LMFDB - Embedded newform 6160.2.a.bs.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:	$4$
Coefficient field:	4.4.111028.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 10x^{2} - 2x + 4 $ x^4 - 10x^2 - 2x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1540)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.766757$ 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)

$f(q)$	$=$	$q-0.766757 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.41208 q^{9} +O(q^{10})$	$q-0.766757 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.41208 q^{9} -1.00000 q^{11} -4.45002 q^{13} -0.766757 q^{15} +2.76676 q^{17} -7.21678 q^{19} -0.766757 q^{21} -7.09535 q^{23} +1.00000 q^{25} +4.14975 q^{27} +2.00000 q^{29} -4.64533 q^{31} +0.766757 q^{33} +1.00000 q^{35} +11.0953 q^{37} +3.41208 q^{39} +8.17884 q^{41} -10.6289 q^{43} -2.41208 q^{45} +4.91651 q^{47} +1.00000 q^{49} -2.12143 q^{51} -10.1624 q^{53} -1.00000 q^{55} +5.53351 q^{57} -3.11181 q^{59} +8.57145 q^{61} -2.41208 q^{63} -4.45002 q^{65} +12.6289 q^{67} +5.44041 q^{69} +7.36653 q^{71} +5.23324 q^{73} -0.766757 q^{75} -1.00000 q^{77} +15.3792 q^{79} +4.05440 q^{81} +5.14090 q^{83} +2.76676 q^{85} -1.53351 q^{87} -14.5074 q^{89} -4.45002 q^{91} +3.56184 q^{93} -7.21678 q^{95} +3.68327 q^{97} +2.41208 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 4 q^{7} + 8 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 + 4 * q^7 + 8 * q^9	$ 4 q + 4 q^{5} + 4 q^{7} + 8 q^{9} - 4 q^{11} + 2 q^{13} + 8 q^{17} - 6 q^{19} + 6 q^{23} + 4 q^{25} + 6 q^{27} + 8 q^{29} - 4 q^{31} + 4 q^{35} + 10 q^{37} - 4 q^{39} + 12 q^{41} - 2 q^{43} + 8 q^{45} + 6 q^{47} + 4 q^{49} - 20 q^{51} + 6 q^{53} - 4 q^{55} + 16 q^{57} - 4 q^{59} + 26 q^{61} + 8 q^{63} + 2 q^{65} + 10 q^{67} - 18 q^{69} - 4 q^{71} + 24 q^{73} - 4 q^{77} - 8 q^{79} + 40 q^{81} + 2 q^{83} + 8 q^{85} - 6 q^{89} + 2 q^{91} - 14 q^{93} - 6 q^{95} - 2 q^{97} - 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^5 + 4 * q^7 + 8 * q^9 - 4 * q^11 + 2 * q^13 + 8 * q^17 - 6 * q^19 + 6 * q^23 + 4 * q^25 + 6 * q^27 + 8 * q^29 - 4 * q^31 + 4 * q^35 + 10 * q^37 - 4 * q^39 + 12 * q^41 - 2 * q^43 + 8 * q^45 + 6 * q^47 + 4 * q^49 - 20 * q^51 + 6 * q^53 - 4 * q^55 + 16 * q^57 - 4 * q^59 + 26 * q^61 + 8 * q^63 + 2 * q^65 + 10 * q^67 - 18 * q^69 - 4 * q^71 + 24 * q^73 - 4 * q^77 - 8 * q^79 + 40 * q^81 + 2 * q^83 + 8 * q^85 - 6 * q^89 + 2 * q^91 - 14 * q^93 - 6 * q^95 - 2 * q^97 - 8 * 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
$3$	−0.766757	−0.442687	−0.221344	−	0.975196i	$-0.571044\pi$
$3$	−0.766757	−0.442687	−0.221344	+	0.975196i	$0.571044\pi$
$4$	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
$9$	−2.41208	−0.804028
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−4.45002	−1.23421	−0.617107	−	0.786879i	$-0.711695\pi$
$13$	−4.45002	−1.23421	−0.617107	+	0.786879i	$0.711695\pi$
$14$	0	0
$15$	−0.766757	−0.197976
$16$	0	0
$17$	2.76676	0.671037	0.335519	−	0.942034i	$-0.391088\pi$
$17$	2.76676	0.671037	0.335519	+	0.942034i	$0.391088\pi$
$18$	0	0
$19$	−7.21678	−1.65564	−0.827821	−	0.560992i	$-0.810420\pi$
$19$	−7.21678	−1.65564	−0.827821	+	0.560992i	$0.810420\pi$
$20$	0	0
$21$	−0.766757	−0.167320
$22$	0	0
$23$	−7.09535	−1.47948	−0.739741	−	0.672891i	$-0.765052\pi$
$23$	−7.09535	−1.47948	−0.739741	+	0.672891i	$0.765052\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.14975	0.798620
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−4.64533	−0.834325	−0.417163	−	0.908832i	$-0.636975\pi$
$31$	−4.64533	−0.834325	−0.417163	+	0.908832i	$0.636975\pi$
$32$	0	0
$33$	0.766757	0.133475
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	11.0953	1.82406	0.912032	−	0.410119i	$-0.134513\pi$
$37$	11.0953	1.82406	0.912032	+	0.410119i	$0.134513\pi$
$38$	0	0
$39$	3.41208	0.546371
$40$	0	0
$41$	8.17884	1.27732	0.638660	−	0.769489i	$-0.279489\pi$
$41$	8.17884	1.27732	0.638660	+	0.769489i	$0.279489\pi$
