\pi$
$740$		0			0
$741$	−36.4537	−	13.0327i	−1.33916	−	0.478770i
$742$		0			0
$743$	34.3439			1.25995			0.629977	−	0.776613i	$-0.283064\pi$
$743$	34.3439			1.25995			0.629977	+	0.776613i	$0.283064\pi$
$744$		0			0
$745$	−8.40207			−0.307828
$746$		0			0
$747$	−15.1346	+	18.4609i	−0.553745	+	0.675450i
$748$		0			0
$749$	13.0946			0.478466
$750$		0			0
$751$			36.1147i			1.31785i	0.752211	+	0.658923i	$0.228988\pi$
$751$			36.1147i			1.31785i	−0.752211	+	0.658923i	$0.771012\pi$
$752$		0			0
$753$	13.7527	+	4.91680i	0.501176	+	0.179178i
$754$		0			0
$755$			19.2713i			0.701356i
$756$		0			0
$757$			46.4814i			1.68940i	0.535244	+	0.844698i	$0.320220\pi$
$757$			46.4814i			1.68940i	−0.535244	+	0.844698i	$0.679780\pi$
$758$		0			0
$759$	−14.3936	+	40.2600i	−0.522453	+	1.46135i
$760$		0			0
$761$	41.1600			1.49205			0.746025	−	0.665918i	$-0.231960\pi$
$761$	41.1600			1.49205			0.746025	+	0.665918i	$0.231960\pi$
$762$		0			0
$763$			4.41217i			0.159731i
$764$		0			0
$765$	−3.78293	+	4.61436i	−0.136772	+	0.166833i
$766$		0			0
$767$			13.8465i			0.499969i
$768$		0			0
$769$	14.3756			0.518396			0.259198	−	0.965824i	$-0.416542\pi$
$769$	14.3756			0.518396			0.259198	+	0.965824i	$0.416542\pi$
$770$		0			0
$771$	−11.6947	+	32.7110i	−0.421173	+	1.17806i
$772$		0			0
$773$	17.9885			0.647003			0.323501	−	0.946228i	$-0.395140\pi$
$773$	17.9885			0.647003			0.323501	+	0.946228i	$0.395140\pi$
$774$		0			0
$775$	−3.57477			−0.128410
$776$		0			0
$777$	−4.70003	+	13.1464i	−0.168613	+	0.471624i
$778$		0			0
$779$	−54.1905			−1.94158
$780$		0			0
$781$			64.2105i			2.29763i
$782$		0			0
$783$	20.4752	−	12.2939i	0.731723	−	0.439347i
$784$		0			0
$785$			3.94029i			0.140635i
$786$		0			0
$787$	2.57947			0.0919482			0.0459741	−	0.998943i	$-0.485361\pi$
$787$	2.57947			0.0919482			0.0459741	+	0.998943i	$0.485361\pi$
$788$		0			0
$789$	−5.68968	+	15.9145i	−0.202558	+	0.566571i
$790$		0			0
$791$			2.46091i			0.0874999i
$792$		0			0
$793$			17.6304i			0.626073i
$794$		0			0
$795$	11.3044	+	4.04150i	0.400926	+	0.143337i
$796$		0			0
$797$		−	47.7714i		−	1.69215i	−0.533063	−	0.846076i	$-0.678959\pi$
$797$		−	47.7714i		−	1.69215i	0.533063	−	0.846076i	$-0.321041\pi$
$798$		0			0
$799$	−7.73841			−0.273765
$800$		0			0
$801$	0.973935	+	0.798448i	0.0344123	+	0.0282118i
$802$		0			0
$803$	−49.3136			−1.74024
$804$		0			0
$805$	−6.48773			−0.228662
$806$		0			0
$807$	16.6724	+	5.96065i	0.586897	+	0.209825i
$808$		0			0
$809$		−	22.4918i		−	0.790771i	−0.918515	−	0.395386i	$-0.870611\pi$
