LMFDB - Embedded newform 6048.2.h.h.2591.11

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			2591.11
Root			$-2.55743 + 2.55743i$ of defining polynomial
Character	$\chi$	$=$	6048.2591
Dual form			6048.2.h.h.2591.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.517638i q^{5} +1.00000i q^{7} +O(q^{10})$	$q+0.517638i q^{5} +1.00000i q^{7} -3.18300 q^{11} -2.64764 q^{13} +5.15854i q^{17} -8.23349i q^{19} +4.38134 q^{23} +4.73205 q^{25} -5.11485i q^{29} +0.548511i q^{31} -0.517638 q^{35} -2.37969 q^{37} +0.473945i q^{41} -0.816460i q^{43} +0.493272 q^{47} -1.00000 q^{49} -0.915903i q^{53} -1.64764i q^{55} +12.7229 q^{59} -3.40231 q^{61} -1.37052i q^{65} +0.585849i q^{67} +6.95020 q^{71} +2.41703 q^{73} -3.18300i q^{77} +12.5787i q^{79} -0.215871 q^{83} -2.67026 q^{85} +10.1525i q^{89} -2.64764i q^{91} +4.26197 q^{95} -7.23349 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	0.517638i	0.231495i	0.993279	+	0.115747i	$0.0369263\pi$
$5$	0.517638i	0.231495i	−0.993279	+	0.115747i	$0.963074\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.18300	−0.959710	−0.479855	−	0.877348i	$-0.659311\pi$
$11$	−3.18300	−0.959710	−0.479855	+	0.877348i	$0.659311\pi$
$12$	0	0
$13$	−2.64764	−0.734324	−0.367162	−	0.930157i	$-0.619670\pi$
$13$	−2.64764	−0.734324	−0.367162	+	0.930157i	$0.619670\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.15854i	1.25113i	0.780172	+	0.625565i	$0.215132\pi$
$17$	5.15854i	1.25113i	−0.780172	+	0.625565i	$0.784868\pi$
$18$	0	0
$19$	− 8.23349i	− 1.88889i	−0.328666	−	0.944446i	$-0.606599\pi$
$19$	− 8.23349i	− 1.88889i	0.328666	−	0.944446i	$-0.393401\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.38134	0.913573	0.456786	−	0.889576i	$-0.349000\pi$
$23$	4.38134	0.913573	0.456786	+	0.889576i	$0.349000\pi$
$24$	0	0
$25$	4.73205	0.946410
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 5.11485i	− 0.949804i	−0.880039	−	0.474902i	$-0.842484\pi$
$29$	− 5.11485i	− 0.949804i	0.880039	−	0.474902i	$-0.157516\pi$
$30$	0	0
$31$	0.548511i	0.0985155i	0.998786	+	0.0492577i	$0.0156856\pi$
$31$	0.548511i	0.0985155i	−0.998786	+	0.0492577i	$0.984314\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.517638	−0.0874968
$36$	0	0
$37$	−2.37969	−0.391219	−0.195609	−	0.980682i	$-0.562669\pi$
$37$	−2.37969	−0.391219	−0.195609	+	0.980682i	$0.562669\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.473945i	0.0740177i	0.999315	+	0.0370089i	$0.0117830\pi$
$41$	0.473945i	0.0740177i	−0.999315	+	0.0370089i	$0.988217\pi$
$42$	0	0
$43$	− 0.816460i	− 0.124509i	−0.998060	−	0.0622545i	$-0.980171\pi$
$43$	− 0.816460i	− 0.124509i	0.998060	−	0.0622545i	$-0.0198290\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.493272	0.0719512	0.0359756	−	0.999353i	$-0.488546\pi$
$47$	0.493272	0.0719512	0.0359756	+	0.999353i	$0.488546\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 0.915903i	− 0.125809i	−0.998020	−	0.0629045i	$-0.979964\pi$
$53$	− 0.915903i	− 0.125809i	0.998020	−	0.0629045i	$-0.0200363\pi$
$54$	0	0
$55$	− 1.64764i	− 0.222168i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.7229	1.65638	0.828189	−	0.560449i	$-0.189371\pi$
$59$	12.7229	1.65638	0.828189	+	0.560449i	$0.189371\pi$
$60$	0	0
$61$	−3.40231	−0.435621	−0.217811	−	0.975991i	$-0.569891\pi$
$61$	−3.40231	−0.435621	−0.217811	+	0.975991i	$0.569891\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 1.37052i	− 0.169992i
$66$	0	0
$67$	0.585849i	0.0715729i	0.999359	+	0.0357864i	$0.0113936\pi$
$67$	0.585849i	0.0715729i	−0.999359	+	0.0357864i	$0.988606\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.95020	0.824837	0.412419	−	0.910994i	$-0.364684\pi$
$71$	6.95020	0.824837	0.412419	+	0.910994i	$0.364684\pi$
$72$	0	0
$73$	2.41703	0.282892	0.141446	−	0.989946i	$-0.454825\pi$
$73$	2.41703	0.282892	0.141446	+	0.989946i	$0.454825\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.18300i	− 0.362736i
$78$	0	0
$79$	12.5787i	1.41522i	0.706605	+	0.707609i	$0.250226\pi$
$79$	12.5787i	1.41522i	−0.706605	+	0.707609i	$0.749774\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.215871	−0.0236949	−0.0118475	−	0.999930i	$-0.503771\pi$
$83$	−0.215871	−0.0236949	−0.0118475	+	0.999930i	$0.503771\pi$
$84$	0	0
$85$	−2.67026	−0.289630
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.1525i	1.07617i	0.842892	+	0.538083i	$0.180851\pi$
$89$	10.1525i	1.07617i	−0.842892	+	0.538083i	$0.819149\pi$
$90$	0	0
$91$	− 2.64764i	− 0.277548i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.26197	0.437269
