LMFDB - Embedded newform 6160.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6160,2,Mod(1,6160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6160, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6160.a (trivial)

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:	yes
Analytic conductor:	$49.1878476451$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	6160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-2.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +6.74456 q^{13} -2.00000 q^{15} -6.74456 q^{17} +6.74456 q^{19} +2.00000 q^{21} +6.74456 q^{23} +1.00000 q^{25} +4.00000 q^{27} +8.74456 q^{29} -4.74456 q^{31} -2.00000 q^{33} -1.00000 q^{35} +0.744563 q^{37} -13.4891 q^{39} -4.00000 q^{41} -4.00000 q^{43} +1.00000 q^{45} -4.74456 q^{47} +1.00000 q^{49} +13.4891 q^{51} -12.7446 q^{53} +1.00000 q^{55} -13.4891 q^{57} +8.74456 q^{59} +1.25544 q^{61} -1.00000 q^{63} +6.74456 q^{65} +4.00000 q^{67} -13.4891 q^{69} +4.00000 q^{71} +10.7446 q^{73} -2.00000 q^{75} -1.00000 q^{77} -6.74456 q^{79} -11.0000 q^{81} -8.00000 q^{83} -6.74456 q^{85} -17.4891 q^{87} +15.4891 q^{89} -6.74456 q^{91} +9.48913 q^{93} +6.74456 q^{95} -16.7446 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 4 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{15} - 2 q^{17} + 2 q^{19} + 4 q^{21} + 2 q^{23} + 2 q^{25} + 8 q^{27} + 6 q^{29} + 2 q^{31} - 4 q^{33} - 2 q^{35} - 10 q^{37} - 4 q^{39} - 8 q^{41} - 8 q^{43} + 2 q^{45} + 2 q^{47} + 2 q^{49} + 4 q^{51} - 14 q^{53} + 2 q^{55} - 4 q^{57} + 6 q^{59} + 14 q^{61} - 2 q^{63} + 2 q^{65} + 8 q^{67} - 4 q^{69} + 8 q^{71} + 10 q^{73} - 4 q^{75} - 2 q^{77} - 2 q^{79} - 22 q^{81} - 16 q^{83} - 2 q^{85} - 12 q^{87} + 8 q^{89} - 2 q^{91} - 4 q^{93} + 2 q^{95} - 22 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 4 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 - 4 * q^15 - 2 * q^17 + 2 * q^19 + 4 * q^21 + 2 * q^23 + 2 * q^25 + 8 * q^27 + 6 * q^29 + 2 * q^31 - 4 * q^33 - 2 * q^35 - 10 * q^37 - 4 * q^39 - 8 * q^41 - 8 * q^43 + 2 * q^45 + 2 * q^47 + 2 * q^49 + 4 * q^51 - 14 * q^53 + 2 * q^55 - 4 * q^57 + 6 * q^59 + 14 * q^61 - 2 * q^63 + 2 * q^65 + 8 * q^67 - 4 * q^69 + 8 * q^71 + 10 * q^73 - 4 * q^75 - 2 * q^77 - 2 * q^79 - 22 * q^81 - 16 * q^83 - 2 * q^85 - 12 * q^87 + 8 * q^89 - 2 * q^91 - 4 * q^93 + 2 * q^95 - 22 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	6.74456	1.87061	0.935303	−	0.353849i	$-0.115127\pi$
$13$	6.74456	1.87061	0.935303	+	0.353849i	$0.115127\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	−6.74456	−1.63580	−0.817898	−	0.575363i	$-0.804861\pi$
$17$	−6.74456	−1.63580	−0.817898	+	0.575363i	$0.804861\pi$
$18$	0	0
$19$	6.74456	1.54731	0.773654	−	0.633608i	$-0.218427\pi$
$19$	6.74456	1.54731	0.773654	+	0.633608i	$0.218427\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	6.74456	1.40634	0.703169	−	0.711022i	$-0.251768\pi$
$23$	6.74456	1.40634	0.703169	+	0.711022i	$0.251768\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	8.74456	1.62382	0.811912	−	0.583779i	$-0.198427\pi$
$29$	8.74456	1.62382	0.811912	+	0.583779i	$0.198427\pi$
$30$	0	0
$31$	−4.74456	−0.852149	−0.426074	−	0.904688i	$-0.640104\pi$
$31$	−4.74456	−0.852149	−0.426074	+	0.904688i	$0.640104\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	0.744563	0.122405	0.0612027	−	0.998125i	$-0.480506\pi$
$37$	0.744563	0.122405	0.0612027	+	0.998125i	$0.480506\pi$
$38$	0	0
$39$	−13.4891	−2.15999
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−4.74456	−0.692066	−0.346033	−	0.938222i	$-0.612471\pi$
$47$	−4.74456	−0.692066	−0.346033	+	0.938222i	$0.612471\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	13.4891	1.88886
$52$	0	0
$53$	−12.7446	−1.75060	−0.875300	−	0.483580i	$-0.839336\pi$
$53$	−12.7446	−1.75060	−0.875300	+	0.483580i	$0.839336\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−13.4891	−1.78668
$58$	0	0
$59$	8.74456	1.13845	0.569223	−	0.822183i	$-0.307244\pi$
$59$	8.74456	1.13845	0.569223	+	0.822183i	$0.307244\pi$
$60$	0	0
$61$	1.25544	0.160742	0.0803711	−	0.996765i	$-0.474389\pi$
$61$	1.25544	0.160742	0.0803711	+	0.996765i	$0.474389\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	6.74456	0.836560
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	−13.4891	−1.62390
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	10.7446	1.25756	0.628778	−	0.777585i	$-0.283555\pi$
