LMFDB - Embedded newform 6048.2.h.h.2591.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(2591,6048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6048.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 242x^{12} + 13297x^{8} + 201600x^{4} + 331776 $ x^16 + 242x^12 + 13297x^8 + 201600*x^4 + 331776
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.3
Root			$-1.53379 + 1.53379i$ of defining polynomial
Character	$\chi$	$=$	6048.2591
Dual form			6048.2.h.h.2591.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.93185i q^{5} +1.00000i q^{7} +O(q^{10})$	$q-1.93185i q^{5} +1.00000i q^{7} -3.58521 q^{11} +5.92609 q^{13} -6.96655i q^{17} -5.33820i q^{19} -2.96713 q^{23} +1.26795 q^{25} -3.06757i q^{29} -1.27006i q^{31} +1.93185 q^{35} +9.65815 q^{37} +8.10227i q^{41} -2.46199i q^{43} -6.45299 q^{47} -1.00000 q^{49} +11.2092i q^{53} +6.92609i q^{55} -6.34847 q^{59} -10.7263 q^{61} -11.4483i q^{65} +6.26430i q^{67} +1.92887 q^{71} -2.12379 q^{73} -3.58521i q^{77} -8.71379i q^{79} -7.96655 q^{83} -13.4583 q^{85} -9.04118i q^{89} +5.92609i q^{91} -10.3126 q^{95} -4.33820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 8 q^{13} + 48 q^{25} + 40 q^{37} - 16 q^{49} - 56 q^{73} - 16 q^{85} - 16 q^{97}+O(q^{100}) $ 16 * q + 8 * q^13 + 48 * q^25 + 40 * q^37 - 16 * q^49 - 56 * q^73 - 16 * q^85 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$.

$n$	$2593$	$3781$	$3809$	$4159$
$\chi(n)$	$1$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 1.93185i	− 0.863950i	−0.901886	−	0.431975i	$-0.857817\pi$
$5$	− 1.93185i	− 0.863950i	0.901886	−	0.431975i	$-0.142183\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.58521	−1.08098	−0.540491	−	0.841350i	$-0.681761\pi$
$11$	−3.58521	−1.08098	−0.540491	+	0.841350i	$0.681761\pi$
$12$	0	0
$13$	5.92609	1.64360	0.821801	−	0.569774i	$-0.192969\pi$
$13$	5.92609	1.64360	0.821801	+	0.569774i	$0.192969\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.96655i	− 1.68964i	−0.535053	−	0.844818i	$-0.679709\pi$
$17$	− 6.96655i	− 1.68964i	0.535053	−	0.844818i	$-0.320291\pi$
$18$	0	0
$19$	− 5.33820i	− 1.22467i	−0.790599	−	0.612334i	$-0.790231\pi$
$19$	− 5.33820i	− 1.22467i	0.790599	−	0.612334i	$-0.209769\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.96713	−0.618689	−0.309344	−	0.950950i	$-0.600110\pi$
$23$	−2.96713	−0.618689	−0.309344	+	0.950950i	$0.600110\pi$
$24$	0	0
$25$	1.26795	0.253590
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 3.06757i	− 0.569634i	−0.958582	−	0.284817i	$-0.908067\pi$
$29$	− 3.06757i	− 0.569634i	0.958582	−	0.284817i	$-0.0919328\pi$
$30$	0	0
$31$	− 1.27006i	− 0.228109i	−0.993474	−	0.114055i	$-0.963616\pi$
$31$	− 1.27006i	− 0.228109i	0.993474	−	0.114055i	$-0.0363839\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.93185	0.326543
$36$	0	0
$37$	9.65815	1.58779	0.793895	−	0.608055i	$-0.208050\pi$
$37$	9.65815	1.58779	0.793895	+	0.608055i	$0.208050\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.10227i	1.26536i	0.774413	+	0.632681i	$0.218046\pi$
$41$	8.10227i	1.26536i	−0.774413	+	0.632681i	$0.781954\pi$
$42$	0	0
$43$	− 2.46199i	− 0.375450i	−0.982222	−	0.187725i	$-0.939889\pi$
$43$	− 2.46199i	− 0.375450i	0.982222	−	0.187725i	$-0.0601114\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.45299	−0.941265	−0.470632	−	0.882329i	$-0.655974\pi$
$47$	−6.45299	−0.941265	−0.470632	+	0.882329i	$0.655974\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.2092i	1.53970i	0.638224	+	0.769850i	$0.279669\pi$
$53$	11.2092i	1.53970i	−0.638224	+	0.769850i	$0.720331\pi$
$54$	0	0
$55$	6.92609i	0.933914i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.34847	−0.826500	−0.413250	−	0.910618i	$-0.635606\pi$
$59$	−6.34847	−0.826500	−0.413250	+	0.910618i	$0.635606\pi$
$60$	0	0
$61$	−10.7263	−1.37336	−0.686680	−	0.726960i	$-0.740933\pi$
$61$	−10.7263	−1.37336	−0.686680	+	0.726960i	$0.740933\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 11.4483i	− 1.41999i
$66$	0	0
$67$	6.26430i	0.765306i	0.923892	+	0.382653i	$0.124989\pi$
$67$	6.26430i	0.765306i	−0.923892	+	0.382653i	$0.875011\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.92887	0.228915	0.114457	−	0.993428i	$-0.463487\pi$
$71$	1.92887	0.228915	0.114457	+	0.993428i	$0.463487\pi$
$72$	0	0
$73$	−2.12379	−0.248571	−0.124285	−	0.992247i	$-0.539664\pi$
$73$	−2.12379	−0.248571	−0.124285	+	0.992247i	$0.539664\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.58521i	− 0.408573i
$78$	0	0
$79$	− 8.71379i	− 0.980378i	−0.871616	−	0.490189i	$-0.836928\pi$
