LMFDB - Embedded newform 2448.2.be.x.1441.1

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			1441.1
Root			$0.222191i$ of defining polynomial
Character	$\chi$	$=$	2448.1441
Dual form			2448.2.be.x.1585.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.57133 - 1.57133i) q^{5} +(-1.63640 + 1.63640i) q^{7} +(1.37930 - 1.37930i) q^{11} -1.32218 q^{13} +(3.88555 - 1.37930i) q^{17} +5.95063i q^{19} +(-3.37930 + 3.37930i) q^{23} -0.0618655i q^{25} +(0.122204 + 0.122204i) q^{29} +(-0.314226 - 0.314226i) q^{31} +5.14265 q^{35} +(3.75861 + 3.75861i) q^{37} +(2.84414 - 2.84414i) q^{41} -8.33468i q^{43} -2.38404 q^{47} +1.64436i q^{49} -0.727190i q^{53} -4.33468 q^{55} -11.7870i q^{59} +(6.14265 - 6.14265i) q^{61} +(2.07757 + 2.07757i) q^{65} +14.1866 q^{67} +(4.31423 + 4.31423i) q^{71} +(7.46483 + 7.46483i) q^{73} +4.51420i q^{77} +(-0.493752 + 0.493752i) q^{79} -8.52971i q^{83} +(-8.27281 - 3.93813i) q^{85} +5.40297 q^{89} +(2.16362 - 2.16362i) q^{91} +(9.35038 - 9.35038i) q^{95} +(3.33468 + 3.33468i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.57133	−	1.57133i	−0.702719	−	0.702719i	0.262275	−	0.964993i	$-0.415527\pi$
$5$	−1.57133	−	1.57133i	−0.702719	−	0.702719i	−0.964993	+	0.262275i	$0.915527\pi$
$6$		0			0
$7$	−1.63640	+	1.63640i	−0.618503	+	0.618503i	−0.945147	−	0.326644i	$-0.894082\pi$
$7$	−1.63640	+	1.63640i	−0.618503	+	0.618503i	0.326644	+	0.945147i	$0.394082\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.37930	−	1.37930i	0.415876	−	0.415876i	−0.467904	−	0.883780i	$-0.654991\pi$
$11$	1.37930	−	1.37930i	0.415876	−	0.415876i	0.883780	+	0.467904i	$0.154991\pi$
$12$		0			0
$13$	−1.32218			−0.366706			−0.183353	−	0.983047i	$-0.558695\pi$
$13$	−1.32218			−0.366706			−0.183353	+	0.983047i	$0.558695\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.88555	−	1.37930i	0.942385	−	0.334530i
$17$	3.88555	−	1.37930i	0.942385	−	0.334530i
$18$		0			0
$19$			5.95063i			1.36517i	0.730807	+	0.682584i	$0.239144\pi$
$19$			5.95063i			1.36517i	−0.730807	+	0.682584i	$0.760856\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.37930	+	3.37930i	−0.704634	+	0.704634i	−0.965402	−	0.260768i	$-0.916024\pi$
$23$	−3.37930	+	3.37930i	−0.704634	+	0.704634i	0.260768	+	0.965402i	$0.416024\pi$
$24$		0			0
$25$		−	0.0618655i		−	0.0123731i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.122204	+	0.122204i	0.0226927	+	0.0226927i	0.718362	−	0.695669i	$-0.244892\pi$
$29$	0.122204	+	0.122204i	0.0226927	+	0.0226927i	−0.695669	+	0.718362i	$0.744892\pi$
$30$		0			0
$31$	−0.314226	−	0.314226i	−0.0564367	−	0.0564367i	0.678325	−	0.734762i	$-0.262706\pi$
$31$	−0.314226	−	0.314226i	−0.0564367	−	0.0564367i	−0.734762	+	0.678325i	$0.762706\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	5.14265			0.869267
$36$		0			0
$37$	3.75861	+	3.75861i	0.617911	+	0.617911i	0.944995	−	0.327084i	$-0.106066\pi$
$37$	3.75861	+	3.75861i	0.617911	+	0.617911i	−0.327084	+	0.944995i	$0.606066\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.84414	−	2.84414i	0.444179	−	0.444179i	−0.449234	−	0.893414i	$-0.648303\pi$
$41$	2.84414	−	2.84414i	0.444179	−	0.444179i	0.893414	+	0.449234i	$0.148303\pi$
$42$		0			0
$43$		−	8.33468i		−	1.27103i	−0.772090	−	0.635513i	$-0.780789\pi$
$43$		−	8.33468i		−	1.27103i	0.772090	−	0.635513i	$-0.219211\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.38404			−0.347749			−0.173874	−	0.984768i	$-0.555629\pi$
$47$	−2.38404			−0.347749			−0.173874	+	0.984768i	$0.555629\pi$
$48$		0			0
$49$			1.64436i			0.234908i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	0.727190i		−	0.0998872i	−0.998752	−	0.0499436i	$-0.984096\pi$
$53$		−	0.727190i		−	0.0998872i	0.998752	−	0.0499436i	$-0.0159042\pi$
$54$		0			0
$55$	−4.33468			−0.584488
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	11.7870i		−	1.53454i	−0.641325	−	0.767269i	$-0.721615\pi$
$59$		−	11.7870i		−	1.53454i	0.641325	−	0.767269i	$-0.278385\pi$
$60$		0			0
$61$	6.14265	−	6.14265i	0.786486	−	0.786486i	−0.194430	−	0.980916i	$-0.562286\pi$
$61$	6.14265	−	6.14265i	0.786486	−	0.786486i	0.980916	+	0.194430i	$0.0622858\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.07757	+	2.07757i	0.257691	+	0.257691i
$66$		0			0
$67$	14.1866			1.73317			0.866583	−	0.499034i	$-0.166312\pi$
$67$	14.1866			1.73317			0.866583	+	0.499034i	$0.166312\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.31423	+	4.31423i	0.512004	+	0.512004i	0.915140	−	0.403136i	$-0.132080\pi$
$71$	4.31423	+	4.31423i	0.512004	+	0.512004i	−0.403136	+	0.915140i	$0.632080\pi$
$72$		0			0
$73$	7.46483	+	7.46483i	0.873693	+	0.873693i	0.992873	−	0.119180i	$-0.0380266\pi$
$73$	7.46483	+	7.46483i	0.873693	+	0.873693i	−0.119180	+	0.992873i	$0.538027\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			4.51420i			0.514441i
$78$		0			0
$79$	−0.493752	+	0.493752i	−0.0555514	+	0.0555514i	−0.734337	−	0.678785i	$-0.762507\pi$
$79$	−0.493752	+	0.493752i	−0.0555514	+	0.0555514i	0.678785	+	0.734337i	$0.262507\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	8.52971i		−	0.936258i	−0.883660	−	0.468129i	$-0.844928\pi$
$83$		−	8.52971i		−	0.936258i	0.883660	−	0.468129i	$-0.155072\pi$
$84$		0			0
$85$	−8.27281	−	3.93813i	−0.897312	−	0.427151i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.40297			0.572713			0.286357	−	0.958123i	$-0.407556\pi$
$89$	5.40297			0.572713			0.286357	+	0.958123i	$0.407556\pi$
$90$		0			0
$91$	2.16362	−	2.16362i	0.226809	−	0.226809i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	9.35038	−	9.35038i	0.959329	−	0.959329i
$96$		0			0
$97$	3.33468	+	3.33468i	0.338585	+	0.338585i	0.855835	−	0.517250i	$-0.173044\pi$
