LMFDB - Embedded newform 6160.2.a.bd.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:	$1$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3080)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.210756$ 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)

$f(q)$	$=$	$q-1.21076 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.53407 q^{9} +O(q^{10})$	$q-1.21076 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.53407 q^{9} +1.00000 q^{11} -1.21076 q^{13} +1.21076 q^{15} +4.27890 q^{17} -7.06814 q^{19} -1.21076 q^{21} +2.95558 q^{23} +1.00000 q^{25} +5.48965 q^{27} -2.00000 q^{29} +6.16634 q^{31} -1.21076 q^{33} -1.00000 q^{35} -4.95558 q^{37} +1.46593 q^{39} +0.166338 q^{41} -6.44523 q^{43} +1.53407 q^{45} +2.70041 q^{47} +1.00000 q^{49} -5.18070 q^{51} +4.11256 q^{53} -1.00000 q^{55} +8.55779 q^{57} +14.7241 q^{59} -14.3915 q^{61} -1.53407 q^{63} +1.21076 q^{65} +9.60221 q^{67} -3.57849 q^{69} -6.13628 q^{71} -12.2789 q^{73} -1.21076 q^{75} +1.00000 q^{77} +12.1126 q^{79} -2.04442 q^{81} +1.48965 q^{83} -4.27890 q^{85} +2.42151 q^{87} +2.42151 q^{89} -1.21076 q^{91} -7.46593 q^{93} +7.06814 q^{95} -7.71477 q^{97} -1.53407 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	$ 3 q - 2 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 2 q^{13} + 2 q^{15} - 4 q^{17} - 6 q^{19} - 2 q^{21} - 2 q^{23} + 3 q^{25} - 2 q^{27} - 6 q^{29} + 6 q^{31} - 2 q^{33} - 3 q^{35} - 4 q^{37} + 12 q^{39} - 12 q^{41} + 10 q^{43} - 3 q^{45} - 12 q^{47} + 3 q^{49} + 4 q^{51} + 8 q^{53} - 3 q^{55} - 8 q^{57} - 2 q^{59} - 22 q^{61} + 3 q^{63} + 2 q^{65} + 6 q^{67} - 14 q^{69} + 12 q^{71} - 20 q^{73} - 2 q^{75} + 3 q^{77} + 32 q^{79} - 17 q^{81} - 14 q^{83} + 4 q^{85} + 4 q^{87} + 4 q^{89} - 2 q^{91} - 30 q^{93} + 6 q^{95} + 4 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 - 2 * q^13 + 2 * q^15 - 4 * q^17 - 6 * q^19 - 2 * q^21 - 2 * q^23 + 3 * q^25 - 2 * q^27 - 6 * q^29 + 6 * q^31 - 2 * q^33 - 3 * q^35 - 4 * q^37 + 12 * q^39 - 12 * q^41 + 10 * q^43 - 3 * q^45 - 12 * q^47 + 3 * q^49 + 4 * q^51 + 8 * q^53 - 3 * q^55 - 8 * q^57 - 2 * q^59 - 22 * q^61 + 3 * q^63 + 2 * q^65 + 6 * q^67 - 14 * q^69 + 12 * q^71 - 20 * q^73 - 2 * q^75 + 3 * q^77 + 32 * q^79 - 17 * q^81 - 14 * q^83 + 4 * q^85 + 4 * q^87 + 4 * q^89 - 2 * q^91 - 30 * q^93 + 6 * q^95 + 4 * q^97 + 3 * 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
$3$	−1.21076	−0.699030	−0.349515	−	0.936931i	$-0.613654\pi$
$3$	−1.21076	−0.699030	−0.349515	+	0.936931i	$0.613654\pi$
$4$	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
$9$	−1.53407	−0.511357
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.21076	−0.335803	−0.167902	−	0.985804i	$-0.553699\pi$
$13$	−1.21076	−0.335803	−0.167902	+	0.985804i	$0.553699\pi$
$14$	0	0
$15$	1.21076	0.312616
$16$	0	0
$17$	4.27890	1.03778	0.518892	−	0.854840i	$-0.326345\pi$
$17$	4.27890	1.03778	0.518892	+	0.854840i	$0.326345\pi$
$18$	0	0
$19$	−7.06814	−1.62154	−0.810771	−	0.585363i	$-0.800952\pi$
$19$	−7.06814	−1.62154	−0.810771	+	0.585363i	$0.800952\pi$
$20$	0	0
$21$	−1.21076	−0.264209
$22$	0	0
$23$	2.95558	0.616281	0.308141	−	0.951341i	$-0.400293\pi$
$23$	2.95558	0.616281	0.308141	+	0.951341i	$0.400293\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.48965	1.05648
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	6.16634	1.10751	0.553753	−	0.832681i	$-0.313195\pi$
$31$	6.16634	1.10751	0.553753	+	0.832681i	$0.313195\pi$
$32$	0	0
$33$	−1.21076	−0.210766
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−4.95558	−0.814693	−0.407346	−	0.913274i	$-0.633546\pi$
$37$	−4.95558	−0.814693	−0.407346	+	0.913274i	$0.633546\pi$
$38$	0	0
$39$	1.46593	0.234737
$40$	0	0
$41$	0.166338	0.0259776	0.0129888	−	0.999916i	$-0.495865\pi$
$41$	0.166338	0.0259776	0.0129888	+	0.999916i	$0.495865\pi$
$42$	0	0
$43$	−6.44523	−0.982889	−0.491444	−	0.870909i	$-0.663531\pi$
$43$	−6.44523	−0.982889	−0.491444	+	0.870909i	$0.663531\pi$
$44$	0	0
$45$	1.53407	0.228686
$46$	0	0
$47$	2.70041	0.393895	0.196947	−	0.980414i	$-0.436897\pi$
$47$	2.70041	0.393895	0.196947	+	0.980414i	$0.436897\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.18070	−0.725443
$52$	0	0
$53$	4.11256	0.564903	0.282452	−	0.959282i	$-0.408852\pi$
$53$	4.11256	0.564903	0.282452	+	0.959282i	$0.408852\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	8.55779	1.13351
$58$	0	0
$59$	14.7241	1.91692	0.958459	−	0.285229i	$-0.0920698\pi$
$59$	14.7241	1.91692	0.958459	+	0.285229i	$0.0920698\pi$
$60$	0	0
$61$	−14.3915	−1.84264	−0.921318	−	0.388809i	$-0.872887\pi$
$61$	−14.3915	−1.84264	−0.921318	+	0.388809i	$0.872887\pi$
$62$	0	0
$63$	−1.53407	−0.193275
$64$	0	0
$65$	1.21076	0.150176
$66$	0	0
$67$	9.60221	1.17310	0.586548	−	0.809914i	$-0.300486\pi$