$79$	− 8.71379i	− 0.980378i	0.871616	−	0.490189i	$-0.163072\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−7.96655	−0.874443	−0.437221	−	0.899354i	$-0.644037\pi$
$83$	−7.96655	−0.874443	−0.437221	+	0.899354i	$0.644037\pi$
$84$	0	0
$85$	−13.4583	−1.45976
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 9.04118i	− 0.958363i	−0.877716	−	0.479181i	$-0.840934\pi$
$89$	− 9.04118i	− 0.958363i	0.877716	−	0.479181i	$-0.159066\pi$
$90$	0	0
$91$	5.92609i	0.621223i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−10.3126	−1.05805
$96$	0	0
$97$	−4.33820	−0.440478	−0.220239	−	0.975446i	$-0.570684\pi$
$97$	−4.33820	−0.440478	−0.220239	+	0.975446i	$0.570684\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.13216i	0.610173i	0.952325	+	0.305086i	$0.0986854\pi$
$101$	6.13216i	0.610173i	−0.952325	+	0.305086i	$0.901315\pi$
$102$	0	0
$103$	− 13.9963i	− 1.37910i	−0.724238	−	0.689551i	$-0.757808\pi$
$103$	− 13.9963i	− 1.37910i	0.724238	−	0.689551i	$-0.242192\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.34251	0.903174	0.451587	−	0.892227i	$-0.350858\pi$
$107$	9.34251	0.903174	0.451587	+	0.892227i	$0.350858\pi$
$108$	0	0
$109$	19.0666	1.82625	0.913125	−	0.407681i	$-0.133662\pi$
$109$	19.0666	1.82625	0.913125	+	0.407681i	$0.133662\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.68863i	0.158853i	0.996841	+	0.0794266i	$0.0253089\pi$
$113$	1.68863i	0.158853i	−0.996841	+	0.0794266i	$0.974691\pi$
$114$	0	0
$115$	5.73205i	0.534516i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.96655	0.638623
$120$	0	0
$121$	1.85373	0.168521
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 12.1087i	− 1.08304i
$126$	0	0
$127$	− 15.2424i	− 1.35254i	−0.736653	−	0.676271i	$-0.763595\pi$
$127$	− 15.2424i	− 1.35254i	0.736653	−	0.676271i	$-0.236405\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.0699	1.40403	0.702017	−	0.712160i	$-0.252283\pi$
$131$	16.0699	1.40403	0.702017	+	0.712160i	$0.252283\pi$
$132$	0	0
$133$	5.33820	0.462881
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.82545i	0.326830i	0.986557	+	0.163415i	$0.0522509\pi$
$137$	3.82545i	0.326830i	−0.986557	+	0.163415i	$0.947749\pi$
$138$	0	0
$139$	− 15.9021i	− 1.34880i	−0.738368	−	0.674398i	$-0.764403\pi$
$139$	− 15.9021i	− 1.34880i	0.738368	−	0.674398i	$-0.235597\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−21.2463	−1.77670
$144$	0	0
$145$	−5.92609	−0.492135
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 8.17987i	− 0.670121i	−0.942197	−	0.335061i	$-0.891243\pi$
$149$	− 8.17987i	− 0.670121i	0.942197	−	0.335061i	$-0.108757\pi$
$150$	0	0
$151$	8.04988i	0.655090i	0.944836	+	0.327545i	$0.106221\pi$
$151$	8.04988i	0.655090i	−0.944836	+	0.327545i	$0.893779\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.45356	−0.197075
$156$	0	0
$157$	−17.6420	−1.40798	−0.703992	−	0.710208i	$-0.748601\pi$
$157$	−17.6420	−1.40798	−0.703992	+	0.710208i	$0.748601\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 2.96713i	− 0.233842i
$162$	0	0
$163$	2.46621i	0.193169i	0.995325	+	0.0965843i	$0.0307917\pi$
$163$	2.46621i	0.193169i	−0.995325	+	0.0965843i	$0.969208\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.1340	−1.24849	−0.624243	−	0.781230i	$-0.714592\pi$
$167$	−16.1340	−1.24849	−0.624243	+	0.781230i	$0.714592\pi$
$168$	0	0
$169$	22.1186	1.70143
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 12.0042i	− 0.912665i	−0.889809	−	0.456332i	$-0.849163\pi$
$173$	− 12.0042i	− 0.912665i	0.889809	−	0.456332i	$-0.150837\pi$
$174$	0	0
$175$	1.26795i	0.0958479i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.4871	1.08282	0.541408	−	0.840760i	$-0.317891\pi$
$179$	14.4871	1.08282	0.541408	+	0.840760i	$0.317891\pi$
$180$	0	0
$181$	−8.72629	−0.648620	−0.324310	−	0.945951i	$-0.605132\pi$
$181$	−8.72629	−0.648620	−0.324310	+	0.945951i	$0.605132\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 18.6581i	− 1.37177i
$186$	0	0
$187$	24.9765i	1.82647i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−12.6628	−0.916245	−0.458122	−	0.888889i	$-0.651478\pi$
$191$	−12.6628	−0.916245	−0.458122	+	0.888889i	$0.651478\pi$
$192$	0	0
$193$	−3.51399	−0.252942	−0.126471	−	0.991970i	$-0.540365\pi$
$193$	−3.51399	−0.252942	−0.126471	+	0.991970i	$0.540365\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 11.3056i	− 0.805488i	−0.915313	−	0.402744i	$-0.868057\pi$
$197$	− 11.3056i	− 0.805488i	0.915313	−	0.402744i	$-0.131943\pi$
$198$	0	0
$199$	− 2.53801i	− 0.179915i	−0.995946	−	0.0899573i	$-0.971327\pi$
$199$	− 2.53801i	− 0.179915i	0.995946	−	0.0899573i	$-0.0286731\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.06757	0.215301