$73$	10.7446	1.25756	0.628778	+	0.777585i	$0.283555\pi$
$74$	0	0
$75$	−2.00000	−0.230940
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−6.74456	−0.758823	−0.379411	−	0.925228i	$-0.623873\pi$
$79$	−6.74456	−0.758823	−0.379411	+	0.925228i	$0.623873\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	−6.74456	−0.731551
$86$	0	0
$87$	−17.4891	−1.87503
$88$	0	0
$89$	15.4891	1.64184	0.820922	−	0.571040i	$-0.193460\pi$
$89$	15.4891	1.64184	0.820922	+	0.571040i	$0.193460\pi$
$90$	0	0
$91$	−6.74456	−0.707022
$92$	0	0
$93$	9.48913	0.983976
$94$	0	0
$95$	6.74456	0.691978
$96$	0	0
$97$	−16.7446	−1.70015	−0.850076	−	0.526659i	$-0.823444\pi$
$97$	−16.7446	−1.70015	−0.850076	+	0.526659i	$0.823444\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	2.74456	0.273094	0.136547	−	0.990634i	$-0.456399\pi$
$101$	2.74456	0.273094	0.136547	+	0.990634i	$0.456399\pi$
$102$	0	0
$103$	0.744563	0.0733639	0.0366820	−	0.999327i	$-0.488321\pi$
$103$	0.744563	0.0733639	0.0366820	+	0.999327i	$0.488321\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−18.2337	−1.74647	−0.873235	−	0.487299i	$-0.837982\pi$
$109$	−18.2337	−1.74647	−0.873235	+	0.487299i	$0.837982\pi$
$110$	0	0
$111$	−1.48913	−0.141342
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	6.74456	0.628934
$116$	0	0
$117$	6.74456	0.623535
$118$	0	0
$119$	6.74456	0.618273
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	8.00000	0.721336
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	2.74456	0.239794	0.119897	−	0.992786i	$-0.461744\pi$
$131$	2.74456	0.239794	0.119897	+	0.992786i	$0.461744\pi$
$132$	0	0
$133$	−6.74456	−0.584828
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	3.48913	0.298096	0.149048	−	0.988830i	$-0.452379\pi$
$137$	3.48913	0.298096	0.149048	+	0.988830i	$0.452379\pi$
$138$	0	0
$139$	−14.7446	−1.25062	−0.625309	−	0.780377i	$-0.715027\pi$
$139$	−14.7446	−1.25062	−0.625309	+	0.780377i	$0.715027\pi$
$140$	0	0
$141$	9.48913	0.799129
$142$	0	0
$143$	6.74456	0.564009
$144$	0	0
$145$	8.74456	0.726196
$146$	0	0
$147$	−2.00000	−0.164957
$148$	0	0
$149$	−0.744563	−0.0609969	−0.0304985	−	0.999535i	$-0.509709\pi$
$149$	−0.744563	−0.0609969	−0.0304985	+	0.999535i	$0.509709\pi$
$150$	0	0
$151$	9.25544	0.753197	0.376598	−	0.926377i	$-0.377094\pi$
$151$	9.25544	0.753197	0.376598	+	0.926377i	$0.377094\pi$
$152$	0	0
$153$	−6.74456	−0.545266
$154$	0	0
$155$	−4.74456	−0.381092
$156$	0	0
$157$	0.510875	0.0407722	0.0203861	−	0.999792i	$-0.493510\pi$
$157$	0.510875	0.0407722	0.0203861	+	0.999792i	$0.493510\pi$
$158$	0	0
$159$	25.4891	2.02142
$160$	0	0
$161$	−6.74456	−0.531546
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	32.4891	2.49916
$170$	0	0
$171$	6.74456	0.515770
$172$	0	0
$173$	20.2337	1.53834	0.769169	−	0.639045i	$-0.220670\pi$
$173$	20.2337	1.53834	0.769169	+	0.639045i	$0.220670\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−17.4891	−1.31456
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−19.4891	−1.44862	−0.724308	−	0.689477i	$-0.757840\pi$
$181$	−19.4891	−1.44862	−0.724308	+	0.689477i	$0.757840\pi$
$182$	0	0
$183$	−2.51087	−0.185609
$184$	0	0
$185$	0.744563	0.0547413
$186$	0	0
$187$	−6.74456	−0.493211
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	−13.4891	−0.965976
$196$	0	0
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	−3.25544	−0.230772	−0.115386	−	0.993321i	$-0.536810\pi$
$199$	−3.25544	−0.230772	−0.115386	+	0.993321i	$0.536810\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	−8.74456	−0.613748
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0	0
$207$	6.74456	0.468780
$208$	0	0
$209$	6.74456	0.466531
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	0	0
$213$	−8.00000	−0.548151
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	4.74456	0.322082
$218$	0	0
$219$	−21.4891	−1.45210
$220$	0	0
$221$	−45.4891	−3.05993
$222$	0	0
$223$	−15.2554	−1.02158	−0.510790	−	0.859706i	$-0.670647\pi$
$223$	−15.2554	−1.02158	−0.510790	+	0.859706i	$0.670647\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	−24.9783	−1.63638	−0.818190	−	0.574948i	$-0.805022\pi$
$233$	−24.9783	−1.63638	−0.818190	+	0.574948i	$0.805022\pi$
$234$	0	0
$235$	−4.74456	−0.309501
$236$	0	0
$237$	13.4891	0.876213
$238$	0	0
$239$	14.7446	0.953746	0.476873	−	0.878972i	$-0.341770\pi$
