LMFDB - Embedded newform 2448.2.be.x.1441.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2448,2,Mod(1441,2448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2448, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2448.1441"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2448 = 2^{4} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2448.be (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,4,0,4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$19.5473784148$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.836829184.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 $ x^8 + 14x^6 + 61x^4 + 84*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 51)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1441.4
Root			$2.06644i$ of defining polynomial
Character	$\chi$	$=$	2448.1441
Dual form			2448.2.be.x.1585.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.87540 + 2.87540i) q^{5} +(-0.652223 + 0.652223i) q^{7} +(1.60525 - 1.60525i) q^{11} -3.57461 q^{13} +(-3.79779 - 1.60525i) q^{17} +1.72984i q^{19} +(-3.60525 + 3.60525i) q^{23} +11.5359i q^{25} +(1.55827 + 1.55827i) q^{29} +(2.92238 + 2.92238i) q^{31} -3.75081 q^{35} +(4.21049 + 4.21049i) q^{37} +(-3.57096 + 3.57096i) q^{41} +5.23146i q^{43} +6.96130 q^{47} +6.14921i q^{49} -2.69555i q^{53} +9.23146 q^{55} -7.39840i q^{59} +(-2.75081 + 2.75081i) q^{61} +(-10.2784 - 10.2784i) q^{65} -12.0419 q^{67} +(1.07762 + 1.07762i) q^{71} +(0.823796 + 0.823796i) q^{73} +2.09396i q^{77} +(-8.40303 + 8.40303i) q^{79} +6.38508i q^{83} +(-6.30445 - 15.5359i) q^{85} +10.3597 q^{89} +(2.33144 - 2.33144i) q^{91} +(-4.97399 + 4.97399i) q^{95} +(-10.2315 - 10.2315i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{5} + 4 q^{7} - 4 q^{13} + 4 q^{17} - 16 q^{23} - 4 q^{29} + 8 q^{31} + 8 q^{35} + 8 q^{37} - 28 q^{41} - 8 q^{47} + 4 q^{55} + 16 q^{61} - 16 q^{65} - 8 q^{67} + 24 q^{71} + 20 q^{73} - 20 q^{79}+ \cdots - 12 q^{97}+O(q^{100}) $ 8 * q + 4 * q^5 + 4 * q^7 - 4 * q^13 + 4 * q^17 - 16 * q^23 - 4 * q^29 + 8 * q^31 + 8 * q^35 + 8 * q^37 - 28 * q^41 - 8 * q^47 + 4 * q^55 + 16 * q^61 - 16 * q^65 - 8 * q^67 + 24 * q^71 + 20 * q^73 - 20 * q^79 - 32 * q^85 + 8 * q^89 + 36 * q^91 + 8 * q^95 - 12 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2448\mathbb{Z}\right)^\times$.

$n$	$613$	$1361$	$1873$	$2143$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.87540	+	2.87540i	1.28592	+	1.28592i	0.937243	+	0.348677i	$0.113369\pi$
$5$	2.87540	+	2.87540i	1.28592	+	1.28592i	0.348677	+	0.937243i	$0.386631\pi$
$6$		0			0
$7$	−0.652223	+	0.652223i	−0.246517	+	0.246517i	−0.819540	−	0.573023i	$-0.805771\pi$
$7$	−0.652223	+	0.652223i	−0.246517	+	0.246517i	0.573023	+	0.819540i	$0.305771\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.60525	−	1.60525i	0.484000	−	0.484000i	−0.422407	−	0.906406i	$-0.638815\pi$
$11$	1.60525	−	1.60525i	0.484000	−	0.484000i	0.906406	+	0.422407i	$0.138815\pi$
$12$		0			0
$13$	−3.57461			−0.991417			−0.495709	−	0.868489i	$-0.665092\pi$
$13$	−3.57461			−0.991417			−0.495709	+	0.868489i	$0.665092\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.79779	−	1.60525i	−0.921099	−	0.389329i
$17$	−3.79779	−	1.60525i	−0.921099	−	0.389329i
$18$		0			0
$19$			1.72984i			0.396853i	0.980116	+	0.198426i	$0.0635830\pi$
$19$			1.72984i			0.396853i	−0.980116	+	0.198426i	$0.936417\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.60525	+	3.60525i	−0.751746	+	0.751746i	−0.974805	−	0.223059i	$-0.928396\pi$
$23$	−3.60525	+	3.60525i	−0.751746	+	0.751746i	0.223059	+	0.974805i	$0.428396\pi$
$24$		0			0
$25$			11.5359i			2.30718i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.55827	+	1.55827i	0.289363	+	0.289363i	0.836828	−	0.547465i	$-0.184407\pi$
$29$	1.55827	+	1.55827i	0.289363	+	0.289363i	−0.547465	+	0.836828i	$0.684407\pi$
$30$		0			0
$31$	2.92238	+	2.92238i	0.524875	+	0.524875i	0.919040	−	0.394165i	$-0.128966\pi$
$31$	2.92238	+	2.92238i	0.524875	+	0.524875i	−0.394165	+	0.919040i	$0.628966\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.75081			−0.634003
$36$		0			0
$37$	4.21049	+	4.21049i	0.692200	+	0.692200i	0.962716	−	0.270515i	$-0.0871941\pi$
$37$	4.21049	+	4.21049i	0.692200	+	0.692200i	−0.270515	+	0.962716i	$0.587194\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.57096	+	3.57096i	−0.557690	+	0.557690i	−0.928649	−	0.370959i	$-0.879029\pi$
$41$	−3.57096	+	3.57096i	−0.557690	+	0.557690i	0.370959	+	0.928649i	$0.379029\pi$
$42$		0			0
$43$			5.23146i			0.797790i	0.916997	+	0.398895i	$0.130606\pi$
$43$			5.23146i			0.797790i	−0.916997	+	0.398895i	$0.869394\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.96130			1.01541			0.507705	−	0.861531i	$-0.330494\pi$
$47$	6.96130			1.01541			0.507705	+	0.861531i	$0.330494\pi$
$48$		0			0
$49$			6.14921i			0.878459i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	2.69555i		−	0.370263i	−0.982714	−	0.185131i	$-0.940729\pi$
$53$		−	2.69555i		−	0.370263i	0.982714	−	0.185131i	$-0.0592711\pi$
$54$		0			0
$55$	9.23146			1.24477
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	7.39840i		−	0.963190i	−0.876394	−	0.481595i	$-0.840058\pi$
$59$		−	7.39840i		−	0.963190i	0.876394	−	0.481595i	$-0.159942\pi$
$60$		0			0
$61$	−2.75081	+	2.75081i	−0.352205	+	0.352205i	−0.860929	−	0.508724i	$-0.830117\pi$
$61$	−2.75081	+	2.75081i	−0.352205	+	0.352205i	0.508724	+	0.860929i	$0.330117\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−10.2784	−	10.2784i	−1.27488	−	1.27488i
$66$		0			0
$67$	−12.0419			−1.47116			−0.735578	−	0.677440i	$-0.763090\pi$
$67$	−12.0419			−1.47116			−0.735578	+	0.677440i	$0.763090\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.07762	+	1.07762i	0.127890	+	0.127890i	0.768154	−	0.640265i	$-0.221175\pi$
$71$	1.07762	+	1.07762i	0.127890	+	0.127890i	−0.640265	+	0.768154i	$0.721175\pi$