$67$	9.60221	1.17310	0.586548	+	0.809914i	$0.300486\pi$
$68$	0	0
$69$	−3.57849	−0.430799
$70$	0	0
$71$	−6.13628	−0.728243	−0.364121	−	0.931352i	$-0.618631\pi$
$71$	−6.13628	−0.728243	−0.364121	+	0.931352i	$0.618631\pi$
$72$	0	0
$73$	−12.2789	−1.43714	−0.718568	−	0.695457i	$-0.755202\pi$
$73$	−12.2789	−1.43714	−0.718568	+	0.695457i	$0.755202\pi$
$74$	0	0
$75$	−1.21076	−0.139806
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	12.1126	1.36277	0.681385	−	0.731926i	$-0.261378\pi$
$79$	12.1126	1.36277	0.681385	+	0.731926i	$0.261378\pi$
$80$	0	0
$81$	−2.04442	−0.227158
$82$	0	0
$83$	1.48965	0.163511	0.0817553	−	0.996652i	$-0.473947\pi$
$83$	1.48965	0.163511	0.0817553	+	0.996652i	$0.473947\pi$
$84$	0	0
$85$	−4.27890	−0.464111
$86$	0	0
$87$	2.42151	0.259613
$88$	0	0
$89$	2.42151	0.256680	0.128340	−	0.991730i	$-0.459035\pi$
$89$	2.42151	0.256680	0.128340	+	0.991730i	$0.459035\pi$
$90$	0	0
$91$	−1.21076	−0.126922
$92$	0	0
$93$	−7.46593	−0.774181
$94$	0	0
$95$	7.06814	0.725176
$96$	0	0
$97$	−7.71477	−0.783316	−0.391658	−	0.920111i	$-0.628098\pi$
$97$	−7.71477	−0.783316	−0.391658	+	0.920111i	$0.628098\pi$
$98$	0	0
$99$	−1.53407	−0.154180
$100$	0	0
$101$	−14.0775	−1.40076	−0.700382	−	0.713768i	$-0.746987\pi$
$101$	−14.0775	−1.40076	−0.700382	+	0.713768i	$0.746987\pi$
$102$	0	0
$103$	13.7685	1.35666	0.678328	−	0.734760i	$-0.262705\pi$
$103$	13.7685	1.35666	0.678328	+	0.734760i	$0.262705\pi$
$104$	0	0
$105$	1.21076	0.118158
$106$	0	0
$107$	5.91116	0.571454	0.285727	−	0.958311i	$-0.407765\pi$
$107$	5.91116	0.571454	0.285727	+	0.958311i	$0.407765\pi$
$108$	0	0
$109$	0.421512	0.0403735	0.0201868	−	0.999796i	$-0.493574\pi$
$109$	0.421512	0.0403735	0.0201868	+	0.999796i	$0.493574\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	−13.7148	−1.29018	−0.645088	−	0.764108i	$-0.723179\pi$
$113$	−13.7148	−1.29018	−0.645088	+	0.764108i	$0.723179\pi$
$114$	0	0
$115$	−2.95558	−0.275609
$116$	0	0
$117$	1.85738	0.171715
$118$	0	0
$119$	4.27890	0.392246
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.201395	−0.0181591
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−9.62593	−0.854163	−0.427082	−	0.904213i	$-0.640458\pi$
$127$	−9.62593	−0.854163	−0.427082	+	0.904213i	$0.640458\pi$
$128$	0	0
$129$	7.80361	0.687069
$130$	0	0
$131$	−14.9793	−1.30875	−0.654374	−	0.756171i	$-0.727068\pi$
$131$	−14.9793	−1.30875	−0.654374	+	0.756171i	$0.727068\pi$
$132$	0	0
$133$	−7.06814	−0.612886
$134$	0	0
$135$	−5.48965	−0.472474
$136$	0	0
$137$	22.0237	1.88161	0.940807	−	0.338943i	$-0.110069\pi$
$137$	22.0237	1.88161	0.940807	+	0.338943i	$0.110069\pi$
$138$	0	0
$139$	−3.26454	−0.276894	−0.138447	−	0.990370i	$-0.544211\pi$
$139$	−3.26454	−0.276894	−0.138447	+	0.990370i	$0.544211\pi$
$140$	0	0
$141$	−3.26953	−0.275345
$142$	0	0
$143$	−1.21076	−0.101248
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	−1.21076	−0.0998615
$148$	0	0
$149$	−1.66732	−0.136593	−0.0682963	−	0.997665i	$-0.521756\pi$
$149$	−1.66732	−0.136593	−0.0682963	+	0.997665i	$0.521756\pi$
$150$	0	0
$151$	−11.8273	−0.962494	−0.481247	−	0.876585i	$-0.659816\pi$
$151$	−11.8273	−0.962494	−0.481247	+	0.876585i	$0.659816\pi$
$152$	0	0
$153$	−6.56413	−0.530678
$154$	0	0
$155$	−6.16634	−0.495292
$156$	0	0
$157$	10.1363	0.808963	0.404482	−	0.914546i	$-0.367452\pi$
$157$	10.1363	0.808963	0.404482	+	0.914546i	$0.367452\pi$
$158$	0	0
$159$	−4.97930	−0.394885
$160$	0	0
$161$	2.95558	0.232932
$162$	0	0
$163$	2.39779	0.187809	0.0939047	−	0.995581i	$-0.470065\pi$
$163$	2.39779	0.187809	0.0939047	+	0.995581i	$0.470065\pi$
$164$	0	0
$165$	1.21076	0.0942572
$166$	0	0
$167$	−13.8223	−1.06960	−0.534802	−	0.844977i	$-0.679614\pi$
$167$	−13.8223	−1.06960	−0.534802	+	0.844977i	$0.679614\pi$
$168$	0	0
$169$	−11.5341	−0.887236
$170$	0	0
$171$	10.8430	0.829187
$172$	0	0
$173$	−18.1901	−1.38296	−0.691482	−	0.722393i	$-0.743042\pi$
$173$	−18.1901	−1.38296	−0.691482	+	0.722393i	$0.743042\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−17.8273	−1.33998
$178$	0	0
$179$	−15.5972	−1.16579	−0.582895	−	0.812547i	$-0.698080\pi$
$179$	−15.5972	−1.16579	−0.582895	+	0.812547i	$0.698080\pi$
$180$	0	0
$181$	19.2044	1.42745	0.713727	−	0.700424i	$-0.247006\pi$
$181$	19.2044	1.42745	0.713727	+	0.700424i	$0.247006\pi$
$182$	0	0
$183$	17.4245	1.28806
$184$	0	0
$185$	4.95558	0.364342
$186$	0	0
$187$	4.27890	0.312904
$188$	0	0
$189$	5.48965	0.399313
$190$	0	0
$191$	14.4215	1.04350	0.521752	−	0.853097i	$-0.325279\pi$
$191$	14.4215	1.04350	0.521752	+	0.853097i	$0.325279\pi$
$192$	0	0
$193$	−4.62291	−0.332764	−0.166382	−	0.986061i	$-0.553208\pi$
$193$	−4.62291	−0.332764	−0.166382	+	0.986061i	$0.553208\pi$
$194$	0	0
$195$	−1.46593	−0.104977
$196$	0	0
