LMFDB - Embedded newform 5616.2.d.j.2159.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5616,2,Mod(2159,5616)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5616, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5616.2159"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5616 = 2^{4} \cdot 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5616.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,14,0,4,0,0,0,0,0,0,0,0,0,-2,0,6,0,0,0,0, 0,0,0,0,0,-8,0,14,0,0,0,0,0,0,0,0,0,26,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(49)] == traces)

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

Self dual:	no
Analytic conductor:	$44.8439857752$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2159.1
Root			$-1.18614 + 1.26217i$ of defining polynomial
Character	$\chi$	$=$	5616.2159
Dual form			5616.2.d.j.2159.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.52434i q^{5} -0.792287i q^{7} +0.627719 q^{11} +1.00000 q^{13} +3.46410i q^{17} -3.37228 q^{23} -1.37228 q^{25} +5.84096i q^{29} +7.42554i q^{31} -2.00000 q^{35} +0.627719 q^{37} +3.16915i q^{41} +9.94987i q^{43} +3.62772 q^{47} +6.37228 q^{49} +11.1846i q^{53} -1.58457i q^{55} -11.7446 q^{59} +4.37228 q^{61} -2.52434i q^{65} -8.21782i q^{67} +9.11684 q^{71} +1.25544 q^{73} -0.497333i q^{77} +0.294954i q^{79} -6.25544 q^{83} +8.74456 q^{85} -3.61158i q^{89} -0.792287i q^{91} +11.4891 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 14 q^{11} + 4 q^{13} - 2 q^{23} + 6 q^{25} - 8 q^{35} + 14 q^{37} + 26 q^{47} + 14 q^{49} - 24 q^{59} + 6 q^{61} + 2 q^{71} + 28 q^{73} - 48 q^{83} + 12 q^{85}+O(q^{100}) $ 4 * q + 14 * q^11 + 4 * q^13 - 2 * q^23 + 6 * q^25 - 8 * q^35 + 14 * q^37 + 26 * q^47 + 14 * q^49 - 24 * q^59 + 6 * q^61 + 2 * q^71 + 28 * q^73 - 48 * q^83 + 12 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5616\mathbb{Z}\right)^\times$.

$n$	$703$	$2081$	$3889$	$4213$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$5$	− 2.52434i	− 1.12892i	−0.825461	−	0.564459i	$-0.809085\pi$
$5$	− 2.52434i	− 1.12892i	0.825461	−	0.564459i	$-0.190915\pi$
$6$	0	0
$7$	− 0.792287i	− 0.299456i	−0.988727	−	0.149728i	$-0.952160\pi$
$7$	− 0.792287i	− 0.299456i	0.988727	−	0.149728i	$-0.0478399\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.627719	0.189264	0.0946322	−	0.995512i	$-0.469833\pi$
$11$	0.627719	0.189264	0.0946322	+	0.995512i	$0.469833\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.46410i	0.840168i	0.907485	+	0.420084i	$0.137999\pi$
$17$	3.46410i	0.840168i	−0.907485	+	0.420084i	$0.862001\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.37228	−0.703169	−0.351585	−	0.936156i	$-0.614357\pi$
$23$	−3.37228	−0.703169	−0.351585	+	0.936156i	$0.614357\pi$
$24$	0	0
$25$	−1.37228	−0.274456
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.84096i	1.08464i	0.840172	+	0.542320i	$0.182454\pi$
$29$	5.84096i	1.08464i	−0.840172	+	0.542320i	$0.817546\pi$
$30$	0	0
$31$	7.42554i	1.33367i	0.745207	+	0.666833i	$0.232351\pi$
$31$	7.42554i	1.33367i	−0.745207	+	0.666833i	$0.767649\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	0.627719	0.103196	0.0515982	−	0.998668i	$-0.483568\pi$
$37$	0.627719	0.103196	0.0515982	+	0.998668i	$0.483568\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.16915i	0.494938i	0.968896	+	0.247469i	$0.0795988\pi$
$41$	3.16915i	0.494938i	−0.968896	+	0.247469i	$0.920401\pi$
$42$	0	0
$43$	9.94987i	1.51734i	0.651474	+	0.758671i	$0.274151\pi$
$43$	9.94987i	1.51734i	−0.651474	+	0.758671i	$0.725849\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.62772	0.529157	0.264579	−	0.964364i	$-0.414767\pi$
$47$	3.62772	0.529157	0.264579	+	0.964364i	$0.414767\pi$
$48$	0	0
$49$	6.37228	0.910326
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.1846i	1.53632i	0.640257	+	0.768161i	$0.278828\pi$
$53$	11.1846i	1.53632i	−0.640257	+	0.768161i	$0.721172\pi$
$54$	0	0
$55$	− 1.58457i	− 0.213664i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.7446	−1.52901	−0.764506	−	0.644617i	$-0.777017\pi$
$59$	−11.7446	−1.52901	−0.764506	+	0.644617i	$0.777017\pi$
$60$	0	0
$61$	4.37228	0.559813	0.279907	−	0.960027i	$-0.409696\pi$
$61$	4.37228	0.559813	0.279907	+	0.960027i	$0.409696\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 2.52434i	− 0.313106i
$66$	0	0
$67$	− 8.21782i	− 1.00397i	−0.864877	−	0.501983i	$-0.832604\pi$
$67$	− 8.21782i	− 1.00397i	0.864877	−	0.501983i	$-0.167396\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.11684	1.08197	0.540985	−	0.841032i	$-0.318052\pi$
