LMFDB - Embedded newform 6048.2.h.e.2591.2

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:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.2
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	6048.2591
Dual form			6048.2.h.e.2591.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{5} -1.00000i q^{7} +O(q^{10})$	$q-2.82843i q^{5} -1.00000i q^{7} +0.635674 q^{11} +0.550510 q^{13} +3.14626i q^{17} +4.00000i q^{19} +3.14626 q^{23} -3.00000 q^{25} -5.19615i q^{29} -10.7980i q^{31} -2.82843 q^{35} -8.89898 q^{37} -3.46410i q^{41} -0.550510i q^{43} +1.27135 q^{47} -1.00000 q^{49} -2.36773i q^{53} -1.79796i q^{55} +11.4887 q^{59} +3.10102 q^{61} -1.55708i q^{65} -1.44949i q^{67} +5.97469 q^{71} -4.00000 q^{73} -0.635674i q^{77} -10.0000i q^{79} -3.46410 q^{83} +8.89898 q^{85} -13.5386i q^{89} -0.550510i q^{91} +11.3137 q^{95} +3.79796 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 24 q^{13} - 24 q^{25} - 32 q^{37} - 8 q^{49} + 64 q^{61} - 32 q^{73} + 32 q^{85} - 48 q^{97}+O(q^{100}) $ 8 * q + 24 * q^13 - 24 * q^25 - 32 * q^37 - 8 * q^49 + 64 * q^61 - 32 * q^73 + 32 * q^85 - 48 * 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
$5$	− 2.82843i	− 1.26491i	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	− 2.82843i	− 1.26491i	0.774597	−	0.632456i	$-0.217953\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.635674	0.191663	0.0958315	−	0.995398i	$-0.469449\pi$
$11$	0.635674	0.191663	0.0958315	+	0.995398i	$0.469449\pi$
$12$	0	0
$13$	0.550510	0.152684	0.0763420	−	0.997082i	$-0.475676\pi$
$13$	0.550510	0.152684	0.0763420	+	0.997082i	$0.475676\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.14626i	0.763081i	0.924352	+	0.381541i	$0.124606\pi$
$17$	3.14626i	0.763081i	−0.924352	+	0.381541i	$0.875394\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.14626	0.656041	0.328021	−	0.944671i	$-0.393618\pi$
$23$	3.14626	0.656041	0.328021	+	0.944671i	$0.393618\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 5.19615i	− 0.964901i	−0.875923	−	0.482451i	$-0.839747\pi$
$29$	− 5.19615i	− 0.964901i	0.875923	−	0.482451i	$-0.160253\pi$
$30$	0	0
$31$	− 10.7980i	− 1.93937i	−0.244357	−	0.969685i	$-0.578577\pi$
$31$	− 10.7980i	− 1.93937i	0.244357	−	0.969685i	$-0.421423\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.82843	−0.478091
$36$	0	0
$37$	−8.89898	−1.46298	−0.731492	−	0.681850i	$-0.761175\pi$
$37$	−8.89898	−1.46298	−0.731492	+	0.681850i	$0.761175\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 3.46410i	− 0.541002i	−0.962720	−	0.270501i	$-0.912811\pi$
$41$	− 3.46410i	− 0.541002i	0.962720	−	0.270501i	$-0.0871893\pi$
$42$	0	0
$43$	− 0.550510i	− 0.0839520i	−0.999119	−	0.0419760i	$-0.986635\pi$
$43$	− 0.550510i	− 0.0839520i	0.999119	−	0.0419760i	$-0.0133653\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.27135	0.185445	0.0927227	−	0.995692i	$-0.470443\pi$
$47$	1.27135	0.185445	0.0927227	+	0.995692i	$0.470443\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 2.36773i	− 0.325232i	−0.986689	−	0.162616i	$-0.948007\pi$
$53$	− 2.36773i	− 0.325232i	0.986689	−	0.162616i	$-0.0519932\pi$
$54$	0	0
$55$	− 1.79796i	− 0.242437i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.4887	1.49570	0.747849	−	0.663868i	$-0.231086\pi$
$59$	11.4887	1.49570	0.747849	+	0.663868i	$0.231086\pi$
$60$	0	0
$61$	3.10102	0.397045	0.198522	−	0.980096i	$-0.436386\pi$
$61$	3.10102	0.397045	0.198522	+	0.980096i	$0.436386\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 1.55708i	− 0.193132i
$66$	0	0
$67$	− 1.44949i	− 0.177083i	−0.996072	−	0.0885417i	$-0.971779\pi$
$67$	− 1.44949i	− 0.177083i	0.996072	−	0.0885417i	$-0.0282206\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.97469	0.709065	0.354533	−	0.935044i	$-0.384640\pi$
$71$	5.97469	0.709065	0.354533	+	0.935044i	$0.384640\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.635674i	− 0.0724418i
$78$	0	0
$79$	− 10.0000i	− 1.12509i	−0.826767	−	0.562544i	$-0.809823\pi$
$79$	− 10.0000i	− 1.12509i	0.826767	−	0.562544i	$-0.190177\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.46410	−0.380235	−0.190117	−	0.981761i	$-0.560887\pi$
$83$	−3.46410	−0.380235	−0.190117	+	0.981761i	$0.560887\pi$
$84$	0	0
$85$	8.89898	0.965230
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 13.5386i	− 1.43509i	−0.696514	−	0.717543i	$-0.745267\pi$
$89$	− 13.5386i	− 1.43509i	0.696514	−	0.717543i	$-0.254733\pi$
$90$	0	0