$72$		0			0
$73$	0.823796	+	0.823796i	0.0964180	+	0.0964180i	0.753670	−	0.657252i	$-0.228281\pi$
$73$	0.823796	+	0.823796i	0.0964180	+	0.0964180i	−0.657252	+	0.753670i	$0.728281\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			2.09396i			0.238628i
$78$		0			0
$79$	−8.40303	+	8.40303i	−0.945415	+	0.945415i	−0.998585	−	0.0531704i	$-0.983067\pi$
$79$	−8.40303	+	8.40303i	−0.945415	+	0.945415i	0.0531704	+	0.998585i	$0.483067\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			6.38508i			0.700854i	0.936590	+	0.350427i	$0.113964\pi$
$83$			6.38508i			0.700854i	−0.936590	+	0.350427i	$0.886036\pi$
$84$		0			0
$85$	−6.30445	−	15.5359i	−0.683813	−	1.68511i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.3597			1.09813			0.549063	−	0.835781i	$-0.314985\pi$
$89$	10.3597			1.09813			0.549063	+	0.835781i	$0.314985\pi$
$90$		0			0
$91$	2.33144	−	2.33144i	0.244401	−	0.244401i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.97399	+	4.97399i	−0.510321	+	0.510321i
$96$		0			0
$97$	−10.2315	−	10.2315i	−1.03885	−	1.03885i	−0.999214	−	0.0396330i	$-0.987381\pi$
$97$	−10.2315	−	10.2315i	−1.03885	−	1.03885i	−0.0396330	−	0.999214i	$-0.512619\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.265746			−0.0264427			−0.0132213	−	0.999913i	$-0.504209\pi$
$101$	−0.265746			−0.0264427			−0.0132213	+	0.999913i	$0.504209\pi$
$102$		0			0
$103$	3.03429			0.298977			0.149489	−	0.988763i	$-0.452237\pi$
$103$	3.03429			0.298977			0.149489	+	0.988763i	$0.452237\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.8964	−	13.8964i	−1.34341	−	1.34341i	−0.892637	−	0.450776i	$-0.851147\pi$
$107$	−13.8964	−	13.8964i	−1.34341	−	1.34341i	−0.450776	−	0.892637i	$-0.648853\pi$
$108$		0			0
$109$	−0.506033	+	0.506033i	−0.0484692	+	0.0484692i	−0.730926	−	0.682457i	$-0.760912\pi$
$109$	−0.506033	+	0.506033i	−0.0484692	+	0.0484692i	0.682457	+	0.730926i	$0.260912\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.33509	−	8.33509i	0.784099	−	0.784099i	−0.196421	−	0.980520i	$-0.562932\pi$
$113$	8.33509	−	8.33509i	0.784099	−	0.784099i	0.980520	+	0.196421i	$0.0629319\pi$
$114$		0			0
$115$	−20.7331			−1.93337
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.52398	−	1.43003i	0.323043	−	0.131090i
$120$		0			0
$121$			5.84638i			0.531489i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−18.7934	+	18.7934i	−1.68093	+	1.68093i
$126$		0			0
$127$		−	2.81048i		−	0.249390i	−0.992195	−	0.124695i	$-0.960205\pi$
$127$		−	2.81048i		−	0.249390i	0.992195	−	0.124695i	$-0.0397952\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.60525	+	5.60525i	0.489733	+	0.489733i	0.908222	−	0.418489i	$-0.137440\pi$
$131$	5.60525	+	5.60525i	0.489733	+	0.489733i	−0.418489	+	0.908222i	$0.637440\pi$
$132$		0			0
$133$	−1.12824	−	1.12824i	−0.0978309	−	0.0978309i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.103218			0.00881851			0.00440926	−	0.999990i	$-0.498596\pi$
$137$	0.103218			0.00881851			0.00440926	+	0.999990i	$0.498596\pi$
$138$		0			0
$139$	10.2151	+	10.2151i	0.866435	+	0.866435i	0.992076	−	0.125641i	$-0.0400986\pi$
$139$	10.2151	+	10.2151i	0.866435	+	0.866435i	−0.125641	+	0.992076i	$0.540099\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.73812	+	5.73812i	−0.479846	+	0.479846i
$144$		0			0
$145$			8.96130i			0.744195i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.59557			0.294561			0.147280	−	0.989095i	$-0.452948\pi$
$149$	3.59557			0.294561			0.147280	+	0.989095i	$0.452948\pi$
$150$		0			0
$151$			7.58828i			0.617526i	0.951139	+	0.308763i	$0.0999149\pi$
$151$			7.58828i			0.617526i	−0.951139	+	0.308763i	$0.900085\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			16.8061i			1.34990i
$156$		0			0
$157$	18.9569			1.51292			0.756462	−	0.654038i	$-0.226926\pi$
$157$	18.9569			1.51292			0.756462	+	0.654038i	$0.226926\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	4.70285i		−	0.370636i
$162$		0			0
$163$	9.49860	−	9.49860i	0.743988	−	0.743988i	−0.229355	−	0.973343i	$-0.573662\pi$
$163$	9.49860	−	9.49860i	0.743988	−	0.743988i	0.973343	+	0.229355i	$0.0736617\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.51129	−	1.51129i	−0.116947	−	0.116947i	0.646211	−	0.763158i	$-0.276352\pi$
$167$	−1.51129	−	1.51129i	−0.116947	−	0.116947i	−0.763158	+	0.646211i	$0.776352\pi$
$168$		0			0
$169$	−0.222197			−0.0170921
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.73349	+	3.73349i	0.283852	+	0.283852i	0.834643	−	0.550791i	$-0.185674\pi$
$173$	3.73349	+	3.73349i	0.283852	+	0.283852i	−0.550791	+	0.834643i	$0.685674\pi$
$174$		0			0
$175$	−7.52398	−	7.52398i	−0.568760	−	0.568760i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	7.77619i		−	0.581220i	−0.956842	−	0.290610i	$-0.906142\pi$
$179$		−	7.77619i		−	0.581220i	0.956842	−	0.290610i	$-0.0938582\pi$
$180$		0			0
$181$	0.647592	−	0.647592i	0.0481351	−	0.0481351i	−0.682630	−	0.730765i	$-0.739164\pi$
$181$	0.647592	−	0.647592i	0.0481351	−	0.0481351i	0.730765	+	0.682630i	$0.239164\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			24.2137i			1.78023i
$186$		0			0
$187$	−8.67319	+	3.51957i	−0.634247	+	0.257376i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	19.1658			1.38679			0.693393	−	0.720560i	$-0.256115\pi$
$191$	19.1658			1.38679			0.693393	+	0.720560i	$0.256115\pi$
$192$		0			0
$193$	3.33144	−	3.33144i	0.239802	−	0.239802i	−0.576966	−	0.816768i	$-0.695763\pi$
$193$	3.33144	−	3.33144i	0.239802	−	0.239802i	0.816768	+	0.576966i	$0.195763\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	10.5323	−	10.5323i	0.750392	−	0.750392i	−0.224160	−	0.974552i	$-0.571964\pi$
$197$	10.5323	−	10.5323i	0.750392	−	0.750392i	0.974552	+	0.224160i	$0.0719638\pi$
$198$		0			0
$199$	−13.2821	−	13.2821i	−0.941542	−	0.941542i	0.0568413	−	0.998383i	$-0.481897\pi$