$239$	14.7446	0.953746	0.476873	+	0.878972i	$0.341770\pi$
$240$	0	0
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	45.4891	2.89440
$248$	0	0
$249$	16.0000	1.01396
$250$	0	0
$251$	26.2337	1.65586	0.827928	−	0.560835i	$-0.189520\pi$
$251$	26.2337	1.65586	0.827928	+	0.560835i	$0.189520\pi$
$252$	0	0
$253$	6.74456	0.424027
$254$	0	0
$255$	13.4891	0.844722
$256$	0	0
$257$	−10.2337	−0.638360	−0.319180	−	0.947694i	$-0.603407\pi$
$257$	−10.2337	−0.638360	−0.319180	+	0.947694i	$0.603407\pi$
$258$	0	0
$259$	−0.744563	−0.0462649
$260$	0	0
$261$	8.74456	0.541275
$262$	0	0
$263$	−26.9783	−1.66355	−0.831775	−	0.555113i	$-0.812675\pi$
$263$	−26.9783	−1.66355	−0.831775	+	0.555113i	$0.812675\pi$
$264$	0	0
$265$	−12.7446	−0.782892
$266$	0	0
$267$	−30.9783	−1.89584
$268$	0	0
$269$	20.9783	1.27907	0.639533	−	0.768763i	$-0.279128\pi$
$269$	20.9783	1.27907	0.639533	+	0.768763i	$0.279128\pi$
$270$	0	0
$271$	−14.9783	−0.909864	−0.454932	−	0.890526i	$-0.650336\pi$
$271$	−14.9783	−0.909864	−0.454932	+	0.890526i	$0.650336\pi$
$272$	0	0
$273$	13.4891	0.816399
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	15.4891	0.930651	0.465326	−	0.885140i	$-0.345937\pi$
$277$	15.4891	0.930651	0.465326	+	0.885140i	$0.345937\pi$
$278$	0	0
$279$	−4.74456	−0.284050
$280$	0	0
$281$	14.0000	0.835170	0.417585	−	0.908638i	$-0.362877\pi$
$281$	14.0000	0.835170	0.417585	+	0.908638i	$0.362877\pi$
$282$	0	0
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	0	0
$285$	−13.4891	−0.799027
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	28.4891	1.67583
$290$	0	0
$291$	33.4891	1.96317
$292$	0	0
$293$	13.2554	0.774391	0.387195	−	0.921998i	$-0.373444\pi$
$293$	13.2554	0.774391	0.387195	+	0.921998i	$0.373444\pi$
$294$	0	0
$295$	8.74456	0.509128
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	45.4891	2.63070
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	−5.48913	−0.315342
$304$	0	0
$305$	1.25544	0.0718861
$306$	0	0
$307$	1.48913	0.0849889	0.0424944	−	0.999097i	$-0.486470\pi$
$307$	1.48913	0.0849889	0.0424944	+	0.999097i	$0.486470\pi$
$308$	0	0
$309$	−1.48913	−0.0847134
$310$	0	0
$311$	20.7446	1.17632	0.588158	−	0.808746i	$-0.299853\pi$
$311$	20.7446	1.17632	0.588158	+	0.808746i	$0.299853\pi$
$312$	0	0
$313$	−2.23369	−0.126256	−0.0631278	−	0.998005i	$-0.520108\pi$
$313$	−2.23369	−0.126256	−0.0631278	+	0.998005i	$0.520108\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	2.23369	0.125456	0.0627282	−	0.998031i	$-0.480020\pi$
$317$	2.23369	0.125456	0.0627282	+	0.998031i	$0.480020\pi$
$318$	0	0
$319$	8.74456	0.489602
$320$	0	0
$321$	−24.0000	−1.33955
$322$	0	0
$323$	−45.4891	−2.53108
$324$	0	0
$325$	6.74456	0.374121
$326$	0	0
$327$	36.4674	2.01665
$328$	0	0
$329$	4.74456	0.261576
$330$	0	0
$331$	−14.9783	−0.823279	−0.411640	−	0.911347i	$-0.635044\pi$
$331$	−14.9783	−0.823279	−0.411640	+	0.911347i	$0.635044\pi$
$332$	0	0
$333$	0.744563	0.0408018
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	0	0
$339$	−20.0000	−1.08625
$340$	0	0
$341$	−4.74456	−0.256932
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−13.4891	−0.726230
$346$	0	0
$347$	22.9783	1.23354	0.616769	−	0.787145i	$-0.288441\pi$
$347$	22.9783	1.23354	0.616769	+	0.787145i	$0.288441\pi$
$348$	0	0
$349$	30.7446	1.64572	0.822859	−	0.568245i	$-0.192377\pi$
$349$	30.7446	1.64572	0.822859	+	0.568245i	$0.192377\pi$
$350$	0	0
$351$	26.9783	1.43999
$352$	0	0
$353$	8.74456	0.465426	0.232713	−	0.972545i	$-0.425240\pi$
$353$	8.74456	0.465426	0.232713	+	0.972545i	$0.425240\pi$
$354$	0	0
$355$	4.00000	0.212298
$356$	0	0
$357$	−13.4891	−0.713920
$358$	0	0
$359$	1.25544	0.0662594	0.0331297	−	0.999451i	$-0.489453\pi$
$359$	1.25544	0.0662594	0.0331297	+	0.999451i	$0.489453\pi$
$360$	0	0
$361$	26.4891	1.39416
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	10.7446	0.562396
$366$	0	0
$367$	8.74456	0.456462	0.228231	−	0.973607i	$-0.426706\pi$
$367$	8.74456	0.456462	0.228231	+	0.973607i	$0.426706\pi$
$368$	0	0
$369$	−4.00000	−0.208232
$370$	0	0
$371$	12.7446	0.661665
$372$	0	0
$373$	−16.9783	−0.879100	−0.439550	−	0.898218i	$-0.644862\pi$
$373$	−16.9783	−0.879100	−0.439550	+	0.898218i	$0.644862\pi$
$374$	0	0
$375$	−2.00000	−0.103280
$376$	0	0
$377$	58.9783	3.03753
$378$	0	0
$379$	37.4891	1.92569	0.962844	−	0.270060i	$-0.0870435\pi$