$204$	0	0
$205$	15.6524	1.09321
$206$	0	0
$207$	0	0
$208$	0	0
$209$	19.1386i	1.32384i
$210$	0	0
$211$	22.8303i	1.57170i	0.618416	+	0.785851i	$0.287775\pi$
$211$	22.8303i	1.57170i	−0.618416	+	0.785851i	$0.712225\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4.75620	−0.324370
$216$	0	0
$217$	1.27006	0.0862172
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 41.2844i	− 2.77709i
$222$	0	0
$223$	− 19.4048i	− 1.29944i	−0.760173	−	0.649721i	$-0.774886\pi$
$223$	− 19.4048i	− 1.29944i	0.760173	−	0.649721i	$-0.225114\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−25.6287	−1.70104	−0.850519	−	0.525944i	$-0.823712\pi$
$227$	−25.6287	−1.70104	−0.850519	+	0.525944i	$0.823712\pi$
$228$	0	0
$229$	−21.8023	−1.44074	−0.720368	−	0.693592i	$-0.756027\pi$
$229$	−21.8023	−1.44074	−0.720368	+	0.693592i	$0.756027\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 28.1070i	− 1.84135i	−0.390327	−	0.920676i	$-0.627638\pi$
$233$	− 28.1070i	− 1.84135i	0.390327	−	0.920676i	$-0.372362\pi$
$234$	0	0
$235$	12.4662i	0.813206i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.79017	0.115796	0.0578982	−	0.998322i	$-0.481560\pi$
$239$	1.79017	0.115796	0.0578982	+	0.998322i	$0.481560\pi$
$240$	0	0
$241$	−19.6785	−1.26760	−0.633802	−	0.773495i	$-0.718507\pi$
$241$	−19.6785	−1.26760	−0.633802	+	0.773495i	$0.718507\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.93185i	0.123421i
$246$	0	0
$247$	− 31.6347i	− 2.01287i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−22.9672	−1.44968	−0.724839	−	0.688918i	$-0.758086\pi$
$251$	−22.9672	−1.44968	−0.724839	+	0.688918i	$0.758086\pi$
$252$	0	0
$253$	10.6378	0.668791
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 28.7563i	− 1.79377i	−0.442265	−	0.896884i	$-0.645825\pi$
$257$	− 28.7563i	− 1.79377i	0.442265	−	0.896884i	$-0.354175\pi$
$258$	0	0
$259$	9.65815i	0.600128i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−30.8415	−1.90177	−0.950883	−	0.309549i	$-0.899822\pi$
$263$	−30.8415	−1.90177	−0.950883	+	0.309549i	$0.899822\pi$
$264$	0	0
$265$	21.6545	1.33022
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8.16635i	0.497911i	0.968515	+	0.248956i	$0.0800873\pi$
$269$	8.16635i	0.497911i	−0.968515	+	0.248956i	$0.919913\pi$
$270$	0	0
$271$	− 1.46832i	− 0.0891941i	−0.999005	−	0.0445970i	$-0.985800\pi$
$271$	− 1.46832i	− 0.0891941i	0.999005	−	0.0445970i	$-0.0142004\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.54586	−0.274126
$276$	0	0
$277$	−6.53225	−0.392485	−0.196242	−	0.980555i	$-0.562874\pi$
$277$	−6.53225	−0.392485	−0.196242	+	0.980555i	$0.562874\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	13.2415i	0.789921i	0.918698	+	0.394960i	$0.129242\pi$
$281$	13.2415i	0.789921i	−0.918698	+	0.394960i	$0.870758\pi$
$282$	0	0
$283$	6.44219i	0.382948i	0.981498	+	0.191474i	$0.0613268\pi$
$283$	6.44219i	0.382948i	−0.981498	+	0.191474i	$0.938673\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.10227	−0.478262
$288$	0	0
$289$	−31.5328	−1.85487
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 13.5623i	− 0.792319i	−0.918182	−	0.396159i	$-0.870343\pi$
$293$	− 13.5623i	− 0.792319i	0.918182	−	0.396159i	$-0.129657\pi$
$294$	0	0
$295$	12.2643i	0.714055i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−17.5835	−1.01688
$300$	0	0
$301$	2.46199	0.141907
$302$	0	0
$303$	0	0
$304$	0	0
$305$	20.7216i	1.18652i
$306$	0	0
$307$	29.8710i	1.70483i	0.522867	+	0.852414i	$0.324862\pi$
$307$	29.8710i	1.70483i	−0.522867	+	0.852414i	$0.675138\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	27.6551	1.56817	0.784087	−	0.620651i	$-0.213131\pi$
$311$	27.6551	1.56817	0.784087	+	0.620651i	$0.213131\pi$
$312$	0	0
$313$	16.7430	0.946371	0.473185	−	0.880963i	$-0.343104\pi$
$313$	16.7430	0.946371	0.473185	+	0.880963i	$0.343104\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.136810i	0.00768401i	0.999993	+	0.00384201i	$0.00122295\pi$
$317$	0.136810i	0.00768401i	−0.999993	+	0.00384201i	$0.998777\pi$
$318$	0	0
$319$	10.9979i	0.615764i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−37.1888	−2.06924
$324$	0	0
$325$	7.51399	0.416801
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 6.45299i	− 0.355765i
$330$	0	0
$331$	21.9927i	1.20883i	0.796670	+	0.604414i	$0.206593\pi$
$331$	21.9927i	1.20883i	−0.796670	+	0.604414i	$0.793407\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12.1017	0.661186
$336$	0	0
$337$	−8.11859	−0.442248	−0.221124	−	0.975246i	$-0.570973\pi$