$97$	3.33468	+	3.33468i	0.338585	+	0.338585i	−0.517250	+	0.855835i	$0.673044\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.11123			0.707594			0.353797	−	0.935322i	$-0.384890\pi$
$101$	7.11123			0.707594			0.353797	+	0.935322i	$0.384890\pi$
$102$		0			0
$103$	9.22344			0.908813			0.454406	−	0.890795i	$-0.349851\pi$
$103$	9.22344			0.908813			0.454406	+	0.890795i	$0.349851\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.66461	+	3.66461i	0.354271	+	0.354271i	0.861696	−	0.507425i	$-0.169402\pi$
$107$	3.66461	+	3.66461i	0.354271	+	0.354271i	−0.507425	+	0.861696i	$0.669402\pi$
$108$		0			0
$109$	14.1247	−	14.1247i	1.35290	−	1.35290i	0.470503	−	0.882399i	$-0.344073\pi$
$109$	14.1247	−	14.1247i	1.35290	−	1.35290i	0.882399	−	0.470503i	$-0.155927\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	12.3299	−	12.3299i	1.15990	−	1.15990i	0.175407	−	0.984496i	$-0.443876\pi$
$113$	12.3299	−	12.3299i	1.15990	−	1.15990i	0.984496	−	0.175407i	$-0.0561240\pi$
$114$		0			0
$115$	10.6200			0.990318
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.10124	+	8.61544i	−0.375960	+	0.789776i
$120$		0			0
$121$			7.19504i			0.654094i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.95384	+	7.95384i	−0.711413	+	0.711413i
$126$		0			0
$127$			9.85189i			0.874214i	0.899409	+	0.437107i	$0.143997\pi$
$127$			9.85189i			0.874214i	−0.899409	+	0.437107i	$0.856003\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.37930	+	5.37930i	0.469992	+	0.469992i	0.901912	−	0.431920i	$-0.142164\pi$
$131$	5.37930	+	5.37930i	0.469992	+	0.469992i	−0.431920	+	0.901912i	$0.642164\pi$
$132$		0			0
$133$	−9.73764	−	9.73764i	−0.844361	−	0.844361i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−22.0723			−1.88577			−0.942883	−	0.333125i	$-0.891897\pi$
$137$	−22.0723			−1.88577			−0.942883	+	0.333125i	$0.891897\pi$
$138$		0			0
$139$	−2.53465	−	2.53465i	−0.214986	−	0.214986i	0.591395	−	0.806382i	$-0.298577\pi$
$139$	−2.53465	−	2.53465i	−0.214986	−	0.214986i	−0.806382	+	0.591395i	$0.798577\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.82369	+	1.82369i	−0.152504	+	0.152504i
$144$		0			0
$145$		−	0.384044i		−	0.0318931i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−11.7711			−0.964327			−0.482163	−	0.876081i	$-0.660149\pi$
$149$	−11.7711			−0.964327			−0.482163	+	0.876081i	$0.660149\pi$
$150$		0			0
$151$		−	16.1037i		−	1.31050i	−0.755411	−	0.655252i	$-0.772563\pi$
$151$		−	16.1037i		−	1.31050i	0.755411	−	0.655252i	$-0.227437\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			0.987504i			0.0793182i
$156$		0			0
$157$	6.45535			0.515193			0.257597	−	0.966253i	$-0.417069\pi$
$157$	6.45535			0.515193			0.257597	+	0.966253i	$0.417069\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	11.0598i		−	0.871636i
$162$		0			0
$163$	11.8314	−	11.8314i	0.926710	−	0.926710i	−0.0707816	−	0.997492i	$-0.522549\pi$
$163$	11.8314	−	11.8314i	0.926710	−	0.926710i	0.997492	+	0.0707816i	$0.0225494\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.13490	+	1.13490i	0.0878209	+	0.0878209i	0.749653	−	0.661832i	$-0.230221\pi$
$167$	1.13490	+	1.13490i	0.0878209	+	0.0878209i	−0.661832	+	0.749653i	$0.730221\pi$
$168$		0			0
$169$	−11.2518			−0.865526
$170$		0			0
$171$		0			0
$172$		0			0
$173$	12.1169	+	12.1169i	0.921234	+	0.921234i	0.997117	−	0.0758824i	$-0.0241774\pi$
$173$	12.1169	+	12.1169i	0.921234	+	0.921234i	−0.0758824	+	0.997117i	$0.524177\pi$
$174$		0			0
$175$	0.101237	+	0.101237i	0.00765280	+	0.00765280i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			11.0753i			0.827809i	0.910320	+	0.413905i	$0.135835\pi$
$179$			11.0753i			0.827809i	−0.910320	+	0.413905i	$0.864165\pi$
$180$		0			0
$181$	13.9297	−	13.9297i	1.03538	−	1.03538i	0.0360332	−	0.999351i	$-0.488528\pi$
$181$	13.9297	−	13.9297i	1.03538	−	1.03538i	0.999351	−	0.0360332i	$-0.0114722\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	11.8120i		−	0.868436i
$186$		0			0
$187$	3.45688	−	7.26184i	0.252792	−	0.531038i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.60953			−0.116461			−0.0582307	−	0.998303i	$-0.518546\pi$
$191$	−1.60953			−0.116461			−0.0582307	+	0.998303i	$0.518546\pi$
$192$		0			0
$193$	3.16362	−	3.16362i	0.227722	−	0.227722i	−0.584018	−	0.811741i	$-0.698520\pi$
$193$	3.16362	−	3.16362i	0.227722	−	0.227722i	0.811741	+	0.584018i	$0.198520\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5.22818	+	5.22818i	−0.372492	+	0.372492i	−0.868384	−	0.495892i	$-0.834841\pi$
$197$	−5.22818	+	5.22818i	−0.372492	+	0.372492i	0.495892	+	0.868384i	$0.334841\pi$
$198$		0			0
$199$	−5.08874	−	5.08874i	−0.360731	−	0.360731i	0.503351	−	0.864082i	$-0.332100\pi$
$199$	−5.08874	−	5.08874i	−0.360731	−	0.360731i	−0.864082	+	0.503351i	$0.832100\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.399950			−0.0280710
$204$		0			0
$205$	−8.93813			−0.624266
$206$		0			0
$207$		0			0
$208$		0			0
$209$	8.20773	+	8.20773i	0.567741	+	0.567741i
$210$		0			0
$211$	9.48830	−	9.48830i	0.653201	−	0.653201i	−0.300561	−	0.953763i	$-0.597174\pi$
$211$	9.48830	−	9.48830i	0.653201	−	0.653201i	0.953763	+	0.300561i	$0.0971740\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−13.0965	+	13.0965i	−0.893174	+	0.893174i
$216$		0			0
$217$	1.02840			0.0698125
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.13740	+	1.82369i	−0.345579	+	0.122674i
$222$		0			0
$223$			21.2643i			1.42397i	0.702197	+	0.711983i	$0.252203\pi$
$223$			21.2643i			1.42397i	−0.702197	+	0.711983i	$0.747797\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.50304	−	3.50304i	0.232505	−	0.232505i	−0.581233	−	0.813737i	$-0.697429\pi$
$227$	3.50304	−	3.50304i	0.232505	−	0.232505i	0.813737	+	0.581233i	$0.197429\pi$