$71$	9.11684	1.08197	0.540985	+	0.841032i	$0.318052\pi$
$72$	0	0
$73$	1.25544	0.146938	0.0734689	−	0.997298i	$-0.476593\pi$
$73$	1.25544	0.146938	0.0734689	+	0.997298i	$0.476593\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.497333i	− 0.0566764i
$78$	0	0
$79$	0.294954i	0.0331849i	0.999862	+	0.0165924i	$0.00528178\pi$
$79$	0.294954i	0.0331849i	−0.999862	+	0.0165924i	$0.994718\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.25544	−0.686623	−0.343312	−	0.939222i	$-0.611549\pi$
$83$	−6.25544	−0.686623	−0.343312	+	0.939222i	$0.611549\pi$
$84$	0	0
$85$	8.74456	0.948481
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 3.61158i	− 0.382827i	−0.981510	−	0.191413i	$-0.938693\pi$
$89$	− 3.61158i	− 0.382827i	0.981510	−	0.191413i	$-0.0613071\pi$
$90$	0	0
$91$	− 0.792287i	− 0.0830542i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.4891	1.16654	0.583272	−	0.812277i	$-0.301772\pi$
$97$	11.4891	1.16654	0.583272	+	0.812277i	$0.301772\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.4742i	1.24123i	0.784115	+	0.620615i	$0.213117\pi$
$101$	12.4742i	1.24123i	−0.784115	+	0.620615i	$0.786883\pi$
$102$	0	0
$103$	2.52434i	0.248730i	0.992237	+	0.124365i	$0.0396894\pi$
$103$	2.52434i	0.248730i	−0.992237	+	0.124365i	$0.960311\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.25544	−0.314715	−0.157358	−	0.987542i	$-0.550297\pi$
$107$	−3.25544	−0.314715	−0.157358	+	0.987542i	$0.550297\pi$
$108$	0	0
$109$	1.25544	0.120249	0.0601245	−	0.998191i	$-0.480850\pi$
$109$	1.25544	0.120249	0.0601245	+	0.998191i	$0.480850\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1.87953i	− 0.176811i	−0.996085	−	0.0884055i	$-0.971823\pi$
$113$	− 1.87953i	− 0.176811i	0.996085	−	0.0884055i	$-0.0281771\pi$
$114$	0	0
$115$	8.51278i	0.793821i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.74456	0.251594
$120$	0	0
$121$	−10.6060	−0.964179
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 9.15759i	− 0.819080i
$126$	0	0
$127$	4.40387i	0.390780i	0.980726	+	0.195390i	$0.0625972\pi$
$127$	4.40387i	0.390780i	−0.980726	+	0.195390i	$0.937403\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−14.2337	−1.24360	−0.621802	−	0.783175i	$-0.713599\pi$
$131$	−14.2337	−1.24360	−0.621802	+	0.783175i	$0.713599\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1.08724i	− 0.0928892i	−0.998921	−	0.0464446i	$-0.985211\pi$
$137$	− 1.08724i	− 0.0928892i	0.998921	−	0.0464446i	$-0.0147891\pi$
$138$	0	0
$139$	− 2.52434i	− 0.214112i	−0.994253	−	0.107056i	$-0.965858\pi$
$139$	− 2.52434i	− 0.214112i	0.994253	−	0.107056i	$-0.0341423\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.627719	0.0524925
$144$	0	0
$145$	14.7446	1.22447
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 19.8448i	− 1.62575i	−0.582436	−	0.812877i	$-0.697900\pi$
$149$	− 19.8448i	− 1.62575i	0.582436	−	0.812877i	$-0.302100\pi$
$150$	0	0
$151$	15.4410i	1.25657i	0.777984	+	0.628285i	$0.216243\pi$
$151$	15.4410i	1.25657i	−0.777984	+	0.628285i	$0.783757\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	18.7446	1.50560
$156$	0	0
$157$	13.1168	1.04684	0.523419	−	0.852075i	$-0.324656\pi$
$157$	13.1168	1.04684	0.523419	+	0.852075i	$0.324656\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.67181i	0.210568i
$162$	0	0
$163$	3.16915i	0.248227i	0.992268	+	0.124113i	$0.0396087\pi$
$163$	3.16915i	0.248227i	−0.992268	+	0.124113i	$0.960391\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.86141	0.453569	0.226785	−	0.973945i	$-0.427179\pi$
$167$	5.86141	0.453569	0.226785	+	0.973945i	$0.427179\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 13.2665i	− 1.00863i	−0.863519	−	0.504317i	$-0.831744\pi$
$173$	− 13.2665i	− 1.00863i	0.863519	−	0.504317i	$-0.168256\pi$
$174$	0	0
$175$	1.08724i	0.0821877i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.74456	−0.354625	−0.177313	−	0.984155i	$-0.556740\pi$
$179$	−4.74456	−0.354625	−0.177313	+	0.984155i	$0.556740\pi$
$180$	0	0
$181$	17.8614	1.32763	0.663814	−	0.747898i	$-0.268937\pi$
$181$	17.8614	1.32763	0.663814	+	0.747898i	$0.268937\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 1.58457i	− 0.116500i
$186$	0	0
$187$	2.17448i	0.159014i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.62772	0.479565	0.239782	−	0.970827i	$-0.422924\pi$
$191$	6.62772	0.479565	0.239782	+	0.970827i	$0.422924\pi$
$192$	0	0
$193$	16.7446	1.20530	0.602650	−	0.798006i	$-0.294112\pi$
$193$	16.7446	1.20530	0.602650	+	0.798006i	$0.294112\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.86797i	0.560569i	0.959917	+	0.280285i	$0.0904289\pi$