$91$	− 0.550510i	− 0.0577092i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	11.3137	1.16076
$96$	0	0
$97$	3.79796	0.385624	0.192812	−	0.981236i	$-0.438239\pi$
$97$	3.79796	0.385624	0.192812	+	0.981236i	$0.438239\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.65685i	0.562878i	0.959579	+	0.281439i	$0.0908117\pi$
$101$	5.65685i	0.562878i	−0.959579	+	0.281439i	$0.909188\pi$
$102$	0	0
$103$	− 5.89898i	− 0.581244i	−0.956838	−	0.290622i	$-0.906138\pi$
$103$	− 5.89898i	− 0.581244i	0.956838	−	0.290622i	$-0.0938622\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.38551	0.423963	0.211981	−	0.977274i	$-0.432008\pi$
$107$	4.38551	0.423963	0.211981	+	0.977274i	$0.432008\pi$
$108$	0	0
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.48528i	0.798228i	0.916901	+	0.399114i	$0.130682\pi$
$113$	8.48528i	0.798228i	−0.916901	+	0.399114i	$0.869318\pi$
$114$	0	0
$115$	− 8.89898i	− 0.829834i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.14626	0.288418
$120$	0	0
$121$	−10.5959	−0.963265
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 5.65685i	− 0.505964i
$126$	0	0
$127$	1.10102i	0.0976998i	0.998806	+	0.0488499i	$0.0155556\pi$
$127$	1.10102i	0.0976998i	−0.998806	+	0.0488499i	$0.984444\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.02458	−0.701111	−0.350555	−	0.936542i	$-0.614007\pi$
$131$	−8.02458	−0.701111	−0.350555	+	0.936542i	$0.614007\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 8.19955i	− 0.700535i	−0.936650	−	0.350268i	$-0.886091\pi$
$137$	− 8.19955i	− 0.700535i	0.936650	−	0.350268i	$-0.113909\pi$
$138$	0	0
$139$	3.79796i	0.322139i	0.986943	+	0.161069i	$0.0514943\pi$
$139$	3.79796i	0.322139i	−0.986943	+	0.161069i	$0.948506\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.349945	0.0292639
$144$	0	0
$145$	−14.6969	−1.22051
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.36773i	0.193972i	0.995286	+	0.0969858i	$0.0309201\pi$
$149$	2.36773i	0.193972i	−0.995286	+	0.0969858i	$0.969080\pi$
$150$	0	0
$151$	− 13.7980i	− 1.12286i	−0.827524	−	0.561431i	$-0.810251\pi$
$151$	− 13.7980i	− 1.12286i	0.827524	−	0.561431i	$-0.189749\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−30.5412	−2.45313
$156$	0	0
$157$	−18.1464	−1.44824	−0.724121	−	0.689673i	$-0.757754\pi$
$157$	−18.1464	−1.44824	−0.724121	+	0.689673i	$0.757754\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 3.14626i	− 0.247960i
$162$	0	0
$163$	7.44949i	0.583489i	0.956496	+	0.291745i	$0.0942357\pi$
$163$	7.44949i	0.583489i	−0.956496	+	0.291745i	$0.905764\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.73545	−0.366440	−0.183220	−	0.983072i	$-0.558652\pi$
$167$	−4.73545	−0.366440	−0.183220	+	0.983072i	$0.558652\pi$
$168$	0	0
$169$	−12.6969	−0.976688
$170$	0	0
$171$	0	0
$172$	0	0
$173$	21.7060i	1.65028i	0.564930	+	0.825139i	$0.308903\pi$
$173$	21.7060i	1.65028i	−0.564930	+	0.825139i	$0.691097\pi$
$174$	0	0
$175$	3.00000i	0.226779i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−16.6848	−1.24708	−0.623542	−	0.781790i	$-0.714307\pi$
$179$	−16.6848	−1.24708	−0.623542	+	0.781790i	$0.714307\pi$
$180$	0	0
$181$	−24.3485	−1.80981	−0.904904	−	0.425616i	$-0.860057\pi$
$181$	−24.3485	−1.80981	−0.904904	+	0.425616i	$0.860057\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	25.1701i	1.85054i
$186$	0	0
$187$	2.00000i	0.146254i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.6848	1.20727	0.603636	−	0.797260i	$-0.293718\pi$
$191$	16.6848	1.20727	0.603636	+	0.797260i	$0.293718\pi$
$192$	0	0
$193$	5.89898	0.424618	0.212309	−	0.977203i	$-0.431902\pi$
$193$	5.89898	0.424618	0.212309	+	0.977203i	$0.431902\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 16.9706i	− 1.20910i	−0.796566	−	0.604551i	$-0.793352\pi$
$197$	− 16.9706i	− 1.20910i	0.796566	−	0.604551i	$-0.206648\pi$
$198$	0	0
$199$	1.20204i	0.0852104i	0.999092	+	0.0426052i	$0.0135658\pi$
$199$	1.20204i	0.0852104i	−0.999092	+	0.0426052i	$0.986434\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.19615	−0.364698
$204$	0	0
$205$	−9.79796	−0.684319
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.54270i	0.175882i
$210$	0	0
$211$	− 20.1464i	− 1.38694i	−0.720487	−	0.693469i	$-0.756082\pi$
$211$	− 20.1464i	− 1.38694i	0.720487	−	0.693469i	$-0.243918\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.55708	−0.106192
$216$	0	0
$217$	−10.7980	−0.733013
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.73205i	0.116510i
$222$	0	0
$223$	− 9.79796i	− 0.656120i	−0.944657	−	0.328060i	$-0.893605\pi$