$228$		0			0
$229$			1.83842i			0.121487i	0.998153	+	0.0607433i	$0.0193471\pi$
$229$			1.83842i			0.121487i	−0.998153	+	0.0607433i	$0.980653\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.05713	+	1.05713i	0.0692546	+	0.0692546i	0.740886	−	0.671631i	$-0.234406\pi$
$233$	1.05713	+	1.05713i	0.0692546	+	0.0692546i	−0.671631	+	0.740886i	$0.734406\pi$
$234$		0			0
$235$	3.74611	+	3.74611i	0.244369	+	0.244369i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−10.0250			−0.648463			−0.324231	−	0.945978i	$-0.605106\pi$
$239$	−10.0250			−0.648463			−0.324231	+	0.945978i	$0.605106\pi$
$240$		0			0
$241$	−8.14265	−	8.14265i	−0.524514	−	0.524514i	0.394417	−	0.918932i	$-0.370947\pi$
$241$	−8.14265	−	8.14265i	−0.524514	−	0.524514i	−0.918932	+	0.394417i	$0.870947\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.58382	−	2.58382i	0.165074	−	0.165074i
$246$		0			0
$247$		−	7.86780i		−	0.500616i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	0.940179			0.0593436			0.0296718	−	0.999560i	$-0.490554\pi$
$251$	0.940179			0.0593436			0.0296718	+	0.999560i	$0.490554\pi$
$252$		0			0
$253$			9.32218i			0.586080i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			2.15515i			0.134435i	0.997738	+	0.0672173i	$0.0214121\pi$
$257$			2.15515i			0.134435i	−0.997738	+	0.0672173i	$0.978588\pi$
$258$		0			0
$259$	−12.3012			−0.764360
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	13.1427i		−	0.810411i	−0.914226	−	0.405205i	$-0.867200\pi$
$263$		−	13.1427i		−	0.810411i	0.914226	−	0.405205i	$-0.132800\pi$
$264$		0			0
$265$	−1.14265	+	1.14265i	−0.0701926	+	0.0701926i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.7299	+	10.7299i	0.654213	+	0.654213i	0.954005	−	0.299792i	$-0.0969173\pi$
$269$	10.7299	+	10.7299i	0.654213	+	0.654213i	−0.299792	+	0.954005i	$0.596917\pi$
$270$		0			0
$271$	−11.8360			−0.718985			−0.359493	−	0.933148i	$-0.617050\pi$
$271$	−11.8360			−0.718985			−0.359493	+	0.933148i	$0.617050\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.0853313	−	0.0853313i	−0.00514567	−	0.00514567i
$276$		0			0
$277$	19.0229	+	19.0229i	1.14298	+	1.14298i	0.987903	+	0.155076i	$0.0495621\pi$
$277$	19.0229	+	19.0229i	1.14298	+	1.14298i	0.155076	+	0.987903i	$0.450438\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			22.2339i			1.32636i	0.748459	+	0.663181i	$0.230794\pi$
$281$			22.2339i			1.32636i	−0.748459	+	0.663181i	$0.769206\pi$
$282$		0			0
$283$	−16.9406	+	16.9406i	−1.00702	+	1.00702i	−0.00704092	+	0.999975i	$0.502241\pi$
$283$	−16.9406	+	16.9406i	−1.00702	+	1.00702i	−0.999975	+	0.00704092i	$0.997759\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			9.30832i			0.549453i
$288$		0			0
$289$	13.1950	−	10.7187i	0.776179	−	0.630513i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	7.44387			0.434875			0.217438	−	0.976074i	$-0.430230\pi$
$293$	7.44387			0.434875			0.217438	+	0.976074i	$0.430230\pi$
$294$		0			0
$295$	−18.5212	+	18.5212i	−1.07835	+	1.07835i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.46804	−	4.46804i	0.258394	−	0.258394i
$300$		0			0
$301$	13.6389	+	13.6389i	0.786133	+	0.786133i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−19.3042			−1.10536
$306$		0			0
$307$	14.2184			0.811486			0.405743	−	0.913987i	$-0.367013\pi$
$307$	14.2184			0.811486			0.405743	+	0.913987i	$0.367013\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−7.32672	−	7.32672i	−0.415460	−	0.415460i	0.468175	−	0.883636i	$-0.344912\pi$
$311$	−7.32672	−	7.32672i	−0.415460	−	0.415460i	−0.883636	+	0.468175i	$0.844912\pi$
$312$		0			0
$313$	−12.3222	+	12.3222i	−0.696491	+	0.696491i	−0.963652	−	0.267161i	$-0.913914\pi$
$313$	−12.3222	+	12.3222i	−0.696491	+	0.696491i	0.267161	+	0.963652i	$0.413914\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−10.0235	+	10.0235i	−0.562974	+	0.562974i	−0.930151	−	0.367177i	$-0.880324\pi$
$317$	−10.0235	+	10.0235i	−0.562974	+	0.562974i	0.367177	+	0.930151i	$0.380324\pi$
$318$		0			0
$319$	0.337113			0.0188747
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8.20773	+	23.1215i	0.456690	+	1.28651i
$324$		0			0
$325$			0.0817972i			0.00453730i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.90126	−	3.90126i	0.215084	−	0.215084i
$330$		0			0
$331$		−	10.8485i		−	0.596287i	−0.954521	−	0.298143i	$-0.903633\pi$
$331$		−	10.8485i		−	0.596287i	0.954521	−	0.298143i	$-0.0963673\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−22.2917	−	22.2917i	−1.21793	−	1.21793i
$336$		0			0
$337$	−6.94216	−	6.94216i	−0.378164	−	0.378164i	0.492276	−	0.870439i	$-0.336165\pi$
$337$	−6.94216	−	6.94216i	−0.378164	−	0.378164i	−0.870439	+	0.492276i	$0.836165\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−0.866827			−0.0469413
$342$		0			0
$343$	−14.1457	−	14.1457i	−0.763794	−	0.763794i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	18.4180	−	18.4180i	0.988728	−	0.988728i	−0.0112092	−	0.999937i	$-0.503568\pi$
$347$	18.4180	−	18.4180i	0.988728	−	0.988728i	0.999937	+	0.0112092i	$0.00356806\pi$
$348$		0			0
$349$		−	18.6609i		−	0.998894i	−0.866344	−	0.499447i	$-0.833537\pi$
$349$		−	18.6609i		−	0.998894i	0.866344	−	0.499447i	$-0.166463\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	11.8339			0.629857			0.314929	−	0.949115i	$-0.398019\pi$
$353$	11.8339			0.629857			0.314929	+	0.949115i	$0.398019\pi$
$354$		0			0
$355$		−	13.5581i		−	0.719590i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			14.1392i			0.746241i	0.927783	+	0.373120i	$0.121712\pi$
$359$			14.1392i			0.746241i	−0.927783	+	0.373120i	$0.878288\pi$
$360$		0			0
$361$	−16.4100			−0.863685