$197$	7.86797i	0.560569i	−0.959917	+	0.280285i	$0.909571\pi$
$198$	0	0
$199$	5.98844i	0.424509i	0.977214	+	0.212255i	$0.0680806\pi$
$199$	5.98844i	0.424509i	−0.977214	+	0.212255i	$0.931919\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.62772	0.324802
$204$	0	0
$205$	8.00000	0.558744
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 8.66025i	− 0.596196i	−0.954535	−	0.298098i	$-0.903648\pi$
$211$	− 8.66025i	− 0.596196i	0.954535	−	0.298098i	$-0.0963523\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	25.1168	1.71295
$216$	0	0
$217$	5.88316	0.399375
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.46410i	0.233021i
$222$	0	0
$223$	14.1514i	0.947645i	0.880620	+	0.473823i	$0.157126\pi$
$223$	14.1514i	0.947645i	−0.880620	+	0.473823i	$0.842874\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−19.1168	−1.26883	−0.634415	−	0.772993i	$-0.718759\pi$
$227$	−19.1168	−1.26883	−0.634415	+	0.772993i	$0.718759\pi$
$228$	0	0
$229$	11.3723	0.751502	0.375751	−	0.926721i	$-0.377385\pi$
$229$	11.3723	0.751502	0.375751	+	0.926721i	$0.377385\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 7.22316i	− 0.473205i	−0.971607	−	0.236602i	$-0.923966\pi$
$233$	− 7.22316i	− 0.473205i	0.971607	−	0.236602i	$-0.0760339\pi$
$234$	0	0
$235$	− 9.15759i	− 0.597375i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5.25544	−0.339946	−0.169973	−	0.985449i	$-0.554368\pi$
$239$	−5.25544	−0.339946	−0.169973	+	0.985449i	$0.554368\pi$
$240$	0	0
$241$	−22.9783	−1.48016	−0.740080	−	0.672519i	$-0.765212\pi$
$241$	−22.9783	−1.48016	−0.740080	+	0.672519i	$0.765212\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 16.0858i	− 1.02768i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.4891	1.23014	0.615071	−	0.788471i	$-0.289127\pi$
$251$	19.4891	1.23014	0.615071	+	0.788471i	$0.289127\pi$
$252$	0	0
$253$	−2.11684	−0.133085
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.2000i	1.19767i	0.800874	+	0.598833i	$0.204369\pi$
$257$	19.2000i	1.19767i	−0.800874	+	0.598833i	$0.795631\pi$
$258$	0	0
$259$	− 0.497333i	− 0.0309028i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4.86141	0.299767	0.149884	−	0.988704i	$-0.452110\pi$
$263$	4.86141	0.299767	0.149884	+	0.988704i	$0.452110\pi$
$264$	0	0
$265$	28.2337	1.73438
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 0.792287i	− 0.0483066i	−0.999708	−	0.0241533i	$-0.992311\pi$
$269$	− 0.792287i	− 0.0483066i	0.999708	−	0.0241533i	$-0.00768898\pi$
$270$	0	0
$271$	10.3923i	0.631288i	0.948878	+	0.315644i	$0.102220\pi$
$271$	10.3923i	0.631288i	−0.948878	+	0.315644i	$0.897780\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.861407	−0.0519448
$276$	0	0
$277$	23.3505	1.40300	0.701499	−	0.712671i	$-0.252515\pi$
$277$	23.3505	1.40300	0.701499	+	0.712671i	$0.252515\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 21.3368i	− 1.27285i	−0.771339	−	0.636425i	$-0.780413\pi$
$281$	− 21.3368i	− 1.27285i	0.771339	−	0.636425i	$-0.219587\pi$
$282$	0	0
$283$	8.07035i	0.479732i	0.970806	+	0.239866i	$0.0771036\pi$
$283$	8.07035i	0.479732i	−0.970806	+	0.239866i	$0.922896\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.51087	0.148212
$288$	0	0
$289$	5.00000	0.294118
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 20.1947i	− 1.17979i	−0.807481	−	0.589894i	$-0.799170\pi$
$293$	− 20.1947i	− 1.17979i	0.807481	−	0.589894i	$-0.200830\pi$
$294$	0	0
$295$	29.6472i	1.72613i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.37228	−0.195024
$300$	0	0
$301$	7.88316	0.454378
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 11.0371i	− 0.631983i
$306$	0	0
$307$	− 13.5615i	− 0.773993i	−0.922081	−	0.386996i	$-0.873513\pi$
$307$	− 13.5615i	− 0.773993i	0.922081	−	0.386996i	$-0.126487\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−10.1168	−0.573674	−0.286837	−	0.957979i	$-0.592604\pi$
$311$	−10.1168	−0.573674	−0.286837	+	0.957979i	$0.592604\pi$
$312$	0	0
$313$	−11.0000	−0.621757	−0.310878	−	0.950450i	$-0.600623\pi$
$313$	−11.0000	−0.621757	−0.310878	+	0.950450i	$0.600623\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.51278i	0.478125i	0.971004	+	0.239063i	$0.0768401\pi$
$317$	8.51278i	0.478125i	−0.971004	+	0.239063i	$0.923160\pi$
$318$	0	0
$319$	3.66648i	0.205284i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−1.37228	−0.0761205
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 2.87419i	− 0.158459i
$330$	0	0
$331$	10.3923i	0.571213i	0.958347	+	0.285606i	$0.0921950\pi$
