LMFDB - Embedded newform 6160.2.a.r.1.1

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			1.1
Root			$3.37228$ 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)

$f(q)$	$=$	$q-2.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-2.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -4.74456 q^{13} -2.00000 q^{15} +4.74456 q^{17} -4.74456 q^{19} +2.00000 q^{21} -4.74456 q^{23} +1.00000 q^{25} +4.00000 q^{27} -2.74456 q^{29} +6.74456 q^{31} -2.00000 q^{33} -1.00000 q^{35} -10.7446 q^{37} +9.48913 q^{39} -4.00000 q^{41} -4.00000 q^{43} +1.00000 q^{45} +6.74456 q^{47} +1.00000 q^{49} -9.48913 q^{51} -1.25544 q^{53} +1.00000 q^{55} +9.48913 q^{57} -2.74456 q^{59} +12.7446 q^{61} -1.00000 q^{63} -4.74456 q^{65} +4.00000 q^{67} +9.48913 q^{69} +4.00000 q^{71} -0.744563 q^{73} -2.00000 q^{75} -1.00000 q^{77} +4.74456 q^{79} -11.0000 q^{81} -8.00000 q^{83} +4.74456 q^{85} +5.48913 q^{87} -7.48913 q^{89} +4.74456 q^{91} -13.4891 q^{93} -4.74456 q^{95} -5.25544 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$13$	−4.74456	−1.31590	−0.657952	−	0.753059i	$-0.728577\pi$
$13$	−4.74456	−1.31590	−0.657952	+	0.753059i	$0.728577\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	4.74456	1.15073	0.575363	−	0.817898i	$-0.304861\pi$
$17$	4.74456	1.15073	0.575363	+	0.817898i	$0.304861\pi$
$18$	0	0
$19$	−4.74456	−1.08848	−0.544239	−	0.838930i	$-0.683181\pi$
$19$	−4.74456	−1.08848	−0.544239	+	0.838930i	$0.683181\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	−4.74456	−0.989310	−0.494655	−	0.869090i	$-0.664706\pi$
$23$	−4.74456	−0.989310	−0.494655	+	0.869090i	$0.664706\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	−2.74456	−0.509652	−0.254826	−	0.966987i	$-0.582018\pi$
$29$	−2.74456	−0.509652	−0.254826	+	0.966987i	$0.582018\pi$
$30$	0	0
$31$	6.74456	1.21136	0.605680	−	0.795709i	$-0.292901\pi$
$31$	6.74456	1.21136	0.605680	+	0.795709i	$0.292901\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−10.7446	−1.76640	−0.883198	−	0.469001i	$-0.844614\pi$
$37$	−10.7446	−1.76640	−0.883198	+	0.469001i	$0.844614\pi$
$38$	0	0
$39$	9.48913	1.51948
$40$	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
$47$	6.74456	0.983796	0.491898	−	0.870653i	$-0.336303\pi$
$47$	6.74456	0.983796	0.491898	+	0.870653i	$0.336303\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−9.48913	−1.32874
$52$	0	0
$53$	−1.25544	−0.172448	−0.0862238	−	0.996276i	$-0.527480\pi$
$53$	−1.25544	−0.172448	−0.0862238	+	0.996276i	$0.527480\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	9.48913	1.25687
$58$	0	0
$59$	−2.74456	−0.357312	−0.178656	−	0.983912i	$-0.557175\pi$
$59$	−2.74456	−0.357312	−0.178656	+	0.983912i	$0.557175\pi$
$60$	0	0
$61$	12.7446	1.63177	0.815887	−	0.578211i	$-0.196249\pi$
$61$	12.7446	1.63177	0.815887	+	0.578211i	$0.196249\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−4.74456	−0.588491
$66$	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
$69$	9.48913	1.14236
$70$	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
$73$	−0.744563	−0.0871445	−0.0435722	−	0.999050i	$-0.513874\pi$
$73$	−0.744563	−0.0871445	−0.0435722	+	0.999050i	$0.513874\pi$
$74$	0	0
$75$	−2.00000	−0.230940
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	4.74456	0.533805	0.266903	−	0.963724i	$-0.414000\pi$
$79$	4.74456	0.533805	0.266903	+	0.963724i	$0.414000\pi$
$80$	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
$85$	4.74456	0.514620
$86$	0	0
$87$	5.48913	0.588496
$88$	0	0
$89$	−7.48913	−0.793846	−0.396923	−	0.917852i	$-0.629922\pi$
$89$	−7.48913	−0.793846	−0.396923	+	0.917852i	$0.629922\pi$
$90$	0	0
$91$	4.74456	0.497365
$92$	0	0
$93$	−13.4891	−1.39876
$94$	0	0
$95$	−4.74456	−0.486782
$96$	0	0
$97$	−5.25544	−0.533609	−0.266804	−	0.963751i	$-0.585968\pi$
$97$	−5.25544	−0.533609	−0.266804	+	0.963751i	$0.585968\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−8.74456	−0.870117	−0.435058	−	0.900402i	$-0.643272\pi$