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	23.4594i		−	1.22792i
$366$		0			0
$367$	−2.63300	+	2.63300i	−0.137441	+	0.137441i	−0.772480	−	0.635039i	$-0.780984\pi$
$367$	−2.63300	+	2.63300i	−0.137441	+	0.137441i	0.635039	+	0.772480i	$0.280984\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.18998	+	1.18998i	0.0617806	+	0.0617806i
$372$		0			0
$373$	1.53618			0.0795403			0.0397701	−	0.999209i	$-0.487337\pi$
$373$	1.53618			0.0795403			0.0397701	+	0.999209i	$0.487337\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.161575	−	0.161575i	−0.00832156	−	0.00832156i
$378$		0			0
$379$	25.5627	+	25.5627i	1.31307	+	1.31307i	0.919144	+	0.393922i	$0.128882\pi$
$379$	25.5627	+	25.5627i	1.31307	+	1.31307i	0.393922	+	0.919144i	$0.371118\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	16.3492i		−	0.835404i	−0.908584	−	0.417702i	$-0.862836\pi$
$383$		−	16.3492i		−	0.835404i	0.908584	−	0.417702i	$-0.137164\pi$
$384$		0			0
$385$	7.09328	−	7.09328i	0.361507	−	0.361507i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	16.4533i		−	0.834216i	−0.908857	−	0.417108i	$-0.863044\pi$
$389$		−	16.4533i		−	0.834216i	0.908857	−	0.417108i	$-0.136956\pi$
$390$		0			0
$391$	−8.46938	+	17.7916i	−0.428315	+	0.899758i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.55169			0.0780740
$396$		0			0
$397$	−25.7816	+	25.7816i	−1.29394	+	1.29394i	−0.361610	+	0.932330i	$0.617773\pi$
$397$	−25.7816	+	25.7816i	−1.29394	+	1.29394i	−0.932330	+	0.361610i	$0.882227\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6.84414	−	6.84414i	0.341780	−	0.341780i	−0.515256	−	0.857036i	$-0.672303\pi$
$401$	6.84414	−	6.84414i	0.341780	−	0.341780i	0.857036	+	0.515256i	$0.172303\pi$
$402$		0			0
$403$	0.415463	+	0.415463i	0.0206957	+	0.0206957i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	10.3685			0.513949
$408$		0			0
$409$	−2.52228			−0.124719			−0.0623593	−	0.998054i	$-0.519862\pi$
$409$	−2.52228			−0.124719			−0.0623593	+	0.998054i	$0.519862\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	19.2883	+	19.2883i	0.949116	+	0.949116i
$414$		0			0
$415$	−13.4030	+	13.4030i	−0.657926	+	0.657926i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−18.6154	+	18.6154i	−0.909424	+	0.909424i	−0.996226	−	0.0868019i	$-0.972335\pi$
$419$	−18.6154	+	18.6154i	−0.909424	+	0.909424i	0.0868019	+	0.996226i	$0.472335\pi$
$420$		0			0
$421$	6.61089			0.322195			0.161098	−	0.986938i	$-0.448497\pi$
$421$	6.61089			0.322195			0.161098	+	0.986938i	$0.448497\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.0853313	−	0.240382i	−0.00413918	−	0.0116602i
$426$		0			0
$427$			20.1037i			0.972888i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	19.8882	−	19.8882i	0.957983	−	0.957983i	−0.0411690	−	0.999152i	$-0.513108\pi$
$431$	19.8882	−	19.8882i	0.957983	−	0.957983i	0.999152	+	0.0411690i	$0.0131082\pi$
$432$		0			0
$433$			17.1497i			0.824162i	0.911147	+	0.412081i	$0.135198\pi$
$433$			17.1497i			0.824162i	−0.911147	+	0.412081i	$0.864802\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−20.1090	−	20.1090i	−0.961944	−	0.961944i
$438$		0			0
$439$	−5.81593	−	5.81593i	−0.277579	−	0.277579i	0.554563	−	0.832142i	$-0.312886\pi$
$439$	−5.81593	−	5.81593i	−0.277579	−	0.277579i	−0.832142	+	0.554563i	$0.812886\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−18.4658			−0.877339			−0.438669	−	0.898649i	$-0.644550\pi$
$443$	−18.4658			−0.877339			−0.438669	+	0.898649i	$0.644550\pi$
$444$		0			0
$445$	−8.48983	−	8.48983i	−0.402456	−	0.402456i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	21.8673	−	21.8673i	1.03198	−	1.03198i	0.0325094	−	0.999471i	$-0.489650\pi$
$449$	21.8673	−	21.8673i	1.03198	−	1.03198i	0.999471	−	0.0325094i	$-0.0103499\pi$
$450$		0			0
$451$		−	7.84586i		−	0.369447i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.79951			−0.318766
$456$		0			0
$457$			15.5337i			0.726638i	0.931665	+	0.363319i	$0.118356\pi$
$457$			15.5337i			0.726638i	−0.931665	+	0.363319i	$0.881644\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	38.9032i		−	1.81190i	−0.423380	−	0.905952i	$-0.639156\pi$
$461$		−	38.9032i		−	1.81190i	0.423380	−	0.905952i	$-0.360844\pi$
$462$		0			0
$463$	−22.7572			−1.05762			−0.528808	−	0.848742i	$-0.677361\pi$
$463$	−22.7572			−1.05762			−0.528808	+	0.848742i	$0.677361\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	12.5551i		−	0.580981i	−0.956878	−	0.290490i	$-0.906182\pi$
$467$		−	12.5551i		−	0.580981i	0.956878	−	0.290490i	$-0.0938184\pi$
$468$		0			0
$469$	−23.2150	+	23.2150i	−1.07197	+	1.07197i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−11.4961	−	11.4961i	−0.528589	−	0.528589i
$474$		0			0
$475$	0.368139			0.0168914
$476$		0			0
$477$		0			0
$478$		0			0
$479$	26.3340	+	26.3340i	1.20323	+	1.20323i	0.973179	+	0.230051i	$0.0738892\pi$
$479$	26.3340	+	26.3340i	1.20323	+	1.20323i	0.230051	+	0.973179i	$0.426111\pi$
$480$		0			0
$481$	−4.96955	−	4.96955i	−0.226592	−	0.226592i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	10.4797i		−	0.475860i
$486$		0			0
$487$	1.16310	−	1.16310i	0.0527052	−	0.0527052i	−0.680263	−	0.732968i	$-0.738134\pi$
$487$	1.16310	−	1.16310i	0.0527052	−	0.0527052i	0.732968	+	0.680263i	$0.238134\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	4.30121i		−	0.194111i	−0.995279	−	0.0970555i	$-0.969058\pi$
$491$		−	4.30121i		−	0.194111i	0.995279	−	0.0970555i	$-0.0309424\pi$
$492$		0			0
$493$	0.643386	+	0.306273i	0.0289766	+	0.0137939i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−14.1196			−0.633352
$498$		0			0
$499$	−1.05732	+	1.05732i	−0.0473322	+	0.0473322i	−0.730377	−	0.683045i	$-0.760655\pi$