$331$	10.3923i	0.571213i	−0.958347	+	0.285606i	$0.907805\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−20.7446	−1.13340
$336$	0	0
$337$	4.25544	0.231808	0.115904	−	0.993260i	$-0.463023\pi$
$337$	4.25544	0.231808	0.115904	+	0.993260i	$0.463023\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.66115i	0.252415i
$342$	0	0
$343$	− 10.5947i	− 0.572059i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	13.2554	0.711589	0.355795	−	0.934564i	$-0.384210\pi$
$347$	13.2554	0.711589	0.355795	+	0.934564i	$0.384210\pi$
$348$	0	0
$349$	−4.23369	−0.226624	−0.113312	−	0.993559i	$-0.536146\pi$
$349$	−4.23369	−0.226624	−0.113312	+	0.993559i	$0.536146\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	17.1730i	0.914028i	0.889459	+	0.457014i	$0.151081\pi$
$353$	17.1730i	0.914028i	−0.889459	+	0.457014i	$0.848919\pi$
$354$	0	0
$355$	− 23.0140i	− 1.22146i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5.62772	0.297020	0.148510	−	0.988911i	$-0.452552\pi$
$359$	5.62772	0.297020	0.148510	+	0.988911i	$0.452552\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 3.16915i	− 0.165881i
$366$	0	0
$367$	29.0024i	1.51391i	0.653464	+	0.756957i	$0.273315\pi$
$367$	29.0024i	1.51391i	−0.653464	+	0.756957i	$0.726685\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.86141	0.460061
$372$	0	0
$373$	2.88316	0.149284	0.0746421	−	0.997210i	$-0.476219\pi$
$373$	2.88316	0.149284	0.0746421	+	0.997210i	$0.476219\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.84096i	0.300825i
$378$	0	0
$379$	8.21782i	0.422121i	0.977473	+	0.211061i	$0.0676917\pi$
$379$	8.21782i	0.422121i	−0.977473	+	0.211061i	$0.932308\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	25.7228	1.31437	0.657187	−	0.753727i	$-0.271746\pi$
$383$	25.7228	1.31437	0.657187	+	0.753727i	$0.271746\pi$
$384$	0	0
$385$	−1.25544	−0.0639830
$386$	0	0
$387$	0	0
$388$	0	0
$389$	32.1716i	1.63116i	0.578642	+	0.815582i	$0.303583\pi$
$389$	32.1716i	1.63116i	−0.578642	+	0.815582i	$0.696417\pi$
$390$	0	0
$391$	− 11.6819i	− 0.590780i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.744563	0.0374630
$396$	0	0
$397$	−9.37228	−0.470381	−0.235191	−	0.971949i	$-0.575571\pi$
$397$	−9.37228	−0.470381	−0.235191	+	0.971949i	$0.575571\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 18.4627i	− 0.921981i	−0.887405	−	0.460990i	$-0.847494\pi$
$401$	− 18.4627i	− 0.921981i	0.887405	−	0.460990i	$-0.152506\pi$
$402$	0	0
$403$	7.42554i	0.369892i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.394031	0.0195314
$408$	0	0
$409$	10.7446	0.531284	0.265642	−	0.964072i	$-0.414416\pi$
$409$	10.7446	0.531284	0.265642	+	0.964072i	$0.414416\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9.30506i	0.457872i
$414$	0	0
$415$	15.7908i	0.775142i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	−8.23369	−0.401285	−0.200643	−	0.979664i	$-0.564303\pi$
$421$	−8.23369	−0.401285	−0.200643	+	0.979664i	$0.564303\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 4.75372i	− 0.230589i
$426$	0	0
$427$	− 3.46410i	− 0.167640i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−37.1168	−1.78786	−0.893928	−	0.448211i	$-0.852061\pi$
$431$	−37.1168	−1.78786	−0.893928	+	0.448211i	$0.852061\pi$
$432$	0	0
$433$	8.11684	0.390071	0.195035	−	0.980796i	$-0.437518\pi$
$433$	8.11684	0.390071	0.195035	+	0.980796i	$0.437518\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	24.8935i	1.18810i	0.804427	+	0.594051i	$0.202472\pi$
$439$	24.8935i	1.18810i	−0.804427	+	0.594051i	$0.797528\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	−9.11684	−0.432180
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11.0371i	0.520874i	0.965491	+	0.260437i	$0.0838666\pi$
$449$	11.0371i	0.520874i	−0.965491	+	0.260437i	$0.916133\pi$
$450$	0	0
$451$	1.98933i	0.0936740i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	11.4891	0.537439	0.268719	−	0.963219i	$-0.413400\pi$
$457$	11.4891	0.537439	0.268719	+	0.963219i	$0.413400\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 1.23472i	− 0.0575065i	−0.999587	−	0.0287533i	$-0.990846\pi$
$461$	− 1.23472i	− 0.0575065i	0.999587	−	0.0287533i	$-0.00915371\pi$
$462$	0	0
$463$	− 5.54601i	− 0.257745i	−0.991661	−	0.128872i	$-0.958864\pi$
$463$	− 5.54601i	− 0.257745i	0.991661	−	0.128872i	$-0.0411358\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−32.9783	−1.52605	−0.763026	−	0.646368i	$-0.776287\pi$
$467$	−32.9783	−1.52605	−0.763026	+	0.646368i	$0.776287\pi$
$468$	0	0
$469$	−6.51087	−0.300644