$96$	0	0
$97$	−7.23349	−0.734450	−0.367225	−	0.930132i	$-0.619692\pi$
$97$	−7.23349	−0.734450	−0.367225	+	0.930132i	$0.619692\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	17.6975i	1.76097i	0.474073	+	0.880486i	$0.342783\pi$
$101$	17.6975i	1.76097i	−0.474073	+	0.880486i	$0.657217\pi$
$102$	0	0
$103$	− 4.85380i	− 0.478259i	−0.970988	−	0.239129i	$-0.923138\pi$
$103$	− 4.85380i	− 0.478259i	0.970988	−	0.239129i	$-0.0768620\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.3360	1.77261	0.886306	−	0.463100i	$-0.153263\pi$
$107$	18.3360	1.77261	0.886306	+	0.463100i	$0.153263\pi$
$108$	0	0
$109$	9.35524	0.896069	0.448035	−	0.894016i	$-0.352124\pi$
$109$	9.35524	0.896069	0.448035	+	0.894016i	$0.352124\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 5.53748i	− 0.520922i	−0.965484	−	0.260461i	$-0.916125\pi$
$113$	− 5.53748i	− 0.520922i	0.965484	−	0.260461i	$-0.0838746\pi$
$114$	0	0
$115$	2.26795i	0.211487i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.15854	−0.472883
$120$	0	0
$121$	−0.868519	−0.0789563
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.03768i	0.450584i
$126$	0	0
$127$	17.4070i	1.54462i	0.635244	+	0.772312i	$0.280900\pi$
$127$	17.4070i	1.54462i	−0.635244	+	0.772312i	$0.719100\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.8911	0.951558	0.475779	−	0.879565i	$-0.342166\pi$
$131$	10.8911	0.951558	0.475779	+	0.879565i	$0.342166\pi$
$132$	0	0
$133$	8.23349	0.713934
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.6707i	1.33884i	0.742886	+	0.669418i	$0.233457\pi$
$137$	15.6707i	1.33884i	−0.742886	+	0.669418i	$0.766543\pi$
$138$	0	0
$139$	14.3600i	1.21800i	0.793172	+	0.608998i	$0.208428\pi$
$139$	14.3600i	1.21800i	−0.793172	+	0.608998i	$0.791572\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.42744	0.704738
$144$	0	0
$145$	2.64764	0.219875
$146$	0	0
$147$	0	0
$148$	0	0
$149$	22.7367i	1.86266i	0.364171	+	0.931332i	$0.381352\pi$
$149$	22.7367i	1.86266i	−0.364171	+	0.931332i	$0.618648\pi$
$150$	0	0
$151$	− 5.06467i	− 0.412157i	−0.978535	−	0.206079i	$-0.933930\pi$
$151$	− 5.06467i	− 0.412157i	0.978535	−	0.206079i	$-0.0660703\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.283930	−0.0228058
$156$	0	0
$157$	17.5069	1.39720	0.698602	−	0.715510i	$-0.253806\pi$
$157$	17.5069	1.39720	0.698602	+	0.715510i	$0.253806\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.38134i	0.345298i
$162$	0	0
$163$	− 9.74466i	− 0.763261i	−0.924315	−	0.381630i	$-0.875363\pi$
$163$	− 9.74466i	− 0.763261i	0.924315	−	0.381630i	$-0.124637\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.4241	−1.50308	−0.751542	−	0.659685i	$-0.770690\pi$
$167$	−19.4241	−1.50308	−0.751542	+	0.659685i	$0.770690\pi$
$168$	0	0
$169$	−5.99000	−0.460769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	17.2673i	1.31281i	0.754410	+	0.656404i	$0.227923\pi$
$173$	17.2673i	1.31281i	−0.754410	+	0.656404i	$0.772077\pi$
$174$	0	0
$175$	4.73205i	0.357709i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−11.2858	−0.843540	−0.421770	−	0.906703i	$-0.638591\pi$
$179$	−11.2858	−0.843540	−0.421770	+	0.906703i	$0.638591\pi$
$180$	0	0
$181$	−1.40231	−0.104233	−0.0521164	−	0.998641i	$-0.516597\pi$
$181$	−1.40231	−0.104233	−0.0521164	+	0.998641i	$0.516597\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 1.23182i	− 0.0905652i
$186$	0	0
$187$	− 16.4196i	− 1.20072i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.00639	0.434607	0.217303	−	0.976104i	$-0.430274\pi$
$191$	6.00639	0.434607	0.217303	+	0.976104i	$0.430274\pi$
$192$	0	0
$193$	16.5288	1.18977	0.594884	−	0.803812i	$-0.297198\pi$
$193$	16.5288	1.18977	0.594884	+	0.803812i	$0.297198\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5.84687i	− 0.416572i	−0.978068	−	0.208286i	$-0.933211\pi$
$197$	− 5.84687i	− 0.416572i	0.978068	−	0.208286i	$-0.0667885\pi$
$198$	0	0
$199$	− 4.18354i	− 0.296563i	−0.988945	−	0.148282i	$-0.952626\pi$
$199$	− 4.18354i	− 0.296563i	0.988945	−	0.148282i	$-0.0473742\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.11485	0.358992
$204$	0	0
$205$	−0.245332	−0.0171347
$206$	0	0
$207$	0	0
$208$	0	0
$209$	26.2072i	1.81279i
$210$	0	0
$211$	− 21.2882i	− 1.46554i	−0.680478	−	0.732769i	$-0.738228\pi$
$211$	− 21.2882i	− 1.46554i	0.680478	−	0.732769i	$-0.261772\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.422631	0.0288232
$216$	0	0
$217$	−0.548511	−0.0372354
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 13.6580i	− 0.918735i