$223$	− 9.79796i	− 0.656120i	0.944657	−	0.328060i	$-0.106395\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.8530	0.720339	0.360170	−	0.932887i	$-0.382719\pi$
$227$	10.8530	0.720339	0.360170	+	0.932887i	$0.382719\pi$
$228$	0	0
$229$	−12.8990	−0.852389	−0.426194	−	0.904632i	$-0.640146\pi$
$229$	−12.8990	−0.852389	−0.426194	+	0.904632i	$0.640146\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.4203i	1.40329i	0.712527	+	0.701645i	$0.247551\pi$
$233$	21.4203i	1.40329i	−0.712527	+	0.701645i	$0.752449\pi$
$234$	0	0
$235$	− 3.59592i	− 0.234572i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.82843	0.182956	0.0914779	−	0.995807i	$-0.470841\pi$
$239$	2.82843	0.182956	0.0914779	+	0.995807i	$0.470841\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.82843i	0.180702i
$246$	0	0
$247$	2.20204i	0.140113i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−16.0492	−1.01301	−0.506507	−	0.862236i	$-0.669064\pi$
$251$	−16.0492	−1.01301	−0.506507	+	0.862236i	$0.669064\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.1633i	1.19537i	0.801729	+	0.597687i	$0.203914\pi$
$257$	19.1633i	1.19537i	−0.801729	+	0.597687i	$0.796086\pi$
$258$	0	0
$259$	8.89898i	0.552956i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−20.1168	−1.24046	−0.620228	−	0.784421i	$-0.712960\pi$
$263$	−20.1168	−1.24046	−0.620228	+	0.784421i	$0.712960\pi$
$264$	0	0
$265$	−6.69694	−0.411390
$266$	0	0
$267$	0	0
$268$	0	0
$269$	20.1489i	1.22850i	0.789111	+	0.614251i	$0.210542\pi$
$269$	20.1489i	1.22850i	−0.789111	+	0.614251i	$0.789458\pi$
$270$	0	0
$271$	− 20.1010i	− 1.22105i	−0.791997	−	0.610525i	$-0.790958\pi$
$271$	− 20.1010i	− 1.22105i	0.791997	−	0.610525i	$-0.209042\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.90702	−0.114998
$276$	0	0
$277$	21.1010	1.26784	0.633919	−	0.773400i	$-0.281445\pi$
$277$	21.1010	1.26784	0.633919	+	0.773400i	$0.281445\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 27.3629i	− 1.63233i	−0.577818	−	0.816166i	$-0.696096\pi$
$281$	− 27.3629i	− 1.63233i	0.577818	−	0.816166i	$-0.303904\pi$
$282$	0	0
$283$	− 13.7980i	− 0.820204i	−0.912040	−	0.410102i	$-0.865493\pi$
$283$	− 13.7980i	− 0.820204i	0.912040	−	0.410102i	$-0.134507\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.46410	−0.204479
$288$	0	0
$289$	7.10102	0.417707
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 7.56388i	− 0.441886i	−0.975287	−	0.220943i	$-0.929086\pi$
$293$	− 7.56388i	− 0.441886i	0.975287	−	0.220943i	$-0.0709136\pi$
$294$	0	0
$295$	− 32.4949i	− 1.89193i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.73205	0.100167
$300$	0	0
$301$	−0.550510	−0.0317309
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 8.77101i	− 0.502227i
$306$	0	0
$307$	− 2.69694i	− 0.153922i	−0.997034	−	0.0769612i	$-0.975478\pi$
$307$	− 2.69694i	− 0.153922i	0.997034	−	0.0769612i	$-0.0245218\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	27.0771	1.53540	0.767702	−	0.640807i	$-0.221400\pi$
$311$	27.0771	1.53540	0.767702	+	0.640807i	$0.221400\pi$
$312$	0	0
$313$	−23.5959	−1.33372	−0.666860	−	0.745183i	$-0.732362\pi$
$313$	−23.5959	−1.33372	−0.666860	+	0.745183i	$0.732362\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.27135i	0.0714061i	0.999362	+	0.0357030i	$0.0113670\pi$
$317$	1.27135i	0.0714061i	−0.999362	+	0.0357030i	$0.988633\pi$
$318$	0	0
$319$	− 3.30306i	− 0.184936i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−12.5851	−0.700251
$324$	0	0
$325$	−1.65153	−0.0916104
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 1.27135i	− 0.0700917i
$330$	0	0
$331$	− 20.3485i	− 1.11845i	−0.829015	−	0.559227i	$-0.811098\pi$
$331$	− 20.3485i	− 1.11845i	0.829015	−	0.559227i	$-0.188902\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.09978	−0.223995
$336$	0	0
$337$	−10.1010	−0.550238	−0.275119	−	0.961410i	$-0.588717\pi$
$337$	−10.1010	−0.550238	−0.275119	+	0.961410i	$0.588717\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 6.86399i	− 0.371706i
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−14.4921	−0.777976	−0.388988	−	0.921243i	$-0.627175\pi$
$347$	−14.4921	−0.777976	−0.388988	+	0.921243i	$0.627175\pi$
$348$	0	0
$349$	−34.8434	−1.86512	−0.932561	−	0.361012i	$-0.882431\pi$
$349$	−34.8434	−1.86512	−0.932561	+	0.361012i	$0.882431\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 15.3814i	− 0.818668i	−0.912385	−	0.409334i	$-0.865761\pi$
$353$	− 15.3814i	− 0.818668i	0.912385	−	0.409334i	$-0.134239\pi$