$337$	−8.11859	−0.442248	−0.221124	+	0.975246i	$0.570973\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.55343i	0.246582i
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.86909	−0.368752	−0.184376	−	0.982856i	$-0.559026\pi$
$347$	−6.86909	−0.368752	−0.184376	+	0.982856i	$0.559026\pi$
$348$	0	0
$349$	−0.140506	−0.00752111	−0.00376055	−	0.999993i	$-0.501197\pi$
$349$	−0.140506	−0.00752111	−0.00376055	+	0.999993i	$0.501197\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	24.6711i	1.31311i	0.754279	+	0.656554i	$0.227987\pi$
$353$	24.6711i	1.31311i	−0.754279	+	0.656554i	$0.772013\pi$
$354$	0	0
$355$	− 3.72629i	− 0.197771i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.8985	1.63076	0.815380	−	0.578926i	$-0.196528\pi$
$359$	30.8985	1.63076	0.815380	+	0.578926i	$0.196528\pi$
$360$	0	0
$361$	−9.49640	−0.499811
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.10285i	0.214753i
$366$	0	0
$367$	− 22.7326i	− 1.18663i	−0.804969	−	0.593316i	$-0.797818\pi$
$367$	− 22.7326i	− 1.18663i	0.804969	−	0.593316i	$-0.202182\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.2092	−0.581952
$372$	0	0
$373$	11.9084	0.616594	0.308297	−	0.951290i	$-0.400241\pi$
$373$	11.9084	0.616594	0.308297	+	0.951290i	$0.400241\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 18.1787i	− 0.936252i
$378$	0	0
$379$	27.2466i	1.39956i	0.714356	+	0.699782i	$0.246720\pi$
$379$	27.2466i	1.39956i	−0.714356	+	0.699782i	$0.753280\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	16.2385	0.829749	0.414875	−	0.909879i	$-0.363825\pi$
$383$	16.2385	0.829749	0.414875	+	0.909879i	$0.363825\pi$
$384$	0	0
$385$	−6.92609	−0.352986
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 11.0002i	− 0.557730i	−0.960330	−	0.278865i	$-0.910042\pi$
$389$	− 11.0002i	− 0.557730i	0.960330	−	0.278865i	$-0.0899582\pi$
$390$	0	0
$391$	20.6706i	1.04536i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−16.8338	−0.846998
$396$	0	0
$397$	−27.5608	−1.38324	−0.691618	−	0.722263i	$-0.743102\pi$
$397$	−27.5608	−1.38324	−0.691618	+	0.722263i	$0.743102\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 5.66929i	− 0.283111i	−0.989930	−	0.141555i	$-0.954790\pi$
$401$	− 5.66929i	− 0.283111i	0.989930	−	0.141555i	$-0.0452103\pi$
$402$	0	0
$403$	− 7.52648i	− 0.374921i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−34.6265	−1.71637
$408$	0	0
$409$	−15.2382	−0.753479	−0.376739	−	0.926319i	$-0.622955\pi$
$409$	−15.2382	−0.753479	−0.376739	+	0.926319i	$0.622955\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 6.34847i	− 0.312388i
$414$	0	0
$415$	15.3902i	0.755475i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	23.7569	1.16060	0.580300	−	0.814403i	$-0.302935\pi$
$419$	23.7569	1.16060	0.580300	+	0.814403i	$0.302935\pi$
$420$	0	0
$421$	33.6010	1.63761	0.818805	−	0.574072i	$-0.194637\pi$
$421$	33.6010	1.63761	0.818805	+	0.574072i	$0.194637\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 8.83323i	− 0.428475i
$426$	0	0
$427$	− 10.7263i	− 0.519082i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3.41960	−0.164716	−0.0823581	−	0.996603i	$-0.526245\pi$
$431$	−3.41960	−0.164716	−0.0823581	+	0.996603i	$0.526245\pi$
$432$	0	0
$433$	9.42250	0.452816	0.226408	−	0.974033i	$-0.427302\pi$
$433$	9.42250	0.452816	0.226408	+	0.974033i	$0.427302\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	15.8391i	0.757688i
$438$	0	0
$439$	− 14.1040i	− 0.673147i	−0.941657	−	0.336573i	$-0.890732\pi$
$439$	− 14.1040i	− 0.673147i	0.941657	−	0.336573i	$-0.109268\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5.06481	0.240636	0.120318	−	0.992735i	$-0.461609\pi$
$443$	5.06481	0.240636	0.120318	+	0.992735i	$0.461609\pi$
$444$	0	0
$445$	−17.4662	−0.827978
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 7.52055i	− 0.354917i	−0.984128	−	0.177458i	$-0.943212\pi$
$449$	− 7.52055i	− 0.354917i	0.984128	−	0.177458i	$-0.0567875\pi$
$450$	0	0
$451$	− 29.0483i	− 1.36783i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	11.4483	0.536706
$456$	0	0
$457$	−12.3620	−0.578268	−0.289134	−	0.957289i	$-0.593367\pi$
$457$	−12.3620	−0.578268	−0.289134	+	0.957289i	$0.593367\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 12.2111i	− 0.568727i	−0.958717	−	0.284363i	$-0.908218\pi$
$461$	− 12.2111i	− 0.568727i	0.958717	−	0.284363i	$-0.0917822\pi$
$462$	0	0
$463$	− 8.00422i	− 0.371988i	−0.982551	−	0.185994i	$-0.940450\pi$
$463$	− 8.00422i	− 0.371988i	0.982551	−	0.185994i	$-0.0595504\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	34.9725	1.61834	0.809168	−	0.587577i	$-0.199918\pi$