$379$	37.4891	1.92569	0.962844	+	0.270060i	$0.0870435\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−19.7228	−1.00779	−0.503894	−	0.863765i	$-0.668100\pi$
$383$	−19.7228	−1.00779	−0.503894	+	0.863765i	$0.668100\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	−7.48913	−0.379714	−0.189857	−	0.981812i	$-0.560802\pi$
$389$	−7.48913	−0.379714	−0.189857	+	0.981812i	$0.560802\pi$
$390$	0	0
$391$	−45.4891	−2.30048
$392$	0	0
$393$	−5.48913	−0.276890
$394$	0	0
$395$	−6.74456	−0.339356
$396$	0	0
$397$	35.4891	1.78115	0.890574	−	0.454838i	$-0.150303\pi$
$397$	35.4891	1.78115	0.890574	+	0.454838i	$0.150303\pi$
$398$	0	0
$399$	13.4891	0.675301
$400$	0	0
$401$	23.4891	1.17299	0.586495	−	0.809953i	$-0.300507\pi$
$401$	23.4891	1.17299	0.586495	+	0.809953i	$0.300507\pi$
$402$	0	0
$403$	−32.0000	−1.59403
$404$	0	0
$405$	−11.0000	−0.546594
$406$	0	0
$407$	0.744563	0.0369066
$408$	0	0
$409$	−21.4891	−1.06257	−0.531284	−	0.847194i	$-0.678290\pi$
$409$	−21.4891	−1.06257	−0.531284	+	0.847194i	$0.678290\pi$
$410$	0	0
$411$	−6.97825	−0.344212
$412$	0	0
$413$	−8.74456	−0.430292
$414$	0	0
$415$	−8.00000	−0.392705
$416$	0	0
$417$	29.4891	1.44409
$418$	0	0
$419$	0.744563	0.0363743	0.0181871	−	0.999835i	$-0.494211\pi$
$419$	0.744563	0.0363743	0.0181871	+	0.999835i	$0.494211\pi$
$420$	0	0
$421$	4.51087	0.219847	0.109923	−	0.993940i	$-0.464939\pi$
$421$	4.51087	0.219847	0.109923	+	0.993940i	$0.464939\pi$
$422$	0	0
$423$	−4.74456	−0.230689
$424$	0	0
$425$	−6.74456	−0.327159
$426$	0	0
$427$	−1.25544	−0.0607549
$428$	0	0
$429$	−13.4891	−0.651261
$430$	0	0
$431$	17.2554	0.831165	0.415583	−	0.909555i	$-0.363578\pi$
$431$	17.2554	0.831165	0.415583	+	0.909555i	$0.363578\pi$
$432$	0	0
$433$	36.7446	1.76583	0.882915	−	0.469532i	$-0.155577\pi$
$433$	36.7446	1.76583	0.882915	+	0.469532i	$0.155577\pi$
$434$	0	0
$435$	−17.4891	−0.838539
$436$	0	0
$437$	45.4891	2.17604
$438$	0	0
$439$	−14.9783	−0.714873	−0.357436	−	0.933937i	$-0.616349\pi$
$439$	−14.9783	−0.714873	−0.357436	+	0.933937i	$0.616349\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	29.4891	1.40107	0.700535	−	0.713618i	$-0.252945\pi$
$443$	29.4891	1.40107	0.700535	+	0.713618i	$0.252945\pi$
$444$	0	0
$445$	15.4891	0.734255
$446$	0	0
$447$	1.48913	0.0704332
$448$	0	0
$449$	−4.97825	−0.234938	−0.117469	−	0.993077i	$-0.537478\pi$
$449$	−4.97825	−0.234938	−0.117469	+	0.993077i	$0.537478\pi$
$450$	0	0
$451$	−4.00000	−0.188353
$452$	0	0
$453$	−18.5109	−0.869717
$454$	0	0
$455$	−6.74456	−0.316190
$456$	0	0
$457$	3.48913	0.163214	0.0816072	−	0.996665i	$-0.473995\pi$
$457$	3.48913	0.163214	0.0816072	+	0.996665i	$0.473995\pi$
$458$	0	0
$459$	−26.9783	−1.25924
$460$	0	0
$461$	33.7228	1.57063	0.785314	−	0.619098i	$-0.212502\pi$
$461$	33.7228	1.57063	0.785314	+	0.619098i	$0.212502\pi$
$462$	0	0
$463$	−1.25544	−0.0583451	−0.0291726	−	0.999574i	$-0.509287\pi$
$463$	−1.25544	−0.0583451	−0.0291726	+	0.999574i	$0.509287\pi$
$464$	0	0
$465$	9.48913	0.440048
$466$	0	0
$467$	−16.9783	−0.785660	−0.392830	−	0.919611i	$-0.628504\pi$
$467$	−16.9783	−0.785660	−0.392830	+	0.919611i	$0.628504\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−1.02175	−0.0470797
$472$	0	0
$473$	−4.00000	−0.183920
$474$	0	0
$475$	6.74456	0.309462
$476$	0	0
$477$	−12.7446	−0.583533
$478$	0	0
$479$	41.4891	1.89569	0.947843	−	0.318737i	$-0.103259\pi$
$479$	41.4891	1.89569	0.947843	+	0.318737i	$0.103259\pi$
$480$	0	0
$481$	5.02175	0.228972
$482$	0	0
$483$	13.4891	0.613776
$484$	0	0
$485$	−16.7446	−0.760331
$486$	0	0
$487$	−9.25544	−0.419404	−0.209702	−	0.977765i	$-0.567249\pi$
$487$	−9.25544	−0.419404	−0.209702	+	0.977765i	$0.567249\pi$
$488$	0	0
$489$	−8.00000	−0.361773
$490$	0	0
$491$	−30.9783	−1.39803	−0.699014	−	0.715108i	$-0.746378\pi$
$491$	−30.9783	−1.39803	−0.699014	+	0.715108i	$0.746378\pi$
$492$	0	0
$493$	−58.9783	−2.65625
$494$	0	0
$495$	1.00000	0.0449467
$496$	0	0
$497$	−4.00000	−0.179425
$498$	0	0
$499$	13.4891	0.603856	0.301928	−	0.953331i	$-0.402370\pi$
$499$	13.4891	0.603856	0.301928	+	0.953331i	$0.402370\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−6.51087	−0.290306	−0.145153	−	0.989409i	$-0.546367\pi$
$503$	−6.51087	−0.290306	−0.145153	+	0.989409i	$0.546367\pi$
$504$	0	0
$505$	2.74456	0.122131
$506$	0	0
$507$	−64.9783	−2.88579
$508$	0	0
$509$	−35.4891	−1.57303	−0.786514	−	0.617573i	$-0.788116\pi$