$101$	−8.74456	−0.870117	−0.435058	+	0.900402i	$0.643272\pi$
$102$	0	0
$103$	−10.7446	−1.05869	−0.529347	−	0.848406i	$-0.677563\pi$
$103$	−10.7446	−1.05869	−0.529347	+	0.848406i	$0.677563\pi$
$104$	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
$109$	16.2337	1.55491	0.777453	−	0.628941i	$-0.216511\pi$
$109$	16.2337	1.55491	0.777453	+	0.628941i	$0.216511\pi$
$110$	0	0
$111$	21.4891	2.03966
$112$	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
$115$	−4.74456	−0.442433
$116$	0	0
$117$	−4.74456	−0.438635
$118$	0	0
$119$	−4.74456	−0.434933
$120$	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
$131$	−8.74456	−0.764016	−0.382008	−	0.924159i	$-0.624767\pi$
$131$	−8.74456	−0.764016	−0.382008	+	0.924159i	$0.624767\pi$
$132$	0	0
$133$	4.74456	0.411406
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	−19.4891	−1.66507	−0.832534	−	0.553974i	$-0.813111\pi$
$137$	−19.4891	−1.66507	−0.832534	+	0.553974i	$0.813111\pi$
$138$	0	0
$139$	−3.25544	−0.276123	−0.138061	−	0.990424i	$-0.544087\pi$
$139$	−3.25544	−0.276123	−0.138061	+	0.990424i	$0.544087\pi$
$140$	0	0
$141$	−13.4891	−1.13599
$142$	0	0
$143$	−4.74456	−0.396760
$144$	0	0
$145$	−2.74456	−0.227924
$146$	0	0
$147$	−2.00000	−0.164957
$148$	0	0
$149$	10.7446	0.880229	0.440114	−	0.897942i	$-0.354938\pi$
$149$	10.7446	0.880229	0.440114	+	0.897942i	$0.354938\pi$
$150$	0	0
$151$	20.7446	1.68817	0.844084	−	0.536211i	$-0.180145\pi$
$151$	20.7446	1.68817	0.844084	+	0.536211i	$0.180145\pi$
$152$	0	0
$153$	4.74456	0.383575
$154$	0	0
$155$	6.74456	0.541736
$156$	0	0
$157$	23.4891	1.87464	0.937318	−	0.348475i	$-0.113300\pi$
$157$	23.4891	1.87464	0.937318	+	0.348475i	$0.113300\pi$
$158$	0	0
$159$	2.51087	0.199125
$160$	0	0
$161$	4.74456	0.373924
$162$	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
$169$	9.51087	0.731606
$170$	0	0
$171$	−4.74456	−0.362826
$172$	0	0
$173$	−14.2337	−1.08217	−0.541084	−	0.840969i	$-0.681986\pi$
$173$	−14.2337	−1.08217	−0.541084	+	0.840969i	$0.681986\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	5.48913	0.412588
$178$	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
$181$	3.48913	0.259345	0.129672	−	0.991557i	$-0.458607\pi$
$181$	3.48913	0.259345	0.129672	+	0.991557i	$0.458607\pi$
$182$	0	0
$183$	−25.4891	−1.88421
$184$	0	0
$185$	−10.7446	−0.789956
$186$	0	0
$187$	4.74456	0.346957
$188$	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
$195$	9.48913	0.679530
$196$	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
$199$	−14.7446	−1.04521	−0.522607	−	0.852574i	$-0.675041\pi$
$199$	−14.7446	−1.04521	−0.522607	+	0.852574i	$0.675041\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	2.74456	0.192631
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0	0
$207$	−4.74456	−0.329770
$208$	0	0
$209$	−4.74456	−0.328188
$210$	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
$217$	−6.74456	−0.457851
$218$	0	0
$219$	1.48913	0.100626
$220$	0	0
$221$	−22.5109	−1.51425
$222$	0	0
$223$	−26.7446	−1.79095	−0.895474	−	0.445113i	$-0.853163\pi$
$223$	−26.7446	−1.79095	−0.895474	+	0.445113i	$0.853163\pi$
$224$	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
$233$	20.9783	1.37433	0.687165	−	0.726501i	$-0.258855\pi$
$233$	20.9783	1.37433	0.687165	+	0.726501i	$0.258855\pi$
$234$	0	0
$235$	6.74456	0.439967
$236$	0	0
$237$	−9.48913	−0.616385
$238$	0	0
$239$	3.25544	0.210577	0.105288	−	0.994442i	$-0.466423\pi$
$239$	3.25544	0.210577	0.105288	+	0.994442i	$0.466423\pi$
$240$	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
$247$	22.5109	1.43233
$248$	0	0
$249$	16.0000	1.01396
$250$	0	0
$251$	−8.23369	−0.519706	−0.259853	−	0.965648i	$-0.583674\pi$
$251$	−8.23369	−0.519706	−0.259853	+	0.965648i	$0.583674\pi$
$252$	0	0
$253$	−4.74456	−0.298288
$254$	0	0
$255$	−9.48913	−0.594232
$256$	0	0
$257$	24.2337	1.51166	0.755828	−	0.654770i	$-0.227235\pi$