$467$	34.9725	1.61834	0.809168	+	0.587577i	$0.199918\pi$
$468$	0	0
$469$	−6.26430	−0.289258
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.82676i	0.405855i
$474$	0	0
$475$	− 6.76857i	− 0.310563i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0.141096	0.00644683	0.00322342	−	0.999995i	$-0.498974\pi$
$479$	0.141096	0.00644683	0.00322342	+	0.999995i	$0.498974\pi$
$480$	0	0
$481$	57.2351	2.60969
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.38076i	0.380551i
$486$	0	0
$487$	− 18.5161i	− 0.839044i	−0.907745	−	0.419522i	$-0.862198\pi$
$487$	− 18.5161i	− 0.839044i	0.907745	−	0.419522i	$-0.137802\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−33.4544	−1.50978	−0.754888	−	0.655854i	$-0.772309\pi$
$491$	−33.4544	−1.50978	−0.754888	+	0.655854i	$0.772309\pi$
$492$	0	0
$493$	−21.3704	−0.962474
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.92887i	0.0865216i
$498$	0	0
$499$	− 37.8647i	− 1.69506i	−0.530751	−	0.847528i	$-0.678090\pi$
$499$	− 37.8647i	− 1.69506i	0.530751	−	0.847528i	$-0.321910\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	19.9336	0.888796	0.444398	−	0.895829i	$-0.353418\pi$
$503$	19.9336	0.888796	0.444398	+	0.895829i	$0.353418\pi$
$504$	0	0
$505$	11.8464	0.527159
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6.33087i	0.280611i	0.990108	+	0.140305i	$0.0448084\pi$
$509$	6.33087i	0.280611i	−0.990108	+	0.140305i	$0.955192\pi$
$510$	0	0
$511$	− 2.12379i	− 0.0939510i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−27.0389	−1.19147
$516$	0	0
$517$	23.1353	1.01749
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 29.1404i	− 1.27666i	−0.769761	−	0.638332i	$-0.779625\pi$
$521$	− 29.1404i	− 1.27666i	0.769761	−	0.638332i	$-0.220375\pi$
$522$	0	0
$523$	7.13474i	0.311981i	0.987759	+	0.155990i	$0.0498569\pi$
$523$	7.13474i	0.311981i	−0.987759	+	0.155990i	$0.950143\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.84792	−0.385422
$528$	0	0
$529$	−14.1962	−0.617224
$530$	0	0
$531$	0	0
$532$	0	0
$533$	48.0148i	2.07975i
$534$	0	0
$535$	− 18.0483i	− 0.780298i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.58521	0.154426
$540$	0	0
$541$	−29.0243	−1.24785	−0.623926	−	0.781483i	$-0.714464\pi$
$541$	−29.0243	−1.24785	−0.623926	+	0.781483i	$0.714464\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 36.8338i	− 1.57779i
$546$	0	0
$547$	− 16.8759i	− 0.721563i	−0.932650	−	0.360782i	$-0.882510\pi$
$547$	− 16.8759i	− 0.721563i	0.932650	−	0.360782i	$-0.117490\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−16.3753	−0.697612
$552$	0	0
$553$	8.71379	0.370548
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.41255i	0.271708i	0.990729	+	0.135854i	$0.0433779\pi$
$557$	6.41255i	0.271708i	−0.990729	+	0.135854i	$0.956622\pi$
$558$	0	0
$559$	− 14.5900i	− 0.617091i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	34.1803	1.44053	0.720263	−	0.693701i	$-0.244021\pi$
$563$	34.1803	1.44053	0.720263	+	0.693701i	$0.244021\pi$
$564$	0	0
$565$	3.26219	0.137241
$566$	0	0
$567$	0	0
$568$	0	0
$569$	16.4841i	0.691050i	0.938409	+	0.345525i	$0.112299\pi$
$569$	16.4841i	0.691050i	−0.938409	+	0.345525i	$0.887701\pi$
$570$	0	0
$571$	23.9105i	1.00062i	0.865845	+	0.500312i	$0.166781\pi$
$571$	23.9105i	1.00062i	−0.865845	+	0.500312i	$0.833219\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.76217	−0.156893
$576$	0	0
$577$	30.0967	1.25294	0.626471	−	0.779445i	$-0.284499\pi$
$577$	30.0967	1.25294	0.626471	+	0.779445i	$0.284499\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 7.96655i	− 0.330508i
$582$	0	0
$583$	− 40.1873i	− 1.66439i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11.6853	0.482304	0.241152	−	0.970487i	$-0.422475\pi$
$587$	11.6853	0.482304	0.241152	+	0.970487i	$0.422475\pi$
$588$	0	0
$589$	−6.77983	−0.279358
$590$	0	0
$591$	0	0
$592$	0	0
$593$	16.6788i	0.684914i	0.939533	+	0.342457i	$0.111259\pi$
$593$	16.6788i	0.684914i	−0.939533	+	0.342457i	$0.888741\pi$
$594$	0	0
$595$	− 13.4583i	− 0.551738i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.64980	0.353421	0.176711	−	0.984263i	$-0.443454\pi$
$599$	8.64980	0.353421	0.176711	+	0.984263i	$0.443454\pi$
$600$	0	0
$601$	−15.6171	−0.637035	−0.318518	−	0.947917i	$-0.603185\pi$
$601$	−15.6171	−0.637035	−0.318518	+	0.947917i	$0.603185\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 3.58114i	− 0.145594i
$606$	0	0
$607$	20.8668i	0.846957i	0.905906	+	0.423479i	$0.139191\pi$
$607$	20.8668i	0.846957i	−0.905906	+	0.423479i	$0.860809\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−38.2410	−1.54707
$612$	0	0