$509$	−35.4891	−1.57303	−0.786514	+	0.617573i	$0.788116\pi$
$510$	0	0
$511$	−10.7446	−0.475311
$512$	0	0
$513$	26.9783	1.19112
$514$	0	0
$515$	0.744563	0.0328094
$516$	0	0
$517$	−4.74456	−0.208666
$518$	0	0
$519$	−40.4674	−1.77632
$520$	0	0
$521$	−2.00000	−0.0876216	−0.0438108	−	0.999040i	$-0.513950\pi$
$521$	−2.00000	−0.0876216	−0.0438108	+	0.999040i	$0.513950\pi$
$522$	0	0
$523$	17.4891	0.764746	0.382373	−	0.924008i	$-0.375107\pi$
$523$	17.4891	0.764746	0.382373	+	0.924008i	$0.375107\pi$
$524$	0	0
$525$	2.00000	0.0872872
$526$	0	0
$527$	32.0000	1.39394
$528$	0	0
$529$	22.4891	0.977788
$530$	0	0
$531$	8.74456	0.379482
$532$	0	0
$533$	−26.9783	−1.16856
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	−8.00000	−0.345225
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	1.76631	0.0759397	0.0379698	−	0.999279i	$-0.487911\pi$
$541$	1.76631	0.0759397	0.0379698	+	0.999279i	$0.487911\pi$
$542$	0	0
$543$	38.9783	1.67272
$544$	0	0
$545$	−18.2337	−0.781045
$546$	0	0
$547$	−14.9783	−0.640424	−0.320212	−	0.947346i	$-0.603754\pi$
$547$	−14.9783	−0.640424	−0.320212	+	0.947346i	$0.603754\pi$
$548$	0	0
$549$	1.25544	0.0535808
$550$	0	0
$551$	58.9783	2.51256
$552$	0	0
$553$	6.74456	0.286808
$554$	0	0
$555$	−1.48913	−0.0632098
$556$	0	0
$557$	−0.978251	−0.0414498	−0.0207249	−	0.999785i	$-0.506597\pi$
$557$	−0.978251	−0.0414498	−0.0207249	+	0.999785i	$0.506597\pi$
$558$	0	0
$559$	−26.9783	−1.14106
$560$	0	0
$561$	13.4891	0.569511
$562$	0	0
$563$	−5.48913	−0.231339	−0.115670	−	0.993288i	$-0.536901\pi$
$563$	−5.48913	−0.231339	−0.115670	+	0.993288i	$0.536901\pi$
$564$	0	0
$565$	10.0000	0.420703
$566$	0	0
$567$	11.0000	0.461957
$568$	0	0
$569$	16.5109	0.692172	0.346086	−	0.938203i	$-0.387511\pi$
$569$	16.5109	0.692172	0.346086	+	0.938203i	$0.387511\pi$
$570$	0	0
$571$	17.4891	0.731897	0.365949	−	0.930635i	$-0.380745\pi$
$571$	17.4891	0.731897	0.365949	+	0.930635i	$0.380745\pi$
$572$	0	0
$573$	−32.0000	−1.33682
$574$	0	0
$575$	6.74456	0.281268
$576$	0	0
$577$	−8.74456	−0.364041	−0.182020	−	0.983295i	$-0.558264\pi$
$577$	−8.74456	−0.364041	−0.182020	+	0.983295i	$0.558264\pi$
$578$	0	0
$579$	4.00000	0.166234
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	−12.7446	−0.527826
$584$	0	0
$585$	6.74456	0.278853
$586$	0	0
$587$	4.97825	0.205474	0.102737	−	0.994709i	$-0.467240\pi$
$587$	4.97825	0.205474	0.102737	+	0.994709i	$0.467240\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	−20.0000	−0.822690
$592$	0	0
$593$	−40.2337	−1.65220	−0.826100	−	0.563524i	$-0.809445\pi$
$593$	−40.2337	−1.65220	−0.826100	+	0.563524i	$0.809445\pi$
$594$	0	0
$595$	6.74456	0.276500
$596$	0	0
$597$	6.51087	0.266472
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−33.4891	−1.36605	−0.683025	−	0.730395i	$-0.739336\pi$
$601$	−33.4891	−1.36605	−0.683025	+	0.730395i	$0.739336\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	17.4891	0.708695
$610$	0	0
$611$	−32.0000	−1.29458
$612$	0	0
$613$	11.4891	0.464041	0.232021	−	0.972711i	$-0.425466\pi$
$613$	11.4891	0.464041	0.232021	+	0.972711i	$0.425466\pi$
$614$	0	0
$615$	8.00000	0.322591
$616$	0	0
$617$	−10.0000	−0.402585	−0.201292	−	0.979531i	$-0.564514\pi$
$617$	−10.0000	−0.402585	−0.201292	+	0.979531i	$0.564514\pi$
$618$	0	0
$619$	−0.744563	−0.0299265	−0.0149632	−	0.999888i	$-0.504763\pi$
$619$	−0.744563	−0.0299265	−0.0149632	+	0.999888i	$0.504763\pi$
$620$	0	0
$621$	26.9783	1.08260
$622$	0	0
$623$	−15.4891	−0.620559
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−13.4891	−0.538704
$628$	0	0
$629$	−5.02175	−0.200230
$630$	0	0
$631$	2.97825	0.118562	0.0592811	−	0.998241i	$-0.481119\pi$
$631$	2.97825	0.118562	0.0592811	+	0.998241i	$0.481119\pi$
$632$	0	0
$633$	−24.0000	−0.953914
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.74456	0.267229
$638$	0	0
$639$	4.00000	0.158238
$640$	0	0
$641$	11.4891	0.453793	0.226897	−	0.973919i	$-0.427142\pi$
$641$	11.4891	0.453793	0.226897	+	0.973919i	$0.427142\pi$
$642$	0	0
$643$	−46.4674	−1.83249	−0.916247	−	0.400613i	$-0.868797\pi$
$643$	−46.4674	−1.83249	−0.916247	+	0.400613i	$0.868797\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	−8.74456	−0.343784	−0.171892	−	0.985116i	$-0.554988\pi$
$647$	−8.74456	−0.343784	−0.171892	+	0.985116i	$0.554988\pi$
$648$	0	0
$649$	8.74456	0.343254
$650$	0	0
$651$	−9.48913	−0.371908
$652$	0	0