$499$	−1.05732	+	1.05732i	−0.0473322	+	0.0473322i	0.683045	+	0.730377i	$0.260655\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.19274	−	1.19274i	0.0531815	−	0.0531815i	−0.680016	−	0.733197i	$-0.738027\pi$
$503$	1.19274	−	1.19274i	0.0531815	−	0.0531815i	0.733197	+	0.680016i	$0.238027\pi$
$504$		0			0
$505$	−11.1741	−	11.1741i	−0.497240	−	0.497240i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6.91070			0.306312			0.153156	−	0.988202i	$-0.451056\pi$
$509$	6.91070			0.306312			0.153156	+	0.988202i	$0.451056\pi$
$510$		0			0
$511$	−24.4310			−1.08076
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−14.4930	−	14.4930i	−0.638640	−	0.638640i
$516$		0			0
$517$	−3.28832	+	3.28832i	−0.144620	+	0.144620i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9.47259	+	9.47259i	−0.415002	+	0.415002i	−0.883477	−	0.468475i	$-0.844804\pi$
$521$	−9.47259	+	9.47259i	−0.415002	+	0.415002i	0.468475	+	0.883477i	$0.344804\pi$
$522$		0			0
$523$	−4.40904			−0.192794			−0.0963969	−	0.995343i	$-0.530732\pi$
$523$	−4.40904			−0.192794			−0.0963969	+	0.995343i	$0.530732\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.65436	−	0.787529i	−0.0720648	−	0.0343053i
$528$		0			0
$529$			0.160604i			0.00698276i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.76046	+	3.76046i	−0.162883	+	0.162883i
$534$		0			0
$535$		−	11.5166i		−	0.497906i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.26807	+	2.26807i	0.0976927	+	0.0976927i
$540$		0			0
$541$	−16.3451	−	16.3451i	−0.702732	−	0.702732i	0.262264	−	0.964996i	$-0.415531\pi$
$541$	−16.3451	−	16.3451i	−0.702732	−	0.702732i	−0.964996	+	0.262264i	$0.915531\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−44.3890			−1.90142
$546$		0			0
$547$	6.70674	+	6.70674i	0.286760	+	0.286760i	0.835797	−	0.549038i	$-0.185006\pi$
$547$	6.70674	+	6.70674i	0.286760	+	0.286760i	−0.549038	+	0.835797i	$0.685006\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−0.727190	+	0.727190i	−0.0309793	+	0.0309793i
$552$		0			0
$553$		−	1.61596i		−	0.0687174i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−38.1670			−1.61719			−0.808593	−	0.588368i	$-0.799771\pi$
$557$	−38.1670			−1.61719			−0.808593	+	0.588368i	$0.799771\pi$
$558$		0			0
$559$			11.0199i			0.466093i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			13.8184i			0.582377i	0.956666	+	0.291189i	$0.0940507\pi$
$563$			13.8184i			0.582377i	−0.956666	+	0.291189i	$0.905949\pi$
$564$		0			0
$565$	−38.7487			−1.63017
$566$		0			0
$567$		0			0
$568$		0			0
$569$			12.9516i			0.542959i	0.962444	+	0.271480i	$0.0875130\pi$
$569$			12.9516i			0.542959i	−0.962444	+	0.271480i	$0.912487\pi$
$570$		0			0
$571$	−4.60596	+	4.60596i	−0.192753	+	0.192753i	−0.796885	−	0.604131i	$-0.793520\pi$
$571$	−4.60596	+	4.60596i	−0.192753	+	0.192753i	0.604131	+	0.796885i	$0.293520\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.209062	+	0.209062i	0.00871850	+	0.00871850i
$576$		0			0
$577$	15.6703			0.652364			0.326182	−	0.945307i	$-0.394238\pi$
$577$	15.6703			0.652364			0.326182	+	0.945307i	$0.394238\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	13.9581	+	13.9581i	0.579078	+	0.579078i
$582$		0			0
$583$	−1.00302	−	1.00302i	−0.0415407	−	0.0415407i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	14.9766i		−	0.618150i	−0.951038	−	0.309075i	$-0.899981\pi$
$587$		−	14.9766i		−	0.618150i	0.951038	−	0.309075i	$-0.100019\pi$
$588$		0			0
$589$	1.86984	−	1.86984i	0.0770456	−	0.0770456i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			46.3701i			1.90419i	0.305801	+	0.952096i	$0.401076\pi$
$593$			46.3701i			1.90419i	−0.305801	+	0.952096i	$0.598924\pi$
$594$		0			0
$595$	19.9820	−	7.09328i	0.819184	−	0.290796i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−28.7068			−1.17293			−0.586464	−	0.809975i	$-0.699480\pi$
$599$	−28.7068			−1.17293			−0.586464	+	0.809975i	$0.699480\pi$
$600$		0			0
$601$	−14.6998	+	14.6998i	−0.599617	+	0.599617i	−0.940211	−	0.340593i	$-0.889372\pi$
$601$	−14.6998	+	14.6998i	−0.599617	+	0.599617i	0.340593	+	0.940211i	$0.389372\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	11.3058	−	11.3058i	0.459644	−	0.459644i
$606$		0			0
$607$	12.1665	+	12.1665i	0.493823	+	0.493823i	0.909509	−	0.415685i	$-0.136458\pi$
$607$	12.1665	+	12.1665i	0.493823	+	0.493823i	−0.415685	+	0.909509i	$0.636458\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.15213			0.127522
$612$		0			0
$613$	−42.7657			−1.72729			−0.863644	−	0.504102i	$-0.831824\pi$
$613$	−42.7657			−1.72729			−0.863644	+	0.504102i	$0.831824\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	27.0678	+	27.0678i	1.08971	+	1.08971i	0.995558	+	0.0941495i	$0.0300132\pi$
$617$	27.0678	+	27.0678i	1.08971	+	1.08971i	0.0941495	+	0.995558i	$0.469987\pi$
$618$		0			0
$619$	−6.55659	+	6.55659i	−0.263532	+	0.263532i	−0.826487	−	0.562956i	$-0.809664\pi$
$619$	−6.55659	+	6.55659i	−0.263532	+	0.263532i	0.562956	+	0.826487i	$0.309664\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−8.84144	+	8.84144i	−0.354225	+	0.354225i
$624$		0			0
$625$	24.6868			0.987474
$626$		0			0
$627$		0			0
$628$		0			0
$629$	19.7885	+	9.42001i	0.789021	+	0.375600i
$630$		0			0
$631$		−	33.8478i		−	1.34746i	−0.738978	−	0.673730i	$-0.764691\pi$
$631$		−	33.8478i		−	1.34746i	0.738978	−	0.673730i	$-0.235309\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	15.4805	−	15.4805i	0.614327	−	0.614327i
$636$		0			0
$637$		−	2.17414i		−	0.0861424i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−7.54594	−	7.54594i	−0.298047	−	0.298047i	0.542202	−	0.840248i	$-0.317591\pi$
$641$	−7.54594	−	7.54594i	−0.298047	−	0.298047i	−0.840248	+	0.542202i	$0.817591\pi$
$642$		0			0