$199$	−13.2821	−	13.2821i	−0.941542	−	0.941542i	−0.998383	+	0.0568413i	$0.981897\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.03268			−0.142666
$204$		0			0
$205$	−20.5359			−1.43429
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.77682	+	2.77682i	0.192077	+	0.192077i
$210$		0			0
$211$	−4.15826	+	4.15826i	−0.286266	+	0.286266i	−0.835602	−	0.549336i	$-0.814881\pi$
$211$	−4.15826	+	4.15826i	−0.286266	+	0.286266i	0.549336	+	0.835602i	$0.314881\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−15.0426	+	15.0426i	−1.02589	+	1.02589i
$216$		0			0
$217$	−3.81209			−0.258781
$218$		0			0
$219$		0			0
$220$		0			0
$221$	13.5756	+	5.73812i	0.913193	+	0.385988i
$222$		0			0
$223$		−	5.58387i		−	0.373923i	−0.982367	−	0.186962i	$-0.940136\pi$
$223$		−	5.58387i		−	0.373923i	0.982367	−	0.186962i	$-0.0598640\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−19.4666	+	19.4666i	−1.29204	+	1.29204i	−0.358519	+	0.933522i	$0.616718\pi$
$227$	−19.4666	+	19.4666i	−1.29204	+	1.29204i	−0.933522	+	0.358519i	$0.883282\pi$
$228$		0			0
$229$		−	3.57019i		−	0.235925i	−0.993018	−	0.117962i	$-0.962364\pi$
$229$		−	3.57019i		−	0.235925i	0.993018	−	0.117962i	$-0.0376362\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.969360	−	0.969360i	−0.0635049	−	0.0635049i	0.674641	−	0.738146i	$-0.264298\pi$
$233$	−0.969360	−	0.969360i	−0.0635049	−	0.0635049i	−0.738146	+	0.674641i	$0.764298\pi$
$234$		0			0
$235$	20.0166	+	20.0166i	1.30574	+	1.30574i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	21.6121			1.39797			0.698986	−	0.715135i	$-0.253635\pi$
$239$	21.6121			1.39797			0.698986	+	0.715135i	$0.253635\pi$
$240$		0			0
$241$	0.750810	+	0.750810i	0.0483639	+	0.0483639i	0.730875	−	0.682511i	$-0.239112\pi$
$241$	0.750810	+	0.750810i	0.0483639	+	0.0483639i	−0.682511	+	0.730875i	$0.739112\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−17.6815	+	17.6815i	−1.12963	+	1.12963i
$246$		0			0
$247$		−	6.18350i		−	0.393446i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7.29715			0.460592			0.230296	−	0.973121i	$-0.426031\pi$
$251$	7.29715			0.460592			0.230296	+	0.973121i	$0.426031\pi$
$252$		0			0
$253$			11.5746i			0.727689i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	22.5569i		−	1.40706i	−0.710666	−	0.703530i	$-0.751606\pi$
$257$		−	22.5569i		−	1.40706i	0.710666	−	0.703530i	$-0.248394\pi$
$258$		0			0
$259$	−5.49236			−0.341278
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	4.24919i		−	0.262016i	−0.991381	−	0.131008i	$-0.958179\pi$
$263$		−	4.24919i		−	0.262016i	0.991381	−	0.131008i	$-0.0418214\pi$
$264$		0			0
$265$	7.75081	−	7.75081i	0.476128	−	0.476128i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	8.36776	+	8.36776i	0.510191	+	0.510191i	0.914585	−	0.404394i	$-0.132517\pi$
$269$	8.36776	+	8.36776i	0.510191	+	0.510191i	−0.404394	+	0.914585i	$0.632517\pi$
$270$		0			0
$271$	11.8045			0.717070			0.358535	−	0.933516i	$-0.383276\pi$
$271$	11.8045			0.717070			0.358535	+	0.933516i	$0.383276\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	18.5180	+	18.5180i	1.11667	+	1.11667i
$276$		0			0
$277$	−7.37338	−	7.37338i	−0.443023	−	0.443023i	0.450003	−	0.893027i	$-0.351423\pi$
$277$	−7.37338	−	7.37338i	−0.443023	−	0.443023i	−0.893027	+	0.450003i	$0.851423\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			5.46697i			0.326132i	0.986615	+	0.163066i	$0.0521384\pi$
$281$			5.46697i			0.326132i	−0.986615	+	0.163066i	$0.947862\pi$
$282$		0			0
$283$	−12.4716	+	12.4716i	−0.741360	+	0.741360i	−0.972840	−	0.231480i	$-0.925643\pi$
$283$	−12.4716	+	12.4716i	−0.741360	+	0.741360i	0.231480	+	0.972840i	$0.425643\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	4.65812i		−	0.274960i
$288$		0			0
$289$	11.8464	+	12.1928i	0.696846	+	0.717221i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−8.25845			−0.482464			−0.241232	−	0.970467i	$-0.577551\pi$
$293$	−8.25845			−0.482464			−0.241232	+	0.970467i	$0.577551\pi$
$294$		0			0
$295$	21.2734	−	21.2734i	1.23859	−	1.23859i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.8873	−	12.8873i	0.745293	−	0.745293i
$300$		0			0
$301$	−3.41208	−	3.41208i	−0.196669	−	0.196669i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−15.8194			−0.905815
$306$		0			0
$307$	9.94601			0.567649			0.283824	−	0.958876i	$-0.408397\pi$
$307$	9.94601			0.567649			0.283824	+	0.958876i	$0.408397\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	11.7284	+	11.7284i	0.665059	+	0.665059i	0.956568	−	0.291509i	$-0.0941573\pi$
$311$	11.7284	+	11.7284i	0.665059	+	0.665059i	−0.291509	+	0.956568i	$0.594157\pi$
$312$		0			0
$313$	−14.5746	+	14.5746i	−0.823805	+	0.823805i	−0.986651	−	0.162846i	$-0.947933\pi$
$313$	−14.5746	+	14.5746i	−0.823805	+	0.823805i	0.162846	+	0.986651i	$0.447933\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.01795	+	3.01795i	−0.169505	+	0.169505i	−0.786762	−	0.617257i	$-0.788244\pi$
$317$	−3.01795	+	3.01795i	−0.169505	+	0.169505i	0.617257	+	0.786762i	$0.288244\pi$
$318$		0			0
$319$	5.00280			0.280103
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.77682	−	6.56957i	0.154506	−	0.365540i
$324$		0			0
$325$		−	41.2363i		−	2.28738i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.54032	+	4.54032i	−0.250316	+	0.250316i
$330$		0			0
$331$			28.6105i			1.57258i	0.617860	+	0.786288i	$0.288000\pi$
$331$			28.6105i			1.57258i	−0.617860	+	0.786288i	$0.712000\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−34.6254	−	34.6254i	−1.89179	−	1.89179i
$336$		0			0
$337$	22.1585	+	22.1585i	1.20705	+	1.20705i	0.971978	+	0.235071i	$0.0755322\pi$
$337$	22.1585	+	22.1585i	1.20705	+	1.20705i	0.235071	+	0.971978i	$0.424468\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	9.38228			0.508079
$342$		0			0
$343$	−8.57622	−	8.57622i	−0.463072	−	0.463072i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.51066	+	8.51066i	−0.456876	+	0.456876i	−0.897629	−	0.440752i	$-0.854712\pi$