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.24572i	0.287179i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	26.6060	1.21566	0.607829	−	0.794068i	$-0.292041\pi$
$479$	26.6060	1.21566	0.607829	+	0.794068i	$0.292041\pi$
$480$	0	0
$481$	0.627719	0.0286215
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 29.0024i	− 1.31693i
$486$	0	0
$487$	− 0.202380i	− 0.00917070i	−0.999989	−	0.00458535i	$-0.998540\pi$
$487$	− 0.202380i	− 0.00917070i	0.999989	−	0.00458535i	$-0.00145957\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15.7228	0.709561	0.354780	−	0.934950i	$-0.384556\pi$
$491$	15.7228	0.709561	0.354780	+	0.934950i	$0.384556\pi$
$492$	0	0
$493$	−20.2337	−0.911279
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 7.22316i	− 0.324003i
$498$	0	0
$499$	− 3.75906i	− 0.168278i	−0.996454	−	0.0841392i	$-0.973186\pi$
$499$	− 3.75906i	− 0.168278i	0.996454	−	0.0841392i	$-0.0268140\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3.88316	−0.173141	−0.0865707	−	0.996246i	$-0.527591\pi$
$503$	−3.88316	−0.173141	−0.0865707	+	0.996246i	$0.527591\pi$
$504$	0	0
$505$	31.4891	1.40125
$506$	0	0
$507$	0	0
$508$	0	0
$509$	23.7137i	1.05109i	0.850765	+	0.525546i	$0.176139\pi$
$509$	23.7137i	1.05109i	−0.850765	+	0.525546i	$0.823861\pi$
$510$	0	0
$511$	− 0.994667i	− 0.0440015i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.37228	0.280796
$516$	0	0
$517$	2.27719	0.100151
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20.4897i	0.897668i	0.893615	+	0.448834i	$0.148161\pi$
$521$	20.4897i	0.897668i	−0.893615	+	0.448834i	$0.851839\pi$
$522$	0	0
$523$	− 23.3089i	− 1.01923i	−0.860403	−	0.509615i	$-0.829788\pi$
$523$	− 23.3089i	− 1.01923i	0.860403	−	0.509615i	$-0.170212\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−25.7228	−1.12050
$528$	0	0
$529$	−11.6277	−0.505553
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.16915i	0.137271i
$534$	0	0
$535$	8.21782i	0.355287i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.00000	0.172292
$540$	0	0
$541$	27.0951	1.16491	0.582455	−	0.812863i	$-0.302092\pi$
$541$	27.0951	1.16491	0.582455	+	0.812863i	$0.302092\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 3.16915i	− 0.135751i
$546$	0	0
$547$	9.30506i	0.397856i	0.980014	+	0.198928i	$0.0637460\pi$
$547$	9.30506i	0.397856i	−0.980014	+	0.198928i	$0.936254\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0.233688	0.00993742
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 31.8766i	− 1.35066i	−0.737517	−	0.675328i	$-0.764002\pi$
$557$	− 31.8766i	− 1.35066i	0.737517	−	0.675328i	$-0.235998\pi$
$558$	0	0
$559$	9.94987i	0.420835i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	36.9783	1.55845	0.779224	−	0.626746i	$-0.215614\pi$
$563$	36.9783	1.55845	0.779224	+	0.626746i	$0.215614\pi$
$564$	0	0
$565$	−4.74456	−0.199605
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.16915i	0.132858i	0.997791	+	0.0664288i	$0.0211605\pi$
$569$	3.16915i	0.132858i	−0.997791	+	0.0664288i	$0.978839\pi$
$570$	0	0
$571$	− 16.3807i	− 0.685513i	−0.939424	−	0.342756i	$-0.888639\pi$
$571$	− 16.3807i	− 0.685513i	0.939424	−	0.342756i	$-0.111361\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.62772	0.192989
$576$	0	0
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4.95610i	0.205614i
$582$	0	0
$583$	7.02078i	0.290771i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.6277	0.521202	0.260601	−	0.965447i	$-0.416079\pi$
$587$	12.6277	0.521202	0.260601	+	0.965447i	$0.416079\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	12.7692i	0.524367i	0.965018	+	0.262183i	$0.0844425\pi$
$593$	12.7692i	0.524367i	−0.965018	+	0.262183i	$0.915557\pi$
$594$	0	0
$595$	− 6.92820i	− 0.284029i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.74456	0.193858	0.0969288	−	0.995291i	$-0.469098\pi$
$599$	4.74456	0.193858	0.0969288	+	0.995291i	$0.469098\pi$
$600$	0	0
$601$	−24.2554	−0.989400	−0.494700	−	0.869064i	$-0.664722\pi$
$601$	−24.2554	−0.989400	−0.494700	+	0.869064i	$0.664722\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	26.7730i	1.08848i
$606$	0	0
$607$	− 1.28962i	− 0.0523441i	−0.999657	−	0.0261720i	$-0.991668\pi$
$607$	− 1.28962i	− 0.0523441i	0.999657	−	0.0261720i	$-0.00833177\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.62772	0.146762
$612$	0	0
$613$	−0.627719	−0.0253533	−0.0126767	−	0.999920i	$-0.504035\pi$
$613$	−0.627719	−0.0253533	−0.0126767	+	0.999920i	$0.504035\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	40.6295i	1.63568i	0.575445	+	0.817840i	$0.304829\pi$