$257$	24.2337	1.51166	0.755828	+	0.654770i	$0.227235\pi$
$258$	0	0
$259$	10.7446	0.667635
$260$	0	0
$261$	−2.74456	−0.169884
$262$	0	0
$263$	18.9783	1.17025	0.585125	−	0.810943i	$-0.301046\pi$
$263$	18.9783	1.17025	0.585125	+	0.810943i	$0.301046\pi$
$264$	0	0
$265$	−1.25544	−0.0771209
$266$	0	0
$267$	14.9783	0.916654
$268$	0	0
$269$	−24.9783	−1.52295	−0.761475	−	0.648194i	$-0.775525\pi$
$269$	−24.9783	−1.52295	−0.761475	+	0.648194i	$0.775525\pi$
$270$	0	0
$271$	30.9783	1.88179	0.940897	−	0.338692i	$-0.109984\pi$
$271$	30.9783	1.88179	0.940897	+	0.338692i	$0.109984\pi$
$272$	0	0
$273$	−9.48913	−0.574308
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	−7.48913	−0.449978	−0.224989	−	0.974361i	$-0.572235\pi$
$277$	−7.48913	−0.449978	−0.224989	+	0.974361i	$0.572235\pi$
$278$	0	0
$279$	6.74456	0.403786
$280$	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
$285$	9.48913	0.562087
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	5.51087	0.324169
$290$	0	0
$291$	10.5109	0.616158
$292$	0	0
$293$	24.7446	1.44559	0.722796	−	0.691061i	$-0.242856\pi$
$293$	24.7446	1.44559	0.722796	+	0.691061i	$0.242856\pi$
$294$	0	0
$295$	−2.74456	−0.159795
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	22.5109	1.30184
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	17.4891	1.00472
$304$	0	0
$305$	12.7446	0.729752
$306$	0	0
$307$	−21.4891	−1.22645	−0.613225	−	0.789909i	$-0.710128\pi$
$307$	−21.4891	−1.22645	−0.613225	+	0.789909i	$0.710128\pi$
$308$	0	0
$309$	21.4891	1.22247
$310$	0	0
$311$	9.25544	0.524828	0.262414	−	0.964955i	$-0.415481\pi$
$311$	9.25544	0.524828	0.262414	+	0.964955i	$0.415481\pi$
$312$	0	0
$313$	32.2337	1.82196	0.910978	−	0.412455i	$-0.135329\pi$
$313$	32.2337	1.82196	0.910978	+	0.412455i	$0.135329\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	−32.2337	−1.81042	−0.905212	−	0.424960i	$-0.860288\pi$
$317$	−32.2337	−1.81042	−0.905212	+	0.424960i	$0.860288\pi$
$318$	0	0
$319$	−2.74456	−0.153666
$320$	0	0
$321$	−24.0000	−1.33955
$322$	0	0
$323$	−22.5109	−1.25254
$324$	0	0
$325$	−4.74456	−0.263181
$326$	0	0
$327$	−32.4674	−1.79545
$328$	0	0
$329$	−6.74456	−0.371840
$330$	0	0
$331$	30.9783	1.70272	0.851359	−	0.524583i	$-0.175779\pi$
$331$	30.9783	1.70272	0.851359	+	0.524583i	$0.175779\pi$
$332$	0	0
$333$	−10.7446	−0.588798
$334$	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
$341$	6.74456	0.365239
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	9.48913	0.510877
$346$	0	0
$347$	−22.9783	−1.23354	−0.616769	−	0.787145i	$-0.711559\pi$
$347$	−22.9783	−1.23354	−0.616769	+	0.787145i	$0.711559\pi$
$348$	0	0
$349$	19.2554	1.03072	0.515360	−	0.856974i	$-0.327658\pi$
$349$	19.2554	1.03072	0.515360	+	0.856974i	$0.327658\pi$
$350$	0	0
$351$	−18.9783	−1.01298
$352$	0	0
$353$	−2.74456	−0.146078	−0.0730392	−	0.997329i	$-0.523270\pi$
$353$	−2.74456	−0.146078	−0.0730392	+	0.997329i	$0.523270\pi$
$354$	0	0
$355$	4.00000	0.212298
$356$	0	0
$357$	9.48913	0.502218
$358$	0	0
$359$	12.7446	0.672632	0.336316	−	0.941749i	$-0.390819\pi$
$359$	12.7446	0.672632	0.336316	+	0.941749i	$0.390819\pi$
$360$	0	0
$361$	3.51087	0.184783
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	−0.744563	−0.0389722
$366$	0	0
$367$	−2.74456	−0.143265	−0.0716325	−	0.997431i	$-0.522821\pi$
$367$	−2.74456	−0.143265	−0.0716325	+	0.997431i	$0.522821\pi$
$368$	0	0
$369$	−4.00000	−0.208232
$370$	0	0
$371$	1.25544	0.0651791
$372$	0	0
$373$	28.9783	1.50044	0.750218	−	0.661190i	$-0.229948\pi$
$373$	28.9783	1.50044	0.750218	+	0.661190i	$0.229948\pi$
$374$	0	0
$375$	−2.00000	−0.103280
$376$	0	0
$377$	13.0217	0.670654
$378$	0	0
$379$	14.5109	0.745374	0.372687	−	0.927957i	$-0.378437\pi$
$379$	14.5109	0.745374	0.372687	+	0.927957i	$0.378437\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	37.7228	1.92755	0.963773	−	0.266724i	$-0.0859413\pi$