$347$	−8.51066	+	8.51066i	−0.456876	+	0.456876i	0.440752	+	0.897629i	$0.354712\pi$
$348$		0			0
$349$			33.3512i			1.78525i	0.450799	+	0.892625i	$0.351139\pi$
$349$			33.3512i			1.78525i	−0.450799	+	0.892625i	$0.648861\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.56570			−0.349457			−0.174729	−	0.984617i	$-0.555905\pi$
$353$	−6.56570			−0.349457			−0.174729	+	0.984617i	$0.555905\pi$
$354$		0			0
$355$			6.19717i			0.328912i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	21.5508i		−	1.13741i	−0.822541	−	0.568705i	$-0.807445\pi$
$359$		−	21.5508i		−	1.13741i	0.822541	−	0.568705i	$-0.192555\pi$
$360$		0			0
$361$	16.0077			0.842508
$362$		0			0
$363$		0			0
$364$		0			0
$365$			4.73749i			0.247972i
$366$		0			0
$367$	25.1478	−	25.1478i	1.31271	−	1.31271i	0.393292	−	0.919414i	$-0.371336\pi$
$367$	25.1478	−	25.1478i	1.31271	−	1.31271i	0.919414	−	0.393292i	$-0.128664\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.75810	+	1.75810i	0.0912761	+	0.0912761i
$372$		0			0
$373$	−35.8452			−1.85599			−0.927997	−	0.372588i	$-0.878471\pi$
$373$	−35.8452			−1.85599			−0.927997	+	0.372588i	$0.878471\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.57019	−	5.57019i	−0.286879	−	0.286879i
$378$		0			0
$379$	−15.5002	−	15.5002i	−0.796193	−	0.796193i	0.186300	−	0.982493i	$-0.440350\pi$
$379$	−15.5002	−	15.5002i	−0.796193	−	0.796193i	−0.982493	+	0.186300i	$0.940350\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	34.3109i		−	1.75321i	−0.481215	−	0.876603i	$-0.659804\pi$
$383$		−	34.3109i		−	1.75321i	0.481215	−	0.876603i	$-0.340196\pi$
$384$		0			0
$385$	−6.02097	+	6.02097i	−0.306857	+	0.306857i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	34.1956i		−	1.73379i	−0.498493	−	0.866894i	$-0.666113\pi$
$389$		−	34.1956i		−	1.73379i	0.498493	−	0.866894i	$-0.333887\pi$
$390$		0			0
$391$	19.4793	−	7.90465i	0.985108	−	0.399755i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−48.3242			−2.43146
$396$		0			0
$397$	0.162887	−	0.162887i	0.00817507	−	0.00817507i	−0.703007	−	0.711183i	$-0.748160\pi$
$397$	0.162887	−	0.162887i	0.00817507	−	0.00817507i	0.711183	+	0.703007i	$0.248160\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.429041	−	0.429041i	0.0214253	−	0.0214253i	−0.696313	−	0.717738i	$-0.745177\pi$
$401$	0.429041	−	0.429041i	0.0214253	−	0.0214253i	0.717738	+	0.696313i	$0.245177\pi$
$402$		0			0
$403$	−10.4464	−	10.4464i	−0.520370	−	0.520370i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	13.5177			0.670049
$408$		0			0
$409$	−1.50925			−0.0746278			−0.0373139	−	0.999304i	$-0.511880\pi$
$409$	−1.50925			−0.0746278			−0.0373139	+	0.999304i	$0.511880\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	4.82541	+	4.82541i	0.237443	+	0.237443i
$414$		0			0
$415$	−18.3597	+	18.3597i	−0.901242	+	0.901242i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.56997	+	8.56997i	−0.418671	+	0.418671i	−0.884745	−	0.466075i	$-0.845668\pi$
$419$	−8.56997	+	8.56997i	−0.418671	+	0.418671i	0.466075	+	0.884745i	$0.345668\pi$
$420$		0			0
$421$	17.8730			0.871078			0.435539	−	0.900170i	$-0.356558\pi$
$421$	17.8730			0.871078			0.435539	+	0.900170i	$0.356558\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	18.5180	−	43.8109i	0.898253	−	2.12514i
$426$		0			0
$427$		−	3.58828i		−	0.173649i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	7.87442	−	7.87442i	0.379297	−	0.379297i	−0.491551	−	0.870849i	$-0.663570\pi$
$431$	7.87442	−	7.87442i	0.379297	−	0.379297i	0.870849	+	0.491551i	$0.163570\pi$
$432$		0			0
$433$		−	29.1182i		−	1.39933i	−0.714471	−	0.699665i	$-0.753333\pi$
$433$		−	29.1182i		−	1.39933i	0.714471	−	0.699665i	$-0.246667\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.23650	−	6.23650i	−0.298332	−	0.298332i
$438$		0			0
$439$	−15.9776	−	15.9776i	−0.762571	−	0.762571i	0.214215	−	0.976786i	$-0.431281\pi$
$439$	−15.9776	−	15.9776i	−0.762571	−	0.762571i	−0.976786	+	0.214215i	$0.931281\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	32.1976			1.52975			0.764877	−	0.644176i	$-0.222800\pi$
$443$	32.1976			1.52975			0.764877	+	0.644176i	$0.222800\pi$
$444$		0			0
$445$	29.7883	+	29.7883i	1.41210	+	1.41210i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.792171	−	0.792171i	0.0373849	−	0.0373849i	−0.688167	−	0.725552i	$-0.741584\pi$
$449$	0.792171	−	0.792171i	0.0373849	−	0.0373849i	0.725552	+	0.688167i	$0.241584\pi$
$450$		0			0
$451$			11.4645i			0.539844i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	13.4077			0.628561
$456$		0			0
$457$		−	40.0795i		−	1.87484i	−0.348203	−	0.937419i	$-0.613208\pi$
$457$		−	40.0795i		−	1.87484i	0.348203	−	0.937419i	$-0.386792\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			4.99595i			0.232684i	0.993209	+	0.116342i	$0.0371169\pi$
$461$			4.99595i			0.232684i	−0.993209	+	0.116342i	$0.962883\pi$
$462$		0			0
$463$	39.0452			1.81458			0.907292	−	0.420502i	$-0.138146\pi$
$463$	39.0452			1.81458			0.907292	+	0.420502i	$0.138146\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			10.5242i			0.487002i	0.969901	+	0.243501i	$0.0782959\pi$
$467$			10.5242i			0.487002i	−0.969901	+	0.243501i	$0.921704\pi$
$468$		0			0
$469$	7.85403	−	7.85403i	0.362665	−	0.362665i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	8.39778	+	8.39778i	0.386130	+	0.386130i
$474$		0			0
$475$	−19.9553			−0.915611
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−18.3593	−	18.3593i	−0.838858	−	0.838858i	0.149851	−	0.988709i	$-0.452121\pi$
$479$	−18.3593	−	18.3593i	−0.838858	−	0.838858i	−0.988709	+	0.149851i	$0.952121\pi$
$480$		0			0
$481$	−15.0508	−	15.0508i	−0.686259	−	0.686259i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	58.8392i		−	2.67175i
$486$		0			0
$487$	−18.0599	+	18.0599i	−0.818372	+	0.818372i	−0.985872	−	0.167500i	$-0.946430\pi$