$617$	40.6295i	1.63568i	−0.575445	+	0.817840i	$0.695171\pi$
$618$	0	0
$619$	− 2.57924i	− 0.103668i	−0.998656	−	0.0518342i	$-0.983493\pi$
$619$	− 2.57924i	− 0.103668i	0.998656	−	0.0518342i	$-0.0165067\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.86141	−0.114640
$624$	0	0
$625$	−29.9783	−1.19913
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.17448i	0.0867022i
$630$	0	0
$631$	− 2.96677i	− 0.118105i	−0.998255	−	0.0590526i	$-0.981192\pi$
$631$	− 2.96677i	− 0.118105i	0.998255	−	0.0590526i	$-0.0188080\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	11.1168	0.441158
$636$	0	0
$637$	6.37228	0.252479
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 26.5330i	− 1.04799i	−0.851721	−	0.523995i	$-0.824441\pi$
$641$	− 26.5330i	− 1.04799i	0.851721	−	0.523995i	$-0.175559\pi$
$642$	0	0
$643$	− 42.2689i	− 1.66692i	−0.552577	−	0.833462i	$-0.686355\pi$
$643$	− 42.2689i	− 1.66692i	0.552577	−	0.833462i	$-0.313645\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24.3505	−0.957318	−0.478659	−	0.878001i	$-0.658877\pi$
$647$	−24.3505	−0.957318	−0.478659	+	0.878001i	$0.658877\pi$
$648$	0	0
$649$	−7.37228	−0.289387
$650$	0	0
$651$	0	0
$652$	0	0
$653$	48.4974i	1.89785i	0.315501	+	0.948925i	$0.397828\pi$
$653$	48.4974i	1.89785i	−0.315501	+	0.948925i	$0.602172\pi$
$654$	0	0
$655$	35.9306i	1.40393i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−44.7446	−1.74300	−0.871500	−	0.490395i	$-0.836853\pi$
$659$	−44.7446	−1.74300	−0.871500	+	0.490395i	$0.836853\pi$
$660$	0	0
$661$	39.0951	1.52062	0.760311	−	0.649559i	$-0.225046\pi$
$661$	39.0951	1.52062	0.760311	+	0.649559i	$0.225046\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 19.6974i	− 0.762685i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.74456	0.105953
$672$	0	0
$673$	0.372281	0.0143504	0.00717520	−	0.999974i	$-0.497716\pi$
$673$	0.372281	0.0143504	0.00717520	+	0.999974i	$0.497716\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.4125i	1.09198i	0.837791	+	0.545991i	$0.183847\pi$
$677$	28.4125i	1.09198i	−0.837791	+	0.545991i	$0.816153\pi$
$678$	0	0
$679$	− 9.10268i	− 0.349329i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−44.4891	−1.70233	−0.851165	−	0.524899i	$-0.824103\pi$
$683$	−44.4891	−1.70233	−0.851165	+	0.524899i	$0.824103\pi$
$684$	0	0
$685$	−2.74456	−0.104864
$686$	0	0
$687$	0	0
$688$	0	0
$689$	11.1846i	0.426099i
$690$	0	0
$691$	3.16915i	0.120560i	0.998182	+	0.0602800i	$0.0191994\pi$
$691$	3.16915i	0.120560i	−0.998182	+	0.0602800i	$0.980801\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.37228	−0.241714
$696$	0	0
$697$	−10.9783	−0.415831
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 21.4843i	− 0.811452i	−0.913995	−	0.405726i	$-0.867019\pi$
$701$	− 21.4843i	− 0.811452i	0.913995	−	0.405726i	$-0.132981\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.88316	0.371694
$708$	0	0
$709$	−25.2554	−0.948488	−0.474244	−	0.880393i	$-0.657279\pi$
$709$	−25.2554	−0.948488	−0.474244	+	0.880393i	$0.657279\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 25.0410i	− 0.937793i
$714$	0	0
$715$	− 1.58457i	− 0.0592597i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−25.3723	−0.946226	−0.473113	−	0.881002i	$-0.656870\pi$
$719$	−25.3723	−0.946226	−0.473113	+	0.881002i	$0.656870\pi$
$720$	0	0
$721$	2.00000	0.0744839
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 8.01544i	− 0.297686i
$726$	0	0
$727$	27.4179i	1.01687i	0.861100	+	0.508436i	$0.169776\pi$
$727$	27.4179i	1.01687i	−0.861100	+	0.508436i	$0.830224\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−34.4674	−1.27482
$732$	0	0
$733$	9.37228	0.346173	0.173087	−	0.984907i	$-0.444626\pi$
$733$	9.37228	0.346173	0.173087	+	0.984907i	$0.444626\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5.15848i	− 0.190015i
$738$	0	0
$739$	− 15.1460i	− 0.557156i	−0.960414	−	0.278578i	$-0.910137\pi$
$739$	− 15.1460i	− 0.557156i	0.960414	−	0.278578i	$-0.0898630\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.13859	0.0784574	0.0392287	−	0.999230i	$-0.487510\pi$
$743$	2.13859	0.0784574	0.0392287	+	0.999230i	$0.487510\pi$
$744$	0	0
$745$	−50.0951	−1.83534
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.57924i	0.0942434i
$750$	0	0
$751$	3.46410i	0.126407i	0.998001	+	0.0632034i	$0.0201317\pi$
$751$	3.46410i	0.126407i	−0.998001	+	0.0632034i	$0.979868\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	38.9783	1.41856
$756$	0	0
$757$	−43.7228	−1.58913	−0.794566	−	0.607177i	$-0.792302\pi$