$653$	15.7228	0.615281	0.307641	−	0.951503i	$-0.400461\pi$
$653$	15.7228	0.615281	0.307641	+	0.951503i	$0.400461\pi$
$654$	0	0
$655$	2.74456	0.107239
$656$	0	0
$657$	10.7446	0.419185
$658$	0	0
$659$	41.4891	1.61619	0.808093	−	0.589054i	$-0.200500\pi$
$659$	41.4891	1.61619	0.808093	+	0.589054i	$0.200500\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	90.9783	3.53330
$664$	0	0
$665$	−6.74456	−0.261543
$666$	0	0
$667$	58.9783	2.28365
$668$	0	0
$669$	30.5109	1.17962
$670$	0	0
$671$	1.25544	0.0484656
$672$	0	0
$673$	20.9783	0.808652	0.404326	−	0.914615i	$-0.367506\pi$
$673$	20.9783	0.808652	0.404326	+	0.914615i	$0.367506\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	36.2337	1.39257	0.696287	−	0.717764i	$-0.254834\pi$
$677$	36.2337	1.39257	0.696287	+	0.717764i	$0.254834\pi$
$678$	0	0
$679$	16.7446	0.642597
$680$	0	0
$681$	40.0000	1.53280
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	3.48913	0.133313
$686$	0	0
$687$	12.0000	0.457829
$688$	0	0
$689$	−85.9565	−3.27468
$690$	0	0
$691$	−1.76631	−0.0671937	−0.0335968	−	0.999435i	$-0.510696\pi$
$691$	−1.76631	−0.0671937	−0.0335968	+	0.999435i	$0.510696\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	−14.7446	−0.559293
$696$	0	0
$697$	26.9783	1.02187
$698$	0	0
$699$	49.9565	1.88953
$700$	0	0
$701$	22.2337	0.839755	0.419877	−	0.907581i	$-0.362073\pi$
$701$	22.2337	0.839755	0.419877	+	0.907581i	$0.362073\pi$
$702$	0	0
$703$	5.02175	0.189399
$704$	0	0
$705$	9.48913	0.357381
$706$	0	0
$707$	−2.74456	−0.103220
$708$	0	0
$709$	0.510875	0.0191863	0.00959315	−	0.999954i	$-0.496946\pi$
$709$	0.510875	0.0191863	0.00959315	+	0.999954i	$0.496946\pi$
$710$	0	0
$711$	−6.74456	−0.252941
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	6.74456	0.252232
$716$	0	0
$717$	−29.4891	−1.10129
$718$	0	0
$719$	−7.72281	−0.288012	−0.144006	−	0.989577i	$-0.545999\pi$
$719$	−7.72281	−0.288012	−0.144006	+	0.989577i	$0.545999\pi$
$720$	0	0
$721$	−0.744563	−0.0277290
$722$	0	0
$723$	−40.0000	−1.48762
$724$	0	0
$725$	8.74456	0.324765
$726$	0	0
$727$	−14.2337	−0.527898	−0.263949	−	0.964537i	$-0.585025\pi$
$727$	−14.2337	−0.527898	−0.263949	+	0.964537i	$0.585025\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	26.9783	0.997827
$732$	0	0
$733$	−16.2337	−0.599605	−0.299802	−	0.954001i	$-0.596921\pi$
$733$	−16.2337	−0.599605	−0.299802	+	0.954001i	$0.596921\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	4.00000	0.147342
$738$	0	0
$739$	30.9783	1.13955	0.569777	−	0.821800i	$-0.307030\pi$
$739$	30.9783	1.13955	0.569777	+	0.821800i	$0.307030\pi$
$740$	0	0
$741$	−90.9783	−3.34217
$742$	0	0
$743$	−26.9783	−0.989736	−0.494868	−	0.868968i	$-0.664784\pi$
$743$	−26.9783	−0.989736	−0.494868	+	0.868968i	$0.664784\pi$
$744$	0	0
$745$	−0.744563	−0.0272787
$746$	0	0
$747$	−8.00000	−0.292705
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	0	0
$753$	−52.4674	−1.91202
$754$	0	0
$755$	9.25544	0.336840
$756$	0	0
$757$	−19.2554	−0.699851	−0.349925	−	0.936778i	$-0.613793\pi$
$757$	−19.2554	−0.699851	−0.349925	+	0.936778i	$0.613793\pi$
$758$	0	0
$759$	−13.4891	−0.489624
$760$	0	0
$761$	−29.4891	−1.06898	−0.534490	−	0.845175i	$-0.679496\pi$
$761$	−29.4891	−1.06898	−0.534490	+	0.845175i	$0.679496\pi$
$762$	0	0
$763$	18.2337	0.660104
$764$	0	0
$765$	−6.74456	−0.243850
$766$	0	0
$767$	58.9783	2.12958
$768$	0	0
$769$	−8.00000	−0.288487	−0.144244	−	0.989542i	$-0.546075\pi$
$769$	−8.00000	−0.288487	−0.144244	+	0.989542i	$0.546075\pi$
$770$	0	0
$771$	20.4674	0.737115
$772$	0	0
$773$	−51.4891	−1.85194	−0.925968	−	0.377603i	$-0.876748\pi$
$773$	−51.4891	−1.85194	−0.925968	+	0.377603i	$0.876748\pi$
$774$	0	0
$775$	−4.74456	−0.170430
$776$	0	0
$777$	1.48913	0.0534221
$778$	0	0
$779$	−26.9783	−0.966596
$780$	0	0
$781$	4.00000	0.143131
$782$	0	0
$783$	34.9783	1.25002
$784$	0	0
$785$	0.510875	0.0182339
$786$	0	0
$787$	−5.48913	−0.195666	−0.0978331	−	0.995203i	$-0.531191\pi$
$787$	−5.48913	−0.195666	−0.0978331	+	0.995203i	$0.531191\pi$
$788$	0	0
$789$	53.9565	1.92090
$790$	0	0
$791$	−10.0000	−0.355559
$792$	0	0
$793$	8.46738	0.300685
$794$	0	0
$795$	25.4891	0.904006
$796$	0	0
$797$	−42.4674	−1.50427	−0.752136	−	0.659008i	$-0.770976\pi$
$797$	−42.4674	−1.50427	−0.752136	+	0.659008i	$0.770976\pi$
$798$	0	0
$799$	32.0000	1.13208
$800$	0	0
$801$	15.4891	0.547281
$802$	0	0
$803$	10.7446	0.379167