$42$	0	0
$43$	−10.6289	−1.62089	−0.810443	−	0.585817i	$-0.800774\pi$
$43$	−10.6289	−1.62089	−0.810443	+	0.585817i	$0.800774\pi$
$44$	0	0
$45$	−2.41208	−0.359572
$46$	0	0
$47$	4.91651	0.717146	0.358573	−	0.933502i	$-0.383263\pi$
$47$	4.91651	0.717146	0.358573	+	0.933502i	$0.383263\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.12143	−0.297060
$52$	0	0
$53$	−10.1624	−1.39591	−0.697955	−	0.716142i	$-0.745906\pi$
$53$	−10.1624	−1.39591	−0.697955	+	0.716142i	$0.745906\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	5.53351	0.732932
$58$	0	0
$59$	−3.11181	−0.405124	−0.202562	−	0.979269i	$-0.564927\pi$
$59$	−3.11181	−0.405124	−0.202562	+	0.979269i	$0.564927\pi$
$60$	0	0
$61$	8.57145	1.09746	0.548731	−	0.835999i	$-0.315111\pi$
$61$	8.57145	1.09746	0.548731	+	0.835999i	$0.315111\pi$
$62$	0	0
$63$	−2.41208	−0.303894
$64$	0	0
$65$	−4.45002	−0.551957
$66$	0	0
$67$	12.6289	1.54286	0.771431	−	0.636313i	$-0.219542\pi$
$67$	12.6289	1.54286	0.771431	+	0.636313i	$0.219542\pi$
$68$	0	0
$69$	5.44041	0.654948
$70$	0	0
$71$	7.36653	0.874246	0.437123	−	0.899402i	$-0.355997\pi$
$71$	7.36653	0.874246	0.437123	+	0.899402i	$0.355997\pi$
$72$	0	0
$73$	5.23324	0.612505	0.306252	−	0.951950i	$-0.400925\pi$
$73$	5.23324	0.612505	0.306252	+	0.951950i	$0.400925\pi$
$74$	0	0
$75$	−0.766757	−0.0885374
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	15.3792	1.73029	0.865145	−	0.501522i	$-0.167226\pi$
$79$	15.3792	1.73029	0.865145	+	0.501522i	$0.167226\pi$
$80$	0	0
$81$	4.05440	0.450489
$82$	0	0
$83$	5.14090	0.564287	0.282144	−	0.959372i	$-0.408955\pi$
$83$	5.14090	0.564287	0.282144	+	0.959372i	$0.408955\pi$
$84$	0	0
$85$	2.76676	0.300097
$86$	0	0
$87$	−1.53351	−0.164410
$88$	0	0
$89$	−14.5074	−1.53778	−0.768892	−	0.639378i	$-0.779192\pi$
$89$	−14.5074	−1.53778	−0.768892	+	0.639378i	$0.779192\pi$
$90$	0	0
$91$	−4.45002	−0.466489
$92$	0	0
$93$	3.56184	0.369345
$94$	0	0
$95$	−7.21678	−0.740426
$96$	0	0
$97$	3.68327	0.373979	0.186989	−	0.982362i	$-0.440127\pi$
$97$	3.68327	0.373979	0.186989	+	0.982362i	$0.440127\pi$
$98$	0	0
$99$	2.41208	0.242424
$100$	0	0
$101$	7.57830	0.754069	0.377035	−	0.926199i	$-0.376944\pi$
$101$	7.57830	0.754069	0.377035	+	0.926199i	$0.376944\pi$
$102$	0	0
$103$	13.6668	1.34663	0.673315	−	0.739356i	$-0.264870\pi$
$103$	13.6668	1.34663	0.673315	+	0.739356i	$0.264870\pi$
$104$	0	0
$105$	−0.766757	−0.0748278
$106$	0	0
$107$	6.82417	0.659717	0.329859	−	0.944030i	$-0.392999\pi$
$107$	6.82417	0.659717	0.329859	+	0.944030i	$0.392999\pi$
$108$	0	0
$109$	0.466487	0.0446813	0.0223407	−	0.999750i	$-0.492888\pi$
$109$	0.466487	0.0446813	0.0223407	+	0.999750i	$0.492888\pi$
$110$	0	0
$111$	−8.50743	−0.807490
$112$	0	0
$113$	18.6572	1.75512	0.877560	−	0.479467i	$-0.159170\pi$
$113$	18.6572	1.75512	0.877560	+	0.479467i	$0.159170\pi$
$114$	0	0
$115$	−7.09535	−0.661645
$116$	0	0
$117$	10.7338	0.992343
$118$	0	0
$119$	2.76676	0.253628
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.27118	−0.565453
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.90004	−0.434809	−0.217404	−	0.976082i	$-0.569759\pi$
$127$	−4.90004	−0.434809	−0.217404	+	0.976082i	$0.569759\pi$
$128$	0	0
$129$	8.14975	0.717546
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	−7.21678	−0.625774
$134$	0	0
$135$	4.14975	0.357154
$136$	0	0
$137$	10.1624	0.868230	0.434115	−	0.900857i	$-0.357061\pi$
$137$	10.1624	0.868230	0.434115	+	0.900857i	$0.357061\pi$
$138$	0	0
$139$	12.9000	1.09417	0.547084	−	0.837078i	$-0.315738\pi$
$139$	12.9000	1.09417	0.547084	+	0.837078i	$0.315738\pi$
$140$	0	0
$141$	−3.76977	−0.317472
$142$	0	0
$143$	4.45002	0.372130
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	−0.766757	−0.0632410
$148$	0	0
$149$	−7.14290	−0.585169	−0.292585	−	0.956240i	$-0.594515\pi$
$149$	−7.14290	−0.585169	−0.292585	+	0.956240i	$0.594515\pi$
$150$	0	0
$151$	−15.1692	−1.23445	−0.617227	−	0.786785i	$-0.711744\pi$