$197$	3.37709	0.240608	0.120304	−	0.992737i	$-0.461613\pi$
$197$	3.37709	0.240608	0.120304	+	0.992737i	$0.461613\pi$
$198$	0	0
$199$	−21.7923	−1.54481	−0.772407	−	0.635128i	$-0.780947\pi$
$199$	−21.7923	−1.54481	−0.772407	+	0.635128i	$0.780947\pi$
$200$	0	0
$201$	−11.6259	−0.820030
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	−0.166338	−0.0116175
$206$	0	0
$207$	−4.53407	−0.315140
$208$	0	0
$209$	−7.06814	−0.488913
$210$	0	0
$211$	25.4483	1.75193	0.875965	−	0.482374i	$-0.160225\pi$
$211$	25.4483	1.75193	0.875965	+	0.482374i	$0.160225\pi$
$212$	0	0
$213$	7.42954	0.509064
$214$	0	0
$215$	6.44523	0.439561
$216$	0	0
$217$	6.16634	0.418598
$218$	0	0
$219$	14.8667	1.00460
$220$	0	0
$221$	−5.18070	−0.348492
$222$	0	0
$223$	−14.0538	−0.941111	−0.470555	−	0.882370i	$-0.655946\pi$
$223$	−14.0538	−0.941111	−0.470555	+	0.882370i	$0.655946\pi$
$224$	0	0
$225$	−1.53407	−0.102271
$226$	0	0
$227$	−25.2933	−1.67877	−0.839386	−	0.543535i	$-0.817085\pi$
$227$	−25.2933	−1.67877	−0.839386	+	0.543535i	$0.817085\pi$
$228$	0	0
$229$	21.3120	1.40834	0.704168	−	0.710034i	$-0.251320\pi$
$229$	21.3120	1.40834	0.704168	+	0.710034i	$0.251320\pi$
$230$	0	0
$231$	−1.21076	−0.0796619
$232$	0	0
$233$	−12.7305	−0.834000	−0.417000	−	0.908906i	$-0.636919\pi$
$233$	−12.7305	−0.834000	−0.417000	+	0.908906i	$0.636919\pi$
$234$	0	0
$235$	−2.70041	−0.176155
$236$	0	0
$237$	−14.6654	−0.952617
$238$	0	0
$239$	20.7829	1.34433	0.672167	−	0.740399i	$-0.265364\pi$
$239$	20.7829	1.34433	0.672167	+	0.740399i	$0.265364\pi$
$240$	0	0
$241$	−23.8811	−1.53832	−0.769159	−	0.639058i	$-0.779324\pi$
$241$	−23.8811	−1.53832	−0.769159	+	0.639058i	$0.779324\pi$
$242$	0	0
$243$	−13.9937	−0.897694
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	8.55779	0.544519
$248$	0	0
$249$	−1.80361	−0.114299
$250$	0	0
$251$	−8.76552	−0.553275	−0.276637	−	0.960974i	$-0.589220\pi$
$251$	−8.76552	−0.553275	−0.276637	+	0.960974i	$0.589220\pi$
$252$	0	0
$253$	2.95558	0.185816
$254$	0	0
$255$	5.18070	0.324428
$256$	0	0
$257$	20.8905	1.30311	0.651556	−	0.758601i	$-0.274117\pi$
$257$	20.8905	1.30311	0.651556	+	0.758601i	$0.274117\pi$
$258$	0	0
$259$	−4.95558	−0.307925
$260$	0	0
$261$	3.06814	0.189913
$262$	0	0
$263$	0.960582	0.0592320	0.0296160	−	0.999561i	$-0.490572\pi$
$263$	0.960582	0.0592320	0.0296160	+	0.999561i	$0.490572\pi$
$264$	0	0
$265$	−4.11256	−0.252632
$266$	0	0
$267$	−2.93186	−0.179427
$268$	0	0
$269$	−28.2726	−1.72381	−0.861904	−	0.507071i	$-0.830728\pi$
$269$	−28.2726	−1.72381	−0.861904	+	0.507071i	$0.830728\pi$
$270$	0	0
$271$	19.5371	1.18679	0.593397	−	0.804910i	$-0.297786\pi$
$271$	19.5371	1.18679	0.593397	+	0.804910i	$0.297786\pi$
$272$	0	0
$273$	1.46593	0.0887221
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	3.04442	0.182921	0.0914607	−	0.995809i	$-0.470846\pi$
$277$	3.04442	0.182921	0.0914607	+	0.995809i	$0.470846\pi$
$278$	0	0
$279$	−9.45960	−0.566331
$280$	0	0
$281$	14.0474	0.838000	0.419000	−	0.907986i	$-0.362381\pi$
$281$	14.0474	0.838000	0.419000	+	0.907986i	$0.362381\pi$
$282$	0	0
$283$	−10.7542	−0.639270	−0.319635	−	0.947541i	$-0.603560\pi$
$283$	−10.7542	−0.639270	−0.319635	+	0.947541i	$0.603560\pi$
$284$	0	0
$285$	−8.55779	−0.506920
$286$	0	0
$287$	0.166338	0.00981861
$288$	0	0
$289$	1.30895	0.0769973
$290$	0	0
$291$	9.34070	0.547562
$292$	0	0
$293$	−21.6323	−1.26377	−0.631885	−	0.775062i	$-0.717719\pi$
$293$	−21.6323	−1.26377	−0.631885	+	0.775062i	$0.717719\pi$
$294$	0	0
$295$	−14.7241	−0.857272
$296$	0	0
$297$	5.48965	0.318542
$298$	0	0
$299$	−3.57849	−0.206949
$300$	0	0
$301$	−6.44523	−0.371497
$302$	0	0
$303$	17.0444	0.979176
$304$	0	0
$305$	14.3915	0.824052
$306$	0	0
$307$	−31.7335	−1.81113	−0.905563	−	0.424212i	$-0.860551\pi$
$307$	−31.7335	−1.81113	−0.905563	+	0.424212i	$0.860551\pi$
$308$	0	0
$309$	−16.6704	−0.948343
$310$	0	0
$311$	−30.6353	−1.73717	−0.868584	−	0.495542i	$-0.834970\pi$
$311$	−30.6353	−1.73717	−0.868584	+	0.495542i	$0.834970\pi$
$312$	0	0
$313$	−17.5972	−0.994653	−0.497327	−	0.867563i	$-0.665685\pi$
$313$	−17.5972	−0.994653	−0.497327	+	0.867563i	$0.665685\pi$
$314$	0	0
$315$	1.53407	0.0864351
$316$	0	0
$317$	−4.97930	−0.279666	−0.139833	−	0.990175i	$-0.544657\pi$
$317$	−4.97930	−0.279666	−0.139833	+	0.990175i	$0.544657\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	0	0
$321$	−7.15698	−0.399463
$322$	0	0
$323$	−30.2438	−1.68281
$324$	0	0
$325$	−1.21076	−0.0671607
$326$	0	0
$327$	−0.510348	−0.0282223
$328$	0	0
$329$	2.70041	0.148878
$330$	0	0
$331$	6.19639	0.340585	0.170292	−	0.985394i	$-0.445529\pi$
$331$	6.19639	0.340585	0.170292	+	0.985394i	$0.445529\pi$
$332$	0	0
$333$	7.60221	0.416599
$334$	0	0
$335$	−9.60221	−0.524625
$336$	0	0
$337$	−25.5134	−1.38980	−0.694901	−	0.719105i	$-0.744552\pi$