$383$	37.7228	1.92755	0.963773	+	0.266724i	$0.0859413\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	15.4891	0.785330	0.392665	−	0.919682i	$-0.371553\pi$
$389$	15.4891	0.785330	0.392665	+	0.919682i	$0.371553\pi$
$390$	0	0
$391$	−22.5109	−1.13842
$392$	0	0
$393$	17.4891	0.882210
$394$	0	0
$395$	4.74456	0.238725
$396$	0	0
$397$	12.5109	0.627903	0.313951	−	0.949439i	$-0.398347\pi$
$397$	12.5109	0.627903	0.313951	+	0.949439i	$0.398347\pi$
$398$	0	0
$399$	−9.48913	−0.475050
$400$	0	0
$401$	0.510875	0.0255119	0.0127559	−	0.999919i	$-0.495940\pi$
$401$	0.510875	0.0255119	0.0127559	+	0.999919i	$0.495940\pi$
$402$	0	0
$403$	−32.0000	−1.59403
$404$	0	0
$405$	−11.0000	−0.546594
$406$	0	0
$407$	−10.7446	−0.532588
$408$	0	0
$409$	1.48913	0.0736325	0.0368163	−	0.999322i	$-0.488278\pi$
$409$	1.48913	0.0736325	0.0368163	+	0.999322i	$0.488278\pi$
$410$	0	0
$411$	38.9783	1.92266
$412$	0	0
$413$	2.74456	0.135051
$414$	0	0
$415$	−8.00000	−0.392705
$416$	0	0
$417$	6.51087	0.318839
$418$	0	0
$419$	−10.7446	−0.524906	−0.262453	−	0.964945i	$-0.584532\pi$
$419$	−10.7446	−0.524906	−0.262453	+	0.964945i	$0.584532\pi$
$420$	0	0
$421$	27.4891	1.33974	0.669869	−	0.742479i	$-0.266350\pi$
$421$	27.4891	1.33974	0.669869	+	0.742479i	$0.266350\pi$
$422$	0	0
$423$	6.74456	0.327932
$424$	0	0
$425$	4.74456	0.230145
$426$	0	0
$427$	−12.7446	−0.616753
$428$	0	0
$429$	9.48913	0.458139
$430$	0	0
$431$	28.7446	1.38458	0.692288	−	0.721621i	$-0.256603\pi$
$431$	28.7446	1.38458	0.692288	+	0.721621i	$0.256603\pi$
$432$	0	0
$433$	25.2554	1.21370	0.606849	−	0.794817i	$-0.292433\pi$
$433$	25.2554	1.21370	0.606849	+	0.794817i	$0.292433\pi$
$434$	0	0
$435$	5.48913	0.263183
$436$	0	0
$437$	22.5109	1.07684
$438$	0	0
$439$	30.9783	1.47851	0.739256	−	0.673425i	$-0.235178\pi$
$439$	30.9783	1.47851	0.739256	+	0.673425i	$0.235178\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	6.51087	0.309341	0.154670	−	0.987966i	$-0.450568\pi$
$443$	6.51087	0.309341	0.154670	+	0.987966i	$0.450568\pi$
$444$	0	0
$445$	−7.48913	−0.355019
$446$	0	0
$447$	−21.4891	−1.01640
$448$	0	0
$449$	40.9783	1.93388	0.966942	−	0.254998i	$-0.0820748\pi$
$449$	40.9783	1.93388	0.966942	+	0.254998i	$0.0820748\pi$
$450$	0	0
$451$	−4.00000	−0.188353
$452$	0	0
$453$	−41.4891	−1.94933
$454$	0	0
$455$	4.74456	0.222429
$456$	0	0
$457$	−19.4891	−0.911663	−0.455831	−	0.890066i	$-0.650658\pi$
$457$	−19.4891	−0.911663	−0.455831	+	0.890066i	$0.650658\pi$
$458$	0	0
$459$	18.9783	0.885829
$460$	0	0
$461$	−23.7228	−1.10488	−0.552441	−	0.833552i	$-0.686303\pi$
$461$	−23.7228	−1.10488	−0.552441	+	0.833552i	$0.686303\pi$
$462$	0	0
$463$	−12.7446	−0.592290	−0.296145	−	0.955143i	$-0.595701\pi$
$463$	−12.7446	−0.592290	−0.296145	+	0.955143i	$0.595701\pi$
$464$	0	0
$465$	−13.4891	−0.625543
$466$	0	0
$467$	28.9783	1.34095	0.670477	−	0.741931i	$-0.266090\pi$
$467$	28.9783	1.34095	0.670477	+	0.741931i	$0.266090\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−46.9783	−2.16464
$472$	0	0
$473$	−4.00000	−0.183920
$474$	0	0
$475$	−4.74456	−0.217695
$476$	0	0
$477$	−1.25544	−0.0574825
$478$	0	0
$479$	18.5109	0.845783	0.422892	−	0.906180i	$-0.361015\pi$
$479$	18.5109	0.845783	0.422892	+	0.906180i	$0.361015\pi$
$480$	0	0
$481$	50.9783	2.32441
$482$	0	0
$483$	−9.48913	−0.431770
$484$	0	0
$485$	−5.25544	−0.238637
$486$	0	0
$487$	−20.7446	−0.940026	−0.470013	−	0.882660i	$-0.655751\pi$
$487$	−20.7446	−0.940026	−0.470013	+	0.882660i	$0.655751\pi$
$488$	0	0
$489$	−8.00000	−0.361773
$490$	0	0
$491$	14.9783	0.675959	0.337979	−	0.941153i	$-0.390257\pi$
$491$	14.9783	0.675959	0.337979	+	0.941153i	$0.390257\pi$
$492$	0	0
$493$	−13.0217	−0.586470
$494$	0	0
$495$	1.00000	0.0449467
$496$	0	0
$497$	−4.00000	−0.179425
$498$	0	0
$499$	−9.48913	−0.424792	−0.212396	−	0.977184i	$-0.568127\pi$
$499$	−9.48913	−0.424792	−0.212396	+	0.977184i	$0.568127\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−29.4891	−1.31486	−0.657428	−	0.753518i	$-0.728355\pi$