$757$	−43.7228	−1.58913	−0.794566	+	0.607177i	$0.792302\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	14.1137i	0.511621i	0.966727	+	0.255810i	$0.0823423\pi$
$761$	14.1137i	0.511621i	−0.966727	+	0.255810i	$0.917658\pi$
$762$	0	0
$763$	− 0.994667i	− 0.0360094i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−11.7446	−0.424072
$768$	0	0
$769$	10.2337	0.369036	0.184518	−	0.982829i	$-0.440928\pi$
$769$	10.2337	0.369036	0.184518	+	0.982829i	$0.440928\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7.27806i	0.261774i	0.991397	+	0.130887i	$0.0417824\pi$
$773$	7.27806i	0.261774i	−0.991397	+	0.130887i	$0.958218\pi$
$774$	0	0
$775$	− 10.1899i	− 0.366033i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	5.72281	0.204778
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 33.1113i	− 1.18179i
$786$	0	0
$787$	− 44.7384i	− 1.59475i	−0.603484	−	0.797375i	$-0.706221\pi$
$787$	− 44.7384i	− 1.59475i	0.603484	−	0.797375i	$-0.293779\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.48913	−0.0529472
$792$	0	0
$793$	4.37228	0.155264
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 28.5051i	− 1.00970i	−0.863206	−	0.504851i	$-0.831547\pi$
$797$	− 28.5051i	− 1.00970i	0.863206	−	0.504851i	$-0.168453\pi$
$798$	0	0
$799$	12.5668i	0.444581i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0.788061	0.0278101
$804$	0	0
$805$	6.74456	0.237715
$806$	0	0
$807$	0	0
$808$	0	0
$809$	34.6410i	1.21791i	0.793204	+	0.608957i	$0.208412\pi$
$809$	34.6410i	1.21791i	−0.793204	+	0.608957i	$0.791588\pi$
$810$	0	0
$811$	24.9484i	0.876058i	0.898961	+	0.438029i	$0.144323\pi$
$811$	24.9484i	0.876058i	−0.898961	+	0.438029i	$0.855677\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	8.00000	0.280228
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 4.16381i	− 0.145318i	−0.997357	−	0.0726591i	$-0.976852\pi$
$821$	− 4.16381i	− 0.145318i	0.997357	−	0.0726591i	$-0.0231485\pi$
$822$	0	0
$823$	15.2009i	0.529871i	0.964266	+	0.264936i	$0.0853507\pi$
$823$	15.2009i	0.529871i	−0.964266	+	0.264936i	$0.914649\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38.1168	1.32545	0.662726	−	0.748862i	$-0.269399\pi$
$827$	38.1168	1.32545	0.662726	+	0.748862i	$0.269399\pi$
$828$	0	0
$829$	−3.11684	−0.108252	−0.0541262	−	0.998534i	$-0.517237\pi$
$829$	−3.11684	−0.108252	−0.0541262	+	0.998534i	$0.517237\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	22.0742i	0.764827i
$834$	0	0
$835$	− 14.7962i	− 0.512043i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	44.8397	1.54804	0.774018	−	0.633163i	$-0.218244\pi$
$839$	44.8397	1.54804	0.774018	+	0.633163i	$0.218244\pi$
$840$	0	0
$841$	−5.11684	−0.176443
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 2.52434i	− 0.0868399i
$846$	0	0
$847$	8.40297i	0.288730i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.11684	−0.0725645
$852$	0	0
$853$	−11.8832	−0.406872	−0.203436	−	0.979088i	$-0.565211\pi$
$853$	−11.8832	−0.406872	−0.203436	+	0.979088i	$0.565211\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31.2867i	1.06873i	0.845253	+	0.534367i	$0.179450\pi$
$857$	31.2867i	1.06873i	−0.845253	+	0.534367i	$0.820550\pi$
$858$	0	0
$859$	− 30.0897i	− 1.02665i	−0.858195	−	0.513323i	$-0.828414\pi$
$859$	− 30.0897i	− 1.02665i	0.858195	−	0.513323i	$-0.171586\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.6060	0.633355	0.316677	−	0.948533i	$-0.397433\pi$
$863$	18.6060	0.633355	0.316677	+	0.948533i	$0.397433\pi$
$864$	0	0
$865$	−33.4891	−1.13866
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0.185148i	0.00628071i
$870$	0	0
$871$	− 8.21782i	− 0.278450i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−7.25544	−0.245279
$876$	0	0
$877$	21.0951	0.712331	0.356165	−	0.934423i	$-0.384084\pi$
$877$	21.0951	0.712331	0.356165	+	0.934423i	$0.384084\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	17.4303i	0.587242i	0.955922	+	0.293621i	$0.0948604\pi$
$881$	17.4303i	0.587242i	−0.955922	+	0.293621i	$0.905140\pi$
$882$	0	0
$883$	− 55.6280i	− 1.87203i	−0.351958	−	0.936016i	$-0.614484\pi$
$883$	− 55.6280i	− 1.87203i	0.351958	−	0.936016i	$-0.385516\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	39.6060	1.32984	0.664919	−	0.746915i	$-0.268466\pi$
$887$	39.6060	1.32984	0.664919	+	0.746915i	$0.268466\pi$
$888$	0	0
$889$	3.48913	0.117022
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	11.9769i	0.400343i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−43.3723	−1.44655
$900$	0	0
$901$	−38.7446	−1.29077
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 45.0882i	− 1.49878i