$354$	0	0
$355$	− 16.8990i	− 0.896905i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.78194	0.199603	0.0998015	−	0.995007i	$-0.468179\pi$
$359$	3.78194	0.199603	0.0998015	+	0.995007i	$0.468179\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	11.3137i	0.592187i
$366$	0	0
$367$	15.2020i	0.793540i	0.917918	+	0.396770i	$0.129869\pi$
$367$	15.2020i	0.793540i	−0.917918	+	0.396770i	$0.870131\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.36773	−0.122926
$372$	0	0
$373$	36.2929	1.87917	0.939586	−	0.342312i	$-0.111210\pi$
$373$	36.2929	1.87917	0.939586	+	0.342312i	$0.111210\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.86054i	− 0.147325i
$378$	0	0
$379$	− 10.0000i	− 0.513665i	−0.966456	−	0.256833i	$-0.917321\pi$
$379$	− 10.0000i	− 0.513665i	0.966456	−	0.256833i	$-0.0826790\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	38.6766	1.97628	0.988140	−	0.153553i	$-0.0490717\pi$
$383$	38.6766	1.97628	0.988140	+	0.153553i	$0.0490717\pi$
$384$	0	0
$385$	−1.79796	−0.0916325
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8.77101i	0.444708i	0.974966	+	0.222354i	$0.0713741\pi$
$389$	8.77101i	0.444708i	−0.974966	+	0.222354i	$0.928626\pi$
$390$	0	0
$391$	9.89898i	0.500613i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−28.2843	−1.42314
$396$	0	0
$397$	−24.8990	−1.24964	−0.624822	−	0.780767i	$-0.714828\pi$
$397$	−24.8990	−1.24964	−0.624822	+	0.780767i	$0.714828\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	22.9774i	1.14743i	0.819053	+	0.573717i	$0.194499\pi$
$401$	22.9774i	1.14743i	−0.819053	+	0.573717i	$0.805501\pi$
$402$	0	0
$403$	− 5.94439i	− 0.296111i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.65685	−0.280400
$408$	0	0
$409$	0.696938	0.0344614	0.0172307	−	0.999852i	$-0.494515\pi$
$409$	0.696938	0.0344614	0.0172307	+	0.999852i	$0.494515\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 11.4887i	− 0.565321i
$414$	0	0
$415$	9.79796i	0.480963i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	20.2597	0.989751	0.494875	−	0.868964i	$-0.335214\pi$
$419$	20.2597	0.989751	0.494875	+	0.868964i	$0.335214\pi$
$420$	0	0
$421$	7.59592	0.370202	0.185101	−	0.982719i	$-0.440739\pi$
$421$	7.59592	0.370202	0.185101	+	0.982719i	$0.440739\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 9.43879i	− 0.457849i
$426$	0	0
$427$	− 3.10102i	− 0.150069i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.8708	−0.619964	−0.309982	−	0.950742i	$-0.600323\pi$
$431$	−12.8708	−0.619964	−0.309982	+	0.950742i	$0.600323\pi$
$432$	0	0
$433$	19.5959	0.941720	0.470860	−	0.882208i	$-0.343944\pi$
$433$	19.5959	0.941720	0.470860	+	0.882208i	$0.343944\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12.5851i	0.602025i
$438$	0	0
$439$	37.4949i	1.78953i	0.446534	+	0.894767i	$0.352658\pi$
$439$	37.4949i	1.78953i	−0.446534	+	0.894767i	$0.647342\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−27.9985	−1.33025	−0.665125	−	0.746732i	$-0.731622\pi$
$443$	−27.9985	−1.33025	−0.665125	+	0.746732i	$0.731622\pi$
$444$	0	0
$445$	−38.2929	−1.81526
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 12.2993i	− 0.580441i	−0.956960	−	0.290221i	$-0.906271\pi$
$449$	− 12.2993i	− 0.580441i	0.956960	−	0.290221i	$-0.0937287\pi$
$450$	0	0
$451$	− 2.20204i	− 0.103690i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.55708	−0.0729969
$456$	0	0
$457$	−39.6969	−1.85694	−0.928472	−	0.371402i	$-0.878877\pi$
$457$	−39.6969	−1.85694	−0.928472	+	0.371402i	$0.878877\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 6.92820i	− 0.322679i	−0.986899	−	0.161339i	$-0.948419\pi$
$461$	− 6.92820i	− 0.322679i	0.986899	−	0.161339i	$-0.0515813\pi$
$462$	0	0
$463$	− 7.59592i	− 0.353012i	−0.984300	−	0.176506i	$-0.943520\pi$
$463$	− 7.59592i	− 0.353012i	0.984300	−	0.176506i	$-0.0564795\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−14.7778	−0.683835	−0.341918	−	0.939730i	$-0.611076\pi$
$467$	−14.7778	−0.683835	−0.341918	+	0.939730i	$0.611076\pi$
$468$	0	0
$469$	−1.44949	−0.0669312
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 0.349945i	− 0.0160905i
$474$	0	0
$475$	− 12.0000i	− 0.550598i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	33.6554	1.53775	0.768877	−	0.639397i	$-0.220816\pi$
$479$	33.6554	1.53775	0.768877	+	0.639397i	$0.220816\pi$
$480$	0	0
$481$	−4.89898	−0.223374
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 10.7423i	− 0.487780i
$486$	0	0