$487$	−18.0599	+	18.0599i	−0.818372	+	0.818372i	0.167500	+	0.985872i	$0.446430\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			2.50764i			0.113168i	0.998398	+	0.0565842i	$0.0180209\pi$
$491$			2.50764i			0.113168i	−0.998398	+	0.0565842i	$0.981979\pi$
$492$		0			0
$493$	−3.41657	−	8.41937i	−0.153874	−	0.379189i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.40569			−0.0630540
$498$		0			0
$499$	−10.7671	+	10.7671i	−0.482004	+	0.482004i	−0.905771	−	0.423767i	$-0.860707\pi$
$499$	−10.7671	+	10.7671i	−0.482004	+	0.482004i	0.423767	+	0.905771i	$0.360707\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	27.6472	−	27.6472i	1.23273	−	1.23273i	0.269815	−	0.962912i	$-0.413038\pi$
$503$	27.6472	−	27.6472i	1.23273	−	1.23273i	0.962912	−	0.269815i	$-0.0869625\pi$
$504$		0			0
$505$	−0.764127	−	0.764127i	−0.0340032	−	0.0340032i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	31.9138			1.41455			0.707277	−	0.706937i	$-0.249924\pi$
$509$	31.9138			1.41455			0.707277	+	0.706937i	$0.249924\pi$
$510$		0			0
$511$	−1.07460			−0.0475374
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8.72480	+	8.72480i	0.384461	+	0.384461i
$516$		0			0
$517$	11.1746	−	11.1746i	0.491458	−	0.491458i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.41572	−	3.41572i	0.149646	−	0.149646i	−0.628314	−	0.777960i	$-0.716255\pi$
$521$	3.41572	−	3.41572i	0.149646	−	0.149646i	0.777960	+	0.628314i	$0.216255\pi$
$522$		0			0
$523$	36.5734			1.59924			0.799622	−	0.600503i	$-0.205033\pi$
$523$	36.5734			1.59924			0.799622	+	0.600503i	$0.205033\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.40745	−	15.7897i	−0.279113	−	0.687811i
$528$		0			0
$529$		−	2.99559i		−	0.130243i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	12.7648	−	12.7648i	0.552904	−	0.552904i
$534$		0			0
$535$		−	79.9154i		−	3.45504i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	9.87099	+	9.87099i	0.425174	+	0.425174i
$540$		0			0
$541$	7.79877	+	7.79877i	0.335295	+	0.335295i	0.854593	−	0.519298i	$-0.173806\pi$
$541$	7.79877	+	7.79877i	0.335295	+	0.335295i	−0.519298	+	0.854593i	$0.673806\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.91010			−0.124655
$546$		0			0
$547$	19.0046	+	19.0046i	0.812579	+	0.812579i	0.985020	−	0.172441i	$-0.0551653\pi$
$547$	19.0046	+	19.0046i	0.812579	+	0.812579i	−0.172441	+	0.985020i	$0.555165\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2.69555	+	2.69555i	−0.114834	+	0.114834i
$552$		0			0
$553$		−	10.9613i		−	0.466122i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−34.9146			−1.47938			−0.739690	−	0.672948i	$-0.765028\pi$
$557$	−34.9146			−1.47938			−0.739690	+	0.672948i	$0.765028\pi$
$558$		0			0
$559$		−	18.7004i		−	0.790943i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			7.91334i			0.333507i	0.985999	+	0.166754i	$0.0533285\pi$
$563$			7.91334i			0.333507i	−0.985999	+	0.166754i	$0.946672\pi$
$564$		0			0
$565$	47.9335			2.01658
$566$		0			0
$567$		0			0
$568$		0			0
$569$			17.2956i			0.725070i	0.931970	+	0.362535i	$0.118089\pi$
$569$			17.2956i			0.725070i	−0.931970	+	0.362535i	$0.881911\pi$
$570$		0			0
$571$	−13.7031	+	13.7031i	−0.573455	+	0.573455i	−0.933092	−	0.359637i	$-0.882901\pi$
$571$	−13.7031	+	13.7031i	−0.573455	+	0.573455i	0.359637	+	0.933092i	$0.382901\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−41.5898	−	41.5898i	−1.73441	−	1.73441i
$576$		0			0
$577$	−2.89714			−0.120610			−0.0603048	−	0.998180i	$-0.519207\pi$
$577$	−2.89714			−0.120610			−0.0603048	+	0.998180i	$0.519207\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4.16450	−	4.16450i	−0.172772	−	0.172772i
$582$		0			0
$583$	−4.32703	−	4.32703i	−0.179207	−	0.179207i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			12.3165i			0.508357i	0.967157	+	0.254178i	$0.0818050\pi$
$587$			12.3165i			0.508357i	−0.967157	+	0.254178i	$0.918195\pi$
$588$		0			0
$589$	−5.05526	+	5.05526i	−0.208298	+	0.208298i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			43.1763i			1.77304i	0.462693	+	0.886519i	$0.346883\pi$
$593$			43.1763i			1.77304i	−0.462693	+	0.886519i	$0.653117\pi$
$594$		0			0
$595$	14.2448	+	6.02097i	0.583979	+	0.246836i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.70607			−0.274003			−0.137001	−	0.990571i	$-0.543746\pi$
$599$	−6.70607			−0.274003			−0.137001	+	0.990571i	$0.543746\pi$
$600$		0			0
$601$	22.5138	−	22.5138i	0.918355	−	0.918355i	−0.0785544	−	0.996910i	$-0.525030\pi$
$601$	22.5138	−	22.5138i	0.918355	−	0.918355i	0.996910	+	0.0785544i	$0.0250304\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−16.8107	+	16.8107i	−0.683452	+	0.683452i
$606$		0			0
$607$	19.7402	+	19.7402i	0.801228	+	0.801228i	0.983288	−	0.182059i	$-0.0582763\pi$
$607$	19.7402	+	19.7402i	0.801228	+	0.801228i	−0.182059	+	0.983288i	$0.558276\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−24.8839			−1.00669
$612$		0			0
$613$	−5.84314			−0.236002			−0.118001	−	0.993013i	$-0.537649\pi$
$613$	−5.84314			−0.236002			−0.118001	+	0.993013i	$0.537649\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	26.1998	+	26.1998i	1.05477	+	1.05477i	0.998411	+	0.0563548i	$0.0179478\pi$
$617$	26.1998	+	26.1998i	1.05477	+	1.05477i	0.0563548	+	0.998411i	$0.482052\pi$
$618$		0			0
$619$	−11.4329	+	11.4329i	−0.459527	+	0.459527i	−0.898500	−	0.438973i	$-0.855342\pi$
$619$	−11.4329	+	11.4329i	−0.459527	+	0.459527i	0.438973	+	0.898500i	$0.355342\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.75683	+	6.75683i	−0.270707	+	0.270707i
$624$		0			0
$625$	−50.3976			−2.01590
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−9.23168	−	22.7494i	−0.368091	−	0.907079i
$630$		0			0
$631$		−	31.6670i		−	1.26064i	−0.776334	−	0.630322i	$-0.782923\pi$
$631$		−	31.6670i		−	1.26064i	0.776334	−	0.630322i	$-0.217077\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.08126	−	8.08126i	0.320695	−	0.320695i