$809$		−	22.4918i		−	0.790771i	0.918515	−	0.395386i	$-0.129389\pi$
$810$		0			0
$811$		−	20.5481i		−	0.721542i	−0.932654	−	0.360771i	$-0.882514\pi$
$811$		−	20.5481i		−	0.721542i	0.932654	−	0.360771i	$-0.117486\pi$
$812$		0			0
$813$	−1.01145	+	2.82910i	−0.0354730	+	0.0992210i
$814$		0			0
$815$		−	5.03186i		−	0.176258i
$816$		0			0
$817$	−24.8851	−	31.6547i	−0.870619	−	1.10746i
$818$		0			0
$819$	−5.47483	+	6.67811i	−0.191306	+	0.233352i
$820$		0			0
$821$			29.4659i			1.02837i	0.857680	+	0.514184i	$0.171905\pi$
$821$			29.4659i			1.02837i	−0.857680	+	0.514184i	$0.828095\pi$
$822$		0			0
$823$	17.9239			0.624786			0.312393	−	0.949953i	$-0.398869\pi$
$823$	17.9239			0.624786			0.312393	+	0.949953i	$0.398869\pi$
$824$		0			0
$825$	20.9623	+	7.49434i	0.729813	+	0.260919i
$826$		0			0
$827$		−	16.0839i		−	0.559291i	−0.960103	−	0.279646i	$-0.909783\pi$
$827$		−	16.0839i		−	0.559291i	0.960103	−	0.279646i	$-0.0902170\pi$
$828$		0			0
$829$		−	46.9875i		−	1.63194i	−0.578091	−	0.815972i	$-0.696202\pi$
$829$		−	46.9875i		−	1.63194i	0.578091	−	0.815972i	$-0.303798\pi$
$830$		0			0
$831$	−43.2245	−	15.4534i	−1.49944	−	0.536073i
$832$		0			0
$833$		−	9.18514i		−	0.318246i
$834$		0			0
$835$		−	17.3580i		−	0.600700i
$836$		0			0
$837$	5.14604	−	3.08983i	0.177873	−	0.106800i
$838$		0			0
$839$	10.4024			0.359131			0.179566	−	0.983746i	$-0.442531\pi$
$839$	10.4024			0.359131			0.179566	+	0.983746i	$0.442531\pi$
$840$		0			0
$841$	−7.87511			−0.271556
$842$		0			0
$843$	7.82484	+	2.79750i	0.269502	+	0.0963510i
$844$		0			0
$845$	0.344773			0.0118606
$846$		0			0
$847$			4.94224i			0.169817i
$848$		0			0
$849$	−7.06990	+	19.7751i	−0.242638	+	0.678680i
$850$		0			0
$851$	60.5824			2.07674
$852$		0			0
$853$	−0.280047			−0.00958862			−0.00479431	−	0.999989i	$-0.501526\pi$
$853$	−0.280047			−0.00958862			−0.00479431	+	0.999989i	$0.501526\pi$
$854$		0			0
$855$	16.1211	−	19.6643i	0.551330	−	0.672504i
$856$		0			0
$857$		−	2.54303i		−	0.0868683i	−0.999056	−	0.0434341i	$-0.986170\pi$
$857$		−	2.54303i		−	0.0868683i	0.999056	−	0.0434341i	$-0.0138299\pi$
$858$		0			0
$859$			2.47112i			0.0843135i	0.999111	+	0.0421567i	$0.0134229\pi$
$859$			2.47112i			0.0843135i	−0.999111	+	0.0421567i	$0.986577\pi$
$860$		0			0
$861$	−4.06932	+	11.3822i	−0.138682	+	0.387906i
$862$		0			0
$863$	33.7475			1.14878			0.574390	−	0.818582i	$-0.305239\pi$
$863$	33.7475			1.14878			0.574390	+	0.818582i	$0.305239\pi$
$864$		0			0
$865$		−	26.9100i		−	0.914966i
$866$		0			0