$906$	0	0
$907$	38.4550i	1.27688i	0.769673	+	0.638438i	$0.220419\pi$
$907$	38.4550i	1.27688i	−0.769673	+	0.638438i	$0.779581\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−39.0951	−1.29528	−0.647639	−	0.761947i	$-0.724244\pi$
$911$	−39.0951	−1.29528	−0.647639	+	0.761947i	$0.724244\pi$
$912$	0	0
$913$	−3.92665	−0.129953
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.2772i	0.372405i
$918$	0	0
$919$	− 52.7161i	− 1.73895i	−0.493981	−	0.869473i	$-0.664459\pi$
$919$	− 52.7161i	− 1.73895i	0.493981	−	0.869473i	$-0.335541\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.11684	0.300085
$924$	0	0
$925$	−0.861407	−0.0283229
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 44.6458i	− 1.46478i	−0.680885	−	0.732390i	$-0.738405\pi$
$929$	− 44.6458i	− 1.46478i	0.680885	−	0.732390i	$-0.261595\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.48913	0.179514
$936$	0	0
$937$	7.97825	0.260638	0.130319	−	0.991472i	$-0.458400\pi$
$937$	7.97825	0.260638	0.130319	+	0.991472i	$0.458400\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 23.9538i	− 0.780870i	−0.920630	−	0.390435i	$-0.872325\pi$
$941$	− 23.9538i	− 0.780870i	0.920630	−	0.390435i	$-0.127675\pi$
$942$	0	0
$943$	− 10.6873i	− 0.348025i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40.7228	1.32331	0.661657	−	0.749807i	$-0.269854\pi$
$947$	40.7228	1.32331	0.661657	+	0.749807i	$0.269854\pi$
$948$	0	0
$949$	1.25544	0.0407532
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 45.7330i	− 1.48144i	−0.671815	−	0.740719i	$-0.734485\pi$
$953$	− 45.7330i	− 1.48144i	0.671815	−	0.740719i	$-0.265515\pi$
$954$	0	0
$955$	− 16.7306i	− 0.541390i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−0.861407	−0.0278163
$960$	0	0
$961$	−24.1386	−0.778664
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 42.2689i	− 1.36069i
$966$	0	0
$967$	0.202380i	0.00650809i	0.999995	+	0.00325405i	$0.00103580\pi$
$967$	0.202380i	0.00650809i	−0.999995	+	0.00325405i	$0.998964\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	30.9783	0.994139	0.497070	−	0.867711i	$-0.334409\pi$
$971$	30.9783	0.994139	0.497070	+	0.867711i	$0.334409\pi$
$972$	0	0
$973$	−2.00000	−0.0641171
$974$	0	0
$975$	0	0
$976$	0	0
$977$	14.4086i	0.460973i	0.973075	+	0.230487i	$0.0740318\pi$
$977$	14.4086i	0.460973i	−0.973075	+	0.230487i	$0.925968\pi$
$978$	0	0
$979$	− 2.26706i	− 0.0724554i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−41.2554	−1.31584	−0.657922	−	0.753086i	$-0.728564\pi$
$983$	−41.2554	−1.31584	−0.657922	+	0.753086i	$0.728564\pi$
$984$	0	0
$985$	19.8614	0.632837
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 33.5538i	− 1.06695i
$990$	0	0
$991$	− 31.8217i	− 1.01085i	−0.862870	−	0.505425i	$-0.831336\pi$
$991$	− 31.8217i	− 1.01085i	0.862870	−	0.505425i	$-0.168664\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	15.1168	0.479236
$996$	0	0
$997$	−15.1168	−0.478755	−0.239378	−	0.970927i	$-0.576943\pi$
$997$	−15.1168	−0.478755	−0.239378	+	0.970927i	$0.576943\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5616.2.d.j.2159.1	yes	4
3.2	odd	2		5616.2.d.e.2159.4	yes	4
4.3	odd	2		5616.2.d.e.2159.1	✓	4
12.11	even	2	inner	5616.2.d.j.2159.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5616.2.d.e.2159.1	✓	4	4.3	odd	2
5616.2.d.e.2159.4	yes	4	3.2	odd	2
5616.2.d.j.2159.1	yes	4	1.1	even	1	trivial
5616.2.d.j.2159.4	yes	4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5616 = 2^{4} \cdot 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5616.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.52434i q^{5} -0.792287i q^{7} +0.627719 q^{11} +1.00000 q^{13} +3.46410i q^{17} -3.37228 q^{23} -1.37228 q^{25} +5.84096i q^{29} +7.42554i q^{31} -2.00000 q^{35} +0.627719 q^{37} +3.16915i q^{41} +9.94987i q^{43} +3.62772 q^{47} +6.37228 q^{49} +11.1846i q^{53} -1.58457i q^{55} -11.7446 q^{59} +4.37228 q^{61} -2.52434i q^{65} -8.21782i q^{67} +9.11684 q^{71} +1.25544 q^{73} -0.497333i q^{77} +0.294954i q^{79} -6.25544 q^{83} +8.74456 q^{85} -3.61158i q^{89} -0.792287i q^{91} +11.4891 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 14 q^{11} + 4 q^{13} - 2 q^{23} + 6 q^{25} - 8 q^{35} + 14 q^{37} + 26 q^{47} + 14 q^{49} - 24 q^{59} + 6 q^{61} + 2 q^{71} + 28 q^{73} - 48 q^{83} + 12 q^{85}+O(q^{100}) \) 4 * q + 14 * q^11 + 4 * q^13 - 2 * q^23 + 6 * q^25 - 8 * q^35 + 14 * q^37 + 26 * q^47 + 14 * q^49 - 24 * q^59 + 6 * q^61 + 2 * q^71 + 28 * q^73 - 48 * q^83 + 12 * q^85

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