$222$	0	0
$223$	− 12.5887i	− 0.843004i	−0.906828	−	0.421502i	$-0.861503\pi$
$223$	− 12.5887i	− 0.843004i	0.906828	−	0.421502i	$-0.138497\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.19330	0.0792024	0.0396012	−	0.999216i	$-0.487391\pi$
$227$	1.19330	0.0792024	0.0396012	+	0.999216i	$0.487391\pi$
$228$	0	0
$229$	−17.7694	−1.17423	−0.587117	−	0.809502i	$-0.699737\pi$
$229$	−17.7694	−1.17423	−0.587117	+	0.809502i	$0.699737\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 14.1284i	− 0.925580i	−0.886468	−	0.462790i	$-0.846848\pi$
$233$	− 14.1284i	− 0.925580i	0.886468	−	0.462790i	$-0.153152\pi$
$234$	0	0
$235$	0.255337i	0.0166563i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	14.1600	0.915932	0.457966	−	0.888970i	$-0.348578\pi$
$239$	14.1600	0.915932	0.457966	+	0.888970i	$0.348578\pi$
$240$	0	0
$241$	−20.1864	−1.30032	−0.650161	−	0.759797i	$-0.725298\pi$
$241$	−20.1864	−1.30032	−0.650161	+	0.759797i	$0.725298\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 0.517638i	− 0.0330707i
$246$	0	0
$247$	21.7993i	1.38706i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15.7352	0.993197	0.496598	−	0.867980i	$-0.334582\pi$
$251$	15.7352	0.993197	0.496598	+	0.867980i	$0.334582\pi$
$252$	0	0
$253$	−13.9458	−0.876765
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 9.72117i	− 0.606390i	−0.952929	−	0.303195i	$-0.901947\pi$
$257$	− 9.72117i	− 0.606390i	0.952929	−	0.303195i	$-0.0980534\pi$
$258$	0	0
$259$	− 2.37969i	− 0.147867i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	19.5487	1.20542	0.602712	−	0.797959i	$-0.294087\pi$
$263$	19.5487	1.20542	0.602712	+	0.797959i	$0.294087\pi$
$264$	0	0
$265$	0.474106	0.0291241
$266$	0	0
$267$	0	0
$268$	0	0
$269$	9.00699i	0.549166i	0.961563	+	0.274583i	$0.0885398\pi$
$269$	9.00699i	0.549166i	−0.961563	+	0.274583i	$0.911460\pi$
$270$	0	0
$271$	16.0252i	0.973463i	0.873552	+	0.486732i	$0.161811\pi$
$271$	16.0252i	0.973463i	−0.873552	+	0.486732i	$0.838189\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−15.0621	−0.908279
$276$	0	0
$277$	−4.31790	−0.259437	−0.129719	−	0.991551i	$-0.541407\pi$
$277$	−4.31790	−0.259437	−0.129719	+	0.991551i	$0.541407\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.06265i	0.480977i	0.970652	+	0.240489i	$0.0773077\pi$
$281$	8.06265i	0.480977i	−0.970652	+	0.240489i	$0.922692\pi$
$282$	0	0
$283$	− 27.4570i	− 1.63215i	−0.577948	−	0.816074i	$-0.696146\pi$
$283$	− 27.4570i	− 1.63215i	0.577948	−	0.816074i	$-0.303854\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.473945	−0.0279761
$288$	0	0
$289$	−9.61057	−0.565328
$290$	0	0
$291$	0	0
$292$	0	0
$293$	10.1282i	0.591694i	0.955235	+	0.295847i	$0.0956018\pi$
$293$	10.1282i	0.591694i	−0.955235	+	0.295847i	$0.904398\pi$
$294$	0	0
$295$	6.58585i	0.383443i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−11.6002	−0.670858
$300$	0	0
$301$	0.816460	0.0470600
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 1.76117i	− 0.100844i
$306$	0	0
$307$	10.8441i	0.618904i	0.950915	+	0.309452i	$0.100146\pi$
$307$	10.8441i	0.618904i	−0.950915	+	0.309452i	$0.899854\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	22.7209	1.28839	0.644193	−	0.764863i	$-0.277193\pi$
$311$	22.7209	1.28839	0.644193	+	0.764863i	$0.277193\pi$
$312$	0	0
$313$	12.8222	0.724755	0.362377	−	0.932031i	$-0.381965\pi$
$313$	12.8222	0.724755	0.362377	+	0.932031i	$0.381965\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.4593i	0.587455i	0.955889	+	0.293727i	$0.0948958\pi$
$317$	10.4593i	0.587455i	−0.955889	+	0.293727i	$0.905104\pi$
$318$	0	0
$319$	16.2806i	0.911536i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	42.4728	2.36325
$324$	0	0
$325$	−12.5288	−0.694971
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.493272i	0.0271950i
$330$	0	0
$331$	3.70760i	0.203788i	0.994795	+	0.101894i	$0.0324902\pi$
$331$	3.70760i	0.203788i	−0.994795	+	0.101894i	$0.967510\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−0.303258	−0.0165688
$336$	0	0
$337$	19.9900	1.08892	0.544462	−	0.838785i	$-0.316734\pi$
$337$	19.9900	1.08892	0.544462	+	0.838785i	$0.316734\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 1.74591i	− 0.0945463i
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.1226	1.18760	0.593801	−	0.804612i	$-0.297627\pi$
$347$	22.1226	1.18760	0.593801	+	0.804612i	$0.297627\pi$
$348$	0	0
$349$	0.997120	0.0533746	0.0266873	−	0.999644i	$-0.491504\pi$