$613$	35.5526	1.43596	0.717978	−	0.696066i	$-0.245068\pi$
$613$	35.5526	1.43596	0.717978	+	0.696066i	$0.245068\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 27.3777i	− 1.10218i	−0.834445	−	0.551091i	$-0.814212\pi$
$617$	− 27.3777i	− 1.10218i	0.834445	−	0.551091i	$-0.185788\pi$
$618$	0	0
$619$	− 27.5800i	− 1.10853i	−0.832339	−	0.554267i	$-0.812999\pi$
$619$	− 27.5800i	− 1.10853i	0.832339	−	0.554267i	$-0.187001\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	9.04118	0.362227
$624$	0	0
$625$	−17.0526	−0.682102
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 67.2839i	− 2.68279i
$630$	0	0
$631$	− 18.1207i	− 0.721374i	−0.932687	−	0.360687i	$-0.882542\pi$
$631$	− 18.1207i	− 0.721374i	0.932687	−	0.360687i	$-0.117458\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−29.4460	−1.16853
$636$	0	0
$637$	−5.92609	−0.234800
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 10.9215i	− 0.431372i	−0.976463	−	0.215686i	$-0.930801\pi$
$641$	− 10.9215i	− 0.431372i	0.976463	−	0.215686i	$-0.0691987\pi$
$642$	0	0
$643$	43.2372i	1.70511i	0.522639	+	0.852554i	$0.324948\pi$
$643$	43.2372i	1.70511i	−0.522639	+	0.852554i	$0.675052\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−19.4472	−0.764547	−0.382274	−	0.924049i	$-0.624859\pi$
$647$	−19.4472	−0.764547	−0.382274	+	0.924049i	$0.624859\pi$
$648$	0	0
$649$	22.7606	0.893431
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27.7057i	1.08421i	0.840311	+	0.542105i	$0.182372\pi$
$653$	27.7057i	1.08421i	−0.840311	+	0.542105i	$0.817628\pi$
$654$	0	0
$655$	− 31.0447i	− 1.21302i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−20.5359	−0.799966	−0.399983	−	0.916523i	$-0.630984\pi$
$659$	−20.5359	−0.799966	−0.399983	+	0.916523i	$0.630984\pi$
$660$	0	0
$661$	0.763677	0.0297036	0.0148518	−	0.999890i	$-0.495272\pi$
$661$	0.763677	0.0297036	0.0148518	+	0.999890i	$0.495272\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 10.3126i	− 0.399906i
$666$	0	0
$667$	9.10188i	0.352426i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	38.4560	1.48458
$672$	0	0
$673$	−26.2944	−1.01357	−0.506787	−	0.862071i	$-0.669167\pi$
$673$	−26.2944	−1.01357	−0.506787	+	0.862071i	$0.669167\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17.6603i	0.678740i	0.940653	+	0.339370i	$0.110214\pi$
$677$	17.6603i	0.678740i	−0.940653	+	0.339370i	$0.889786\pi$
$678$	0	0
$679$	− 4.33820i	− 0.166485i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	41.9049	1.60345	0.801723	−	0.597695i	$-0.203917\pi$
$683$	41.9049	1.60345	0.801723	+	0.597695i	$0.203917\pi$
$684$	0	0
$685$	7.39020	0.282365
$686$	0	0
$687$	0	0
$688$	0	0
$689$	66.4267i	2.53066i
$690$	0	0
$691$	32.1831i	1.22430i	0.790741	+	0.612151i	$0.209696\pi$
$691$	32.1831i	1.22430i	−0.790741	+	0.612151i	$0.790304\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−30.7204	−1.16529
$696$	0	0
$697$	56.4449	2.13800
$698$	0	0
$699$	0	0
$700$	0	0
$701$	40.8362i	1.54236i	0.636615	+	0.771182i	$0.280334\pi$
$701$	40.8362i	1.54236i	−0.636615	+	0.771182i	$0.719666\pi$
$702$	0	0
$703$	− 51.5571i	− 1.94451i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.13216	−0.230624
$708$	0	0
$709$	24.3986	0.916310	0.458155	−	0.888872i	$-0.348510\pi$
$709$	24.3986	0.916310	0.458155	+	0.888872i	$0.348510\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.76842i	0.141129i
$714$	0	0
$715$	41.0447i	1.53498i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	13.9008	0.518413	0.259206	−	0.965822i	$-0.416539\pi$
$719$	13.9008	0.518413	0.259206	+	0.965822i	$0.416539\pi$
$720$	0	0
$721$	13.9963	0.521251
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 3.88953i	− 0.144453i
$726$	0	0
$727$	− 2.98652i	− 0.110764i	−0.998465	−	0.0553820i	$-0.982362\pi$
$727$	− 2.98652i	− 0.110764i	0.998465	−	0.0553820i	$-0.0176377\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−17.1516	−0.634375
$732$	0	0
$733$	−2.16874	−0.0801044	−0.0400522	−	0.999198i	$-0.512752\pi$
$733$	−2.16874	−0.0801044	−0.0400522	+	0.999198i	$0.512752\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 22.4588i	− 0.827281i
$738$	0	0
$739$	19.1144i	0.703134i	0.936163	+	0.351567i	$0.114351\pi$
$739$	19.1144i	0.703134i	−0.936163	+	0.351567i	$0.885649\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−42.1582	−1.54663	−0.773317	−	0.634020i	$-0.781404\pi$
$743$	−42.1582	−1.54663	−0.773317	+	0.634020i	$0.781404\pi$
$744$	0	0
$745$	−15.8023	−0.578952
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.34251i	0.341368i
$750$	0	0
$751$	19.2016i	0.700678i	0.936623	+	0.350339i	$0.113934\pi$