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3.74983	0.169227	0.0846137	−	0.996414i	$-0.473034\pi$
$491$	3.74983	0.169227	0.0846137	+	0.996414i	$0.473034\pi$
$492$	0	0
$493$	16.3485	0.736298
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 5.97469i	− 0.268002i
$498$	0	0
$499$	16.2020i	0.725303i	0.931925	+	0.362651i	$0.118128\pi$
$499$	16.2020i	0.725303i	−0.931925	+	0.362651i	$0.881872\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	22.3417	0.996167	0.498083	−	0.867129i	$-0.334037\pi$
$503$	22.3417	0.996167	0.498083	+	0.867129i	$0.334037\pi$
$504$	0	0
$505$	16.0000	0.711991
$506$	0	0
$507$	0	0
$508$	0	0
$509$	27.9985i	1.24101i	0.784201	+	0.620507i	$0.213073\pi$
$509$	27.9985i	1.24101i	−0.784201	+	0.620507i	$0.786927\pi$
$510$	0	0
$511$	4.00000i	0.176950i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−16.6848	−0.735222
$516$	0	0
$517$	0.808164	0.0355430
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 8.16744i	− 0.357822i	−0.983865	−	0.178911i	$-0.942743\pi$
$521$	− 8.16744i	− 0.357822i	0.983865	−	0.178911i	$-0.0572575\pi$
$522$	0	0
$523$	0.202041i	0.00883464i	0.999990	+	0.00441732i	$0.00140608\pi$
$523$	0.202041i	0.00883464i	−0.999990	+	0.00441732i	$0.998594\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	33.9732	1.47990
$528$	0	0
$529$	−13.1010	−0.569610
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 1.90702i	− 0.0826024i
$534$	0	0
$535$	− 12.4041i	− 0.536275i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.635674	−0.0273804
$540$	0	0
$541$	−8.49490	−0.365224	−0.182612	−	0.983185i	$-0.558455\pi$
$541$	−8.49490	−0.365224	−0.182612	+	0.983185i	$0.558455\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	22.6274i	0.969252i
$546$	0	0
$547$	− 43.3939i	− 1.85539i	−0.373341	−	0.927694i	$-0.621788\pi$
$547$	− 43.3939i	− 1.85539i	0.373341	−	0.927694i	$-0.378212\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	20.7846	0.885454
$552$	0	0
$553$	−10.0000	−0.425243
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 38.8515i	− 1.64619i	−0.567902	−	0.823096i	$-0.692245\pi$
$557$	− 38.8515i	− 1.64619i	0.567902	−	0.823096i	$-0.307755\pi$
$558$	0	0
$559$	− 0.303062i	− 0.0128181i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14.6028	0.615436	0.307718	−	0.951478i	$-0.400435\pi$
$563$	14.6028	0.615436	0.307718	+	0.951478i	$0.400435\pi$
$564$	0	0
$565$	24.0000	1.00969
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 9.12096i	− 0.382370i	−0.981554	−	0.191185i	$-0.938767\pi$
$569$	− 9.12096i	− 0.382370i	0.981554	−	0.191185i	$-0.0612331\pi$
$570$	0	0
$571$	− 13.2474i	− 0.554388i	−0.960814	−	0.277194i	$-0.910595\pi$
$571$	− 13.2474i	− 0.554388i	0.960814	−	0.277194i	$-0.0894046\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−9.43879	−0.393625
$576$	0	0
$577$	34.6969	1.44445	0.722226	−	0.691657i	$-0.243119\pi$
$577$	34.6969	1.44445	0.722226	+	0.691657i	$0.243119\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.46410i	0.143715i
$582$	0	0
$583$	− 1.50510i	− 0.0623350i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18.7026	−0.771939	−0.385970	−	0.922511i	$-0.626133\pi$
$587$	−18.7026	−0.771939	−0.385970	+	0.922511i	$0.626133\pi$
$588$	0	0
$589$	43.1918	1.77969
$590$	0	0
$591$	0	0
$592$	0	0
$593$	19.1633i	0.786943i	0.919337	+	0.393472i	$0.128726\pi$
$593$	19.1633i	0.786943i	−0.919337	+	0.393472i	$0.871274\pi$
$594$	0	0
$595$	− 8.89898i	− 0.364823i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	20.4026	0.833626	0.416813	−	0.908992i	$-0.363147\pi$
$599$	20.4026	0.833626	0.416813	+	0.908992i	$0.363147\pi$
$600$	0	0
$601$	−7.10102	−0.289657	−0.144828	−	0.989457i	$-0.546263\pi$
$601$	−7.10102	−0.289657	−0.144828	+	0.989457i	$0.546263\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	29.9698i	1.21844i
$606$	0	0
$607$	19.0000i	0.771186i	0.922669	+	0.385593i	$0.126003\pi$
$607$	19.0000i	0.771186i	−0.922669	+	0.385593i	$0.873997\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.699891	0.0283145
$612$	0	0
$613$	−20.0000	−0.807792	−0.403896	−	0.914805i	$-0.632344\pi$
$613$	−20.0000	−0.807792	−0.403896	+	0.914805i	$0.632344\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 12.2993i	− 0.495152i	−0.968868	−	0.247576i	$-0.920366\pi$
$617$	− 12.2993i	− 0.495152i	0.968868	−	0.247576i	$-0.0796340\pi$
$618$	0	0
$619$	− 9.59592i	− 0.385693i	−0.981229	−	0.192846i	$-0.938228\pi$
$619$	− 9.59592i	− 0.385693i	0.981229	−	0.192846i	$-0.0617719\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−13.5386	−0.542411