$804$	0	0
$805$	−6.74456	−0.237715
$806$	0	0
$807$	−41.9565	−1.47694
$808$	0	0
$809$	34.4674	1.21181	0.605904	−	0.795538i	$-0.292811\pi$
$809$	34.4674	1.21181	0.605904	+	0.795538i	$0.292811\pi$
$810$	0	0
$811$	18.7446	0.658211	0.329105	−	0.944293i	$-0.393253\pi$
$811$	18.7446	0.658211	0.329105	+	0.944293i	$0.393253\pi$
$812$	0	0
$813$	29.9565	1.05062
$814$	0	0
$815$	4.00000	0.140114
$816$	0	0
$817$	−26.9783	−0.943850
$818$	0	0
$819$	−6.74456	−0.235674
$820$	0	0
$821$	7.72281	0.269528	0.134764	−	0.990878i	$-0.456972\pi$
$821$	7.72281	0.269528	0.134764	+	0.990878i	$0.456972\pi$
$822$	0	0
$823$	−7.76631	−0.270717	−0.135358	−	0.990797i	$-0.543219\pi$
$823$	−7.76631	−0.270717	−0.135358	+	0.990797i	$0.543219\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	−4.00000	−0.139094	−0.0695468	−	0.997579i	$-0.522155\pi$
$827$	−4.00000	−0.139094	−0.0695468	+	0.997579i	$0.522155\pi$
$828$	0	0
$829$	14.4674	0.502473	0.251236	−	0.967926i	$-0.419163\pi$
$829$	14.4674	0.502473	0.251236	+	0.967926i	$0.419163\pi$
$830$	0	0
$831$	−30.9783	−1.07462
$832$	0	0
$833$	−6.74456	−0.233685
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−18.9783	−0.655984
$838$	0	0
$839$	3.25544	0.112390	0.0561951	−	0.998420i	$-0.482103\pi$
$839$	3.25544	0.112390	0.0561951	+	0.998420i	$0.482103\pi$
$840$	0	0
$841$	47.4674	1.63681
$842$	0	0
$843$	−28.0000	−0.964371
$844$	0	0
$845$	32.4891	1.11766
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	56.0000	1.92192
$850$	0	0
$851$	5.02175	0.172143
$852$	0	0
$853$	−22.7446	−0.778759	−0.389379	−	0.921077i	$-0.627311\pi$
$853$	−22.7446	−0.778759	−0.389379	+	0.921077i	$0.627311\pi$
$854$	0	0
$855$	6.74456	0.230659
$856$	0	0
$857$	−43.2119	−1.47609	−0.738046	−	0.674751i	$-0.764251\pi$
$857$	−43.2119	−1.47609	−0.738046	+	0.674751i	$0.764251\pi$
$858$	0	0
$859$	29.2119	0.996698	0.498349	−	0.866976i	$-0.333940\pi$
$859$	29.2119	0.996698	0.498349	+	0.866976i	$0.333940\pi$
$860$	0	0
$861$	−8.00000	−0.272639
$862$	0	0
$863$	40.2337	1.36957	0.684785	−	0.728745i	$-0.259896\pi$
$863$	40.2337	1.36957	0.684785	+	0.728745i	$0.259896\pi$
$864$	0	0
$865$	20.2337	0.687966
$866$	0	0
$867$	−56.9783	−1.93508
$868$	0	0
$869$	−6.74456	−0.228794
$870$	0	0
$871$	26.9783	0.914123
$872$	0	0
$873$	−16.7446	−0.566718
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−26.4674	−0.893740	−0.446870	−	0.894599i	$-0.647461\pi$
$877$	−26.4674	−0.893740	−0.446870	+	0.894599i	$0.647461\pi$
$878$	0	0
$879$	−26.5109	−0.894190
$880$	0	0
$881$	−55.4891	−1.86948	−0.934738	−	0.355338i	$-0.884366\pi$
$881$	−55.4891	−1.86948	−0.934738	+	0.355338i	$0.884366\pi$
$882$	0	0
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	0	0
$885$	−17.4891	−0.587891
$886$	0	0
$887$	41.4891	1.39307	0.696534	−	0.717524i	$-0.254724\pi$
$887$	41.4891	1.39307	0.696534	+	0.717524i	$0.254724\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−11.0000	−0.368514
$892$	0	0
$893$	−32.0000	−1.07084
$894$	0	0
$895$	4.00000	0.133705
$896$	0	0
$897$	−90.9783	−3.03768
$898$	0	0
$899$	−41.4891	−1.38374
$900$	0	0
$901$	85.9565	2.86363
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	−19.4891	−0.647840
$906$	0	0
$907$	14.5109	0.481826	0.240913	−	0.970547i	$-0.422553\pi$
$907$	14.5109	0.481826	0.240913	+	0.970547i	$0.422553\pi$
$908$	0	0
$909$	2.74456	0.0910314
$910$	0	0
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	−2.51087	−0.0830070
$916$	0	0
$917$	−2.74456	−0.0906334
$918$	0	0
$919$	55.2119	1.82127	0.910637	−	0.413207i	$-0.135592\pi$
$919$	55.2119	1.82127	0.910637	+	0.413207i	$0.135592\pi$
$920$	0	0
$921$	−2.97825	−0.0981367
$922$	0	0
$923$	26.9783	0.888000
$924$	0	0
$925$	0.744563	0.0244811
$926$	0	0
$927$	0.744563	0.0244546
$928$	0	0
$929$	−28.9783	−0.950746	−0.475373	−	0.879784i	$-0.657687\pi$
$929$	−28.9783	−0.950746	−0.475373	+	0.879784i	$0.657687\pi$
$930$	0	0
$931$	6.74456	0.221044
$932$	0	0
$933$	−41.4891	−1.35829
$934$	0	0
$935$	−6.74456	−0.220571
$936$	0	0
$937$	−18.7446	−0.612358	−0.306179	−	0.951974i	$-0.599051\pi$
$937$	−18.7446	−0.612358	−0.306179	+	0.951974i	$0.599051\pi$
$938$	0	0
$939$	4.46738	0.145787
$940$	0	0
$941$	12.2337	0.398807	0.199403	−	0.979917i	$-0.436100\pi$
$941$	12.2337	0.398807	0.199403	+	0.979917i	$0.436100\pi$
$942$	0	0
$943$	−26.9783	−0.878533
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	8.00000	0.259965	0.129983	−	0.991516i	$-0.458508\pi$