$337$	−25.5134	−1.38980	−0.694901	+	0.719105i	$0.744552\pi$
$338$	0	0
$339$	16.6052	0.901873
$340$	0	0
$341$	6.16634	0.333926
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	3.57849	0.192659
$346$	0	0
$347$	−11.4659	−0.615523	−0.307762	−	0.951463i	$-0.599580\pi$
$347$	−11.4659	−0.615523	−0.307762	+	0.951463i	$0.599580\pi$
$348$	0	0
$349$	0.812966	0.0435171	0.0217585	−	0.999763i	$-0.493073\pi$
$349$	0.812966	0.0435171	0.0217585	+	0.999763i	$0.493073\pi$
$350$	0	0
$351$	−6.64663	−0.354771
$352$	0	0
$353$	−0.754187	−0.0401413	−0.0200707	−	0.999799i	$-0.506389\pi$
$353$	−0.754187	−0.0401413	−0.0200707	+	0.999799i	$0.506389\pi$
$354$	0	0
$355$	6.13628	0.325680
$356$	0	0
$357$	−5.18070	−0.274192
$358$	0	0
$359$	−30.0712	−1.58710	−0.793548	−	0.608508i	$-0.791768\pi$
$359$	−30.0712	−1.58710	−0.793548	+	0.608508i	$0.791768\pi$
$360$	0	0
$361$	30.9586	1.62940
$362$	0	0
$363$	−1.21076	−0.0635482
$364$	0	0
$365$	12.2789	0.642707
$366$	0	0
$367$	−8.23145	−0.429678	−0.214839	−	0.976649i	$-0.568923\pi$
$367$	−8.23145	−0.429678	−0.214839	+	0.976649i	$0.568923\pi$
$368$	0	0
$369$	−0.255174	−0.0132838
$370$	0	0
$371$	4.11256	0.213513
$372$	0	0
$373$	−24.2201	−1.25407	−0.627035	−	0.778991i	$-0.715732\pi$
$373$	−24.2201	−1.25407	−0.627035	+	0.778991i	$0.715732\pi$
$374$	0	0
$375$	1.21076	0.0625232
$376$	0	0
$377$	2.42151	0.124714
$378$	0	0
$379$	0.843024	0.0433032	0.0216516	−	0.999766i	$-0.493108\pi$
$379$	0.843024	0.0433032	0.0216516	+	0.999766i	$0.493108\pi$
$380$	0	0
$381$	11.6547	0.597086
$382$	0	0
$383$	−31.2869	−1.59869	−0.799344	−	0.600874i	$-0.794819\pi$
$383$	−31.2869	−1.59869	−0.799344	+	0.600874i	$0.794819\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	9.88744	0.502607
$388$	0	0
$389$	−25.0869	−1.27195	−0.635977	−	0.771708i	$-0.719403\pi$
$389$	−25.0869	−1.27195	−0.635977	+	0.771708i	$0.719403\pi$
$390$	0	0
$391$	12.6466	0.639568
$392$	0	0
$393$	18.1363	0.914854
$394$	0	0
$395$	−12.1126	−0.609449
$396$	0	0
$397$	−3.77488	−0.189456	−0.0947280	−	0.995503i	$-0.530198\pi$
$397$	−3.77488	−0.189456	−0.0947280	+	0.995503i	$0.530198\pi$
$398$	0	0
$399$	8.55779	0.428425
$400$	0	0
$401$	−8.31395	−0.415179	−0.207589	−	0.978216i	$-0.566562\pi$
$401$	−8.31395	−0.415179	−0.207589	+	0.978216i	$0.566562\pi$
$402$	0	0
$403$	−7.46593	−0.371904
$404$	0	0
$405$	2.04442	0.101588
$406$	0	0
$407$	−4.95558	−0.245639
$408$	0	0
$409$	26.2138	1.29619	0.648094	−	0.761560i	$-0.275566\pi$
$409$	26.2138	1.29619	0.648094	+	0.761560i	$0.275566\pi$
$410$	0	0
$411$	−26.6654	−1.31530
$412$	0	0
$413$	14.7241	0.724527
$414$	0	0
$415$	−1.48965	−0.0731241
$416$	0	0
$417$	3.95256	0.193557
$418$	0	0
$419$	24.8417	1.21360	0.606798	−	0.794856i	$-0.292454\pi$
$419$	24.8417	1.21360	0.606798	+	0.794856i	$0.292454\pi$
$420$	0	0
$421$	−14.1126	−0.687804	−0.343902	−	0.939006i	$-0.611749\pi$
$421$	−14.1126	−0.687804	−0.343902	+	0.939006i	$0.611749\pi$
$422$	0	0
$423$	−4.14262	−0.201421
$424$	0	0
$425$	4.27890	0.207557
$426$	0	0
$427$	−14.3915	−0.696451
$428$	0	0
$429$	1.46593	0.0707758
$430$	0	0
$431$	33.2281	1.60054	0.800272	−	0.599638i	$-0.204689\pi$
$431$	33.2281	1.60054	0.800272	+	0.599638i	$0.204689\pi$
$432$	0	0
$433$	−27.4008	−1.31680	−0.658400	−	0.752669i	$-0.728766\pi$
$433$	−27.4008	−1.31680	−0.658400	+	0.752669i	$0.728766\pi$
$434$	0	0
$435$	−2.42151	−0.116103
$436$	0	0
$437$	−20.8905	−0.999327
$438$	0	0
$439$	38.9192	1.85751	0.928756	−	0.370692i	$-0.120879\pi$
$439$	38.9192	1.85751	0.928756	+	0.370692i	$0.120879\pi$
$440$	0	0
$441$	−1.53407	−0.0730510
$442$	0	0
$443$	−5.04442	−0.239668	−0.119834	−	0.992794i	$-0.538236\pi$
$443$	−5.04442	−0.239668	−0.119834	+	0.992794i	$0.538236\pi$
$444$	0	0
$445$	−2.42151	−0.114791
$446$	0	0
$447$	2.01872	0.0954823
$448$	0	0
$449$	20.8667	0.984763	0.492381	−	0.870380i	$-0.336127\pi$
$449$	20.8667	0.984763	0.492381	+	0.870380i	$0.336127\pi$
$450$	0	0
$451$	0.166338	0.00783254
$452$	0	0
$453$	14.3200	0.672813
$454$	0	0
$455$	1.21076	0.0567611
$456$	0	0
$457$	−3.77988	−0.176815	−0.0884077	−	0.996084i	$-0.528178\pi$
$457$	−3.77988	−0.176815	−0.0884077	+	0.996084i	$0.528178\pi$
$458$	0	0
$459$	23.4897	1.09640
$460$	0	0
$461$	−17.7923	−0.828669	−0.414334	−	0.910125i	$-0.635986\pi$
$461$	−17.7923	−0.828669	−0.414334	+	0.910125i	$0.635986\pi$
$462$	0	0
$463$	22.0524	1.02486	0.512432	−	0.858728i	$-0.328745\pi$
$463$	22.0524	1.02486	0.512432	+	0.858728i	$0.328745\pi$
$464$	0	0
$465$	7.46593	0.346224
$466$	0	0
$467$	−28.9255	−1.33851	−0.669257	−	0.743031i	$-0.733387\pi$
$467$	−28.9255	−1.33851	−0.669257	+	0.743031i	$0.733387\pi$
$468$	0	0
$469$	9.60221	0.443389
$470$	0	0
$471$	−12.2726	−0.565490
$472$	0	0
$473$	−6.44523	−0.296352
$474$	0	0
$475$	−7.06814	−0.324309
$476$	0	0
$477$	−6.30895	−0.288867