$503$	−29.4891	−1.31486	−0.657428	+	0.753518i	$0.728355\pi$
$504$	0	0
$505$	−8.74456	−0.389128
$506$	0	0
$507$	−19.0217	−0.844786
$508$	0	0
$509$	−12.5109	−0.554535	−0.277267	−	0.960793i	$-0.589429\pi$
$509$	−12.5109	−0.554535	−0.277267	+	0.960793i	$0.589429\pi$
$510$	0	0
$511$	0.744563	0.0329375
$512$	0	0
$513$	−18.9783	−0.837910
$514$	0	0
$515$	−10.7446	−0.473462
$516$	0	0
$517$	6.74456	0.296626
$518$	0	0
$519$	28.4674	1.24958
$520$	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
$523$	−5.48913	−0.240023	−0.120011	−	0.992773i	$-0.538293\pi$
$523$	−5.48913	−0.240023	−0.120011	+	0.992773i	$0.538293\pi$
$524$	0	0
$525$	2.00000	0.0872872
$526$	0	0
$527$	32.0000	1.39394
$528$	0	0
$529$	−0.489125	−0.0212663
$530$	0	0
$531$	−2.74456	−0.119104
$532$	0	0
$533$	18.9783	0.822039
$534$	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
$541$	36.2337	1.55781	0.778904	−	0.627143i	$-0.215776\pi$
$541$	36.2337	1.55781	0.778904	+	0.627143i	$0.215776\pi$
$542$	0	0
$543$	−6.97825	−0.299465
$544$	0	0
$545$	16.2337	0.695375
$546$	0	0
$547$	30.9783	1.32453	0.662267	−	0.749268i	$-0.269594\pi$
$547$	30.9783	1.32453	0.662267	+	0.749268i	$0.269594\pi$
$548$	0	0
$549$	12.7446	0.543925
$550$	0	0
$551$	13.0217	0.554745
$552$	0	0
$553$	−4.74456	−0.201759
$554$	0	0
$555$	21.4891	0.912163
$556$	0	0
$557$	44.9783	1.90579	0.952895	−	0.303301i	$-0.0980887\pi$
$557$	44.9783	1.90579	0.952895	+	0.303301i	$0.0980887\pi$
$558$	0	0
$559$	18.9783	0.802694
$560$	0	0
$561$	−9.48913	−0.400631
$562$	0	0
$563$	17.4891	0.737079	0.368539	−	0.929612i	$-0.379858\pi$
$563$	17.4891	0.737079	0.368539	+	0.929612i	$0.379858\pi$
$564$	0	0
$565$	10.0000	0.420703
$566$	0	0
$567$	11.0000	0.461957
$568$	0	0
$569$	39.4891	1.65547	0.827735	−	0.561119i	$-0.189629\pi$
$569$	39.4891	1.65547	0.827735	+	0.561119i	$0.189629\pi$
$570$	0	0
$571$	−5.48913	−0.229713	−0.114856	−	0.993382i	$-0.536641\pi$
$571$	−5.48913	−0.229713	−0.114856	+	0.993382i	$0.536641\pi$
$572$	0	0
$573$	−32.0000	−1.33682
$574$	0	0
$575$	−4.74456	−0.197862
$576$	0	0
$577$	2.74456	0.114258	0.0571288	−	0.998367i	$-0.481805\pi$
$577$	2.74456	0.114258	0.0571288	+	0.998367i	$0.481805\pi$
$578$	0	0
$579$	4.00000	0.166234
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	−1.25544	−0.0519949
$584$	0	0
$585$	−4.74456	−0.196164
$586$	0	0
$587$	−40.9783	−1.69135	−0.845677	−	0.533696i	$-0.820803\pi$
$587$	−40.9783	−1.69135	−0.845677	+	0.533696i	$0.820803\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	−20.0000	−0.822690
$592$	0	0
$593$	−5.76631	−0.236794	−0.118397	−	0.992966i	$-0.537776\pi$
$593$	−5.76631	−0.236794	−0.118397	+	0.992966i	$0.537776\pi$
$594$	0	0
$595$	−4.74456	−0.194508
$596$	0	0
$597$	29.4891	1.20691
$598$	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
$601$	−10.5109	−0.428748	−0.214374	−	0.976752i	$-0.568771\pi$
$601$	−10.5109	−0.428748	−0.214374	+	0.976752i	$0.568771\pi$
$602$	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
$609$	−5.48913	−0.222431
$610$	0	0
$611$	−32.0000	−1.29458
$612$	0	0
$613$	−11.4891	−0.464041	−0.232021	−	0.972711i	$-0.574534\pi$
$613$	−11.4891	−0.464041	−0.232021	+	0.972711i	$0.574534\pi$
$614$	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
$619$	10.7446	0.431860	0.215930	−	0.976409i	$-0.430722\pi$
$619$	10.7446	0.431860	0.215930	+	0.976409i	$0.430722\pi$
$620$	0	0
$621$	−18.9783	−0.761571
$622$	0	0
$623$	7.48913	0.300045
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	9.48913	0.378959
$628$	0	0
$629$	−50.9783	−2.03264
$630$	0	0
$631$	−42.9783	−1.71094	−0.855469	−	0.517855i	$-0.826731\pi$
$631$	−42.9783	−1.71094	−0.855469	+	0.517855i	$0.826731\pi$
$632$	0	0
$633$	−24.0000	−0.953914
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−4.74456	−0.187986
$638$	0	0
$639$	4.00000	0.158238
$640$	0	0
$641$	−11.4891	−0.453793	−0.226897	−	0.973919i	$-0.572858\pi$
$641$	−11.4891	−0.453793	−0.226897	+	0.973919i	$0.572858\pi$