$151$	−15.1692	−1.23445	−0.617227	+	0.786785i	$0.711744\pi$
$152$	0	0
$153$	−6.67365	−0.539533
$154$	0	0
$155$	−4.64533	−0.373122
$156$	0	0
$157$	−16.5833	−1.32349	−0.661746	−	0.749728i	$-0.730184\pi$
$157$	−16.5833	−1.32349	−0.661746	+	0.749728i	$0.730184\pi$
$158$	0	0
$159$	7.79207	0.617951
$160$	0	0
$161$	−7.09535	−0.559192
$162$	0	0
$163$	19.4530	1.52368	0.761839	−	0.647766i	$-0.224296\pi$
$163$	19.4530	1.52368	0.761839	+	0.647766i	$0.224296\pi$
$164$	0	0
$165$	0.766757	0.0596919
$166$	0	0
$167$	−13.6503	−1.05629	−0.528147	−	0.849153i	$-0.677113\pi$
$167$	−13.6503	−1.05629	−0.528147	+	0.849153i	$0.677113\pi$
$168$	0	0
$169$	6.80270	0.523284
$170$	0	0
$171$	17.4075	1.33118
$172$	0	0
$173$	−10.3761	−0.788884	−0.394442	−	0.918921i	$-0.629062\pi$
$173$	−10.3761	−0.788884	−0.394442	+	0.918921i	$0.629062\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	2.38600	0.179343
$178$	0	0
$179$	−24.6243	−1.84050	−0.920252	−	0.391327i	$-0.872016\pi$
$179$	−24.6243	−1.84050	−0.920252	+	0.391327i	$0.872016\pi$
$180$	0	0
$181$	0.283805	0.0210951	0.0105475	−	0.999944i	$-0.496643\pi$
$181$	0.283805	0.0210951	0.0105475	+	0.999944i	$0.496643\pi$
$182$	0	0
$183$	−6.57222	−0.485832
$184$	0	0
$185$	11.0953	0.815746
$186$	0	0
$187$	−2.76676	−0.202325
$188$	0	0
$189$	4.14975	0.301850
$190$	0	0
$191$	−13.8330	−1.00092	−0.500461	−	0.865759i	$-0.666836\pi$
$191$	−13.8330	−1.00092	−0.500461	+	0.865759i	$0.666836\pi$
$192$	0	0
$193$	−13.6959	−0.985852	−0.492926	−	0.870071i	$-0.664073\pi$
$193$	−13.6959	−0.985852	−0.492926	+	0.870071i	$0.664073\pi$
$194$	0	0
$195$	3.41208	0.244344
$196$	0	0
$197$	−5.99539	−0.427154	−0.213577	−	0.976926i	$-0.568511\pi$
$197$	−5.99539	−0.427154	−0.213577	+	0.976926i	$0.568511\pi$
$198$	0	0
$199$	4.49558	0.318683	0.159341	−	0.987224i	$-0.449063\pi$
$199$	4.49558	0.318683	0.159341	+	0.987224i	$0.449063\pi$
$200$	0	0
$201$	−9.68327	−0.683005
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	8.17884	0.571235
$206$	0	0
$207$	17.1146	1.18955
$208$	0	0
$209$	7.21678	0.499195
$210$	0	0
$211$	−9.83302	−0.676933	−0.338466	−	0.940978i	$-0.609908\pi$
$211$	−9.83302	−0.676933	−0.338466	+	0.940978i	$0.609908\pi$
$212$	0	0
$213$	−5.64834	−0.387018
$214$	0	0
$215$	−10.6289	−0.724882
$216$	0	0
$217$	−4.64533	−0.315345
$218$	0	0
$219$	−4.01262	−0.271148
$220$	0	0
$221$	−12.3121	−0.828203
$222$	0	0
$223$	16.8816	1.13047	0.565237	−	0.824928i	$-0.308785\pi$
$223$	16.8816	1.13047	0.565237	+	0.824928i	$0.308785\pi$
$224$	0	0
$225$	−2.41208	−0.160806
$226$	0	0
$227$	−7.36653	−0.488934	−0.244467	−	0.969658i	$-0.578613\pi$
$227$	−7.36653	−0.488934	−0.244467	+	0.969658i	$0.578613\pi$
$228$	0	0
$229$	20.1578	1.33206	0.666031	−	0.745924i	$-0.267992\pi$
$229$	20.1578	1.33206	0.666031	+	0.745924i	$0.267992\pi$
$230$	0	0
$231$	0.766757	0.0504489
$232$	0	0
$233$	12.8525	0.841995	0.420997	−	0.907062i	$-0.361680\pi$
$233$	12.8525	0.841995	0.420997	+	0.907062i	$0.361680\pi$
$234$	0	0
$235$	4.91651	0.320718
$236$	0	0
$237$	−11.7921	−0.765977
$238$	0	0
$239$	22.5484	1.45853	0.729267	−	0.684230i	$-0.239861\pi$
$239$	22.5484	1.45853	0.729267	+	0.684230i	$0.239861\pi$
$240$	0	0
$241$	30.2218	1.94676	0.973378	−	0.229205i	$-0.0736126\pi$
$241$	30.2218	1.94676	0.973378	+	0.229205i	$0.0736126\pi$
$242$	0	0
$243$	−15.5580	−0.998046
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	32.1148	2.04342
$248$	0	0
$249$	−3.94182	−0.249803
$250$	0	0
$251$	16.6124	1.04857	0.524283	−	0.851544i	$-0.324334\pi$
$251$	16.6124	1.04857	0.524283	+	0.851544i	$0.324334\pi$
$252$	0	0
$253$	7.09535	0.446081
$254$	0	0
$255$	−2.12143	−0.132849
$256$	0	0
$257$	−4.97392	−0.310265	−0.155132	−	0.987894i	$-0.549580\pi$
$257$	−4.97392	−0.310265	−0.155132	+	0.987894i	$0.549580\pi$
$258$	0	0
$259$	11.0953	0.689431
$260$	0	0
$261$	−4.82417	−0.298609
$262$	0	0
$263$	−20.9000	−1.28875	−0.644376	−	0.764709i	$-0.722883\pi$
$263$	−20.9000	−1.28875	−0.644376	+	0.764709i	$0.722883\pi$