$751$	19.2016i	0.700678i	−0.936623	+	0.350339i	$0.886066\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	15.5512	0.565966
$756$	0	0
$757$	−3.84797	−0.139857	−0.0699284	−	0.997552i	$-0.522277\pi$
$757$	−3.84797	−0.139857	−0.0699284	+	0.997552i	$0.522277\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 45.3580i	− 1.64423i	−0.569325	−	0.822113i	$-0.692795\pi$
$761$	− 45.3580i	− 1.64423i	0.569325	−	0.822113i	$-0.307205\pi$
$762$	0	0
$763$	19.0666i	0.690257i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−37.6216	−1.35844
$768$	0	0
$769$	−5.83239	−0.210321	−0.105161	−	0.994455i	$-0.533536\pi$
$769$	−5.83239	−0.210321	−0.105161	+	0.994455i	$0.533536\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 34.9259i	− 1.25620i	−0.778133	−	0.628099i	$-0.783833\pi$
$773$	− 34.9259i	− 1.25620i	0.778133	−	0.628099i	$-0.216167\pi$
$774$	0	0
$775$	− 1.61037i	− 0.0578462i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	43.2516	1.54965
$780$	0	0
$781$	−6.91540	−0.247453
$782$	0	0
$783$	0	0
$784$	0	0
$785$	34.0817i	1.21643i
$786$	0	0
$787$	51.7877i	1.84603i	0.384762	+	0.923016i	$0.374284\pi$
$787$	51.7877i	1.84603i	−0.384762	+	0.923016i	$0.625716\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.68863	−0.0600409
$792$	0	0
$793$	−63.5650	−2.25726
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 43.7027i	− 1.54803i	−0.633167	−	0.774015i	$-0.718246\pi$
$797$	− 43.7027i	− 1.54803i	0.633167	−	0.774015i	$-0.281754\pi$
$798$	0	0
$799$	44.9550i	1.59040i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.61424	0.268701
$804$	0	0
$805$	−5.73205	−0.202028
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 19.6717i	− 0.691620i	−0.938304	−	0.345810i	$-0.887604\pi$
$809$	− 19.6717i	− 0.691620i	0.938304	−	0.345810i	$-0.112396\pi$
$810$	0	0
$811$	3.16694i	0.111206i	0.998453	+	0.0556031i	$0.0177082\pi$
$811$	3.16694i	0.111206i	−0.998453	+	0.0556031i	$0.982292\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.76435	0.166888
$816$	0	0
$817$	−13.1426	−0.459802
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7.54805i	0.263429i	0.991288	+	0.131714i	$0.0420482\pi$
$821$	7.54805i	0.263429i	−0.991288	+	0.131714i	$0.957952\pi$
$822$	0	0
$823$	− 25.0280i	− 0.872420i	−0.899845	−	0.436210i	$-0.856320\pi$
$823$	− 25.0280i	− 0.872420i	0.899845	−	0.436210i	$-0.143680\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.9477	0.937062	0.468531	−	0.883447i	$-0.344783\pi$
$827$	26.9477	0.937062	0.468531	+	0.883447i	$0.344783\pi$
$828$	0	0
$829$	−19.2466	−0.668462	−0.334231	−	0.942491i	$-0.608477\pi$
$829$	−19.2466	−0.668462	−0.334231	+	0.942491i	$0.608477\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.96655i	0.241377i
$834$	0	0
$835$	31.1685i	1.07863i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	25.4979	0.880285	0.440142	−	0.897928i	$-0.354928\pi$
$839$	25.4979	0.880285	0.440142	+	0.897928i	$0.354928\pi$
$840$	0	0
$841$	19.5900	0.675517
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 42.7298i	− 1.46995i
$846$	0	0
$847$	1.85373i	0.0636950i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−28.6570	−0.982348
$852$	0	0
$853$	16.4464	0.563114	0.281557	−	0.959544i	$-0.409149\pi$
$853$	16.4464	0.563114	0.281557	+	0.959544i	$0.409149\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	37.0695i	1.26627i	0.774041	+	0.633135i	$0.218232\pi$
$857$	37.0695i	1.26627i	−0.774041	+	0.633135i	$0.781768\pi$
$858$	0	0
$859$	− 17.6284i	− 0.601472i	−0.953707	−	0.300736i	$-0.902768\pi$
$859$	− 17.6284i	− 0.601472i	0.953707	−	0.300736i	$-0.0972323\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	21.5272	0.732794	0.366397	−	0.930459i	$-0.380591\pi$
$863$	21.5272	0.732794	0.366397	+	0.930459i	$0.380591\pi$
$864$	0	0
$865$	−23.1904	−0.788497
$866$	0	0
$867$	0	0
$868$	0	0
$869$	31.2408i	1.05977i
$870$	0	0
$871$	37.1228i	1.25786i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12.1087	0.409350
$876$	0	0
$877$	−28.1862	−0.951779	−0.475890	−	0.879505i	$-0.657874\pi$
$877$	−28.1862	−0.951779	−0.475890	+	0.879505i	$0.657874\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	15.3854i	0.518346i	0.965831	+	0.259173i	$0.0834500\pi$
$881$	15.3854i	0.518346i	−0.965831	+	0.259173i	$0.916550\pi$
$882$	0	0
$883$	− 29.4069i	− 0.989621i	−0.869001	−	0.494811i	$-0.835237\pi$
$883$	− 29.4069i	− 0.989621i	0.869001	−	0.494811i	$-0.164763\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28.1982	0.946801	0.473401	−	0.880847i	$-0.343026\pi$
$887$	28.1982	0.946801	0.473401	+	0.880847i	$0.343026\pi$
$888$	0	0
$889$	15.2424	0.511213