$349$	0.997120	0.0533746	0.0266873	+	0.999644i	$0.491504\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 20.5403i	− 1.09325i	−0.837378	−	0.546624i	$-0.815913\pi$
$353$	− 20.5403i	− 1.09325i	0.837378	−	0.546624i	$-0.184087\pi$
$354$	0	0
$355$	3.59769i	0.190946i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.28121	−0.331509	−0.165755	−	0.986167i	$-0.553006\pi$
$359$	−6.28121	−0.331509	−0.165755	+	0.986167i	$0.553006\pi$
$360$	0	0
$361$	−48.7904	−2.56791
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.25115i	0.0654880i
$366$	0	0
$367$	0.439376i	0.0229352i	0.999934	+	0.0114676i	$0.00365034\pi$
$367$	0.439376i	0.0229352i	−0.999934	+	0.0114676i	$0.996350\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.915903	0.0475513
$372$	0	0
$373$	−34.2016	−1.77089	−0.885447	−	0.464741i	$-0.846148\pi$
$373$	−34.2016	−1.77089	−0.885447	+	0.464741i	$0.846148\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13.5423i	0.697463i
$378$	0	0
$379$	− 15.9681i	− 0.820229i	−0.912034	−	0.410114i	$-0.865489\pi$
$379$	− 15.9681i	− 0.820229i	0.912034	−	0.410114i	$-0.134511\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	31.6537	1.61743	0.808715	−	0.588200i	$-0.200163\pi$
$383$	31.6537	1.61743	0.808715	+	0.588200i	$0.200163\pi$
$384$	0	0
$385$	1.64764	0.0839716
$386$	0	0
$387$	0	0
$388$	0	0
$389$	25.3751i	1.28657i	0.765627	+	0.643285i	$0.222429\pi$
$389$	25.3751i	1.28657i	−0.765627	+	0.643285i	$0.777571\pi$
$390$	0	0
$391$	22.6013i	1.14300i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6.51123	−0.327615
$396$	0	0
$397$	34.4470	1.72884	0.864422	−	0.502767i	$-0.167685\pi$
$397$	34.4470	1.72884	0.864422	+	0.502767i	$0.167685\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	36.2881i	1.81214i	0.423126	+	0.906071i	$0.360933\pi$
$401$	36.2881i	1.81214i	−0.423126	+	0.906071i	$0.639067\pi$
$402$	0	0
$403$	− 1.45226i	− 0.0723423i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.57456	0.375457
$408$	0	0
$409$	6.84590	0.338508	0.169254	−	0.985572i	$-0.445864\pi$
$409$	6.84590	0.338508	0.169254	+	0.985572i	$0.445864\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	12.7229i	0.626052i
$414$	0	0
$415$	− 0.111743i	− 0.00548525i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	5.12396	0.250322	0.125161	−	0.992136i	$-0.460055\pi$
$419$	5.12396	0.250322	0.125161	+	0.992136i	$0.460055\pi$
$420$	0	0
$421$	16.3926	0.798925	0.399463	−	0.916749i	$-0.369197\pi$
$421$	16.3926	0.798925	0.399463	+	0.916749i	$0.369197\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	24.4105i	1.18408i
$426$	0	0
$427$	− 3.40231i	− 0.164649i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	25.0475	1.20650	0.603248	−	0.797554i	$-0.293873\pi$
$431$	25.0475	1.20650	0.603248	+	0.797554i	$0.293873\pi$
$432$	0	0
$433$	40.1427	1.92914	0.964568	−	0.263834i	$-0.0849870\pi$
$433$	40.1427	1.92914	0.964568	+	0.263834i	$0.0849870\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 36.0737i	− 1.72564i
$438$	0	0
$439$	22.6905i	1.08296i	0.840715	+	0.541478i	$0.182135\pi$
$439$	22.6905i	1.08296i	−0.840715	+	0.541478i	$0.817865\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−26.8137	−1.27396	−0.636979	−	0.770881i	$-0.719816\pi$
$443$	−26.8137	−1.27396	−0.636979	+	0.770881i	$0.719816\pi$
$444$	0	0
$445$	−5.25534	−0.249127
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.12725i	0.289163i	0.989493	+	0.144581i	$0.0461836\pi$
$449$	6.12725i	0.289163i	−0.989493	+	0.144581i	$0.953816\pi$
$450$	0	0
$451$	− 1.50856i	− 0.0710355i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.37052	0.0642510
$456$	0	0
$457$	14.2629	0.667192	0.333596	−	0.942716i	$-0.391738\pi$
$457$	14.2629	0.667192	0.333596	+	0.942716i	$0.391738\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	13.2106i	0.615279i	0.951503	+	0.307639i	$0.0995390\pi$
$461$	13.2106i	0.615279i	−0.951503	+	0.307639i	$0.900461\pi$
$462$	0	0
$463$	2.56112i	0.119026i	0.998228	+	0.0595128i	$0.0189547\pi$
$463$	2.56112i	0.119026i	−0.998228	+	0.0595128i	$0.981045\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.8012	−1.05511	−0.527557	−	0.849519i	$-0.676892\pi$
$467$	−22.8012	−1.05511	−0.527557	+	0.849519i	$0.676892\pi$
$468$	0	0
$469$	−0.585849	−0.0270520
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.59879i	0.119493i
$474$	0	0
$475$	− 38.9613i	− 1.78767i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−36.9525	−1.68840	−0.844201	−	0.536027i	$-0.819924\pi$
$479$	−36.9525	−1.68840	−0.844201	+	0.536027i	$0.819924\pi$
$480$	0	0
$481$	6.30057	0.287281