$264$	0	0
$265$	−10.1624	−0.624270
$266$	0	0
$267$	11.1237	0.680758
$268$	0	0
$269$	12.8262	0.782025	0.391013	−	0.920385i	$-0.372125\pi$
$269$	12.8262	0.782025	0.391013	+	0.920385i	$0.372125\pi$
$270$	0	0
$271$	−0.600540	−0.0364802	−0.0182401	−	0.999834i	$-0.505806\pi$
$271$	−0.600540	−0.0364802	−0.0182401	+	0.999834i	$0.505806\pi$
$272$	0	0
$273$	3.41208	0.206509
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−3.83762	−0.230580	−0.115290	−	0.993332i	$-0.536780\pi$
$277$	−3.83762	−0.230580	−0.115290	+	0.993332i	$0.536780\pi$
$278$	0	0
$279$	11.2049	0.670821
$280$	0	0
$281$	26.9582	1.60819	0.804096	−	0.594499i	$-0.202650\pi$
$281$	26.9582	1.60819	0.804096	+	0.594499i	$0.202650\pi$
$282$	0	0
$283$	3.60939	0.214556	0.107278	−	0.994229i	$-0.465787\pi$
$283$	3.60939	0.214556	0.107278	+	0.994229i	$0.465787\pi$
$284$	0	0
$285$	5.53351	0.327777
$286$	0	0
$287$	8.17884	0.482782
$288$	0	0
$289$	−9.34506	−0.549709
$290$	0	0
$291$	−2.82417	−0.165556
$292$	0	0
$293$	6.37615	0.372498	0.186249	−	0.982503i	$-0.440367\pi$
$293$	6.37615	0.372498	0.186249	+	0.982503i	$0.440367\pi$
$294$	0	0
$295$	−3.11181	−0.181177
$296$	0	0
$297$	−4.14975	−0.240793
$298$	0	0
$299$	31.5745	1.82600
$300$	0	0
$301$	−10.6289	−0.612637
$302$	0	0
$303$	−5.81071	−0.333817
$304$	0	0
$305$	8.57145	0.490800
$306$	0	0
$307$	7.45964	0.425744	0.212872	−	0.977080i	$-0.431718\pi$
$307$	7.45964	0.425744	0.212872	+	0.977080i	$0.431718\pi$
$308$	0	0
$309$	−10.4791	−0.596136
$310$	0	0
$311$	23.2286	1.31718	0.658588	−	0.752504i	$-0.271154\pi$
$311$	23.2286	1.31718	0.658588	+	0.752504i	$0.271154\pi$
$312$	0	0
$313$	−28.8242	−1.62924	−0.814619	−	0.579996i	$-0.803054\pi$
$313$	−28.8242	−1.62924	−0.814619	+	0.579996i	$0.803054\pi$
$314$	0	0
$315$	−2.41208	−0.135906
$316$	0	0
$317$	−30.0819	−1.68957	−0.844784	−	0.535108i	$-0.820271\pi$
$317$	−30.0819	−1.68957	−0.844784	+	0.535108i	$0.820271\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	0	0
$321$	−5.23248	−0.292048
$322$	0	0
$323$	−19.9671	−1.11100
$324$	0	0
$325$	−4.45002	−0.246843
$326$	0	0
$327$	−0.357682	−0.0197798
$328$	0	0
$329$	4.91651	0.271056
$330$	0	0
$331$	5.77637	0.317498	0.158749	−	0.987319i	$-0.449254\pi$
$331$	5.77637	0.317498	0.158749	+	0.987319i	$0.449254\pi$
$332$	0	0
$333$	−26.7629	−1.46660
$334$	0	0
$335$	12.6289	0.689988
$336$	0	0
$337$	−8.73767	−0.475971	−0.237986	−	0.971269i	$-0.576487\pi$
$337$	−8.73767	−0.475971	−0.237986	+	0.971269i	$0.576487\pi$
$338$	0	0
$339$	−14.3055	−0.776969
$340$	0	0
$341$	4.64533	0.251559
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	5.44041	0.292902
$346$	0	0
$347$	10.8717	0.583625	0.291812	−	0.956476i	$-0.405742\pi$
$347$	10.8717	0.583625	0.291812	+	0.956476i	$0.405742\pi$
$348$	0	0
$349$	−15.7533	−0.843255	−0.421627	−	0.906769i	$-0.638541\pi$
$349$	−15.7533	−0.843255	−0.421627	+	0.906769i	$0.638541\pi$
$350$	0	0
$351$	−18.4665	−0.985668
$352$	0	0
$353$	1.60939	0.0856591	0.0428296	−	0.999082i	$-0.486363\pi$
$353$	1.60939	0.0856591	0.0428296	+	0.999082i	$0.486363\pi$
$354$	0	0
$355$	7.36653	0.390975
$356$	0	0
$357$	−2.12143	−0.112278
$358$	0	0
$359$	−0.945598	−0.0499067	−0.0249534	−	0.999689i	$-0.507944\pi$
$359$	−0.945598	−0.0499067	−0.0249534	+	0.999689i	$0.507944\pi$
$360$	0	0
$361$	33.0819	1.74115
$362$	0	0
$363$	−0.766757	−0.0402443
$364$	0	0
$365$	5.23324	0.273920
$366$	0	0
$367$	11.4412	0.597224	0.298612	−	0.954375i	$-0.403476\pi$
$367$	11.4412	0.597224	0.298612	+	0.954375i	$0.403476\pi$
$368$	0	0
$369$	−19.7281	−1.02700
$370$	0	0
$371$	−10.1624	−0.527604
$372$	0	0
$373$	21.2102	1.09822	0.549111	−	0.835750i	$-0.314967\pi$
$373$	21.2102	1.09822	0.549111	+	0.835750i	$0.314967\pi$
$374$	0	0
$375$	−0.766757	−0.0395951
$376$	0	0
$377$	−8.90004	−0.458376
$378$	0	0
$379$	−13.8001	−0.708863	−0.354431	−	0.935082i	$-0.615326\pi$
$379$	−13.8001	−0.708863	−0.354431	+	0.935082i	$0.615326\pi$
$380$	0	0