$636$		0			0
$637$		−	21.9810i		−	0.870919i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−11.2637	−	11.2637i	−0.444890	−	0.444890i	0.448762	−	0.893651i	$-0.351865\pi$
$641$	−11.2637	−	11.2637i	−0.444890	−	0.444890i	−0.893651	+	0.448762i	$0.851865\pi$
$642$		0			0
$643$	11.7990	+	11.7990i	0.465307	+	0.465307i	0.900390	−	0.435083i	$-0.143281\pi$
$643$	11.7990	+	11.7990i	0.465307	+	0.465307i	−0.435083	+	0.900390i	$0.643281\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.35044			−0.328290			−0.164145	−	0.986436i	$-0.552486\pi$
$647$	−8.35044			−0.328290			−0.164145	+	0.986436i	$0.552486\pi$
$648$		0			0
$649$	−11.8762	−	11.8762i	−0.466184	−	0.466184i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	20.2923	−	20.2923i	0.794100	−	0.794100i	−0.188058	−	0.982158i	$-0.560219\pi$
$653$	20.2923	−	20.2923i	0.794100	−	0.794100i	0.982158	+	0.188058i	$0.0602192\pi$
$654$		0			0
$655$			32.2347i			1.25951i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	40.3690			1.57255			0.786276	−	0.617876i	$-0.212007\pi$
$659$	40.3690			1.57255			0.786276	+	0.617876i	$0.212007\pi$
$660$		0			0
$661$			39.4706i			1.53523i	0.640913	+	0.767614i	$0.278556\pi$
$661$			39.4706i			1.53523i	−0.640913	+	0.767614i	$0.721444\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	6.48830i		−	0.251606i
$666$		0			0
$667$	−11.2359			−0.435055
$668$		0			0
$669$		0			0
$670$		0			0
$671$			8.83145i			0.340934i
$672$		0			0
$673$	15.2775	−	15.2775i	0.588903	−	0.588903i	−0.348432	−	0.937334i	$-0.613286\pi$
$673$	15.2775	−	15.2775i	0.588903	−	0.588903i	0.937334	+	0.348432i	$0.113286\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.77542	−	1.77542i	−0.0682351	−	0.0682351i	0.672166	−	0.740401i	$-0.265364\pi$
$677$	−1.77542	−	1.77542i	−0.0682351	−	0.0682351i	−0.740401	+	0.672166i	$0.765364\pi$
$678$		0			0
$679$	13.3464			0.512187
$680$		0			0
$681$		0			0
$682$		0			0
$683$	13.2528	+	13.2528i	0.507106	+	0.507106i	0.913637	−	0.406531i	$-0.133262\pi$
$683$	13.2528	+	13.2528i	0.507106	+	0.507106i	−0.406531	+	0.913637i	$0.633262\pi$
$684$		0			0
$685$	0.296794	+	0.296794i	0.0113399	+	0.0113399i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			9.63554i			0.367085i
$690$		0			0
$691$	27.2733	−	27.2733i	1.03752	−	1.03752i	0.0382556	−	0.999268i	$-0.487820\pi$
$691$	27.2733	−	27.2733i	1.03752	−	1.03752i	0.999268	−	0.0382556i	$-0.0121801\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			58.7452i			2.22833i
$696$		0			0
$697$	19.2940	−	7.82948i	0.730813	−	0.296563i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−28.7827			−1.08711			−0.543553	−	0.839375i	$-0.682921\pi$
$701$	−28.7827			−1.08711			−0.543553	+	0.839375i	$0.682921\pi$
$702$		0			0
$703$	−7.28348	+	7.28348i	−0.274702	+	0.274702i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.173325	−	0.173325i	0.00651858	−	0.00651858i
$708$		0			0
$709$	3.63869	+	3.63869i	0.136654	+	0.136654i	0.772125	−	0.635471i	$-0.219194\pi$
$709$	3.63869	+	3.63869i	0.136654	+	0.136654i	−0.635471	+	0.772125i	$0.719194\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−21.0718			−0.789146
$714$		0			0
$715$	−32.9988			−1.23409
$716$		0			0
$717$		0			0
$718$		0			0
$719$	22.2754	+	22.2754i	0.830733	+	0.830733i	0.987617	−	0.156884i	$-0.0501449\pi$
$719$	22.2754	+	22.2754i	0.830733	+	0.830733i	−0.156884	+	0.987617i	$0.550145\pi$
$720$		0			0
$721$	−1.97903	+	1.97903i	−0.0737030	+	0.0737030i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−17.9760	+	17.9760i	−0.667613	+	0.667613i
$726$		0			0
$727$	31.2775			1.16002			0.580010	−	0.814610i	$-0.303049\pi$
$727$	31.2775			1.16002			0.580010	+	0.814610i	$0.303049\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	8.39778	−	19.8680i	0.310603	−	0.734843i
$732$		0			0
$733$		−	18.8153i		−	0.694960i	−0.937687	−	0.347480i	$-0.887037\pi$
$733$		−	18.8153i		−	0.694960i	0.937687	−	0.347480i	$-0.112963\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−19.3303	+	19.3303i	−0.712039	+	0.712039i
$738$		0			0
$739$		−	6.94762i		−	0.255572i	−0.991802	−	0.127786i	$-0.959213\pi$
$739$		−	6.94762i		−	0.255572i	0.991802	−	0.127786i	$-0.0407871\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	24.0107	+	24.0107i	0.880869	+	0.880869i	0.993623	−	0.112754i	$-0.0359672\pi$
$743$	24.0107	+	24.0107i	0.880869	+	0.880869i	−0.112754	+	0.993623i	$0.535967\pi$
$744$		0			0
$745$	10.3387	+	10.3387i	0.378782	+	0.378782i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	18.1271			0.662349
$750$		0			0
$751$	4.88971	+	4.88971i	0.178428	+	0.178428i	0.790670	−	0.612242i	$-0.209732\pi$
$751$	4.88971	+	4.88971i	0.178428	+	0.178428i	−0.612242	+	0.790670i	$0.709732\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−21.8194	+	21.8194i	−0.794089	+	0.794089i
$756$		0			0
$757$			1.86166i			0.0676633i	0.999428	+	0.0338316i	$0.0107710\pi$
$757$			1.86166i			0.0676633i	−0.999428	+	0.0338316i	$0.989229\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	49.2335			1.78471			0.892357	−	0.451330i	$-0.149050\pi$
$761$	49.2335			1.78471			0.892357	+	0.451330i	$0.149050\pi$
$762$		0			0
$763$		−	0.660093i		−	0.0238970i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			26.4464i			0.954923i
$768$		0			0
$769$	−50.9303			−1.83659			−0.918296	−	0.395895i	$-0.870435\pi$
$769$	−50.9303			−1.83659			−0.918296	+	0.395895i	$0.870435\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	2.91614i		−	0.104886i	−0.998624	−	0.0524431i	$-0.983299\pi$
$773$		−	2.91614i		−	0.104886i	0.998624	−	0.0524431i	$-0.0167008\pi$
$774$		0			0
$775$	−33.7123	+	33.7123i	−1.21098	+	1.21098i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6.17719	−	6.17719i	−0.221321	−	0.221321i
$780$		0			0
$781$	3.45968			0.123797
$782$		0			0
$783$		0			0
$784$		0			0