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 3.74433i	− 0.170021i
$486$	0	0
$487$	6.80934i	0.308560i	0.988027	+	0.154280i	$0.0493059\pi$
$487$	6.80934i	0.308560i	−0.988027	+	0.154280i	$0.950694\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	14.1702	0.639492	0.319746	−	0.947503i	$-0.396402\pi$
$491$	14.1702	0.639492	0.319746	+	0.947503i	$0.396402\pi$
$492$	0	0
$493$	26.3852	1.18833
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.95020i	0.311759i
$498$	0	0
$499$	− 34.6858i	− 1.55275i	−0.630273	−	0.776374i	$-0.717057\pi$
$499$	− 34.6858i	− 1.55275i	0.630273	−	0.776374i	$-0.282943\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9.85580	0.439448	0.219724	−	0.975562i	$-0.429484\pi$
$503$	9.85580	0.439448	0.219724	+	0.975562i	$0.429484\pi$
$504$	0	0
$505$	−9.16092	−0.407656
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16.2874i	0.721926i	0.932580	+	0.360963i	$0.117552\pi$
$509$	16.2874i	0.721926i	−0.932580	+	0.360963i	$0.882448\pi$
$510$	0	0
$511$	2.41703i	0.106923i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.51251	0.110714
$516$	0	0
$517$	−1.57008	−0.0690523
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 27.9338i	− 1.22380i	−0.790935	−	0.611900i	$-0.790405\pi$
$521$	− 27.9338i	− 1.22380i	0.790935	−	0.611900i	$-0.209595\pi$
$522$	0	0
$523$	2.13724i	0.0934550i	0.998908	+	0.0467275i	$0.0148792\pi$
$523$	2.13724i	0.0934550i	−0.998908	+	0.0467275i	$0.985121\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.82952	−0.123256
$528$	0	0
$529$	−3.80385	−0.165385
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 1.25484i	− 0.0543529i
$534$	0	0
$535$	9.49144i	0.410351i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.18300	0.137101
$540$	0	0
$541$	20.2037	0.868627	0.434313	−	0.900762i	$-0.356991\pi$
$541$	20.2037	0.868627	0.434313	+	0.900762i	$0.356991\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.84263i	0.207435i
$546$	0	0
$547$	29.7917i	1.27380i	0.770946	+	0.636901i	$0.219784\pi$
$547$	29.7917i	1.27380i	−0.770946	+	0.636901i	$0.780216\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−42.1131	−1.79408
$552$	0	0
$553$	−12.5787	−0.534902
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 4.18984i	− 0.177529i	−0.996053	−	0.0887645i	$-0.971708\pi$
$557$	− 4.18984i	− 0.177529i	0.996053	−	0.0887645i	$-0.0282919\pi$
$558$	0	0
$559$	2.16169i	0.0914299i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	36.2275	1.52681	0.763405	−	0.645921i	$-0.223526\pi$
$563$	36.2275	1.52681	0.763405	+	0.645921i	$0.223526\pi$
$564$	0	0
$565$	2.86641	0.120591
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6.93088i	0.290557i	0.989391	+	0.145279i	$0.0464079\pi$
$569$	6.93088i	0.290557i	−0.989391	+	0.145279i	$0.953592\pi$
$570$	0	0
$571$	− 27.4822i	− 1.15009i	−0.818120	−	0.575047i	$-0.804984\pi$
$571$	− 27.4822i	− 1.15009i	0.818120	−	0.575047i	$-0.195016\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	20.7327	0.864615
$576$	0	0
$577$	−24.9829	−1.04005	−0.520025	−	0.854151i	$-0.674077\pi$
$577$	−24.9829	−1.04005	−0.520025	+	0.854151i	$0.674077\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 0.215871i	− 0.00895583i
$582$	0	0
$583$	2.91532i	0.120740i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−16.7456	−0.691164	−0.345582	−	0.938389i	$-0.612319\pi$
$587$	−16.7456	−0.691164	−0.345582	+	0.938389i	$0.612319\pi$
$588$	0	0
$589$	4.51616	0.186085
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.78732i	0.114461i	0.998361	+	0.0572307i	$0.0182270\pi$
$593$	2.78732i	0.114461i	−0.998361	+	0.0572307i	$0.981773\pi$
$594$	0	0
$595$	− 2.67026i	− 0.109470i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	26.5145	1.08335	0.541677	−	0.840587i	$-0.317790\pi$
$599$	26.5145	1.08335	0.541677	+	0.840587i	$0.317790\pi$
$600$	0	0
$601$	−21.5198	−0.877811	−0.438906	−	0.898533i	$-0.644634\pi$
$601$	−21.5198	−0.877811	−0.438906	+	0.898533i	$0.644634\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 0.449579i	− 0.0182780i
$606$	0	0
$607$	12.4052i	0.503511i	0.967791	+	0.251756i	$0.0810079\pi$
$607$	12.4052i	0.503511i	−0.967791	+	0.251756i	$0.918992\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.30601	−0.0528354
$612$	0	0
$613$	45.8840	1.85324	0.926619	−	0.376002i	$-0.122701\pi$
$613$	45.8840	1.85324	0.926619	+	0.376002i	$0.122701\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 37.1405i	− 1.49522i	−0.664139	−	0.747610i	$-0.731202\pi$
$617$	− 37.1405i	− 1.49522i	0.664139	−	0.747610i	$-0.268798\pi$