$381$	3.75714	0.192484
$382$	0	0
$383$	−2.05741	−0.105129	−0.0525644	−	0.998618i	$-0.516739\pi$
$383$	−2.05741	−0.105129	−0.0525644	+	0.998618i	$0.516739\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	25.6377	1.30324
$388$	0	0
$389$	33.4247	1.69470	0.847350	−	0.531035i	$-0.178197\pi$
$389$	33.4247	1.69470	0.847350	+	0.531035i	$0.178197\pi$
$390$	0	0
$391$	−19.6311	−0.992788
$392$	0	0
$393$	3.06703	0.154711
$394$	0	0
$395$	15.3792	0.773809
$396$	0	0
$397$	−34.1750	−1.71519	−0.857597	−	0.514322i	$-0.828044\pi$
$397$	−34.1750	−1.71519	−0.857597	+	0.514322i	$0.828044\pi$
$398$	0	0
$399$	5.53351	0.277022
$400$	0	0
$401$	6.78522	0.338838	0.169419	−	0.985544i	$-0.445811\pi$
$401$	6.78522	0.338838	0.169419	+	0.985544i	$0.445811\pi$
$402$	0	0
$403$	20.6718	1.02974
$404$	0	0
$405$	4.05440	0.201465
$406$	0	0
$407$	−11.0953	−0.549976
$408$	0	0
$409$	−28.0199	−1.38549	−0.692747	−	0.721181i	$-0.743600\pi$
$409$	−28.0199	−1.38549	−0.692747	+	0.721181i	$0.743600\pi$
$410$	0	0
$411$	−7.79207	−0.384354
$412$	0	0
$413$	−3.11181	−0.153122
$414$	0	0
$415$	5.14090	0.252357
$416$	0	0
$417$	−9.89120	−0.484374
$418$	0	0
$419$	37.1198	1.81342	0.906711	−	0.421752i	$-0.138585\pi$
$419$	37.1198	1.81342	0.906711	+	0.421752i	$0.138585\pi$
$420$	0	0
$421$	15.7308	0.766673	0.383337	−	0.923609i	$-0.374775\pi$
$421$	15.7308	0.766673	0.383337	+	0.923609i	$0.374775\pi$
$422$	0	0
$423$	−11.8590	−0.576606
$424$	0	0
$425$	2.76676	0.134207
$426$	0	0
$427$	8.57145	0.414802
$428$	0	0
$429$	−3.41208	−0.164737
$430$	0	0
$431$	12.0126	0.578628	0.289314	−	0.957234i	$-0.406573\pi$
$431$	12.0126	0.578628	0.289314	+	0.957234i	$0.406573\pi$
$432$	0	0
$433$	−1.09996	−0.0528605	−0.0264303	−	0.999651i	$-0.508414\pi$
$433$	−1.09996	−0.0528605	−0.0264303	+	0.999651i	$0.508414\pi$
$434$	0	0
$435$	−1.53351	−0.0735263
$436$	0	0
$437$	51.2056	2.44949
$438$	0	0
$439$	−19.5765	−0.934333	−0.467167	−	0.884169i	$-0.654725\pi$
$439$	−19.5765	−0.934333	−0.467167	+	0.884169i	$0.654725\pi$
$440$	0	0
$441$	−2.41208	−0.114861
$442$	0	0
$443$	−0.514040	−0.0244228	−0.0122114	−	0.999925i	$-0.503887\pi$
$443$	−0.514040	−0.0244228	−0.0122114	+	0.999925i	$0.503887\pi$
$444$	0	0
$445$	−14.5074	−0.687718
$446$	0	0
$447$	5.47687	0.259047
$448$	0	0
$449$	18.6461	0.879964	0.439982	−	0.898007i	$-0.354985\pi$
$449$	18.6461	0.879964	0.439982	+	0.898007i	$0.354985\pi$
$450$	0	0
$451$	−8.17884	−0.385127
$452$	0	0
$453$	11.6311	0.546477
$454$	0	0
$455$	−4.45002	−0.208620
$456$	0	0
$457$	−13.1283	−0.614115	−0.307057	−	0.951691i	$-0.599344\pi$
$457$	−13.1283	−0.614115	−0.307057	+	0.951691i	$0.599344\pi$
$458$	0	0
$459$	11.4814	0.535904
$460$	0	0
$461$	40.8942	1.90463	0.952316	−	0.305112i	$-0.0986939\pi$
$461$	40.8942	1.90463	0.952316	+	0.305112i	$0.0986939\pi$
$462$	0	0
$463$	28.7806	1.33755	0.668774	−	0.743466i	$-0.266819\pi$
$463$	28.7806	1.33755	0.668774	+	0.743466i	$0.266819\pi$
$464$	0	0
$465$	3.56184	0.165176
$466$	0	0
$467$	−5.09919	−0.235962	−0.117981	−	0.993016i	$-0.537642\pi$
$467$	−5.09919	−0.235962	−0.117981	+	0.993016i	$0.537642\pi$
$468$	0	0
$469$	12.6289	0.583147
$470$	0	0
$471$	12.7154	0.585893
$472$	0	0
$473$	10.6289	0.488716
$474$	0	0
$475$	−7.21678	−0.331129
$476$	0	0
$477$	24.5125	1.12235
$478$	0	0
$479$	40.1750	1.83564	0.917821	−	0.396994i	$-0.129947\pi$
$479$	40.1750	1.83564	0.917821	+	0.396994i	$0.129947\pi$
$480$	0	0
$481$	−49.3745	−2.25129
$482$	0	0
$483$	5.44041	0.247547
$484$	0	0
$485$	3.68327	0.167248
$486$	0	0
$487$	−6.94760	−0.314826	−0.157413	−	0.987533i	$-0.550315\pi$
$487$	−6.94760	−0.314826	−0.157413	+	0.987533i	$0.550315\pi$
$488$	0	0
$489$	−14.9157	−0.674513
$490$	0	0
$491$	−37.2451	−1.68085	−0.840424	−	0.541930i	$-0.817694\pi$
$491$	−37.2451	−1.68085	−0.840424	+	0.541930i	$0.817694\pi$
$492$	0	0
$493$	5.53351	0.249217
$494$	0	0
$495$	2.41208	0.108415
$496$	0	0
$497$	7.36653	0.330434
$498$	0	0
$499$	−13.8912	−0.621855	−0.310928	−	0.950434i	$-0.600640\pi$