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 27.9985i	− 1.11637i
$630$	0	0
$631$	− 12.4949i	− 0.497414i	−0.968579	−	0.248707i	$-0.919994\pi$
$631$	− 12.4949i	− 0.497414i	0.968579	−	0.248707i	$-0.0800056\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.11416	0.123582
$636$	0	0
$637$	−0.550510	−0.0218120
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 14.4921i	− 0.572403i	−0.958169	−	0.286201i	$-0.907607\pi$
$641$	− 14.4921i	− 0.572403i	0.958169	−	0.286201i	$-0.0923926\pi$
$642$	0	0
$643$	0.494897i	0.0195168i	0.999952	+	0.00975842i	$0.00310625\pi$
$643$	0.494897i	0.0195168i	−0.999952	+	0.00975842i	$0.996894\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	33.6554	1.32313	0.661565	−	0.749888i	$-0.269893\pi$
$647$	33.6554	1.32313	0.661565	+	0.749888i	$0.269893\pi$
$648$	0	0
$649$	7.30306	0.286670
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 19.6882i	− 0.770460i	−0.922821	−	0.385230i	$-0.874122\pi$
$653$	− 19.6882i	− 0.770460i	0.922821	−	0.385230i	$-0.125878\pi$
$654$	0	0
$655$	22.6969i	0.886843i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	20.7846	0.809653	0.404827	−	0.914393i	$-0.367332\pi$
$659$	20.7846	0.809653	0.404827	+	0.914393i	$0.367332\pi$
$660$	0	0
$661$	3.10102	0.120616	0.0603079	−	0.998180i	$-0.480792\pi$
$661$	3.10102	0.120616	0.0603079	+	0.998180i	$0.480792\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 11.3137i	− 0.438727i
$666$	0	0
$667$	− 16.3485i	− 0.633015i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.97124	0.0760989
$672$	0	0
$673$	10.5959	0.408443	0.204221	−	0.978925i	$-0.434534\pi$
$673$	10.5959	0.408443	0.204221	+	0.978925i	$0.434534\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 34.0053i	− 1.30693i	−0.756956	−	0.653466i	$-0.773314\pi$
$677$	− 34.0053i	− 1.30693i	0.756956	−	0.653466i	$-0.226686\pi$
$678$	0	0
$679$	− 3.79796i	− 0.145752i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24.1845	0.925394	0.462697	−	0.886516i	$-0.346882\pi$
$683$	24.1845	0.925394	0.462697	+	0.886516i	$0.346882\pi$
$684$	0	0
$685$	−23.1918	−0.886115
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1.30346i	− 0.0496578i
$690$	0	0
$691$	47.7980i	1.81832i	0.416447	+	0.909160i	$0.363275\pi$
$691$	47.7980i	1.81832i	−0.416447	+	0.909160i	$0.636725\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	10.7423	0.407477
$696$	0	0
$697$	10.8990	0.412828
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 5.65685i	− 0.213656i	−0.994277	−	0.106828i	$-0.965931\pi$
$701$	− 5.65685i	− 0.213656i	0.994277	−	0.106828i	$-0.0340695\pi$
$702$	0	0
$703$	− 35.5959i	− 1.34253i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.65685	0.212748
$708$	0	0
$709$	−2.40408	−0.0902872	−0.0451436	−	0.998981i	$-0.514375\pi$
$709$	−2.40408	−0.0902872	−0.0451436	+	0.998981i	$0.514375\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 33.9732i	− 1.27231i
$714$	0	0
$715$	− 0.989795i	− 0.0370162i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−23.2631	−0.867567	−0.433783	−	0.901017i	$-0.642822\pi$
$719$	−23.2631	−0.867567	−0.433783	+	0.901017i	$0.642822\pi$
$720$	0	0
$721$	−5.89898	−0.219689
$722$	0	0
$723$	0	0
$724$	0	0
$725$	15.5885i	0.578941i
$726$	0	0
$727$	39.4949i	1.46478i	0.680883	+	0.732392i	$0.261596\pi$
$727$	39.4949i	1.46478i	−0.680883	+	0.732392i	$0.738404\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.73205	0.0640622
$732$	0	0
$733$	−14.3485	−0.529973	−0.264986	−	0.964252i	$-0.585367\pi$
$733$	−14.3485	−0.529973	−0.264986	+	0.964252i	$0.585367\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 0.921404i	− 0.0339403i
$738$	0	0
$739$	− 24.2020i	− 0.890286i	−0.895459	−	0.445143i	$-0.853153\pi$
$739$	− 24.2020i	− 0.890286i	0.895459	−	0.445143i	$-0.146847\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−9.08885	−0.333437	−0.166719	−	0.986004i	$-0.553317\pi$
$743$	−9.08885	−0.333437	−0.166719	+	0.986004i	$0.553317\pi$
$744$	0	0
$745$	6.69694	0.245357
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 4.38551i	− 0.160243i
$750$	0	0
$751$	35.1010i	1.28085i	0.768019	+	0.640427i	$0.221243\pi$
$751$	35.1010i	1.28085i	−0.768019	+	0.640427i	$0.778757\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−39.0265	−1.42032
$756$	0	0
$757$	45.1918	1.64253	0.821263	−	0.570550i	$-0.193270\pi$
$757$	45.1918	1.64253	0.821263	+	0.570550i	$0.193270\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 16.9385i	− 0.614019i	−0.951707	−	0.307009i	$-0.900672\pi$