$890$	0	0
$891$	0	0
$892$	0	0
$893$	34.4473i	1.15274i
$894$	0	0
$895$	− 27.9869i	− 0.935500i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.89599	−0.129939
$900$	0	0
$901$	78.0894	2.60153
$902$	0	0
$903$	0	0
$904$	0	0
$905$	16.8579i	0.560375i
$906$	0	0
$907$	− 15.0718i	− 0.500451i	−0.968188	−	0.250225i	$-0.919495\pi$
$907$	− 15.0718i	− 0.500451i	0.968188	−	0.250225i	$-0.0805047\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.49292	−0.181989	−0.0909943	−	0.995851i	$-0.529004\pi$
$911$	−5.49292	−0.181989	−0.0909943	+	0.995851i	$0.529004\pi$
$912$	0	0
$913$	28.5618	0.945256
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.0699i	0.530675i
$918$	0	0
$919$	18.6898i	0.616519i	0.951302	+	0.308259i	$0.0997464\pi$
$919$	18.6898i	0.616519i	−0.951302	+	0.308259i	$0.900254\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	11.4307	0.376245
$924$	0	0
$925$	12.2460	0.402647
$926$	0	0
$927$	0	0
$928$	0	0
$929$	35.6186i	1.16861i	0.811534	+	0.584305i	$0.198633\pi$
$929$	35.6186i	1.16861i	−0.811534	+	0.584305i	$0.801367\pi$
$930$	0	0
$931$	5.33820i	0.174953i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	48.2510	1.57798
$936$	0	0
$937$	−36.3017	−1.18592	−0.592962	−	0.805230i	$-0.702042\pi$
$937$	−36.3017	−1.18592	−0.592962	+	0.805230i	$0.702042\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 31.7581i	− 1.03529i	−0.855597	−	0.517643i	$-0.826810\pi$
$941$	− 31.7581i	− 1.03529i	0.855597	−	0.517643i	$-0.173190\pi$
$942$	0	0
$943$	− 24.0405i	− 0.782865i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.0513	0.976536	0.488268	−	0.872694i	$-0.337629\pi$
$947$	30.0513	0.976536	0.488268	+	0.872694i	$0.337629\pi$
$948$	0	0
$949$	−12.5858	−0.408552
$950$	0	0
$951$	0	0
$952$	0	0
$953$	52.4957i	1.70050i	0.526376	+	0.850252i	$0.323550\pi$
$953$	52.4957i	1.70050i	−0.526376	+	0.850252i	$0.676450\pi$
$954$	0	0
$955$	24.4626i	0.791590i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.82545	−0.123530
$960$	0	0
$961$	29.3870	0.947966
$962$	0	0
$963$	0	0
$964$	0	0
$965$	6.78850i	0.218530i
$966$	0	0
$967$	− 47.4411i	− 1.52560i	−0.646634	−	0.762801i	$-0.723824\pi$
$967$	− 47.4411i	− 1.52560i	0.646634	−	0.762801i	$-0.276176\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	33.9492	1.08948	0.544741	−	0.838605i	$-0.316628\pi$
$971$	33.9492	1.08948	0.544741	+	0.838605i	$0.316628\pi$
$972$	0	0
$973$	15.9021	0.509797
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 12.2747i	− 0.392701i	−0.980534	−	0.196351i	$-0.937091\pi$
$977$	− 12.2747i	− 0.392701i	0.980534	−	0.196351i	$-0.0629090\pi$
$978$	0	0
$979$	32.4145i	1.03597i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	5.43320	0.173292	0.0866461	−	0.996239i	$-0.472385\pi$
$983$	5.43320	0.173292	0.0866461	+	0.996239i	$0.472385\pi$
$984$	0	0
$985$	−21.8407	−0.695901
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7.30505i	0.232287i
$990$	0	0
$991$	− 49.4370i	− 1.57042i	−0.619231	−	0.785209i	$-0.712556\pi$
$991$	− 49.4370i	− 1.57042i	0.619231	−	0.785209i	$-0.287444\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−4.90305	−0.155437
$996$	0	0
$997$	28.0925	0.889697	0.444849	−	0.895606i	$-0.353257\pi$
$997$	28.0925	0.889697	0.444849	+	0.895606i	$0.353257\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.h.h.2591.3	yes	16
3.2	odd	2	inner	6048.2.h.h.2591.16	yes	16
4.3	odd	2	inner	6048.2.h.h.2591.2	✓	16
12.11	even	2	inner	6048.2.h.h.2591.13	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.h.h.2591.2	✓	16	4.3	odd	2	inner
6048.2.h.h.2591.3	yes	16	1.1	even	1	trivial
6048.2.h.h.2591.13	yes	16	12.11	even	2	inner
6048.2.h.h.2591.16	yes	16	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.93185i q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q-1.93185i q^{5} +1.00000i q^{7} -3.58521 q^{11} +5.92609 q^{13} -6.96655i q^{17} -5.33820i q^{19} -2.96713 q^{23} +1.26795 q^{25} -3.06757i q^{29} -1.27006i q^{31} +1.93185 q^{35} +9.65815 q^{37} +8.10227i q^{41} -2.46199i q^{43} -6.45299 q^{47} -1.00000 q^{49} +11.2092i q^{53} +6.92609i q^{55} -6.34847 q^{59} -10.7263 q^{61} -11.4483i q^{65} +6.26430i q^{67} +1.92887 q^{71} -2.12379 q^{73} -3.58521i q^{77} -8.71379i q^{79} -7.96655 q^{83} -13.4583 q^{85} -9.04118i q^{89} +5.92609i q^{91} -10.3126 q^{95} -4.33820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 8 q^{13} + 48 q^{25} + 40 q^{37} - 16 q^{49} - 56 q^{73} - 16 q^{85} - 16 q^{97}+O(q^{100}) \) 16 * q + 8 * q^13 + 48 * q^25 + 40 * q^37 - 16 * q^49 - 56 * q^73 - 16 * q^85 - 16 * q^97

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