$867$	−8.70195	+	24.3401i	−0.295534	+	0.826633i
$868$		0			0
$869$		−	31.6054i		−	1.07214i
$870$		0			0
$871$	1.87511			0.0635358
$872$		0			0
$873$	26.6791	+	21.8720i	0.902950	+	0.740253i
$874$		0			0
$875$			8.83583i			0.298706i
$876$		0			0
$877$	16.1661			0.545889			0.272945	−	0.962030i	$-0.412002\pi$
$877$	16.1661			0.545889			0.272945	+	0.962030i	$0.412002\pi$
$878$		0			0
$879$	29.1518	+	10.4222i	0.983265	+	0.351532i
$880$		0			0
$881$			16.8803i			0.568713i	0.958719	+	0.284356i	$0.0917799\pi$
$881$			16.8803i			0.568713i	−0.958719	+	0.284356i	$0.908220\pi$
$882$		0			0
$883$	−44.9229			−1.51178			−0.755889	−	0.654700i	$-0.772795\pi$
$883$	−44.9229			−1.51178			−0.755889	+	0.654700i	$0.772795\pi$
$884$		0			0
$885$	−8.56387	−	3.06171i	−0.287872	−	0.102918i
$886$		0			0
$887$	−11.3609			−0.381462			−0.190731	−	0.981642i	$-0.561086\pi$
$887$	−11.3609			−0.381462			−0.190731	+	0.981642i	$0.561086\pi$
$888$		0			0
$889$			8.14093i			0.273038i
$890$		0			0
$891$	−36.6538	+	7.33017i	−1.22795	+	0.245570i
$892$		0			0
$893$	32.9776			1.10355
$894$		0			0
$895$	−5.07901			−0.169773
$896$		0			0
$897$	35.2847	+	12.6148i	1.17812	+	0.421197i
$898$		0			0
$899$	5.30933			0.177076
$900$		0			0
$901$	−7.23509			−0.241036
$902$		0			0
$903$	−8.51749	+	2.84986i	−0.283444	+	0.0948373i
$904$		0			0
$905$	15.0116			0.499001
$906$		0			0
$907$	−38.7825			−1.28775			−0.643875	−	0.765131i	$-0.722674\pi$
$907$	−38.7825			−1.28775			−0.643875	+	0.765131i	$0.722674\pi$
$908$		0			0
$909$	11.7296	−	14.3076i	0.389048	−	0.474554i
$910$		0			0
$911$	−23.7070			−0.785449			−0.392724	−	0.919656i	$-0.628467\pi$
$911$	−23.7070			−0.785449			−0.392724	+	0.919656i	$0.628467\pi$
$912$		0			0
$913$	33.0487			1.09375
$914$		0			0
$915$	−10.9041	−	3.89839i	−0.360479	−	0.128877i
$916$		0			0
$917$			16.7215i			0.552194i
$918$		0			0
$919$	−48.6950			−1.60630			−0.803149	−	0.595778i	$-0.796844\pi$
$919$	−48.6950			−1.60630			−0.803149	+	0.595778i	$0.796844\pi$
$920$		0			0
$921$	−7.10521	+	19.8739i	−0.234125	+	0.654866i
$922$		0			0
$923$	56.2754			1.85233
$924$		0			0
$925$		−	31.5436i		−	1.03715i
$926$		0			0
$927$	−3.58538	−	2.93935i	−0.117759	−	0.0965411i
$928$		0			0
$929$	−2.58626			−0.0848523			−0.0424262	−	0.999100i	$-0.513509\pi$
$929$	−2.58626			−0.0848523			−0.0424262	+	0.999100i	$0.513509\pi$
$930$		0			0
$931$			39.1429i			1.28286i
$932$		0			0
$933$	−6.77381	−	2.42174i	−0.221765	−	0.0792842i
$934$		0			0
$935$	8.26062			0.270151
$936$		0			0