$618$	0	0
$619$	− 17.5338i	− 0.704743i	−0.935860	−	0.352371i	$-0.885375\pi$
$619$	− 17.5338i	− 0.704743i	0.935860	−	0.352371i	$-0.114625\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−10.1525	−0.406753
$624$	0	0
$625$	21.0526	0.842102
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 12.2757i	− 0.489466i
$630$	0	0
$631$	15.2706i	0.607911i	0.952686	+	0.303956i	$0.0983075\pi$
$631$	15.2706i	0.607911i	−0.952686	+	0.303956i	$0.901693\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−9.01054	−0.357572
$636$	0	0
$637$	2.64764	0.104903
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.3657i	0.488417i	0.969723	+	0.244208i	$0.0785281\pi$
$641$	12.3657i	0.488417i	−0.969723	+	0.244208i	$0.921472\pi$
$642$	0	0
$643$	− 12.9800i	− 0.511881i	−0.966693	−	0.255940i	$-0.917615\pi$
$643$	− 12.9800i	− 0.511881i	0.966693	−	0.255940i	$-0.0823851\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.183885	0.00722926	0.00361463	−	0.999993i	$-0.498849\pi$
$647$	0.183885	0.00722926	0.00361463	+	0.999993i	$0.498849\pi$
$648$	0	0
$649$	−40.4969	−1.58964
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 35.9300i	− 1.40605i	−0.711165	−	0.703025i	$-0.751832\pi$
$653$	− 35.9300i	− 1.40605i	0.711165	−	0.703025i	$-0.248168\pi$
$654$	0	0
$655$	5.63764i	0.220281i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	20.0211	0.779913	0.389956	−	0.920833i	$-0.372490\pi$
$659$	20.0211	0.779913	0.389956	+	0.920833i	$0.372490\pi$
$660$	0	0
$661$	−33.6434	−1.30858	−0.654288	−	0.756245i	$-0.727032\pi$
$661$	−33.6434	−1.30858	−0.654288	+	0.756245i	$0.727032\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.26197i	0.165272i
$666$	0	0
$667$	− 22.4099i	− 0.867715i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	10.8295	0.418070
$672$	0	0
$673$	24.7523	0.954129	0.477065	−	0.878868i	$-0.341701\pi$
$673$	24.7523	0.954129	0.477065	+	0.878868i	$0.341701\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16.2599i	0.624920i	0.949931	+	0.312460i	$0.101153\pi$
$677$	16.2599i	0.624920i	−0.949931	+	0.312460i	$0.898847\pi$
$678$	0	0
$679$	− 7.23349i	− 0.277596i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−8.52039	−0.326024	−0.163012	−	0.986624i	$-0.552121\pi$
$683$	−8.52039	−0.326024	−0.163012	+	0.986624i	$0.552121\pi$
$684$	0	0
$685$	−8.11174	−0.309934
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.42498i	0.0923845i
$690$	0	0
$691$	− 0.354195i	− 0.0134742i	−0.999977	−	0.00673710i	$-0.997855\pi$
$691$	− 0.354195i	− 0.0134742i	0.999977	−	0.00673710i	$-0.00214450\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.43326	−0.281960
$696$	0	0
$697$	−2.44486	−0.0926058
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 13.0877i	− 0.494315i	−0.968975	−	0.247157i	$-0.920504\pi$
$701$	− 13.0877i	− 0.494315i	0.968975	−	0.247157i	$-0.0794964\pi$
$702$	0	0
$703$	19.5932i	0.738971i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−17.6975	−0.665585
$708$	0	0
$709$	−12.2340	−0.459457	−0.229729	−	0.973255i	$-0.573784\pi$
$709$	−12.2340	−0.459457	−0.229729	+	0.973255i	$0.573784\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.40321i	0.0900011i
$714$	0	0
$715$	4.36236i	0.163143i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−8.54681	−0.318742	−0.159371	−	0.987219i	$-0.550947\pi$
$719$	−8.54681	−0.318742	−0.159371	+	0.987219i	$0.550947\pi$
$720$	0	0
$721$	4.85380	0.180765
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 24.2037i	− 0.898904i
$726$	0	0
$727$	45.1151i	1.67323i	0.547793	+	0.836614i	$0.315468\pi$
$727$	45.1151i	1.67323i	−0.547793	+	0.836614i	$0.684532\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.21175	0.155777
$732$	0	0
$733$	−12.1541	−0.448921	−0.224460	−	0.974483i	$-0.572062\pi$
$733$	−12.1541	−0.448921	−0.224460	+	0.974483i	$0.572062\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.86476i	− 0.0686892i
$738$	0	0
$739$	1.57113i	0.0577949i	0.999582	+	0.0288974i	$0.00919962\pi$
$739$	1.57113i	0.0577949i	−0.999582	+	0.0288974i	$0.990800\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.7028	0.576080	0.288040	−	0.957618i	$-0.406996\pi$
$743$	15.7028	0.576080	0.288040	+	0.957618i	$0.406996\pi$
$744$	0	0
$745$	−11.7694	−0.431197
$746$	0	0
$747$	0	0
$748$	0	0
$749$	18.3360i	0.669984i
$750$	0	0
$751$	− 38.5392i	− 1.40632i	−0.711033	−	0.703159i	$-0.751772\pi$
$751$	− 38.5392i	− 1.40632i	0.711033	−	0.703159i	$-0.248228\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.62167	0.0954122
$756$	0	0