$643$	−27.7990	−	27.7990i	−1.09629	−	1.09629i	−0.994841	−	0.101444i	$-0.967654\pi$
$643$	−27.7990	−	27.7990i	−1.09629	−	1.09629i	−0.101444	−	0.994841i	$-0.532346\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−27.9895			−1.10038			−0.550190	−	0.835039i	$-0.685445\pi$
$647$	−27.9895			−1.10038			−0.550190	+	0.835039i	$0.685445\pi$
$648$		0			0
$649$	−16.2579	−	16.2579i	−0.638177	−	0.638177i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−28.9574	+	28.9574i	−1.13319	+	1.13319i	−0.143545	+	0.989644i	$0.545850\pi$
$653$	−28.9574	+	28.9574i	−1.13319	+	1.13319i	−0.989644	+	0.143545i	$0.954150\pi$
$654$		0			0
$655$		−	16.9053i		−	0.660544i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	10.8164			0.421349			0.210674	−	0.977556i	$-0.432434\pi$
$659$	10.8164			0.421349			0.210674	+	0.977556i	$0.432434\pi$
$660$		0			0
$661$		−	20.0794i		−	0.780997i	−0.920604	−	0.390498i	$-0.872303\pi$
$661$		−	20.0794i		−	0.780997i	0.920604	−	0.390498i	$-0.127697\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			30.6020i			1.18670i
$666$		0			0
$667$	−0.825928			−0.0319801
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	16.9452i		−	0.654161i
$672$		0			0
$673$	19.3820	−	19.3820i	0.747121	−	0.747121i	−0.226816	−	0.973938i	$-0.572832\pi$
$673$	19.3820	−	19.3820i	0.747121	−	0.747121i	0.973938	+	0.226816i	$0.0728317\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	16.0696	+	16.0696i	0.617606	+	0.617606i	0.944917	−	0.327311i	$-0.106142\pi$
$677$	16.0696	+	16.0696i	0.617606	+	0.617606i	−0.327311	+	0.944917i	$0.606142\pi$
$678$		0			0
$679$	−10.9138			−0.418832
$680$		0			0
$681$		0			0
$682$		0			0
$683$	26.3090	+	26.3090i	1.00668	+	1.00668i	0.999978	+	0.00670720i	$0.00213498\pi$
$683$	26.3090	+	26.3090i	1.00668	+	1.00668i	0.00670720	+	0.999978i	$0.497865\pi$
$684$		0			0
$685$	34.6828	+	34.6828i	1.32516	+	1.32516i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			0.961475i			0.0366293i
$690$		0			0
$691$	12.7675	−	12.7675i	0.485700	−	0.485700i	−0.421246	−	0.906946i	$-0.638407\pi$
$691$	12.7675	−	12.7675i	0.485700	−	0.485700i	0.906946	+	0.421246i	$0.138407\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			7.96553i			0.302150i
$696$		0			0
$697$	7.12811	−	14.9740i	0.269996	−	0.567180i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	9.99896			0.377656			0.188828	−	0.982010i	$-0.439531\pi$
$701$	9.99896			0.377656			0.188828	+	0.982010i	$0.439531\pi$
$702$		0			0
$703$	−22.3661	+	22.3661i	−0.843553	+	0.843553i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−11.6369	+	11.6369i	−0.437649	+	0.437649i
$708$		0			0
$709$	0.773543	+	0.773543i	0.0290510	+	0.0290510i	0.721483	−	0.692432i	$-0.243461\pi$
$709$	0.773543	+	0.773543i	0.0290510	+	0.0290510i	−0.692432	+	0.721483i	$0.743461\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.12373			0.0795344
$714$		0			0
$715$	5.73122			0.214335
$716$		0			0
$717$		0			0
$718$		0			0
$719$	30.0392	+	30.0392i	1.12027	+	1.12027i	0.991700	+	0.128572i	$0.0410393\pi$
$719$	30.0392	+	30.0392i	1.12027	+	1.12027i	0.128572	+	0.991700i	$0.458961\pi$
$720$		0			0
$721$	−15.0933	+	15.0933i	−0.562103	+	0.562103i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	0.00756020	−	0.00756020i	0.000280779	−	0.000280779i
$726$		0			0
$727$	45.2169			1.67700			0.838501	−	0.544900i	$-0.183432\pi$
$727$	45.2169			1.67700			0.838501	+	0.544900i	$0.183432\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−11.4961	−	32.3848i	−0.425197	−	1.19780i
$732$		0			0
$733$			21.5990i			0.797778i	0.916999	+	0.398889i	$0.130604\pi$
$733$			21.5990i			0.797778i	−0.916999	+	0.398889i	$0.869396\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	19.5676	−	19.5676i	0.720782	−	0.720782i
$738$		0			0
$739$		−	19.0419i		−	0.700467i	−0.936663	−	0.350233i	$-0.886102\pi$
$739$		−	19.0419i		−	0.700467i	0.936663	−	0.350233i	$-0.113898\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−18.6918	−	18.6918i	−0.685737	−	0.685737i	0.275550	−	0.961287i	$-0.411140\pi$
$743$	−18.6918	−	18.6918i	−0.685737	−	0.685737i	−0.961287	+	0.275550i	$0.911140\pi$
$744$		0			0
$745$	18.4963	+	18.4963i	0.677650	+	0.677650i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−11.9936			−0.438236
$750$		0			0
$751$	3.28582	+	3.28582i	0.119901	+	0.119901i	0.764512	−	0.644610i	$-0.222980\pi$
$751$	3.28582	+	3.28582i	0.119901	+	0.119901i	−0.644610	+	0.764512i	$0.722980\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−25.3042	+	25.3042i	−0.920915	+	0.920915i
$756$		0			0
$757$		−	10.4074i		−	0.378263i	−0.981952	−	0.189131i	$-0.939433\pi$
$757$		−	10.4074i		−	0.378263i	0.981952	−	0.189131i	$-0.0605672\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−38.6365			−1.40057			−0.700286	−	0.713862i	$-0.746944\pi$
$761$	−38.6365			−1.40057			−0.700286	+	0.713862i	$0.746944\pi$
$762$		0			0
$763$			46.2275i			1.67355i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			15.5845i			0.562725i
$768$		0			0
$769$	0.178098			0.00642238			0.00321119	−	0.999995i	$-0.498978\pi$
$769$	0.178098			0.00642238			0.00321119	+	0.999995i	$0.498978\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	4.15554i		−	0.149464i	−0.997204	−	0.0747322i	$-0.976190\pi$
$773$		−	4.15554i		−	0.149464i	0.997204	−	0.0747322i	$-0.0238102\pi$
$774$		0			0
$775$	−0.0194398	+	0.0194398i	−0.000698296	+	0.000698296i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	16.9244	+	16.9244i	0.606380	+	0.606380i
$780$		0			0
$781$	11.9013			0.425861
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−10.1435	−	10.1435i	−0.362036	−	0.362036i
$786$		0			0
$787$	−1.22941	−	1.22941i	−0.0438238	−	0.0438238i	0.684855	−	0.728679i	$-0.259865\pi$