$499$	−13.8912	−0.621855	−0.310928	+	0.950434i	$0.600640\pi$
$500$	0	0
$501$	10.4665	0.467608
$502$	0	0
$503$	4.99315	0.222634	0.111317	−	0.993785i	$-0.464493\pi$
$503$	4.99315	0.222634	0.111317	+	0.993785i	$0.464493\pi$
$504$	0	0
$505$	7.57830	0.337230
$506$	0	0
$507$	−5.21601	−0.231651
$508$	0	0
$509$	−25.0341	−1.10962	−0.554808	−	0.831978i	$-0.687208\pi$
$509$	−25.0341	−1.10962	−0.554808	+	0.831978i	$0.687208\pi$
$510$	0	0
$511$	5.23324	0.231505
$512$	0	0
$513$	−29.9478	−1.32223
$514$	0	0
$515$	13.6668	0.602231
$516$	0	0
$517$	−4.91651	−0.216228
$518$	0	0
$519$	7.95598	0.349229
$520$	0	0
$521$	30.5656	1.33910	0.669552	−	0.742765i	$-0.266486\pi$
$521$	30.5656	1.33910	0.669552	+	0.742765i	$0.266486\pi$
$522$	0	0
$523$	17.7414	0.775779	0.387890	−	0.921706i	$-0.373204\pi$
$523$	17.7414	0.775779	0.387890	+	0.921706i	$0.373204\pi$
$524$	0	0
$525$	−0.766757	−0.0334640
$526$	0	0
$527$	−12.8525	−0.559863
$528$	0	0
$529$	27.3440	1.18887
$530$	0	0
$531$	7.50596	0.325731
$532$	0	0
$533$	−36.3960	−1.57649
$534$	0	0
$535$	6.82417	0.295035
$536$	0	0
$537$	18.8808	0.814767
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−12.0759	−0.519182	−0.259591	−	0.965719i	$-0.583588\pi$
$541$	−12.0759	−0.519182	−0.259591	+	0.965719i	$0.583588\pi$
$542$	0	0
$543$	−0.217610	−0.00933853
$544$	0	0
$545$	0.466487	0.0199821
$546$	0	0
$547$	−41.8583	−1.78973	−0.894865	−	0.446337i	$-0.852728\pi$
$547$	−41.8583	−1.78973	−0.894865	+	0.446337i	$0.852728\pi$
$548$	0	0
$549$	−20.6751	−0.882390
$550$	0	0
$551$	−14.4336	−0.614890
$552$	0	0
$553$	15.3792	0.653988
$554$	0	0
$555$	−8.50743	−0.361120
$556$	0	0
$557$	−17.2431	−0.730614	−0.365307	−	0.930887i	$-0.619036\pi$
$557$	−17.2431	−0.730614	−0.365307	+	0.930887i	$0.619036\pi$
$558$	0	0
$559$	47.2987	2.00052
$560$	0	0
$561$	2.12143	0.0895668
$562$	0	0
$563$	22.1578	0.933839	0.466919	−	0.884300i	$-0.345364\pi$
$563$	22.1578	0.933839	0.466919	+	0.884300i	$0.345364\pi$
$564$	0	0
$565$	18.6572	0.784914
$566$	0	0
$567$	4.05440	0.170269
$568$	0	0
$569$	13.2517	0.555540	0.277770	−	0.960648i	$-0.410405\pi$
$569$	13.2517	0.555540	0.277770	+	0.960648i	$0.410405\pi$
$570$	0	0
$571$	33.2814	1.39278	0.696392	−	0.717661i	$-0.254787\pi$
$571$	33.2814	1.39278	0.696392	+	0.717661i	$0.254787\pi$
$572$	0	0
$573$	10.6066	0.443095
$574$	0	0
$575$	−7.09535	−0.295897
$576$	0	0
$577$	−34.3597	−1.43041	−0.715206	−	0.698914i	$-0.753667\pi$
$577$	−34.3597	−1.43041	−0.715206	+	0.698914i	$0.753667\pi$
$578$	0	0
$579$	10.5014	0.436424
$580$	0	0
$581$	5.14090	0.213281
$582$	0	0
$583$	10.1624	0.420883
$584$	0	0
$585$	10.7338	0.443789
$586$	0	0
$587$	1.60862	0.0663950	0.0331975	−	0.999449i	$-0.489431\pi$
$587$	1.60862	0.0663950	0.0331975	+	0.999449i	$0.489431\pi$
$588$	0	0
$589$	33.5243	1.38134
$590$	0	0
$591$	4.59701	0.189096
$592$	0	0
$593$	−20.5319	−0.843145	−0.421572	−	0.906795i	$-0.638522\pi$
$593$	−20.5319	−0.843145	−0.421572	+	0.906795i	$0.638522\pi$
$594$	0	0
$595$	2.76676	0.113426
$596$	0	0
$597$	−3.44701	−0.141077
$598$	0	0
$599$	19.4232	0.793609	0.396805	−	0.917903i	$-0.370119\pi$
$599$	19.4232	0.793609	0.396805	+	0.917903i	$0.370119\pi$
$600$	0	0
$601$	−10.3114	−0.420609	−0.210305	−	0.977636i	$-0.567446\pi$
$601$	−10.3114	−0.420609	−0.210305	+	0.977636i	$0.567446\pi$
$602$	0	0
$603$	−30.4619	−1.24050
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	31.9321	1.29609	0.648043	−	0.761604i	$-0.275588\pi$
$607$	31.9321	1.29609	0.648043	+	0.761604i	$0.275588\pi$
$608$	0	0
$609$	−1.53351	−0.0621411
$610$	0	0
$611$	−21.8786	−0.885112
$612$	0	0
$613$	46.0191	1.85869	0.929347	−	0.369207i	$-0.120371\pi$
$613$	46.0191	1.85869	0.929347	+	0.369207i	$0.120371\pi$
$614$	0	0
$615$	−6.27118	−0.252878
$616$	0	0
$617$	10.5277	0.423831	0.211915	−	0.977288i	$-0.432030\pi$
$617$	10.5277	0.423831	0.211915	+	0.977288i	$0.432030\pi$
$618$	0	0
$619$	−31.3764	−1.26112	−0.630562	−	0.776139i	$-0.717175\pi$