$642$	0	0
$643$	22.4674	0.886027	0.443013	−	0.896515i	$-0.353909\pi$
$643$	22.4674	0.886027	0.443013	+	0.896515i	$0.353909\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	2.74456	0.107900	0.0539499	−	0.998544i	$-0.482819\pi$
$647$	2.74456	0.107900	0.0539499	+	0.998544i	$0.482819\pi$
$648$	0	0
$649$	−2.74456	−0.107734
$650$	0	0
$651$	13.4891	0.528681
$652$	0	0
$653$	−41.7228	−1.63274	−0.816370	−	0.577529i	$-0.804017\pi$
$653$	−41.7228	−1.63274	−0.816370	+	0.577529i	$0.804017\pi$
$654$	0	0
$655$	−8.74456	−0.341678
$656$	0	0
$657$	−0.744563	−0.0290482
$658$	0	0
$659$	18.5109	0.721081	0.360541	−	0.932744i	$-0.382592\pi$
$659$	18.5109	0.721081	0.360541	+	0.932744i	$0.382592\pi$
$660$	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
$663$	45.0217	1.74850
$664$	0	0
$665$	4.74456	0.183986
$666$	0	0
$667$	13.0217	0.504204
$668$	0	0
$669$	53.4891	2.06801
$670$	0	0
$671$	12.7446	0.491998
$672$	0	0
$673$	−24.9783	−0.962841	−0.481420	−	0.876490i	$-0.659879\pi$
$673$	−24.9783	−0.962841	−0.481420	+	0.876490i	$0.659879\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	1.76631	0.0678849	0.0339424	−	0.999424i	$-0.489194\pi$
$677$	1.76631	0.0678849	0.0339424	+	0.999424i	$0.489194\pi$
$678$	0	0
$679$	5.25544	0.201685
$680$	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
$685$	−19.4891	−0.744641
$686$	0	0
$687$	12.0000	0.457829
$688$	0	0
$689$	5.95650	0.226925
$690$	0	0
$691$	−36.2337	−1.37839	−0.689197	−	0.724574i	$-0.742037\pi$
$691$	−36.2337	−1.37839	−0.689197	+	0.724574i	$0.742037\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	−3.25544	−0.123486
$696$	0	0
$697$	−18.9783	−0.718853
$698$	0	0
$699$	−41.9565	−1.58694
$700$	0	0
$701$	−12.2337	−0.462060	−0.231030	−	0.972947i	$-0.574210\pi$
$701$	−12.2337	−0.462060	−0.231030	+	0.972947i	$0.574210\pi$
$702$	0	0
$703$	50.9783	1.92268
$704$	0	0
$705$	−13.4891	−0.508030
$706$	0	0
$707$	8.74456	0.328873
$708$	0	0
$709$	23.4891	0.882153	0.441076	−	0.897470i	$-0.354597\pi$
$709$	23.4891	0.882153	0.441076	+	0.897470i	$0.354597\pi$
$710$	0	0
$711$	4.74456	0.177935
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	−4.74456	−0.177437
$716$	0	0
$717$	−6.51087	−0.243153
$718$	0	0
$719$	49.7228	1.85435	0.927174	−	0.374631i	$-0.122231\pi$
$719$	49.7228	1.85435	0.927174	+	0.374631i	$0.122231\pi$
$720$	0	0
$721$	10.7446	0.400148
$722$	0	0
$723$	−40.0000	−1.48762
$724$	0	0
$725$	−2.74456	−0.101930
$726$	0	0
$727$	20.2337	0.750426	0.375213	−	0.926939i	$-0.377570\pi$
$727$	20.2337	0.750426	0.375213	+	0.926939i	$0.377570\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−18.9783	−0.701936
$732$	0	0
$733$	18.2337	0.673477	0.336738	−	0.941598i	$-0.390676\pi$
$733$	18.2337	0.673477	0.336738	+	0.941598i	$0.390676\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	4.00000	0.147342
$738$	0	0
$739$	−14.9783	−0.550984	−0.275492	−	0.961303i	$-0.588841\pi$
$739$	−14.9783	−0.550984	−0.275492	+	0.961303i	$0.588841\pi$
$740$	0	0
$741$	−45.0217	−1.65392
$742$	0	0
$743$	18.9783	0.696244	0.348122	−	0.937449i	$-0.386819\pi$
$743$	18.9783	0.696244	0.348122	+	0.937449i	$0.386819\pi$
$744$	0	0
$745$	10.7446	0.393650
$746$	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
$753$	16.4674	0.600105
$754$	0	0
$755$	20.7446	0.754972
$756$	0	0
$757$	−30.7446	−1.11743	−0.558715	−	0.829360i	$-0.688705\pi$
$757$	−30.7446	−1.11743	−0.558715	+	0.829360i	$0.688705\pi$
$758$	0	0
$759$	9.48913	0.344433
$760$	0	0
$761$	−6.51087	−0.236019	−0.118010	−	0.993012i	$-0.537651\pi$
$761$	−6.51087	−0.236019	−0.118010	+	0.993012i	$0.537651\pi$
$762$	0	0
$763$	−16.2337	−0.587699
$764$	0	0
$765$	4.74456	0.171540
$766$	0	0
$767$	13.0217	0.470188
$768$	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
$771$	−48.4674	−1.74551
$772$	0	0
$773$	−28.5109	−1.02546	−0.512732	−	0.858548i	$-0.671367\pi$
$773$	−28.5109	−1.02546	−0.512732	+	0.858548i	$0.671367\pi$
$774$	0	0