$478$	0	0
$479$	14.8430	0.678195	0.339098	−	0.940751i	$-0.389878\pi$
$479$	14.8430	0.678195	0.339098	+	0.940751i	$0.389878\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	−3.57849	−0.162827
$484$	0	0
$485$	7.71477	0.350310
$486$	0	0
$487$	−6.55279	−0.296935	−0.148468	−	0.988917i	$-0.547434\pi$
$487$	−6.55279	−0.296935	−0.148468	+	0.988917i	$0.547434\pi$
$488$	0	0
$489$	−2.90314	−0.131284
$490$	0	0
$491$	−1.58349	−0.0714618	−0.0357309	−	0.999361i	$-0.511376\pi$
$491$	−1.58349	−0.0714618	−0.0357309	+	0.999361i	$0.511376\pi$
$492$	0	0
$493$	−8.55779	−0.385424
$494$	0	0
$495$	1.53407	0.0689513
$496$	0	0
$497$	−6.13628	−0.275250
$498$	0	0
$499$	8.61791	0.385790	0.192895	−	0.981219i	$-0.438212\pi$
$499$	8.61791	0.385790	0.192895	+	0.981219i	$0.438212\pi$
$500$	0	0
$501$	16.7355	0.747685
$502$	0	0
$503$	−23.3534	−1.04128	−0.520638	−	0.853778i	$-0.674306\pi$
$503$	−23.3534	−1.04128	−0.520638	+	0.853778i	$0.674306\pi$
$504$	0	0
$505$	14.0775	0.626441
$506$	0	0
$507$	13.9649	0.620205
$508$	0	0
$509$	23.8510	1.05718	0.528590	−	0.848878i	$-0.322721\pi$
$509$	23.8510	1.05718	0.528590	+	0.848878i	$0.322721\pi$
$510$	0	0
$511$	−12.2789	−0.543186
$512$	0	0
$513$	−38.8016	−1.71313
$514$	0	0
$515$	−13.7685	−0.606715
$516$	0	0
$517$	2.70041	0.118764
$518$	0	0
$519$	22.0237	0.966734
$520$	0	0
$521$	−36.7128	−1.60842	−0.804208	−	0.594347i	$-0.797410\pi$
$521$	−36.7128	−1.60842	−0.804208	+	0.594347i	$0.797410\pi$
$522$	0	0
$523$	21.5972	0.944380	0.472190	−	0.881497i	$-0.343464\pi$
$523$	21.5972	0.944380	0.472190	+	0.881497i	$0.343464\pi$
$524$	0	0
$525$	−1.21076	−0.0528417
$526$	0	0
$527$	26.3851	1.14935
$528$	0	0
$529$	−14.2645	−0.620197
$530$	0	0
$531$	−22.5878	−0.980229
$532$	0	0
$533$	−0.201395	−0.00872336
$534$	0	0
$535$	−5.91116	−0.255562
$536$	0	0
$537$	18.8844	0.814923
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−5.48965	−0.236019	−0.118009	−	0.993012i	$-0.537651\pi$
$541$	−5.48965	−0.236019	−0.118009	+	0.993012i	$0.537651\pi$
$542$	0	0
$543$	−23.2519	−0.997833
$544$	0	0
$545$	−0.421512	−0.0180556
$546$	0	0
$547$	−27.7148	−1.18500	−0.592499	−	0.805571i	$-0.701859\pi$
$547$	−27.7148	−1.18500	−0.592499	+	0.805571i	$0.701859\pi$
$548$	0	0
$549$	22.0775	0.942245
$550$	0	0
$551$	14.1363	0.602226
$552$	0	0
$553$	12.1126	0.515078
$554$	0	0
$555$	−6.00000	−0.254686
$556$	0	0
$557$	−5.18070	−0.219513	−0.109757	−	0.993958i	$-0.535007\pi$
$557$	−5.18070	−0.219513	−0.109757	+	0.993958i	$0.535007\pi$
$558$	0	0
$559$	7.80361	0.330057
$560$	0	0
$561$	−5.18070	−0.218729
$562$	0	0
$563$	−8.96058	−0.377643	−0.188822	−	0.982011i	$-0.560467\pi$
$563$	−8.96058	−0.377643	−0.188822	+	0.982011i	$0.560467\pi$
$564$	0	0
$565$	13.7148	0.576985
$566$	0	0
$567$	−2.04442	−0.0858575
$568$	0	0
$569$	−19.4008	−0.813325	−0.406662	−	0.913579i	$-0.633307\pi$
$569$	−19.4008	−0.813325	−0.406662	+	0.913579i	$0.633307\pi$
$570$	0	0
$571$	−35.5845	−1.48917	−0.744583	−	0.667529i	$-0.767352\pi$
$571$	−35.5845	−1.48917	−0.744583	+	0.667529i	$0.767352\pi$
$572$	0	0
$573$	−17.4609	−0.729441
$574$	0	0
$575$	2.95558	0.123256
$576$	0	0
$577$	11.6673	0.485717	0.242859	−	0.970062i	$-0.421915\pi$
$577$	11.6673	0.485717	0.242859	+	0.970062i	$0.421915\pi$
$578$	0	0
$579$	5.59721	0.232612
$580$	0	0
$581$	1.48965	0.0618012
$582$	0	0
$583$	4.11256	0.170325
$584$	0	0
$585$	−1.85738	−0.0767934
$586$	0	0
$587$	−10.3865	−0.428695	−0.214347	−	0.976757i	$-0.568762\pi$
$587$	−10.3865	−0.428695	−0.214347	+	0.976757i	$0.568762\pi$
$588$	0	0
$589$	−43.5845	−1.79587
$590$	0	0
$591$	−4.08884	−0.168192
$592$	0	0
$593$	−0.367732	−0.0151010	−0.00755048	−	0.999971i	$-0.502403\pi$
$593$	−0.367732	−0.0151010	−0.00755048	+	0.999971i	$0.502403\pi$
$594$	0	0
$595$	−4.27890	−0.175418
$596$	0	0
$597$	26.3851	1.07987
$598$	0	0
$599$	−2.93186	−0.119793	−0.0598963	−	0.998205i	$-0.519077\pi$
$599$	−2.93186	−0.119793	−0.0598963	+	0.998205i	$0.519077\pi$
$600$	0	0
$601$	−2.58785	−0.105561	−0.0527803	−	0.998606i	$-0.516808\pi$
$601$	−2.58785	−0.105561	−0.0527803	+	0.998606i	$0.516808\pi$
$602$	0	0
$603$	−14.7305	−0.599871
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−6.27256	−0.254595	−0.127298	−	0.991865i	$-0.540630\pi$
$607$	−6.27256	−0.254595	−0.127298	+	0.991865i	$0.540630\pi$
$608$	0	0
$609$	2.42151	0.0981246
$610$	0	0
$611$	−3.26953	−0.132271
$612$	0	0
$613$	−5.84605	−0.236120	−0.118060	−	0.993006i	$-0.537667\pi$
$613$	−5.84605	−0.236120	−0.118060	+	0.993006i	$0.537667\pi$
$614$	0	0
$615$	0.201395	0.00812101
$616$	0	0
$617$	14.8667	0.598513	0.299256	−	0.954173i	$-0.403261\pi$
$617$	14.8667	0.598513	0.299256	+	0.954173i	$0.403261\pi$
$618$	0	0
$619$	−16.6166	−0.667876	−0.333938	−	0.942595i	$-0.608378\pi$
$619$	−16.6166	−0.667876	−0.333938	+	0.942595i	$0.608378\pi$