$757$	2.73416	0.0993747	0.0496873	−	0.998765i	$-0.484178\pi$
$757$	2.73416	0.0993747	0.0496873	+	0.998765i	$0.484178\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 9.21155i	− 0.333919i	−0.985964	−	0.166959i	$-0.946605\pi$
$761$	− 9.21155i	− 0.333919i	0.985964	−	0.166959i	$-0.0533949\pi$
$762$	0	0
$763$	9.35524i	0.338682i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−33.6856	−1.21632
$768$	0	0
$769$	43.5687	1.57113	0.785564	−	0.618780i	$-0.212373\pi$
$769$	43.5687	1.57113	0.785564	+	0.618780i	$0.212373\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 38.5831i	− 1.38774i	−0.720101	−	0.693869i	$-0.755905\pi$
$773$	− 38.5831i	− 1.38774i	0.720101	−	0.693869i	$-0.244095\pi$
$774$	0	0
$775$	2.59558i	0.0932361i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.90222	0.139811
$780$	0	0
$781$	−22.1225	−0.791605
$782$	0	0
$783$	0	0
$784$	0	0
$785$	9.06225i	0.323446i
$786$	0	0
$787$	11.1846i	0.398687i	0.979930	+	0.199344i	$0.0638810\pi$
$787$	11.1846i	0.398687i	−0.979930	+	0.199344i	$0.936119\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.53748	0.196890
$792$	0	0
$793$	9.00810	0.319887
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 23.1800i	− 0.821079i	−0.911843	−	0.410539i	$-0.865340\pi$
$797$	− 23.1800i	− 0.821079i	0.911843	−	0.410539i	$-0.134660\pi$
$798$	0	0
$799$	2.54457i	0.0900203i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−7.69340	−0.271494
$804$	0	0
$805$	−2.26795	−0.0799347
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 23.6450i	− 0.831313i	−0.909522	−	0.415657i	$-0.863552\pi$
$809$	− 23.6450i	− 0.831313i	0.909522	−	0.415657i	$-0.136448\pi$
$810$	0	0
$811$	− 24.5971i	− 0.863721i	−0.901940	−	0.431860i	$-0.857857\pi$
$811$	− 24.5971i	− 0.863721i	0.901940	−	0.431860i	$-0.142143\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.04421	0.176691
$816$	0	0
$817$	−6.72232	−0.235184
$818$	0	0
$819$	0	0
$820$	0	0
$821$	51.5879i	1.80043i	0.435446	+	0.900215i	$0.356591\pi$
$821$	51.5879i	1.80043i	−0.435446	+	0.900215i	$0.643409\pi$
$822$	0	0
$823$	15.0575i	0.524873i	0.964949	+	0.262437i	$0.0845260\pi$
$823$	15.0575i	0.524873i	−0.964949	+	0.262437i	$0.915474\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.65940	0.127250	0.0636249	−	0.997974i	$-0.479734\pi$
$827$	3.65940	0.127250	0.0636249	+	0.997974i	$0.479734\pi$
$828$	0	0
$829$	23.9681	0.832448	0.416224	−	0.909262i	$-0.363353\pi$
$829$	23.9681	0.832448	0.416224	+	0.909262i	$0.363353\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 5.15854i	− 0.178733i
$834$	0	0
$835$	− 10.0547i	− 0.347956i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−26.5883	−0.917931	−0.458965	−	0.888454i	$-0.651780\pi$
$839$	−26.5883	−0.917931	−0.458965	+	0.888454i	$0.651780\pi$
$840$	0	0
$841$	2.83831	0.0978726
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 3.10065i	− 0.106666i
$846$	0	0
$847$	− 0.868519i	− 0.0298427i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−10.4262	−0.357407
$852$	0	0
$853$	−28.0181	−0.959321	−0.479661	−	0.877454i	$-0.659240\pi$
$853$	−28.0181	−0.959321	−0.479661	+	0.877454i	$0.659240\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.9852i	0.580205i	0.956996	+	0.290102i	$0.0936893\pi$
$857$	16.9852i	0.580205i	−0.956996	+	0.290102i	$0.906311\pi$
$858$	0	0
$859$	19.9576i	0.680946i	0.940254	+	0.340473i	$0.110587\pi$
$859$	19.9576i	0.680946i	−0.940254	+	0.340473i	$0.889413\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−42.3884	−1.44292	−0.721459	−	0.692457i	$-0.756528\pi$
$863$	−42.3884	−1.44292	−0.721459	+	0.692457i	$0.756528\pi$
$864$	0	0
$865$	−8.93821	−0.303908
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 40.0381i	− 1.35820i
$870$	0	0
$871$	− 1.55112i	− 0.0525577i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−5.03768	−0.170305
$876$	0	0
$877$	−24.4993	−0.827284	−0.413642	−	0.910440i	$-0.635743\pi$
$877$	−24.4993	−0.827284	−0.413642	+	0.910440i	$0.635743\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	24.5364i	0.826652i	0.910583	+	0.413326i	$0.135633\pi$
$881$	24.5364i	0.826652i	−0.910583	+	0.413326i	$0.864367\pi$
$882$	0	0
$883$	− 17.3082i	− 0.582466i	−0.956652	−	0.291233i	$-0.905934\pi$
$883$	− 17.3082i	− 0.582466i	0.956652	−	0.291233i	$-0.0940655\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−40.8260	−1.37080	−0.685401	−	0.728166i	$-0.740373\pi$
$887$	−40.8260	−1.37080	−0.685401	+	0.728166i	$0.740373\pi$
$888$	0	0
$889$	−17.4070	−0.583813
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 4.06135i	− 0.135908i