$619$	−31.3764	−1.26112	−0.630562	+	0.776139i	$0.717175\pi$
$620$	0	0
$621$	−29.4439	−1.18154
$622$	0	0
$623$	−14.5074	−0.581228
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−5.53351	−0.220987
$628$	0	0
$629$	30.6981	1.22401
$630$	0	0
$631$	43.7479	1.74158	0.870789	−	0.491657i	$-0.163609\pi$
$631$	43.7479	1.74158	0.870789	+	0.491657i	$0.163609\pi$
$632$	0	0
$633$	7.53953	0.299669
$634$	0	0
$635$	−4.90004	−0.194452
$636$	0	0
$637$	−4.45002	−0.176316
$638$	0	0
$639$	−17.7687	−0.702919
$640$	0	0
$641$	−8.51204	−0.336205	−0.168103	−	0.985769i	$-0.553764\pi$
$641$	−8.51204	−0.336205	−0.168103	+	0.985769i	$0.553764\pi$
$642$	0	0
$643$	−1.72698	−0.0681054	−0.0340527	−	0.999420i	$-0.510841\pi$
$643$	−1.72698	−0.0681054	−0.0340527	+	0.999420i	$0.510841\pi$
$644$	0	0
$645$	8.14975	0.320896
$646$	0	0
$647$	29.2742	1.15089	0.575444	−	0.817841i	$-0.304829\pi$
$647$	29.2742	1.15089	0.575444	+	0.817841i	$0.304829\pi$
$648$	0	0
$649$	3.11181	0.122149
$650$	0	0
$651$	3.56184	0.139599
$652$	0	0
$653$	−7.22480	−0.282728	−0.141364	−	0.989958i	$-0.545149\pi$
$653$	−7.22480	−0.282728	−0.141364	+	0.989958i	$0.545149\pi$
$654$	0	0
$655$	−4.00000	−0.156293
$656$	0	0
$657$	−12.6230	−0.492471
$658$	0	0
$659$	−28.7609	−1.12037	−0.560183	−	0.828369i	$-0.689269\pi$
$659$	−28.7609	−1.12037	−0.560183	+	0.828369i	$0.689269\pi$
$660$	0	0
$661$	27.7652	1.07994	0.539970	−	0.841684i	$-0.318436\pi$
$661$	27.7652	1.07994	0.539970	+	0.841684i	$0.318436\pi$
$662$	0	0
$663$	9.44041	0.366635
$664$	0	0
$665$	−7.21678	−0.279855
$666$	0	0
$667$	−14.1907	−0.549466
$668$	0	0
$669$	−12.9441	−0.500446
$670$	0	0
$671$	−8.57145	−0.330897
$672$	0	0
$673$	4.19531	0.161717	0.0808586	−	0.996726i	$-0.474234\pi$
$673$	4.19531	0.161717	0.0808586	+	0.996726i	$0.474234\pi$
$674$	0	0
$675$	4.14975	0.159724
$676$	0	0
$677$	−2.07311	−0.0796761	−0.0398380	−	0.999206i	$-0.512684\pi$
$677$	−2.07311	−0.0796761	−0.0398380	+	0.999206i	$0.512684\pi$
$678$	0	0
$679$	3.68327	0.141351
$680$	0	0
$681$	5.64834	0.216445
$682$	0	0
$683$	−32.3860	−1.23922	−0.619608	−	0.784911i	$-0.712708\pi$
$683$	−32.3860	−1.23922	−0.619608	+	0.784911i	$0.712708\pi$
$684$	0	0
$685$	10.1624	0.388284
$686$	0	0
$687$	−15.4561	−0.589687
$688$	0	0
$689$	45.2228	1.72285
$690$	0	0
$691$	−3.20492	−0.121921	−0.0609605	−	0.998140i	$-0.519416\pi$
$691$	−3.20492	−0.121921	−0.0609605	+	0.998140i	$0.519416\pi$
$692$	0	0
$693$	2.41208	0.0916275
$694$	0	0
$695$	12.9000	0.489327
$696$	0	0
$697$	22.6289	0.857129
$698$	0	0
$699$	−9.85473	−0.372740
$700$	0	0
$701$	42.7391	1.61423	0.807116	−	0.590392i	$-0.201027\pi$
$701$	42.7391	1.61423	0.807116	+	0.590392i	$0.201027\pi$
$702$	0	0
$703$	−80.0727	−3.02000
$704$	0	0
$705$	−3.76977	−0.141978
$706$	0	0
$707$	7.57830	0.285011
$708$	0	0
$709$	−18.5676	−0.697321	−0.348661	−	0.937249i	$-0.613363\pi$
$709$	−18.5676	−0.697321	−0.348661	+	0.937249i	$0.613363\pi$
$710$	0	0
$711$	−37.0958	−1.39120
$712$	0	0
$713$	32.9602	1.23437
$714$	0	0
$715$	4.45002	0.166421
$716$	0	0
$717$	−17.2891	−0.645674
$718$	0	0
$719$	19.2616	0.718335	0.359168	−	0.933273i	$-0.383061\pi$
$719$	19.2616	0.718335	0.359168	+	0.933273i	$0.383061\pi$
$720$	0	0
$721$	13.6668	0.508978
$722$	0	0
$723$	−23.1728	−0.861804
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	18.6929	0.693281	0.346640	−	0.937998i	$-0.387322\pi$
$727$	18.6929	0.693281	0.346640	+	0.937998i	$0.387322\pi$
$728$	0	0
$729$	−0.234010	−0.00866703
$730$	0	0
$731$	−29.4075	−1.08767
$732$	0	0
$733$	−37.6491	−1.39060	−0.695301	−	0.718719i	$-0.744729\pi$
$733$	−37.6491	−1.39060	−0.695301	+	0.718719i	$0.744729\pi$
$734$	0	0
$735$	−0.766757	−0.0282822
$736$	0	0
$737$	−12.6289	−0.465190
$738$	0	0
$739$	−39.0012	−1.43468	−0.717341	−	0.696723i	$-0.754641\pi$
$739$	−39.0012	−1.43468	−0.717341	+	0.696723i	$0.754641\pi$
$740$	0	0
$741$	−24.6243	−0.904595
$742$	0	0
$743$	47.2056	1.73180	0.865902	−	0.500213i	$-0.166745\pi$