$620$	0	0
$621$	16.2251	0.651092
$622$	0	0
$623$	2.42151	0.0970158
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	8.55779	0.341765
$628$	0	0
$629$	−21.2044	−0.845476
$630$	0	0
$631$	47.2519	1.88107	0.940534	−	0.339701i	$-0.110326\pi$
$631$	47.2519	1.88107	0.940534	+	0.339701i	$0.110326\pi$
$632$	0	0
$633$	−30.8116	−1.22465
$634$	0	0
$635$	9.62593	0.381993
$636$	0	0
$637$	−1.21076	−0.0479719
$638$	0	0
$639$	9.41349	0.372392
$640$	0	0
$641$	−14.7305	−0.581818	−0.290909	−	0.956751i	$-0.593958\pi$
$641$	−14.7305	−0.581818	−0.290909	+	0.956751i	$0.593958\pi$
$642$	0	0
$643$	−11.5909	−0.457100	−0.228550	−	0.973532i	$-0.573398\pi$
$643$	−11.5909	−0.457100	−0.228550	+	0.973532i	$0.573398\pi$
$644$	0	0
$645$	−7.80361	−0.307267
$646$	0	0
$647$	−20.6116	−0.810325	−0.405162	−	0.914245i	$-0.632785\pi$
$647$	−20.6116	−0.810325	−0.405162	+	0.914245i	$0.632785\pi$
$648$	0	0
$649$	14.7241	0.577973
$650$	0	0
$651$	−7.46593	−0.292613
$652$	0	0
$653$	8.24384	0.322606	0.161303	−	0.986905i	$-0.448430\pi$
$653$	8.24384	0.322606	0.161303	+	0.986905i	$0.448430\pi$
$654$	0	0
$655$	14.9793	0.585290
$656$	0	0
$657$	18.8367	0.734889
$658$	0	0
$659$	−29.9536	−1.16683	−0.583413	−	0.812175i	$-0.698283\pi$
$659$	−29.9536	−1.16683	−0.583413	+	0.812175i	$0.698283\pi$
$660$	0	0
$661$	34.2913	1.33378	0.666888	−	0.745158i	$-0.267626\pi$
$661$	34.2913	1.33378	0.666888	+	0.745158i	$0.267626\pi$
$662$	0	0
$663$	6.27256	0.243606
$664$	0	0
$665$	7.06814	0.274091
$666$	0	0
$667$	−5.91116	−0.228881
$668$	0	0
$669$	17.0157	0.657865
$670$	0	0
$671$	−14.3915	−0.555576
$672$	0	0
$673$	26.8541	1.03515	0.517574	−	0.855638i	$-0.326835\pi$
$673$	26.8541	1.03515	0.517574	+	0.855638i	$0.326835\pi$
$674$	0	0
$675$	5.48965	0.211297
$676$	0	0
$677$	40.5227	1.55742	0.778708	−	0.627387i	$-0.215876\pi$
$677$	40.5227	1.55742	0.778708	+	0.627387i	$0.215876\pi$
$678$	0	0
$679$	−7.71477	−0.296066
$680$	0	0
$681$	30.6240	1.17351
$682$	0	0
$683$	−42.8855	−1.64097	−0.820483	−	0.571670i	$-0.806296\pi$
$683$	−42.8855	−1.64097	−0.820483	+	0.571670i	$0.806296\pi$
$684$	0	0
$685$	−22.0237	−0.841483
$686$	0	0
$687$	−25.8036	−0.984469
$688$	0	0
$689$	−4.97930	−0.189696
$690$	0	0
$691$	1.63727	0.0622846	0.0311423	−	0.999515i	$-0.490085\pi$
$691$	1.63727	0.0622846	0.0311423	+	0.999515i	$0.490085\pi$
$692$	0	0
$693$	−1.53407	−0.0582745
$694$	0	0
$695$	3.26454	0.123831
$696$	0	0
$697$	0.711742	0.0269592
$698$	0	0
$699$	15.4135	0.582992
$700$	0	0
$701$	27.8036	1.05013	0.525064	−	0.851063i	$-0.324041\pi$
$701$	27.8036	1.05013	0.525064	+	0.851063i	$0.324041\pi$
$702$	0	0
$703$	35.0267	1.32106
$704$	0	0
$705$	3.26953	0.123138
$706$	0	0
$707$	−14.0775	−0.529439
$708$	0	0
$709$	−33.9172	−1.27379	−0.636894	−	0.770951i	$-0.719781\pi$
$709$	−33.9172	−1.27379	−0.636894	+	0.770951i	$0.719781\pi$
$710$	0	0
$711$	−18.5815	−0.696861
$712$	0	0
$713$	18.2251	0.682536
$714$	0	0
$715$	1.21076	0.0452797
$716$	0	0
$717$	−25.1630	−0.939731
$718$	0	0
$719$	−28.6640	−1.06899	−0.534494	−	0.845172i	$-0.679498\pi$
$719$	−28.6640	−1.06899	−0.534494	+	0.845172i	$0.679498\pi$
$720$	0	0
$721$	13.7685	0.512768
$722$	0	0
$723$	28.9142	1.07533
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	20.0538	0.743754	0.371877	−	0.928282i	$-0.378714\pi$
$727$	20.0538	0.743754	0.371877	+	0.928282i	$0.378714\pi$
$728$	0	0
$729$	23.0762	0.854673
$730$	0	0
$731$	−27.5785	−1.02003
$732$	0	0
$733$	−19.8160	−0.731920	−0.365960	−	0.930631i	$-0.619259\pi$
$733$	−19.8160	−0.731920	−0.365960	+	0.930631i	$0.619259\pi$
$734$	0	0
$735$	1.21076	0.0446594
$736$	0	0
$737$	9.60221	0.353702
$738$	0	0
$739$	−6.80163	−0.250202	−0.125101	−	0.992144i	$-0.539926\pi$
$739$	−6.80163	−0.250202	−0.125101	+	0.992144i	$0.539926\pi$
$740$	0	0
$741$	−10.3614	−0.380635
$742$	0	0
$743$	−8.79558	−0.322678	−0.161339	−	0.986899i	$-0.551581\pi$
$743$	−8.79558	−0.322678	−0.161339	+	0.986899i	$0.551581\pi$
$744$	0	0
$745$	1.66732	0.0610860
$746$	0	0
$747$	−2.28523	−0.0836122
$748$	0	0
$749$	5.91116	0.215989
$750$	0	0
$751$	31.9873	1.16723	0.583617	−	0.812029i	$-0.301637\pi$
$751$	31.9873	1.16723	0.583617	+	0.812029i	$0.301637\pi$
$752$	0	0
$753$	10.6129	0.386756
$754$	0	0
$755$	11.8273	0.430441
$756$	0	0
$757$	11.5310	0.419103	0.209551	−	0.977798i	$-0.432800\pi$
$757$	11.5310	0.419103	0.209551	+	0.977798i	$0.432800\pi$
$758$	0	0
$759$	−3.57849	−0.129891
$760$	0	0
$761$	−18.7716	−0.680469	−0.340235	−	0.940341i	$-0.610506\pi$
$761$	−18.7716	−0.680469	−0.340235	+	0.940341i	$0.610506\pi$
$762$	0	0
$763$	0.421512	0.0152598
$764$	0	0
$765$	6.56413	0.237327
$766$	0	0
$767$	−17.8273	−0.643707
$768$	0	0
$769$	−15.2759	−0.550862	−0.275431	−	0.961321i	$-0.588821\pi$
$769$	−15.2759	−0.550862	−0.275431	+	0.961321i	$0.588821\pi$