$761$	− 16.9385i	− 0.614019i	0.951707	−	0.307009i	$-0.0993283\pi$
$762$	0	0
$763$	8.00000i	0.289619i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.32464	0.228369
$768$	0	0
$769$	4.89898	0.176662	0.0883309	−	0.996091i	$-0.471847\pi$
$769$	4.89898	0.176662	0.0883309	+	0.996091i	$0.471847\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 3.17837i	− 0.114318i	−0.998365	−	0.0571591i	$-0.981796\pi$
$773$	− 3.17837i	− 0.114318i	0.998365	−	0.0571591i	$-0.0182042\pi$
$774$	0	0
$775$	32.3939i	1.16362i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	13.8564	0.496457
$780$	0	0
$781$	3.79796	0.135902
$782$	0	0
$783$	0	0
$784$	0	0
$785$	51.3258i	1.83190i
$786$	0	0
$787$	17.5959i	0.627227i	0.949551	+	0.313613i	$0.101540\pi$
$787$	17.5959i	0.627227i	−0.949551	+	0.313613i	$0.898460\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8.48528	0.301702
$792$	0	0
$793$	1.70714	0.0606224
$794$	0	0
$795$	0	0
$796$	0	0
$797$	25.4558i	0.901692i	0.892602	+	0.450846i	$0.148878\pi$
$797$	25.4558i	0.901692i	−0.892602	+	0.450846i	$0.851122\pi$
$798$	0	0
$799$	4.00000i	0.141510i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2.54270	−0.0897299
$804$	0	0
$805$	−8.89898	−0.313648
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 34.5768i	− 1.21566i	−0.794069	−	0.607828i	$-0.792041\pi$
$809$	− 34.5768i	− 1.21566i	0.794069	−	0.607828i	$-0.207959\pi$
$810$	0	0
$811$	24.4949i	0.860132i	0.902797	+	0.430066i	$0.141510\pi$
$811$	24.4949i	0.860132i	−0.902797	+	0.430066i	$0.858490\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	21.0703	0.738062
$816$	0	0
$817$	2.20204	0.0770397
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 42.6656i	− 1.48904i	−0.667600	−	0.744520i	$-0.732679\pi$
$821$	− 42.6656i	− 1.48904i	0.667600	−	0.744520i	$-0.267321\pi$
$822$	0	0
$823$	− 30.8990i	− 1.07707i	−0.842603	−	0.538535i	$-0.818978\pi$
$823$	− 30.8990i	− 1.07707i	0.842603	−	0.538535i	$-0.181022\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.81405	−0.132627	−0.0663137	−	0.997799i	$-0.521124\pi$
$827$	−3.81405	−0.132627	−0.0663137	+	0.997799i	$0.521124\pi$
$828$	0	0
$829$	−17.3031	−0.600960	−0.300480	−	0.953788i	$-0.597147\pi$
$829$	−17.3031	−0.600960	−0.300480	+	0.953788i	$0.597147\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 3.14626i	− 0.109012i
$834$	0	0
$835$	13.3939i	0.463514i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.46410	0.119594	0.0597970	−	0.998211i	$-0.480955\pi$
$839$	3.46410	0.119594	0.0597970	+	0.998211i	$0.480955\pi$
$840$	0	0
$841$	2.00000	0.0689655
$842$	0	0
$843$	0	0
$844$	0	0
$845$	35.9124i	1.23542i
$846$	0	0
$847$	10.5959i	0.364080i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−27.9985	−0.959778
$852$	0	0
$853$	−46.3485	−1.58694	−0.793471	−	0.608608i	$-0.791728\pi$
$853$	−46.3485	−1.58694	−0.793471	+	0.608608i	$0.791728\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 13.5386i	− 0.462469i	−0.972898	−	0.231234i	$-0.925724\pi$
$857$	− 13.5386i	− 0.462469i	0.972898	−	0.231234i	$-0.0742764\pi$
$858$	0	0
$859$	26.2929i	0.897101i	0.893758	+	0.448550i	$0.148060\pi$
$859$	26.2929i	0.897101i	−0.893758	+	0.448550i	$0.851940\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−34.2590	−1.16619	−0.583094	−	0.812404i	$-0.698158\pi$
$863$	−34.2590	−1.16619	−0.583094	+	0.812404i	$0.698158\pi$
$864$	0	0
$865$	61.3939	2.08745
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 6.35674i	− 0.215638i
$870$	0	0
$871$	− 0.797959i	− 0.0270378i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−5.65685	−0.191237
$876$	0	0
$877$	38.8990	1.31353	0.656763	−	0.754097i	$-0.271925\pi$
$877$	38.8990	1.31353	0.656763	+	0.754097i	$0.271925\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	10.4244i	0.351207i	0.984461	+	0.175604i	$0.0561877\pi$
$881$	10.4244i	0.351207i	−0.984461	+	0.175604i	$0.943812\pi$
$882$	0	0
$883$	− 16.3485i	− 0.550170i	−0.961420	−	0.275085i	$-0.911294\pi$
$883$	− 16.3485i	− 0.550170i	0.961420	−	0.275085i	$-0.0887060\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	39.0265	1.31038	0.655191	−	0.755463i	$-0.272588\pi$
$887$	39.0265	1.31038	0.655191	+	0.755463i	$0.272588\pi$
$888$	0	0
$889$	1.10102	0.0369270
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.08540i	0.170176i
$894$	0	0
$895$	47.1918i	1.57745i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−56.1078	−1.87130
$900$	0	0
$901$	7.44949	0.248178
$902$	0	0
$903$	0	0
$904$	0	0