$785$	54.5087	+	54.5087i	1.94550	+	1.94550i
$786$		0			0
$787$	−35.9869	−	35.9869i	−1.28279	−	1.28279i	−0.939071	−	0.343724i	$-0.888312\pi$
$787$	−35.9869	−	35.9869i	−1.28279	−	1.28279i	−0.343724	−	0.939071i	$-0.611688\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			10.8727i			0.386588i
$792$		0			0
$793$	9.83306	−	9.83306i	0.349182	−	0.349182i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	13.6061i		−	0.481952i	−0.970531	−	0.240976i	$-0.922532\pi$
$797$		−	13.6061i		−	0.481952i	0.970531	−	0.240976i	$-0.0774676\pi$
$798$		0			0
$799$	−26.4375	−	11.1746i	−0.935293	−	0.395329i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.64479			0.0933326
$804$		0			0
$805$	13.5226	−	13.5226i	0.476609	−	0.476609i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	12.2992	−	12.2992i	0.432416	−	0.432416i	−0.457033	−	0.889450i	$-0.651088\pi$
$809$	12.2992	−	12.2992i	0.432416	−	0.432416i	0.889450	+	0.457033i	$0.151088\pi$
$810$		0			0
$811$	−23.2841	−	23.2841i	−0.817614	−	0.817614i	0.168148	−	0.985762i	$-0.446221\pi$
$811$	−23.2841	−	23.2841i	−0.817614	−	0.817614i	−0.985762	+	0.168148i	$0.946221\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	54.6246			1.91342
$816$		0			0
$817$	−9.04959			−0.316605
$818$		0			0
$819$		0			0
$820$		0			0
$821$	30.9266	+	30.9266i	1.07935	+	1.07935i	0.996568	+	0.0827780i	$0.0263792\pi$
$821$	30.9266	+	30.9266i	1.07935	+	1.07935i	0.0827780	+	0.996568i	$0.473621\pi$
$822$		0			0
$823$	1.18371	−	1.18371i	0.0412617	−	0.0412617i	−0.686175	−	0.727437i	$-0.740712\pi$
$823$	1.18371	−	1.18371i	0.0412617	−	0.0412617i	0.727437	+	0.686175i	$0.240712\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.60525	+	1.60525i	−0.0558198	+	0.0558198i	−0.734466	−	0.678646i	$-0.762567\pi$
$827$	−1.60525	+	1.60525i	−0.0558198	+	0.0558198i	0.678646	+	0.734466i	$0.262567\pi$
$828$		0			0
$829$	−19.2597			−0.668918			−0.334459	−	0.942410i	$-0.608554\pi$
$829$	−19.2597			−0.668918			−0.334459	+	0.942410i	$0.608554\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9.87099	−	23.3534i	0.342010	−	0.809147i
$834$		0			0
$835$		−	8.69114i		−	0.300769i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	10.4694	−	10.4694i	0.361443	−	0.361443i	−0.502901	−	0.864344i	$-0.667734\pi$
$839$	10.4694	−	10.4694i	0.361443	−	0.361443i	0.864344	+	0.502901i	$0.167734\pi$
$840$		0			0
$841$		−	24.1436i		−	0.832538i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.638906	−	0.638906i	−0.0219790	−	0.0219790i
$846$		0			0
$847$	−3.81314	−	3.81314i	−0.131021	−	0.131021i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−30.3597			−1.04072
$852$		0			0
$853$	4.49558	+	4.49558i	0.153926	+	0.153926i	0.779869	−	0.625943i	$-0.215286\pi$
$853$	4.49558	+	4.49558i	0.153926	+	0.153926i	−0.625943	+	0.779869i	$0.715286\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−28.8720	+	28.8720i	−0.986248	+	0.986248i	−0.999907	−	0.0136588i	$-0.995652\pi$
$857$	−28.8720	+	28.8720i	−0.986248	+	0.986248i	0.0136588	+	0.999907i	$0.495652\pi$
$858$		0			0
$859$		−	22.1464i		−	0.755626i	−0.925882	−	0.377813i	$-0.876676\pi$
$859$		−	22.1464i		−	0.755626i	0.925882	−	0.377813i	$-0.123324\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−17.3423			−0.590339			−0.295170	−	0.955445i	$-0.595376\pi$
$863$	−17.3423			−0.590339			−0.295170	+	0.955445i	$0.595376\pi$
$864$		0			0
$865$			21.4706i			0.730021i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			26.9779i			0.915161i
$870$		0			0
$871$	43.0452			1.45853
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	24.5149i		−	0.828756i
$876$		0			0
$877$	−21.3763	+	21.3763i	−0.721825	+	0.721825i	−0.968977	−	0.247152i	$-0.920505\pi$
$877$	−21.3763	+	21.3763i	−0.721825	+	0.721825i	0.247152	+	0.968977i	$0.420505\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−7.47035	−	7.47035i	−0.251683	−	0.251683i	0.569978	−	0.821660i	$-0.306952\pi$
$881$	−7.47035	−	7.47035i	−0.251683	−	0.251683i	−0.821660	+	0.569978i	$0.806952\pi$
$882$		0			0
$883$	−13.9150			−0.468276			−0.234138	−	0.972203i	$-0.575227\pi$
$883$	−13.9150			−0.468276			−0.234138	+	0.972203i	$0.575227\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	25.4307	+	25.4307i	0.853878	+	0.853878i	0.990608	−	0.136730i	$-0.0436593\pi$
$887$	25.4307	+	25.4307i	0.853878	+	0.853878i	−0.136730	+	0.990608i	$0.543659\pi$
$888$		0			0
$889$	1.83306	+	1.83306i	0.0614788	+	0.0614788i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			12.0419i			0.402968i
$894$		0			0
$895$	22.3597	−	22.3597i	0.747402	−	0.747402i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			9.10771i			0.303759i
$900$		0			0
$901$	−4.32703	+	10.2371i	−0.144154	+	0.341049i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	3.72418			0.123796
$906$		0			0
$907$	27.7075	−	27.7075i	0.920012	−	0.920012i	−0.0770180	−	0.997030i	$-0.524540\pi$
$907$	27.7075	−	27.7075i	0.920012	−	0.920012i	0.997030	+	0.0770180i	$0.0245399\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−7.02020	+	7.02020i	−0.232590	+	0.232590i	−0.813773	−	0.581183i	$-0.802590\pi$
$911$	−7.02020	+	7.02020i	−0.232590	+	0.232590i	0.581183	+	0.813773i	$0.302590\pi$
$912$		0			0
$913$	10.2496	+	10.2496i	0.339213	+	0.339213i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−7.31174			−0.241455
$918$		0			0
$919$	−42.8795			−1.41446			−0.707232	−	0.706982i	$-0.750056\pi$
$919$	−42.8795			−1.41446			−0.707232	+	0.706982i	$0.750056\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−3.85206	−	3.85206i	−0.126792	−	0.126792i
$924$		0			0
$925$	−48.5718	+	48.5718i	−1.59703	+	1.59703i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	12.8427	−	12.8427i	0.421356	−	0.421356i	−0.464314	−	0.885671i	$-0.653699\pi$
$929$	12.8427	−	12.8427i	0.421356	−	0.421356i	0.885671	+	0.464314i	$0.153699\pi$
$930$		0			0
$931$	−10.6372			−0.348619