$775$	6.74456	0.242272
$776$	0	0
$777$	−21.4891	−0.770918
$778$	0	0
$779$	18.9783	0.679966
$780$	0	0
$781$	4.00000	0.143131
$782$	0	0
$783$	−10.9783	−0.392331
$784$	0	0
$785$	23.4891	0.838363
$786$	0	0
$787$	17.4891	0.623420	0.311710	−	0.950177i	$-0.399098\pi$
$787$	17.4891	0.623420	0.311710	+	0.950177i	$0.399098\pi$
$788$	0	0
$789$	−37.9565	−1.35129
$790$	0	0
$791$	−10.0000	−0.355559
$792$	0	0
$793$	−60.4674	−2.14726
$794$	0	0
$795$	2.51087	0.0890515
$796$	0	0
$797$	26.4674	0.937523	0.468761	−	0.883325i	$-0.344700\pi$
$797$	26.4674	0.937523	0.468761	+	0.883325i	$0.344700\pi$
$798$	0	0
$799$	32.0000	1.13208
$800$	0	0
$801$	−7.48913	−0.264615
$802$	0	0
$803$	−0.744563	−0.0262750
$804$	0	0
$805$	4.74456	0.167224
$806$	0	0
$807$	49.9565	1.75855
$808$	0	0
$809$	−34.4674	−1.21181	−0.605904	−	0.795538i	$-0.707189\pi$
$809$	−34.4674	−1.21181	−0.605904	+	0.795538i	$0.707189\pi$
$810$	0	0
$811$	7.25544	0.254773	0.127386	−	0.991853i	$-0.459341\pi$
$811$	7.25544	0.254773	0.127386	+	0.991853i	$0.459341\pi$
$812$	0	0
$813$	−61.9565	−2.17291
$814$	0	0
$815$	4.00000	0.140114
$816$	0	0
$817$	18.9783	0.663965
$818$	0	0
$819$	4.74456	0.165788
$820$	0	0
$821$	−49.7228	−1.73534	−0.867669	−	0.497142i	$-0.834383\pi$
$821$	−49.7228	−1.73534	−0.867669	+	0.497142i	$0.834383\pi$
$822$	0	0
$823$	−42.2337	−1.47217	−0.736087	−	0.676887i	$-0.763329\pi$
$823$	−42.2337	−1.47217	−0.736087	+	0.676887i	$0.763329\pi$
$824$	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
$829$	−54.4674	−1.89173	−0.945865	−	0.324560i	$-0.894784\pi$
$829$	−54.4674	−1.89173	−0.945865	+	0.324560i	$0.894784\pi$
$830$	0	0
$831$	14.9783	0.519590
$832$	0	0
$833$	4.74456	0.164389
$834$	0	0
$835$	0	0
$836$	0	0
$837$	26.9783	0.932505
$838$	0	0
$839$	14.7446	0.509039	0.254519	−	0.967068i	$-0.418083\pi$
$839$	14.7446	0.509039	0.254519	+	0.967068i	$0.418083\pi$
$840$	0	0
$841$	−21.4674	−0.740254
$842$	0	0
$843$	−28.0000	−0.964371
$844$	0	0
$845$	9.51087	0.327184
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	56.0000	1.92192
$850$	0	0
$851$	50.9783	1.74751
$852$	0	0
$853$	−11.2554	−0.385379	−0.192689	−	0.981260i	$-0.561721\pi$
$853$	−11.2554	−0.385379	−0.192689	+	0.981260i	$0.561721\pi$
$854$	0	0
$855$	−4.74456	−0.162261
$856$	0	0
$857$	37.2119	1.27114	0.635568	−	0.772045i	$-0.280766\pi$
$857$	37.2119	1.27114	0.635568	+	0.772045i	$0.280766\pi$
$858$	0	0
$859$	−51.2119	−1.74733	−0.873664	−	0.486529i	$-0.838263\pi$
$859$	−51.2119	−1.74733	−0.873664	+	0.486529i	$0.838263\pi$
$860$	0	0
$861$	−8.00000	−0.272639
$862$	0	0
$863$	5.76631	0.196288	0.0981438	−	0.995172i	$-0.468709\pi$
$863$	5.76631	0.196288	0.0981438	+	0.995172i	$0.468709\pi$
$864$	0	0
$865$	−14.2337	−0.483960
$866$	0	0
$867$	−11.0217	−0.374318
$868$	0	0
$869$	4.74456	0.160948
$870$	0	0
$871$	−18.9783	−0.643053
$872$	0	0
$873$	−5.25544	−0.177870
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	42.4674	1.43402	0.717011	−	0.697062i	$-0.245510\pi$
$877$	42.4674	1.43402	0.717011	+	0.697062i	$0.245510\pi$
$878$	0	0
$879$	−49.4891	−1.66923
$880$	0	0
$881$	−32.5109	−1.09532	−0.547660	−	0.836701i	$-0.684481\pi$
$881$	−32.5109	−1.09532	−0.547660	+	0.836701i	$0.684481\pi$
$882$	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
$885$	5.48913	0.184515
$886$	0	0
$887$	18.5109	0.621534	0.310767	−	0.950486i	$-0.399414\pi$
$887$	18.5109	0.621534	0.310767	+	0.950486i	$0.399414\pi$
$888$	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
$897$	−45.0217	−1.50323
$898$	0	0
$899$	−18.5109	−0.617372
$900$	0	0
$901$	−5.95650	−0.198440
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	3.48913	0.115982
$906$	0	0
$907$	37.4891	1.24481	0.622403	−	0.782697i	$-0.286157\pi$
$907$	37.4891	1.24481	0.622403	+	0.782697i	$0.286157\pi$
$908$	0	0
$909$	−8.74456	−0.290039
$910$	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
$915$	−25.4891	−0.842644
$916$	0	0
$917$	8.74456	0.288771