$787$	−1.22941	−	1.22941i	−0.0438238	−	0.0438238i	−0.728679	+	0.684855i	$0.759865\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			40.3535i			1.43481i
$792$		0			0
$793$	−8.12169	+	8.12169i	−0.288410	+	0.288410i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	22.2085i		−	0.786666i	−0.919396	−	0.393333i	$-0.871322\pi$
$797$		−	22.2085i		−	0.786666i	0.919396	−	0.393333i	$-0.128678\pi$
$798$		0			0
$799$	−9.26333	+	3.28832i	−0.327713	+	0.116333i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	20.5926			0.726695
$804$		0			0
$805$	−17.3786	+	17.3786i	−0.612515	+	0.612515i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	2.28300	−	2.28300i	0.0802661	−	0.0802661i	−0.665834	−	0.746100i	$-0.731924\pi$
$809$	2.28300	−	2.28300i	0.0802661	−	0.0802661i	0.746100	+	0.665834i	$0.231924\pi$
$810$		0			0
$811$	17.8304	+	17.8304i	0.626110	+	0.626110i	0.947087	−	0.320977i	$-0.104011\pi$
$811$	17.8304	+	17.8304i	0.626110	+	0.626110i	−0.320977	+	0.947087i	$0.604011\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−37.1821			−1.30243
$816$		0			0
$817$	49.5966			1.73516
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−24.3444	−	24.3444i	−0.849626	−	0.849626i	0.140460	−	0.990086i	$-0.455142\pi$
$821$	−24.3444	−	24.3444i	−0.849626	−	0.849626i	−0.990086	+	0.140460i	$0.955142\pi$
$822$		0			0
$823$	−12.5861	+	12.5861i	−0.438723	+	0.438723i	−0.891582	−	0.452859i	$-0.850404\pi$
$823$	−12.5861	+	12.5861i	−0.438723	+	0.438723i	0.452859	+	0.891582i	$0.350404\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.37930	+	1.37930i	−0.0479631	+	0.0479631i	−0.730682	−	0.682718i	$-0.760798\pi$
$827$	−1.37930	+	1.37930i	−0.0479631	+	0.0479631i	0.682718	+	0.730682i	$0.260798\pi$
$828$		0			0
$829$	−0.904671			−0.0314205			−0.0157103	−	0.999877i	$-0.505001\pi$
$829$	−0.904671			−0.0314205			−0.0157103	+	0.999877i	$0.505001\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.26807	+	6.38924i	0.0785840	+	0.221374i
$834$		0			0
$835$		−	3.56659i		−	0.123427i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−17.1659	+	17.1659i	−0.592633	+	0.592633i	−0.938342	−	0.345708i	$-0.887639\pi$
$839$	−17.1659	+	17.1659i	−0.592633	+	0.592633i	0.345708	+	0.938342i	$0.387639\pi$
$840$		0			0
$841$		−	28.9701i		−	0.998970i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	17.6803	+	17.6803i	0.608222	+	0.608222i
$846$		0			0
$847$	−11.7740	−	11.7740i	−0.404559	−	0.404559i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−25.4030			−0.870802
$852$		0			0
$853$	26.9482	+	26.9482i	0.922689	+	0.922689i	0.997219	−	0.0745302i	$-0.0237457\pi$
$853$	26.9482	+	26.9482i	0.922689	+	0.922689i	−0.0745302	+	0.997219i	$0.523746\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−4.80850	+	4.80850i	−0.164255	+	0.164255i	−0.784449	−	0.620194i	$-0.787054\pi$
$857$	−4.80850	+	4.80850i	−0.164255	+	0.164255i	0.620194	+	0.784449i	$0.287054\pi$
$858$		0			0
$859$		−	22.3072i		−	0.761113i	−0.924758	−	0.380557i	$-0.875732\pi$
$859$		−	22.3072i		−	0.761113i	0.924758	−	0.380557i	$-0.124268\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.400593			−0.0136363			−0.00681816	−	0.999977i	$-0.502170\pi$
$863$	−0.400593			−0.0136363			−0.00681816	+	0.999977i	$0.502170\pi$
$864$		0			0
$865$		−	38.0794i		−	1.29474i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1.36207i			0.0462050i
$870$		0			0
$871$	−18.7572			−0.635563
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	26.0314i		−	0.880023i
$876$		0			0
$877$	−0.149079	+	0.149079i	−0.00503404	+	0.00503404i	−0.709619	−	0.704585i	$-0.751133\pi$
$877$	−0.149079	+	0.149079i	−0.00503404	+	0.00503404i	0.704585	+	0.709619i	$0.251133\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	36.6255	+	36.6255i	1.23394	+	1.23394i	0.962436	+	0.271508i	$0.0875224\pi$
$881$	36.6255	+	36.6255i	1.23394	+	1.23394i	0.271508	+	0.962436i	$0.412478\pi$
$882$		0			0
$883$	23.5757			0.793385			0.396692	−	0.917952i	$-0.370158\pi$
$883$	23.5757			0.793385			0.396692	+	0.917952i	$0.370158\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−11.5500	−	11.5500i	−0.387810	−	0.387810i	0.486096	−	0.873906i	$-0.338421\pi$
$887$	−11.5500	−	11.5500i	−0.387810	−	0.387810i	−0.873906	+	0.486096i	$0.838421\pi$
$888$		0			0
$889$	−16.1217	−	16.1217i	−0.540704	−	0.540704i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	14.1866i		−	0.474735i
$894$		0			0
$895$	17.4030	−	17.4030i	0.581717	−	0.581717i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	0.0767993i		−	0.00256140i
$900$		0			0
$901$	−1.00302	−	2.82554i	−0.0334153	−	0.0941322i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−43.7761			−1.45517
$906$		0			0
$907$	21.7666	−	21.7666i	0.722747	−	0.722747i	−0.246417	−	0.969164i	$-0.579253\pi$
$907$	21.7666	−	21.7666i	0.722747	−	0.722747i	0.969164	+	0.246417i	$0.0792533\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	5.08757	−	5.08757i	0.168559	−	0.168559i	−0.617787	−	0.786346i	$-0.711971\pi$
$911$	5.08757	−	5.08757i	0.168559	−	0.168559i	0.786346	+	0.617787i	$0.211971\pi$
$912$		0			0
$913$	−11.7651	−	11.7651i	−0.389367	−	0.389367i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−17.6054			−0.581383
$918$		0			0
$919$	−11.6873			−0.385527			−0.192764	−	0.981245i	$-0.561745\pi$
$919$	−11.6873			−0.385527			−0.192764	+	0.981245i	$0.561745\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5.70418	−	5.70418i	−0.187755	−	0.187755i
$924$		0			0
$925$	0.232528	−	0.232528i	0.00764548	−	0.00764548i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	10.0287	−	10.0287i	0.329032	−	0.329032i	−0.523186	−	0.852218i	$-0.675257\pi$