$937$			46.6280i			1.52327i	0.648005	+	0.761636i	$0.275603\pi$
$937$			46.6280i			1.52327i	−0.648005	+	0.761636i	$0.724397\pi$
$938$		0			0
$939$	9.15017	+	3.27132i	0.298605	+	0.106756i
$940$		0			0
$941$			38.0240i			1.23955i	0.784781	+	0.619773i	$0.212775\pi$
$941$			38.0240i			1.23955i	−0.784781	+	0.619773i	$0.787225\pi$
$942$		0			0
$943$	52.4528			1.70810
$944$		0			0
$945$	−2.91973	−	4.86275i	−0.0949788	−	0.158185i
$946$		0			0
$947$		−	40.7364i		−	1.32375i	−0.749612	−	0.661877i	$-0.769760\pi$
$947$		−	40.7364i		−	1.32375i	0.749612	−	0.661877i	$-0.230240\pi$
$948$		0			0
$949$			43.2195i			1.40296i
$950$		0			0
$951$	33.7806	+	12.0771i	1.09541	+	0.391627i
$952$		0			0
$953$	−33.7229			−1.09239			−0.546196	−	0.837658i	$-0.683924\pi$
$953$	−33.7229			−1.09239			−0.546196	+	0.837658i	$0.683924\pi$
$954$		0			0
$955$	5.90826			0.191187
$956$		0			0
$957$	−31.1337	−	11.1308i	−1.00641	−	0.359806i
$958$		0			0
$959$		−	9.42606i		−	0.304383i
$960$		0			0
$961$	−29.6656			−0.956955
$962$		0			0
$963$	31.4948	−	38.4169i	1.01491	−	1.23797i
$964$		0			0
$965$	−19.6483			−0.632500
$966$		0			0
$967$	20.3241			0.653579			0.326789	−	0.945097i	$-0.394033\pi$
$967$	20.3241			0.653579			0.326789	+	0.945097i	$0.394033\pi$
$968$		0			0
$969$	−5.15894	+	14.4300i	−0.165729	+	0.463558i
$970$		0			0
$971$			14.4736i			0.464481i	0.972658	+	0.232240i	$0.0746056\pi$
$971$			14.4736i			0.464481i	−0.972658	+	0.232240i	$0.925394\pi$
$972$		0			0
$973$		−	11.6127i		−	0.372286i
$974$		0			0
$975$	6.56819	−	18.3718i	0.210350	−	0.588368i
$976$		0			0
$977$		−	10.2661i		−	0.328443i	−0.986424	−	0.164221i	$-0.947489\pi$
$977$		−	10.2661i		−	0.328443i	0.986424	−	0.164221i	$-0.0525112\pi$
$978$		0			0
$979$		−	1.74354i		−	0.0557237i
$980$		0			0
$981$	12.9444	+	10.6121i	0.413284	+	0.338817i
$982$		0			0
$983$	−1.68775			−0.0538309			−0.0269154	−	0.999638i	$-0.508568\pi$
$983$	−1.68775			−0.0538309			−0.0269154	+	0.999638i	$0.508568\pi$
$984$		0			0
$985$			27.5657i			0.878317i
$986$		0			0
$987$	2.47638	−	6.92665i	0.0788241	−	0.220478i
$988$		0			0
$989$	24.0871	+	30.6396i	0.765925	+	0.974283i
$990$		0			0
$991$		−	45.2324i		−	1.43685i	−0.695603	−	0.718427i	$-0.744863\pi$
$991$		−	45.2324i		−	1.43685i	0.695603	−	0.718427i	$-0.255137\pi$
$992$		0			0
$993$	−12.4331	−	4.44501i	−0.394552	−	0.141058i
$994$		0			0
$995$			12.9455i			0.410400i
$996$		0			0
$997$			19.1641i			0.606932i	0.952842	+	0.303466i	$0.0981439\pi$
$997$			19.1641i			0.606932i	−0.952842	+	0.303466i	$0.901856\pi$
$998$		0			0