$770$	0	0
$771$	−25.2933	−0.910914
$772$	0	0
$773$	−17.0080	−0.611736	−0.305868	−	0.952074i	$-0.598947\pi$
$773$	−17.0080	−0.611736	−0.305868	+	0.952074i	$0.598947\pi$
$774$	0	0
$775$	6.16634	0.221501
$776$	0	0
$777$	6.00000	0.215249
$778$	0	0
$779$	−1.17570	−0.0421238
$780$	0	0
$781$	−6.13628	−0.219573
$782$	0	0
$783$	−10.9793	−0.392368
$784$	0	0
$785$	−10.1363	−0.361779
$786$	0	0
$787$	39.5845	1.41104	0.705518	−	0.708692i	$-0.250714\pi$
$787$	39.5845	1.41104	0.705518	+	0.708692i	$0.250714\pi$
$788$	0	0
$789$	−1.16303	−0.0414050
$790$	0	0
$791$	−13.7148	−0.487641
$792$	0	0
$793$	17.4245	0.618764
$794$	0	0
$795$	4.97930	0.176598
$796$	0	0
$797$	−26.1363	−0.925795	−0.462897	−	0.886412i	$-0.653190\pi$
$797$	−26.1363	−0.925795	−0.462897	+	0.886412i	$0.653190\pi$
$798$	0	0
$799$	11.5548	0.408778
$800$	0	0
$801$	−3.71477	−0.131255
$802$	0	0
$803$	−12.2789	−0.433313
$804$	0	0
$805$	−2.95558	−0.104171
$806$	0	0
$807$	34.2312	1.20499
$808$	0	0
$809$	28.3614	0.997134	0.498567	−	0.866851i	$-0.333860\pi$
$809$	28.3614	0.997134	0.498567	+	0.866851i	$0.333860\pi$
$810$	0	0
$811$	−0.421512	−0.0148013	−0.00740064	−	0.999973i	$-0.502356\pi$
$811$	−0.421512	−0.0148013	−0.00740064	+	0.999973i	$0.502356\pi$
$812$	0	0
$813$	−23.6547	−0.829605
$814$	0	0
$815$	−2.39779	−0.0839909
$816$	0	0
$817$	45.5558	1.59380
$818$	0	0
$819$	1.85738	0.0649023
$820$	0	0
$821$	−9.61988	−0.335736	−0.167868	−	0.985809i	$-0.553688\pi$
$821$	−9.61988	−0.335736	−0.167868	+	0.985809i	$0.553688\pi$
$822$	0	0
$823$	46.3150	1.61444	0.807220	−	0.590251i	$-0.200971\pi$
$823$	46.3150	1.61444	0.807220	+	0.590251i	$0.200971\pi$
$824$	0	0
$825$	−1.21076	−0.0421531
$826$	0	0
$827$	−15.1206	−0.525794	−0.262897	−	0.964824i	$-0.584678\pi$
$827$	−15.1206	−0.525794	−0.262897	+	0.964824i	$0.584678\pi$
$828$	0	0
$829$	17.4295	0.605353	0.302676	−	0.953093i	$-0.402120\pi$
$829$	17.4295	0.605353	0.302676	+	0.953093i	$0.402120\pi$
$830$	0	0
$831$	−3.68605	−0.127868
$832$	0	0
$833$	4.27890	0.148255
$834$	0	0
$835$	13.8223	0.478341
$836$	0	0
$837$	33.8510	1.17006
$838$	0	0
$839$	20.3313	0.701916	0.350958	−	0.936391i	$-0.385856\pi$
$839$	20.3313	0.701916	0.350958	+	0.936391i	$0.385856\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−17.0080	−0.585788
$844$	0	0
$845$	11.5341	0.396784
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	13.0207	0.446869
$850$	0	0
$851$	−14.6466	−0.502080
$852$	0	0
$853$	5.31831	0.182096	0.0910478	−	0.995847i	$-0.470978\pi$
$853$	5.31831	0.182096	0.0910478	+	0.995847i	$0.470978\pi$
$854$	0	0
$855$	−10.8430	−0.370824
$856$	0	0
$857$	41.5495	1.41930	0.709652	−	0.704553i	$-0.248852\pi$
$857$	41.5495	1.41930	0.709652	+	0.704553i	$0.248852\pi$
$858$	0	0
$859$	37.8110	1.29009	0.645047	−	0.764143i	$-0.276838\pi$
$859$	37.8110	1.29009	0.645047	+	0.764143i	$0.276838\pi$
$860$	0	0
$861$	−0.201395	−0.00686351
$862$	0	0
$863$	1.26953	0.0432155	0.0216077	−	0.999767i	$-0.493122\pi$
$863$	1.26953	0.0432155	0.0216077	+	0.999767i	$0.493122\pi$
$864$	0	0
$865$	18.1901	0.618481
$866$	0	0
$867$	−1.58482	−0.0538234
$868$	0	0
$869$	12.1126	0.410890
$870$	0	0
$871$	−11.6259	−0.393930
$872$	0	0
$873$	11.8350	0.400554
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	56.4800	1.90719	0.953597	−	0.301085i	$-0.0973489\pi$
$877$	56.4800	1.90719	0.953597	+	0.301085i	$0.0973489\pi$
$878$	0	0
$879$	26.1914	0.883414
$880$	0	0
$881$	4.04744	0.136362	0.0681809	−	0.997673i	$-0.478280\pi$
$881$	4.04744	0.136362	0.0681809	+	0.997673i	$0.478280\pi$
$882$	0	0
$883$	16.9843	0.571567	0.285784	−	0.958294i	$-0.407746\pi$
$883$	16.9843	0.571567	0.285784	+	0.958294i	$0.407746\pi$
$884$	0	0
$885$	17.8273	0.599259
$886$	0	0
$887$	−18.8717	−0.633651	−0.316826	−	0.948484i	$-0.602617\pi$
$887$	−18.8717	−0.633651	−0.316826	+	0.948484i	$0.602617\pi$
$888$	0	0
$889$	−9.62593	−0.322843
$890$	0	0
$891$	−2.04442	−0.0684906
$892$	0	0
$893$	−19.0869	−0.638718
$894$	0	0
$895$	15.5972	0.521357
$896$	0	0
$897$	4.33268	0.144664
$898$	0	0
$899$	−12.3327	−0.411318
$900$	0	0
$901$	17.5972	0.586248
$902$	0	0
$903$	7.80361	0.259688
$904$	0	0
$905$	−19.2044	−0.638377
$906$	0	0
$907$	0.426511	0.0141621	0.00708104	−	0.999975i	$-0.497746\pi$
$907$	0.426511	0.0141621	0.00708104	+	0.999975i	$0.497746\pi$
$908$	0	0
$909$	21.5959	0.716290
$910$	0	0
$911$	35.1944	1.16604	0.583022	−	0.812457i	$-0.301870\pi$
$911$	35.1944	1.16604	0.583022	+	0.812457i	$0.301870\pi$
$912$	0	0
$913$	1.48965	0.0493003
$914$	0	0
$915$	−17.4245	−0.576037
$916$	0	0
$917$	−14.9793	−0.494660
$918$	0	0
$919$	44.7829	1.47725	0.738626	−	0.674116i	$-0.235475\pi$
$919$	44.7829	1.47725	0.738626	+	0.674116i	$0.235475\pi$
$920$	0	0
$921$	38.4215	1.26603
$922$	0	0
$923$	7.42954	0.244546
$924$	0	0