$929$	10.0287	−	10.0287i	0.329032	−	0.329032i	0.852218	+	0.523186i	$0.175257\pi$
$930$		0			0
$931$	−9.78497			−0.320689
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−16.8426	+	5.97884i	−0.550812	+	0.195529i
$936$		0			0
$937$		−	25.4693i		−	0.832044i	−0.909355	−	0.416022i	$-0.863424\pi$
$937$		−	25.4693i		−	0.832044i	0.909355	−	0.416022i	$-0.136576\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	41.4250	−	41.4250i	1.35042	−	1.35042i	0.465223	−	0.885193i	$-0.345974\pi$
$941$	41.4250	−	41.4250i	1.35042	−	1.35042i	0.885193	−	0.465223i	$-0.154026\pi$
$942$		0			0
$943$			19.2224i			0.625968i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	13.4664	+	13.4664i	0.437598	+	0.437598i	0.891203	−	0.453605i	$-0.149862\pi$
$947$	13.4664	+	13.4664i	0.437598	+	0.437598i	−0.453605	+	0.891203i	$0.649862\pi$
$948$		0			0
$949$	−9.86984	−	9.86984i	−0.320389	−	0.320389i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−19.8922			−0.644371			−0.322185	−	0.946677i	$-0.604417\pi$
$953$	−19.8922			−0.644371			−0.322185	+	0.946677i	$0.604417\pi$
$954$		0			0
$955$	2.52910	+	2.52910i	0.0818397	+	0.0818397i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	36.1192	−	36.1192i	1.16635	−	1.16635i
$960$		0			0
$961$		−	30.8025i		−	0.993630i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−9.94216			−0.320049
$966$		0			0
$967$			53.0310i			1.70536i	0.522433	+	0.852680i	$0.325025\pi$
$967$			53.0310i			1.70536i	−0.522433	+	0.852680i	$0.674975\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	3.02535i		−	0.0970880i	−0.998821	−	0.0485440i	$-0.984542\pi$
$971$		−	3.02535i		−	0.0970880i	0.998821	−	0.0485440i	$-0.0154581\pi$
$972$		0			0
$973$	8.29543			0.265939
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	11.2664i		−	0.360444i	−0.983626	−	0.180222i	$-0.942318\pi$
$977$		−	11.2664i		−	0.360444i	0.983626	−	0.180222i	$-0.0576815\pi$
$978$		0			0
$979$	7.45234	−	7.45234i	0.238178	−	0.238178i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	34.1943	+	34.1943i	1.09063	+	1.09063i	0.995461	+	0.0951684i	$0.0303390\pi$
$983$	34.1943	+	34.1943i	1.09063	+	1.09063i	0.0951684	+	0.995461i	$0.469661\pi$
$984$		0			0
$985$	16.4304			0.523515
$986$		0			0
$987$		0			0
$988$		0			0
$989$	28.1654	+	28.1654i	0.895608	+	0.895608i
$990$		0			0
$991$	−16.9991	−	16.9991i	−0.539994	−	0.539994i	0.383533	−	0.923527i	$-0.374707\pi$
$991$	−16.9991	−	16.9991i	−0.539994	−	0.539994i	−0.923527	+	0.383533i	$0.874707\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			15.9921i			0.506985i
$996$		0			0
$997$	−18.6250	+	18.6250i	−0.589861	+	0.589861i	−0.937594	−	0.347733i	$-0.886952\pi$
$997$	−18.6250	+	18.6250i	−0.589861	+	0.589861i	0.347733	+	0.937594i	$0.386952\pi$
$998$		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.1441.1		8
3.2	odd	2		816.2.bd.e.625.4		8
4.3	odd	2		153.2.f.b.64.1		8
12.11	even	2		51.2.e.a.13.4	yes	8
17.4	even	4	inner	2448.2.be.x.1585.1		8
51.38	odd	4		816.2.bd.e.769.4		8
68.15	odd	8		2601.2.a.be.1.4		4
68.19	odd	8		2601.2.a.bf.1.4		4
68.55	odd	4		153.2.f.b.55.4		8
204.11	odd	16		867.2.h.i.757.2		16
204.23	odd	16		867.2.h.i.757.1		16
204.47	even	4		867.2.e.g.616.1		8
204.59	even	8		867.2.d.f.577.8		8
204.71	odd	16		867.2.h.k.712.3		16
204.83	even	8		867.2.a.l.1.1		4
204.95	odd	16		867.2.h.k.688.4		16
204.107	odd	16		867.2.h.i.733.2		16
204.131	odd	16		867.2.h.i.733.1		16
204.143	odd	16		867.2.h.k.688.3		16
204.155	even	8		867.2.a.k.1.1		4
204.167	odd	16		867.2.h.k.712.4		16
204.179	even	8		867.2.d.f.577.7		8
204.191	even	4		51.2.e.a.4.1	✓	8
204.203	even	2		867.2.e.g.829.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.e.a.4.1	✓	8	204.191	even	4
51.2.e.a.13.4	yes	8	12.11	even	2
153.2.f.b.55.4		8	68.55	odd	4
153.2.f.b.64.1		8	4.3	odd	2
816.2.bd.e.625.4		8	3.2	odd	2
816.2.bd.e.769.4		8	51.38	odd	4
867.2.a.k.1.1		4	204.155	even	8
867.2.a.l.1.1		4	204.83	even	8
867.2.d.f.577.7		8	204.179	even	8
867.2.d.f.577.8		8	204.59	even	8
867.2.e.g.616.1		8	204.47	even	4
867.2.e.g.829.4		8	204.203	even	2
867.2.h.i.733.1		16	204.131	odd	16
867.2.h.i.733.2		16	204.107	odd	16
867.2.h.i.757.1		16	204.23	odd	16
867.2.h.i.757.2		16	204.11	odd	16
867.2.h.k.688.3		16	204.143	odd	16
867.2.h.k.688.4		16	204.95	odd	16
867.2.h.k.712.3		16	204.71	odd	16
867.2.h.k.712.4		16	204.167	odd	16
2448.2.be.x.1441.1		8	1.1	even	1	trivial
2448.2.be.x.1585.1		8	17.4	even	4	inner
2601.2.a.be.1.4		4	68.15	odd	8
2601.2.a.bf.1.4		4	68.19	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)

\(f(q)\)	\(=\)	\(q+(-1.57133 - 1.57133i) q^{5} +(-1.63640 + 1.63640i) q^{7} +(1.37930 - 1.37930i) q^{11} -1.32218 q^{13} +(3.88555 - 1.37930i) q^{17} +5.95063i q^{19} +(-3.37930 + 3.37930i) q^{23} -0.0618655i q^{25} +(0.122204 + 0.122204i) q^{29} +(-0.314226 - 0.314226i) q^{31} +5.14265 q^{35} +(3.75861 + 3.75861i) q^{37} +(2.84414 - 2.84414i) q^{41} -8.33468i q^{43} -2.38404 q^{47} +1.64436i q^{49} -0.727190i q^{53} -4.33468 q^{55} -11.7870i q^{59} +(6.14265 - 6.14265i) q^{61} +(2.07757 + 2.07757i) q^{65} +14.1866 q^{67} +(4.31423 + 4.31423i) q^{71} +(7.46483 + 7.46483i) q^{73} +4.51420i q^{77} +(-0.493752 + 0.493752i) q^{79} -8.52971i q^{83} +(-8.27281 - 3.93813i) q^{85} +5.40297 q^{89} +(2.16362 - 2.16362i) q^{91} +(9.35038 - 9.35038i) q^{95} +(3.33468 + 3.33468i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{5} + 4 q^{7} - 4 q^{13} + 4 q^{17} - 16 q^{23} - 4 q^{29} + 8 q^{31} + 8 q^{35} + 8 q^{37} - 28 q^{41} - 8 q^{47} + 4 q^{55} + 16 q^{61} - 16 q^{65} - 8 q^{67} + 24 q^{71} + 20 q^{73} - 20 q^{79}+ \cdots - 12 q^{97}+O(q^{100}) \) 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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