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−35.0591	−	14.8188i	−1.14656	−	0.484625i
$936$		0			0
$937$		−	1.60243i		−	0.0523492i	−0.999657	−	0.0261746i	$-0.991667\pi$
$937$		−	1.60243i		−	0.0523492i	0.999657	−	0.0261746i	$-0.00833258\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−22.8780	+	22.8780i	−0.745802	+	0.745802i	−0.973688	−	0.227886i	$-0.926819\pi$
$941$	−22.8780	+	22.8780i	−0.745802	+	0.745802i	0.227886	+	0.973688i	$0.426819\pi$
$942$		0			0
$943$		−	25.7484i		−	0.838482i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−17.8063	−	17.8063i	−0.578626	−	0.578626i	0.355898	−	0.934525i	$-0.384175\pi$
$947$	−17.8063	−	17.8063i	−0.578626	−	0.578626i	−0.934525	+	0.355898i	$0.884175\pi$
$948$		0			0
$949$	−2.94474	−	2.94474i	−0.0955905	−	0.0955905i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−54.0658			−1.75136			−0.875681	−	0.482889i	$-0.839587\pi$
$953$	−54.0658			−1.75136			−0.875681	+	0.482889i	$0.839587\pi$
$954$		0			0
$955$	55.1093	+	55.1093i	1.78330	+	1.78330i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−0.0673212	+	0.0673212i	−0.00217391	+	0.00217391i
$960$		0			0
$961$		−	13.9194i		−	0.449012i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	19.1585			0.616733
$966$		0			0
$967$		−	2.17495i		−	0.0699418i	−0.999388	−	0.0349709i	$-0.988866\pi$
$967$		−	2.17495i		−	0.0699418i	0.999388	−	0.0349709i	$-0.0111338\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	47.4481i		−	1.52268i	−0.648352	−	0.761340i	$-0.724542\pi$
$971$		−	47.4481i		−	1.52268i	0.648352	−	0.761340i	$-0.275458\pi$
$972$		0			0
$973$	−13.3251			−0.427182
$974$		0			0
$975$		0			0
$976$		0			0
$977$			20.8226i			0.666175i	0.942896	+	0.333087i	$0.108090\pi$
$977$			20.8226i			0.666175i	−0.942896	+	0.333087i	$0.891910\pi$
$978$		0			0
$979$	16.6299	−	16.6299i	0.531493	−	0.531493i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.71854	+	1.71854i	0.0548130	+	0.0548130i	0.733982	−	0.679169i	$-0.237660\pi$
$983$	1.71854	+	1.71854i	0.0548130	+	0.0548130i	−0.679169	+	0.733982i	$0.737660\pi$
$984$		0			0
$985$	60.5690			1.92989
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−18.8607	−	18.8607i	−0.599735	−	0.599735i
$990$		0			0
$991$	25.8643	+	25.8643i	0.821608	+	0.821608i	0.986339	−	0.164731i	$-0.0526755\pi$
$991$	25.8643	+	25.8643i	0.821608	+	0.821608i	−0.164731	+	0.986339i	$0.552676\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	76.3827i		−	2.42150i
$996$		0			0
$997$	14.6448	−	14.6448i	0.463806	−	0.463806i	−0.436095	−	0.899901i	$-0.643639\pi$
$997$	14.6448	−	14.6448i	0.463806	−	0.463806i	0.899901	+	0.436095i	$0.143639\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2448.2.be.x.1441.4		8
3.2	odd	2		816.2.bd.e.625.1		8
4.3	odd	2		153.2.f.b.64.2		8
12.11	even	2		51.2.e.a.13.3	yes	8
17.4	even	4	inner	2448.2.be.x.1585.4		8
51.38	odd	4		816.2.bd.e.769.1		8
68.15	odd	8		2601.2.a.bf.1.3		4
68.19	odd	8		2601.2.a.be.1.3		4
68.55	odd	4		153.2.f.b.55.3		8
204.11	odd	16		867.2.h.k.757.1		16
204.23	odd	16		867.2.h.k.757.2		16
204.47	even	4		867.2.e.g.616.2		8
204.59	even	8		867.2.d.f.577.5		8
204.71	odd	16		867.2.h.i.712.4		16
204.83	even	8		867.2.a.k.1.2		4
204.95	odd	16		867.2.h.i.688.3		16
204.107	odd	16		867.2.h.k.733.1		16
204.131	odd	16		867.2.h.k.733.2		16
204.143	odd	16		867.2.h.i.688.4		16
204.155	even	8		867.2.a.l.1.2		4
204.167	odd	16		867.2.h.i.712.3		16
204.179	even	8		867.2.d.f.577.6		8
204.191	even	4		51.2.e.a.4.2	✓	8
204.203	even	2		867.2.e.g.829.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.e.a.4.2	✓	8	204.191	even	4
51.2.e.a.13.3	yes	8	12.11	even	2
153.2.f.b.55.3		8	68.55	odd	4
153.2.f.b.64.2		8	4.3	odd	2
816.2.bd.e.625.1		8	3.2	odd	2
816.2.bd.e.769.1		8	51.38	odd	4
867.2.a.k.1.2		4	204.83	even	8
867.2.a.l.1.2		4	204.155	even	8
867.2.d.f.577.5		8	204.59	even	8
867.2.d.f.577.6		8	204.179	even	8
867.2.e.g.616.2		8	204.47	even	4
867.2.e.g.829.3		8	204.203	even	2
867.2.h.i.688.3		16	204.95	odd	16
867.2.h.i.688.4		16	204.143	odd	16
867.2.h.i.712.3		16	204.167	odd	16
867.2.h.i.712.4		16	204.71	odd	16
867.2.h.k.733.1		16	204.107	odd	16
867.2.h.k.733.2		16	204.131	odd	16
867.2.h.k.757.1		16	204.11	odd	16
867.2.h.k.757.2		16	204.23	odd	16
2448.2.be.x.1441.4		8	1.1	even	1	trivial
2448.2.be.x.1585.4		8	17.4	even	4	inner
2601.2.a.be.1.3		4	68.19	odd	8
2601.2.a.bf.1.3		4	68.15	odd	8

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2448.be (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.87540 + 2.87540i) q^{5} +(-0.652223 + 0.652223i) q^{7} +(1.60525 - 1.60525i) q^{11} -3.57461 q^{13} +(-3.79779 - 1.60525i) q^{17} +1.72984i q^{19} +(-3.60525 + 3.60525i) q^{23} +11.5359i q^{25} +(1.55827 + 1.55827i) q^{29} +(2.92238 + 2.92238i) q^{31} -3.75081 q^{35} +(4.21049 + 4.21049i) q^{37} +(-3.57096 + 3.57096i) q^{41} +5.23146i q^{43} +6.96130 q^{47} +6.14921i q^{49} -2.69555i q^{53} +9.23146 q^{55} -7.39840i q^{59} +(-2.75081 + 2.75081i) q^{61} +(-10.2784 - 10.2784i) q^{65} -12.0419 q^{67} +(1.07762 + 1.07762i) q^{71} +(0.823796 + 0.823796i) q^{73} +2.09396i q^{77} +(-8.40303 + 8.40303i) q^{79} +6.38508i q^{83} +(-6.30445 - 15.5359i) q^{85} +10.3597 q^{89} +(2.33144 - 2.33144i) q^{91} +(-4.97399 + 4.97399i) q^{95} +(-10.2315 - 10.2315i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{5} + 4 q^{7} - 4 q^{13} + 4 q^{17} - 16 q^{23} - 4 q^{29} + 8 q^{31} + 8 q^{35} + 8 q^{37} - 28 q^{41} - 8 q^{47} + 4 q^{55} + 16 q^{61} - 16 q^{65} - 8 q^{67} + 24 q^{71} + 20 q^{73} - 20 q^{79}+ \cdots - 12 q^{97}+O(q^{100}) \) 8 * q + 4 * q^5 + 4 * q^7 - 4 * q^13 + 4 * q^17 - 16 * q^23 - 4 * q^29 + 8 * q^31 + 8 * q^35 + 8 * q^37 - 28 * q^41 - 8 * q^47 + 4 * q^55 + 16 * q^61 - 16 * q^65 - 8 * q^67 + 24 * q^71 + 20 * q^73 - 20 * q^79 - 32 * q^85 + 8 * q^89 + 36 * q^91 + 8 * q^95 - 12 * q^97

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