$999$	27.2645	+	45.4084i	0.862609	+	1.43666i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2064.2.l.h.257.6		12
3.2	odd	2	inner	2064.2.l.h.257.8		12
4.3	odd	2		129.2.d.a.128.1	✓	12
12.11	even	2		129.2.d.a.128.11	yes	12
43.42	odd	2	inner	2064.2.l.h.257.7		12
129.128	even	2	inner	2064.2.l.h.257.5		12
172.171	even	2		129.2.d.a.128.12	yes	12
516.515	odd	2		129.2.d.a.128.2	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.2.d.a.128.1	✓	12	4.3	odd	2
129.2.d.a.128.2	yes	12	516.515	odd	2
129.2.d.a.128.11	yes	12	12.11	even	2
129.2.d.a.128.12	yes	12	172.171	even	2
2064.2.l.h.257.5		12	129.128	even	2	inner
2064.2.l.h.257.6		12	1.1	even	1	trivial
2064.2.l.h.257.7		12	43.42	odd	2	inner
2064.2.l.h.257.8		12	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 2064 = 2^{4} \cdot 3 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2064.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.583090 + 1.63095i) q^{3} +1.38036 q^{5} -0.790787i q^{7} +(-2.32001 - 1.90198i) q^{9} +O(q^{10})\)	\(q+(-0.583090 + 1.63095i) q^{3} +1.38036 q^{5} -0.790787i q^{7} +(-2.32001 - 1.90198i) q^{9} +4.15328i q^{11} +3.64002 q^{13} +(-0.804874 + 2.25130i) q^{15} -1.44088i q^{17} +6.14039i q^{19} +(1.28974 + 0.461100i) q^{21} -5.94348i q^{23} -3.09461 q^{25} +(4.45482 - 2.67480i) q^{27} +4.59618 q^{29} +1.15516 q^{31} +(-6.77381 - 2.42174i) q^{33} -1.09157i q^{35} +10.1931i q^{37} +(-2.12246 + 5.93671i) q^{39} +8.82525i q^{41} +(-5.15516 + 4.05269i) q^{43} +(-3.20245 - 2.62542i) q^{45} -5.37060i q^{47} +6.37466 q^{49} +(2.35001 + 0.840166i) q^{51} -5.02128i q^{53} +5.73302i q^{55} +(-10.0147 - 3.58040i) q^{57} +3.80397i q^{59} +4.84348i q^{61} +(-1.50406 + 1.83463i) q^{63} +5.02454 q^{65} +0.515138 q^{67} +(9.69354 + 3.46559i) q^{69} +15.4602 q^{71} +11.8734i q^{73} +(1.80444 - 5.04716i) q^{75} +3.28436 q^{77} -7.60975 q^{79} +(1.76491 + 8.82525i) q^{81} -7.95725i q^{83} -1.98894i q^{85} +(-2.67999 + 7.49615i) q^{87} -0.419798 q^{89} -2.87848i q^{91} +(-0.673563 + 1.88401i) q^{93} +8.47594i q^{95} -11.4995 q^{97} +(7.89948 - 9.63567i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^9	\( 12 q - 12 q^{9} + 12 q^{13} + 8 q^{15} - 4 q^{21} - 16 q^{31} - 32 q^{43} - 24 q^{49} + 12 q^{57} + 8 q^{67} - 56 q^{79} - 44 q^{81} - 48 q^{87} - 4 q^{97} + 40 q^{99}+O(q^{100}) \) 12 * q - 12 * q^9 + 12 * q^13 + 8 * q^15 - 4 * q^21 - 16 * q^31 - 32 * q^43 - 24 * q^49 + 12 * q^57 + 8 * q^67 - 56 * q^79 - 44 * q^81 - 48 * q^87 - 4 * q^97 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more