$947$	8.00000	0.259965	0.129983	+	0.991516i	$0.458508\pi$
$948$	0	0
$949$	72.4674	2.35239
$950$	0	0
$951$	−4.46738	−0.144865
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	16.0000	0.517748
$956$	0	0
$957$	−17.4891	−0.565343
$958$	0	0
$959$	−3.48913	−0.112670
$960$	0	0
$961$	−8.48913	−0.273843
$962$	0	0
$963$	12.0000	0.386695
$964$	0	0
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	50.9783	1.63935	0.819675	−	0.572829i	$-0.194154\pi$
$967$	50.9783	1.63935	0.819675	+	0.572829i	$0.194154\pi$
$968$	0	0
$969$	90.9783	2.92264
$970$	0	0
$971$	−19.7228	−0.632935	−0.316468	−	0.948603i	$-0.602497\pi$
$971$	−19.7228	−0.632935	−0.316468	+	0.948603i	$0.602497\pi$
$972$	0	0
$973$	14.7446	0.472689
$974$	0	0
$975$	−13.4891	−0.431998
$976$	0	0
$977$	12.9783	0.415211	0.207606	−	0.978213i	$-0.433433\pi$
$977$	12.9783	0.415211	0.207606	+	0.978213i	$0.433433\pi$
$978$	0	0
$979$	15.4891	0.495035
$980$	0	0
$981$	−18.2337	−0.582157
$982$	0	0
$983$	−2.23369	−0.0712436	−0.0356218	−	0.999365i	$-0.511341\pi$
$983$	−2.23369	−0.0712436	−0.0356218	+	0.999365i	$0.511341\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	0	0
$987$	−9.48913	−0.302042
$988$	0	0
$989$	−26.9783	−0.857858
$990$	0	0
$991$	−1.48913	−0.0473036	−0.0236518	−	0.999720i	$-0.507529\pi$
$991$	−1.48913	−0.0473036	−0.0236518	+	0.999720i	$0.507529\pi$
$992$	0	0
$993$	29.9565	0.950641
$994$	0	0
$995$	−3.25544	−0.103204
$996$	0	0
$997$	39.2119	1.24185	0.620927	−	0.783868i	$-0.286756\pi$
$997$	39.2119	1.24185	0.620927	+	0.783868i	$0.286756\pi$
$998$	0	0
$999$	2.97825	0.0942277

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.r.1.2		2
4.3	odd	2		770.2.a.k.1.2	✓	2
12.11	even	2		6930.2.a.bo.1.2		2
20.3	even	4		3850.2.c.y.1849.2		4
20.7	even	4		3850.2.c.y.1849.3		4
20.19	odd	2		3850.2.a.bc.1.1		2
28.27	even	2		5390.2.a.bq.1.1		2
44.43	even	2		8470.2.a.bu.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.k.1.2	✓	2	4.3	odd	2
3850.2.a.bc.1.1		2	20.19	odd	2
3850.2.c.y.1849.2		4	20.3	even	4
3850.2.c.y.1849.3		4	20.7	even	4
5390.2.a.bq.1.1		2	28.27	even	2
6160.2.a.r.1.2		2	1.1	even	1	trivial
6930.2.a.bo.1.2		2	12.11	even	2
8470.2.a.bu.1.1		2	44.43	even	2

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:	\( N \)	\(=\)	\( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6160.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +6.74456 q^{13} -2.00000 q^{15} -6.74456 q^{17} +6.74456 q^{19} +2.00000 q^{21} +6.74456 q^{23} +1.00000 q^{25} +4.00000 q^{27} +8.74456 q^{29} -4.74456 q^{31} -2.00000 q^{33} -1.00000 q^{35} +0.744563 q^{37} -13.4891 q^{39} -4.00000 q^{41} -4.00000 q^{43} +1.00000 q^{45} -4.74456 q^{47} +1.00000 q^{49} +13.4891 q^{51} -12.7446 q^{53} +1.00000 q^{55} -13.4891 q^{57} +8.74456 q^{59} +1.25544 q^{61} -1.00000 q^{63} +6.74456 q^{65} +4.00000 q^{67} -13.4891 q^{69} +4.00000 q^{71} +10.7446 q^{73} -2.00000 q^{75} -1.00000 q^{77} -6.74456 q^{79} -11.0000 q^{81} -8.00000 q^{83} -6.74456 q^{85} -17.4891 q^{87} +15.4891 q^{89} -6.74456 q^{91} +9.48913 q^{93} +6.74456 q^{95} -16.7446 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 4 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{15} - 2 q^{17} + 2 q^{19} + 4 q^{21} + 2 q^{23} + 2 q^{25} + 8 q^{27} + 6 q^{29} + 2 q^{31} - 4 q^{33} - 2 q^{35} - 10 q^{37} - 4 q^{39} - 8 q^{41} - 8 q^{43} + 2 q^{45} + 2 q^{47} + 2 q^{49} + 4 q^{51} - 14 q^{53} + 2 q^{55} - 4 q^{57} + 6 q^{59} + 14 q^{61} - 2 q^{63} + 2 q^{65} + 8 q^{67} - 4 q^{69} + 8 q^{71} + 10 q^{73} - 4 q^{75} - 2 q^{77} - 2 q^{79} - 22 q^{81} - 16 q^{83} - 2 q^{85} - 12 q^{87} + 8 q^{89} - 2 q^{91} - 4 q^{93} + 2 q^{95} - 22 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 4 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 - 4 * q^15 - 2 * q^17 + 2 * q^19 + 4 * q^21 + 2 * q^23 + 2 * q^25 + 8 * q^27 + 6 * q^29 + 2 * q^31 - 4 * q^33 - 2 * q^35 - 10 * q^37 - 4 * q^39 - 8 * q^41 - 8 * q^43 + 2 * q^45 + 2 * q^47 + 2 * q^49 + 4 * q^51 - 14 * q^53 + 2 * q^55 - 4 * q^57 + 6 * q^59 + 14 * q^61 - 2 * q^63 + 2 * q^65 + 8 * q^67 - 4 * q^69 + 8 * q^71 + 10 * q^73 - 4 * q^75 - 2 * q^77 - 2 * q^79 - 22 * q^81 - 16 * q^83 - 2 * q^85 - 12 * q^87 + 8 * q^89 - 2 * q^91 - 4 * q^93 + 2 * q^95 - 22 * q^97 + 2 * q^99

Introduction

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