$918$	0	0
$919$	−25.2119	−0.831665	−0.415833	−	0.909441i	$-0.636510\pi$
$919$	−25.2119	−0.831665	−0.415833	+	0.909441i	$0.636510\pi$
$920$	0	0
$921$	42.9783	1.41618
$922$	0	0
$923$	−18.9783	−0.624677
$924$	0	0
$925$	−10.7446	−0.353279
$926$	0	0
$927$	−10.7446	−0.352898
$928$	0	0
$929$	16.9783	0.557038	0.278519	−	0.960431i	$-0.410156\pi$
$929$	16.9783	0.557038	0.278519	+	0.960431i	$0.410156\pi$
$930$	0	0
$931$	−4.74456	−0.155497
$932$	0	0
$933$	−18.5109	−0.606019
$934$	0	0
$935$	4.74456	0.155164
$936$	0	0
$937$	−7.25544	−0.237025	−0.118512	−	0.992953i	$-0.537813\pi$
$937$	−7.25544	−0.237025	−0.118512	+	0.992953i	$0.537813\pi$
$938$	0	0
$939$	−64.4674	−2.10381
$940$	0	0
$941$	−22.2337	−0.724798	−0.362399	−	0.932023i	$-0.618042\pi$
$941$	−22.2337	−0.724798	−0.362399	+	0.932023i	$0.618042\pi$
$942$	0	0
$943$	18.9783	0.618017
$944$	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
$949$	3.53262	0.114674
$950$	0	0
$951$	64.4674	2.09050
$952$	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
$957$	5.48913	0.177438
$958$	0	0
$959$	19.4891	0.629337
$960$	0	0
$961$	14.4891	0.467391
$962$	0	0
$963$	12.0000	0.386695
$964$	0	0
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	5.02175	0.161489	0.0807443	−	0.996735i	$-0.474270\pi$
$967$	5.02175	0.161489	0.0807443	+	0.996735i	$0.474270\pi$
$968$	0	0
$969$	45.0217	1.44631
$970$	0	0
$971$	37.7228	1.21058	0.605291	−	0.796004i	$-0.293057\pi$
$971$	37.7228	1.21058	0.605291	+	0.796004i	$0.293057\pi$
$972$	0	0
$973$	3.25544	0.104365
$974$	0	0
$975$	9.48913	0.303895
$976$	0	0
$977$	−32.9783	−1.05507	−0.527534	−	0.849534i	$-0.676883\pi$
$977$	−32.9783	−1.05507	−0.527534	+	0.849534i	$0.676883\pi$
$978$	0	0
$979$	−7.48913	−0.239353
$980$	0	0
$981$	16.2337	0.518302
$982$	0	0
$983$	32.2337	1.02809	0.514047	−	0.857762i	$-0.328146\pi$
$983$	32.2337	1.02809	0.514047	+	0.857762i	$0.328146\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	0	0
$987$	13.4891	0.429364
$988$	0	0
$989$	18.9783	0.603473
$990$	0	0
$991$	21.4891	0.682625	0.341312	−	0.939950i	$-0.389129\pi$
$991$	21.4891	0.682625	0.341312	+	0.939950i	$0.389129\pi$
$992$	0	0
$993$	−61.9565	−1.96613
$994$	0	0
$995$	−14.7446	−0.467434
$996$	0	0
$997$	−41.2119	−1.30520	−0.652598	−	0.757705i	$-0.726321\pi$
$997$	−41.2119	−1.30520	−0.652598	+	0.757705i	$0.726321\pi$
$998$	0	0
$999$	−42.9783	−1.35977

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.r.1.1		2
4.3	odd	2		770.2.a.k.1.1	✓	2
12.11	even	2		6930.2.a.bo.1.1		2
20.3	even	4		3850.2.c.y.1849.1		4
20.7	even	4		3850.2.c.y.1849.4		4
20.19	odd	2		3850.2.a.bc.1.2		2
28.27	even	2		5390.2.a.bq.1.2		2
44.43	even	2		8470.2.a.bu.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.k.1.1	✓	2	4.3	odd	2
3850.2.a.bc.1.2		2	20.19	odd	2
3850.2.c.y.1849.1		4	20.3	even	4
3850.2.c.y.1849.4		4	20.7	even	4
5390.2.a.bq.1.2		2	28.27	even	2
6160.2.a.r.1.1		2	1.1	even	1	trivial
6930.2.a.bo.1.1		2	12.11	even	2
8470.2.a.bu.1.2		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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -4.74456 q^{13} -2.00000 q^{15} +4.74456 q^{17} -4.74456 q^{19} +2.00000 q^{21} -4.74456 q^{23} +1.00000 q^{25} +4.00000 q^{27} -2.74456 q^{29} +6.74456 q^{31} -2.00000 q^{33} -1.00000 q^{35} -10.7446 q^{37} +9.48913 q^{39} -4.00000 q^{41} -4.00000 q^{43} +1.00000 q^{45} +6.74456 q^{47} +1.00000 q^{49} -9.48913 q^{51} -1.25544 q^{53} +1.00000 q^{55} +9.48913 q^{57} -2.74456 q^{59} +12.7446 q^{61} -1.00000 q^{63} -4.74456 q^{65} +4.00000 q^{67} +9.48913 q^{69} +4.00000 q^{71} -0.744563 q^{73} -2.00000 q^{75} -1.00000 q^{77} +4.74456 q^{79} -11.0000 q^{81} -8.00000 q^{83} +4.74456 q^{85} +5.48913 q^{87} -7.48913 q^{89} +4.74456 q^{91} -13.4891 q^{93} -4.74456 q^{95} -5.25544 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 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

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