$925$	−4.95558	−0.162939
$926$	0	0
$927$	−21.1219	−0.693735
$928$	0	0
$929$	−43.6921	−1.43349	−0.716746	−	0.697335i	$-0.754369\pi$
$929$	−43.6921	−1.43349	−0.716746	+	0.697335i	$0.754369\pi$
$930$	0	0
$931$	−7.06814	−0.231649
$932$	0	0
$933$	37.0919	1.21433
$934$	0	0
$935$	−4.27890	−0.139935
$936$	0	0
$937$	−38.5989	−1.26097	−0.630486	−	0.776201i	$-0.717144\pi$
$937$	−38.5989	−1.26097	−0.630486	+	0.776201i	$0.717144\pi$
$938$	0	0
$939$	21.3059	0.695293
$940$	0	0
$941$	−8.25517	−0.269111	−0.134555	−	0.990906i	$-0.542961\pi$
$941$	−8.25517	−0.269111	−0.134555	+	0.990906i	$0.542961\pi$
$942$	0	0
$943$	0.491625	0.0160095
$944$	0	0
$945$	−5.48965	−0.178578
$946$	0	0
$947$	−35.8935	−1.16638	−0.583191	−	0.812335i	$-0.698196\pi$
$947$	−35.8935	−1.16638	−0.583191	+	0.812335i	$0.698196\pi$
$948$	0	0
$949$	14.8667	0.482595
$950$	0	0
$951$	6.02872	0.195495
$952$	0	0
$953$	50.9616	1.65081	0.825405	−	0.564542i	$-0.190947\pi$
$953$	50.9616	1.65081	0.825405	+	0.564542i	$0.190947\pi$
$954$	0	0
$955$	−14.4215	−0.466669
$956$	0	0
$957$	2.42151	0.0782764
$958$	0	0
$959$	22.0237	0.711183
$960$	0	0
$961$	7.02372	0.226572
$962$	0	0
$963$	−9.06814	−0.292217
$964$	0	0
$965$	4.62291	0.148817
$966$	0	0
$967$	−6.62291	−0.212978	−0.106489	−	0.994314i	$-0.533961\pi$
$967$	−6.62291	−0.212978	−0.106489	+	0.994314i	$0.533961\pi$
$968$	0	0
$969$	36.6179	1.17634
$970$	0	0
$971$	−26.2552	−0.842569	−0.421284	−	0.906929i	$-0.638421\pi$
$971$	−26.2552	−0.842569	−0.421284	+	0.906929i	$0.638421\pi$
$972$	0	0
$973$	−3.26454	−0.104656
$974$	0	0
$975$	1.46593	0.0469473
$976$	0	0
$977$	57.3594	1.83509	0.917545	−	0.397631i	$-0.130168\pi$
$977$	57.3594	1.83509	0.917545	+	0.397631i	$0.130168\pi$
$978$	0	0
$979$	2.42151	0.0773919
$980$	0	0
$981$	−0.646629	−0.0206453
$982$	0	0
$983$	−14.3577	−0.457941	−0.228970	−	0.973433i	$-0.573536\pi$
$983$	−14.3577	−0.457941	−0.228970	+	0.973433i	$0.573536\pi$
$984$	0	0
$985$	−3.37709	−0.107603
$986$	0	0
$987$	−3.26953	−0.104070
$988$	0	0
$989$	−19.0494	−0.605736
$990$	0	0
$991$	30.1964	0.959220	0.479610	−	0.877482i	$-0.340778\pi$
$991$	30.1964	0.959220	0.479610	+	0.877482i	$0.340778\pi$
$992$	0	0
$993$	−7.50232	−0.238079
$994$	0	0
$995$	21.7923	0.690861
$996$	0	0
$997$	14.9730	0.474199	0.237099	−	0.971485i	$-0.423803\pi$
$997$	14.9730	0.474199	0.237099	+	0.971485i	$0.423803\pi$
$998$	0	0
$999$	−27.2044	−0.860710

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bd.1.2		3
4.3	odd	2		3080.2.a.m.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3080.2.a.m.1.2	✓	3	4.3	odd	2
6160.2.a.bd.1.2		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.21076 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.53407 q^{9} +O(q^{10})\)	\(q-1.21076 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.53407 q^{9} +1.00000 q^{11} -1.21076 q^{13} +1.21076 q^{15} +4.27890 q^{17} -7.06814 q^{19} -1.21076 q^{21} +2.95558 q^{23} +1.00000 q^{25} +5.48965 q^{27} -2.00000 q^{29} +6.16634 q^{31} -1.21076 q^{33} -1.00000 q^{35} -4.95558 q^{37} +1.46593 q^{39} +0.166338 q^{41} -6.44523 q^{43} +1.53407 q^{45} +2.70041 q^{47} +1.00000 q^{49} -5.18070 q^{51} +4.11256 q^{53} -1.00000 q^{55} +8.55779 q^{57} +14.7241 q^{59} -14.3915 q^{61} -1.53407 q^{63} +1.21076 q^{65} +9.60221 q^{67} -3.57849 q^{69} -6.13628 q^{71} -12.2789 q^{73} -1.21076 q^{75} +1.00000 q^{77} +12.1126 q^{79} -2.04442 q^{81} +1.48965 q^{83} -4.27890 q^{85} +2.42151 q^{87} +2.42151 q^{89} -1.21076 q^{91} -7.46593 q^{93} +7.06814 q^{95} -7.71477 q^{97} -1.53407 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	\( 3 q - 2 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 2 q^{13} + 2 q^{15} - 4 q^{17} - 6 q^{19} - 2 q^{21} - 2 q^{23} + 3 q^{25} - 2 q^{27} - 6 q^{29} + 6 q^{31} - 2 q^{33} - 3 q^{35} - 4 q^{37} + 12 q^{39} - 12 q^{41} + 10 q^{43} - 3 q^{45} - 12 q^{47} + 3 q^{49} + 4 q^{51} + 8 q^{53} - 3 q^{55} - 8 q^{57} - 2 q^{59} - 22 q^{61} + 3 q^{63} + 2 q^{65} + 6 q^{67} - 14 q^{69} + 12 q^{71} - 20 q^{73} - 2 q^{75} + 3 q^{77} + 32 q^{79} - 17 q^{81} - 14 q^{83} + 4 q^{85} + 4 q^{87} + 4 q^{89} - 2 q^{91} - 30 q^{93} + 6 q^{95} + 4 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 - 2 * q^13 + 2 * q^15 - 4 * q^17 - 6 * q^19 - 2 * q^21 - 2 * q^23 + 3 * q^25 - 2 * q^27 - 6 * q^29 + 6 * q^31 - 2 * q^33 - 3 * q^35 - 4 * q^37 + 12 * q^39 - 12 * q^41 + 10 * q^43 - 3 * q^45 - 12 * q^47 + 3 * q^49 + 4 * q^51 + 8 * q^53 - 3 * q^55 - 8 * q^57 - 2 * q^59 - 22 * q^61 + 3 * q^63 + 2 * q^65 + 6 * q^67 - 14 * q^69 + 12 * q^71 - 20 * q^73 - 2 * q^75 + 3 * q^77 + 32 * q^79 - 17 * q^81 - 14 * q^83 + 4 * q^85 + 4 * q^87 + 4 * q^89 - 2 * q^91 - 30 * q^93 + 6 * q^95 + 4 * q^97 + 3 * q^99

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