$894$	0	0
$895$	− 5.84196i	− 0.195275i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.80555	0.0935704
$900$	0	0
$901$	4.72473	0.157403
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 0.725889i	− 0.0241294i
$906$	0	0
$907$	− 28.9282i	− 0.960545i	−0.877119	−	0.480273i	$-0.840538\pi$
$907$	− 28.9282i	− 0.960545i	0.877119	−	0.480273i	$-0.159462\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−9.90248	−0.328084	−0.164042	−	0.986453i	$-0.552453\pi$
$911$	−9.90248	−0.328084	−0.164042	+	0.986453i	$0.552453\pi$
$912$	0	0
$913$	0.687117	0.0227402
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.8911i	0.359655i
$918$	0	0
$919$	− 24.2910i	− 0.801287i	−0.916234	−	0.400644i	$-0.868787\pi$
$919$	− 24.2910i	− 0.801287i	0.916234	−	0.400644i	$-0.131213\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−18.4016	−0.605698
$924$	0	0
$925$	−11.2608	−0.370254
$926$	0	0
$927$	0	0
$928$	0	0
$929$	30.4046i	0.997544i	0.866733	+	0.498772i	$0.166216\pi$
$929$	30.4046i	0.997544i	−0.866733	+	0.498772i	$0.833784\pi$
$930$	0	0
$931$	8.23349i	0.269842i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8.49943	0.277961
$936$	0	0
$937$	−3.54014	−0.115651	−0.0578257	−	0.998327i	$-0.518417\pi$
$937$	−3.54014	−0.115651	−0.0578257	+	0.998327i	$0.518417\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	26.1731i	0.853220i	0.904436	+	0.426610i	$0.140292\pi$
$941$	26.1731i	0.853220i	−0.904436	+	0.426610i	$0.859708\pi$
$942$	0	0
$943$	2.07651i	0.0676206i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.3342	−0.920739	−0.460369	−	0.887727i	$-0.652283\pi$
$947$	−28.3342	−0.920739	−0.460369	+	0.887727i	$0.652283\pi$
$948$	0	0
$949$	−6.39943	−0.207734
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 55.0723i	− 1.78397i	−0.452069	−	0.891983i	$-0.649314\pi$
$953$	− 55.0723i	− 1.78397i	0.452069	−	0.891983i	$-0.350686\pi$
$954$	0	0
$955$	3.10914i	0.100609i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−15.6707	−0.506033
$960$	0	0
$961$	30.6991	0.990295
$962$	0	0
$963$	0	0
$964$	0	0
$965$	8.55592i	0.275425i
$966$	0	0
$967$	− 25.0733i	− 0.806304i	−0.915133	−	0.403152i	$-0.867915\pi$
$967$	− 25.0733i	− 0.806304i	0.915133	−	0.403152i	$-0.132085\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−48.8089	−1.56635	−0.783176	−	0.621800i	$-0.786402\pi$
$971$	−48.8089	−1.56635	−0.783176	+	0.621800i	$0.786402\pi$
$972$	0	0
$973$	−14.3600	−0.460359
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 61.2645i	− 1.96002i	−0.198940	−	0.980012i	$-0.563750\pi$
$977$	− 61.2645i	− 1.96002i	0.198940	−	0.980012i	$-0.436250\pi$
$978$	0	0
$979$	− 32.3155i	− 1.03281i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.74007	0.0873948	0.0436974	−	0.999045i	$-0.486086\pi$
$983$	2.74007	0.0873948	0.0436974	+	0.999045i	$0.486086\pi$
$984$	0	0
$985$	3.02656	0.0964343
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 3.57719i	− 0.113748i
$990$	0	0
$991$	8.02994i	0.255079i	0.991833	+	0.127540i	$0.0407080\pi$
$991$	8.02994i	0.255079i	−0.991833	+	0.127540i	$0.959292\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2.16556	0.0686529
$996$	0	0
$997$	−16.4217	−0.520082	−0.260041	−	0.965598i	$-0.583736\pi$
$997$	−16.4217	−0.520082	−0.260041	+	0.965598i	$0.583736\pi$
$998$	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.2591.11	yes	16
3.2	odd	2	inner	6048.2.h.h.2591.8	yes	16
4.3	odd	2	inner	6048.2.h.h.2591.10	yes	16
12.11	even	2	inner	6048.2.h.h.2591.5	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.h.h.2591.5	✓	16	12.11	even	2	inner
6048.2.h.h.2591.8	yes	16	3.2	odd	2	inner
6048.2.h.h.2591.10	yes	16	4.3	odd	2	inner
6048.2.h.h.2591.11	yes	16	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+0.517638i q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q+0.517638i q^{5} +1.00000i q^{7} -3.18300 q^{11} -2.64764 q^{13} +5.15854i q^{17} -8.23349i q^{19} +4.38134 q^{23} +4.73205 q^{25} -5.11485i q^{29} +0.548511i q^{31} -0.517638 q^{35} -2.37969 q^{37} +0.473945i q^{41} -0.816460i q^{43} +0.493272 q^{47} -1.00000 q^{49} -0.915903i q^{53} -1.64764i q^{55} +12.7229 q^{59} -3.40231 q^{61} -1.37052i q^{65} +0.585849i q^{67} +6.95020 q^{71} +2.41703 q^{73} -3.18300i q^{77} +12.5787i q^{79} -0.215871 q^{83} -2.67026 q^{85} +10.1525i q^{89} -2.64764i q^{91} +4.26197 q^{95} -7.23349 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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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