$905$	68.8679i	2.28925i
$906$	0	0
$907$	37.1918i	1.23493i	0.786597	+	0.617467i	$0.211841\pi$
$907$	37.1918i	1.23493i	−0.786597	+	0.617467i	$0.788159\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	36.1981	1.19930	0.599648	−	0.800264i	$-0.295307\pi$
$911$	36.1981	1.19930	0.599648	+	0.800264i	$0.295307\pi$
$912$	0	0
$913$	−2.20204	−0.0728769
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.02458i	0.264995i
$918$	0	0
$919$	17.5959i	0.580436i	0.956961	+	0.290218i	$0.0937278\pi$
$919$	17.5959i	0.580436i	−0.956961	+	0.290218i	$0.906272\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.28913	0.108263
$924$	0	0
$925$	26.6969	0.877790
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 22.2775i	− 0.730900i	−0.930831	−	0.365450i	$-0.880915\pi$
$929$	− 22.2775i	− 0.730900i	0.930831	−	0.365450i	$-0.119085\pi$
$930$	0	0
$931$	− 4.00000i	− 0.131095i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.65685	0.184999
$936$	0	0
$937$	46.6969	1.52552	0.762761	−	0.646680i	$-0.223843\pi$
$937$	46.6969	1.52552	0.762761	+	0.646680i	$0.223843\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	27.0771i	0.882689i	0.897338	+	0.441345i	$0.145498\pi$
$941$	27.0771i	0.882689i	−0.897338	+	0.441345i	$0.854502\pi$
$942$	0	0
$943$	− 10.8990i	− 0.354920i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−18.2419	−0.592782	−0.296391	−	0.955067i	$-0.595783\pi$
$947$	−18.2419	−0.592782	−0.296391	+	0.955067i	$0.595783\pi$
$948$	0	0
$949$	−2.20204	−0.0714813
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 20.4347i	− 0.661944i	−0.943641	−	0.330972i	$-0.892623\pi$
$953$	− 20.4347i	− 0.661944i	0.943641	−	0.330972i	$-0.107377\pi$
$954$	0	0
$955$	− 47.1918i	− 1.52709i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.19955	−0.264777
$960$	0	0
$961$	−85.5959	−2.76116
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 16.6848i	− 0.537104i
$966$	0	0
$967$	− 28.8990i	− 0.929329i	−0.885487	−	0.464664i	$-0.846175\pi$
$967$	− 28.8990i	− 0.929329i	0.885487	−	0.464664i	$-0.153825\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.92480	0.125953	0.0629765	−	0.998015i	$-0.479941\pi$
$971$	3.92480	0.125953	0.0629765	+	0.998015i	$0.479941\pi$
$972$	0	0
$973$	3.79796	0.121757
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.4415i	0.845937i	0.906144	+	0.422969i	$0.139012\pi$
$977$	26.4415i	0.845937i	−0.906144	+	0.422969i	$0.860988\pi$
$978$	0	0
$979$	− 8.60612i	− 0.275053i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−9.75663	−0.311188	−0.155594	−	0.987821i	$-0.549729\pi$
$983$	−9.75663	−0.311188	−0.155594	+	0.987821i	$0.549729\pi$
$984$	0	0
$985$	−48.0000	−1.52941
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1.73205i	− 0.0550760i
$990$	0	0
$991$	− 2.49490i	− 0.0792530i	−0.999215	−	0.0396265i	$-0.987383\pi$
$991$	− 2.49490i	− 0.0792530i	0.999215	−	0.0396265i	$-0.0126168\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.39989	0.107784
$996$	0	0
$997$	26.5505	0.840863	0.420432	−	0.907324i	$-0.361879\pi$
$997$	26.5505	0.840863	0.420432	+	0.907324i	$0.361879\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.h.e.2591.2	✓	8
3.2	odd	2	inner	6048.2.h.e.2591.5	yes	8
4.3	odd	2	inner	6048.2.h.e.2591.3	yes	8
12.11	even	2	inner	6048.2.h.e.2591.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.h.e.2591.2	✓	8	1.1	even	1	trivial
6048.2.h.e.2591.3	yes	8	4.3	odd	2	inner
6048.2.h.e.2591.5	yes	8	3.2	odd	2	inner
6048.2.h.e.2591.8	yes	8	12.11	even	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)

\(f(q)\)	\(=\)	\(q-2.82843i q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q-2.82843i q^{5} -1.00000i q^{7} +0.635674 q^{11} +0.550510 q^{13} +3.14626i q^{17} +4.00000i q^{19} +3.14626 q^{23} -3.00000 q^{25} -5.19615i q^{29} -10.7980i q^{31} -2.82843 q^{35} -8.89898 q^{37} -3.46410i q^{41} -0.550510i q^{43} +1.27135 q^{47} -1.00000 q^{49} -2.36773i q^{53} -1.79796i q^{55} +11.4887 q^{59} +3.10102 q^{61} -1.55708i q^{65} -1.44949i q^{67} +5.97469 q^{71} -4.00000 q^{73} -0.635674i q^{77} -10.0000i q^{79} -3.46410 q^{83} +8.89898 q^{85} -13.5386i q^{89} -0.550510i q^{91} +11.3137 q^{95} +3.79796 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 24 q^{13} - 24 q^{25} - 32 q^{37} - 8 q^{49} + 64 q^{61} - 32 q^{73} + 32 q^{85} - 48 q^{97}+O(q^{100}) \) 8 * q + 24 * q^13 - 24 * q^25 - 32 * q^37 - 8 * q^49 + 64 * q^61 - 32 * q^73 + 32 * q^85 - 48 * q^97

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