$743$	47.2056	1.73180	0.865902	+	0.500213i	$0.166745\pi$
$744$	0	0
$745$	−7.14290	−0.261696
$746$	0	0
$747$	−12.4003	−0.453703
$748$	0	0
$749$	6.82417	0.249350
$750$	0	0
$751$	28.9000	1.05458	0.527289	−	0.849686i	$-0.323209\pi$
$751$	28.9000	1.05458	0.527289	+	0.849686i	$0.323209\pi$
$752$	0	0
$753$	−12.7377	−0.464186
$754$	0	0
$755$	−15.1692	−0.552065
$756$	0	0
$757$	−27.9149	−1.01458	−0.507292	−	0.861774i	$-0.669354\pi$
$757$	−27.9149	−1.01458	−0.507292	+	0.861774i	$0.669354\pi$
$758$	0	0
$759$	−5.44041	−0.197474
$760$	0	0
$761$	−19.3956	−0.703091	−0.351545	−	0.936171i	$-0.614344\pi$
$761$	−19.3956	−0.703091	−0.351545	+	0.936171i	$0.614344\pi$
$762$	0	0
$763$	0.466487	0.0168879
$764$	0	0
$765$	−6.67365	−0.241286
$766$	0	0
$767$	13.8476	0.500009
$768$	0	0
$769$	50.0938	1.80643	0.903213	−	0.429192i	$-0.141202\pi$
$769$	50.0938	1.80643	0.903213	+	0.429192i	$0.141202\pi$
$770$	0	0
$771$	3.81379	0.137350
$772$	0	0
$773$	−9.33160	−0.335634	−0.167817	−	0.985818i	$-0.553672\pi$
$773$	−9.33160	−0.335634	−0.167817	+	0.985818i	$0.553672\pi$
$774$	0	0
$775$	−4.64533	−0.166865
$776$	0	0
$777$	−8.50743	−0.305202
$778$	0	0
$779$	−59.0249	−2.11479
$780$	0	0
$781$	−7.36653	−0.263595
$782$	0	0
$783$	8.29950	0.296600
$784$	0	0
$785$	−16.5833	−0.591884
$786$	0	0
$787$	26.0060	0.927014	0.463507	−	0.886093i	$-0.346591\pi$
$787$	26.0060	0.927014	0.463507	+	0.886093i	$0.346591\pi$
$788$	0	0
$789$	16.0252	0.570514
$790$	0	0
$791$	18.6572	0.663373
$792$	0	0
$793$	−38.1431	−1.35450
$794$	0	0
$795$	7.79207	0.276356
$796$	0	0
$797$	30.0839	1.06563	0.532813	−	0.846233i	$-0.321135\pi$
$797$	30.0839	1.06563	0.532813	+	0.846233i	$0.321135\pi$
$798$	0	0
$799$	13.6028	0.481232
$800$	0	0
$801$	34.9932	1.23642
$802$	0	0
$803$	−5.23324	−0.184677
$804$	0	0
$805$	−7.09535	−0.250078
$806$	0	0
$807$	−9.83455	−0.346193
$808$	0	0
$809$	30.6243	1.07669	0.538346	−	0.842724i	$-0.319049\pi$
$809$	30.6243	1.07669	0.538346	+	0.842724i	$0.319049\pi$
$810$	0	0
$811$	−3.15860	−0.110913	−0.0554567	−	0.998461i	$-0.517661\pi$
$811$	−3.15860	−0.110913	−0.0554567	+	0.998461i	$0.517661\pi$
$812$	0	0
$813$	0.460468	0.0161493
$814$	0	0
$815$	19.4530	0.681410
$816$	0	0
$817$	76.7061	2.68361
$818$	0	0
$819$	10.7338	0.375070
$820$	0	0
$821$	−1.06549	−0.0371860	−0.0185930	−	0.999827i	$-0.505919\pi$
$821$	−1.06549	−0.0371860	−0.0185930	+	0.999827i	$0.505919\pi$
$822$	0	0
$823$	−15.0624	−0.525043	−0.262521	−	0.964926i	$-0.584554\pi$
$823$	−15.0624	−0.525043	−0.262521	+	0.964926i	$0.584554\pi$
$824$	0	0
$825$	0.766757	0.0266950
$826$	0	0
$827$	−38.6541	−1.34414	−0.672068	−	0.740490i	$-0.734594\pi$
$827$	−38.6541	−1.34414	−0.672068	+	0.740490i	$0.734594\pi$
$828$	0	0
$829$	7.26694	0.252391	0.126196	−	0.992005i	$-0.459723\pi$
$829$	7.26694	0.252391	0.126196	+	0.992005i	$0.459723\pi$
$830$	0	0
$831$	2.94252	0.102075
$832$	0	0
$833$	2.76676	0.0958624
$834$	0	0
$835$	−13.6503	−0.472389
$836$	0	0
$837$	−19.2770	−0.666309
$838$	0	0
$839$	46.4105	1.60227	0.801134	−	0.598485i	$-0.204230\pi$
$839$	46.4105	1.60227	0.801134	+	0.598485i	$0.204230\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−20.6704	−0.711926
$844$	0	0
$845$	6.80270	0.234020
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−2.76752	−0.0949812
$850$	0	0
$851$	−78.7254	−2.69867
$852$	0	0
$853$	−27.8166	−0.952421	−0.476210	−	0.879331i	$-0.657990\pi$
$853$	−27.8166	−0.952421	−0.476210	+	0.879331i	$0.657990\pi$
$854$	0	0
$855$	17.4075	0.595323
$856$	0	0
$857$	17.7756	0.607203	0.303602	−	0.952799i	$-0.401811\pi$
$857$	17.7756	0.607203	0.303602	+	0.952799i	$0.401811\pi$
$858$	0	0
$859$	6.72120	0.229324	0.114662	−	0.993405i	$-0.463421\pi$
$859$	6.72120	0.229324	0.114662	+	0.993405i	$0.463421\pi$
$860$	0	0
$861$	−6.27118	−0.213721
$862$	0	0
$863$	−12.6618	−0.431012	−0.215506	−	0.976502i	$-0.569140\pi$
$863$	−12.6618	−0.431012	−0.215506	+	0.976502i	$0.569140\pi$
$864$	0	0
$865$	−10.3761	−0.352800
$866$	0	0
$867$	7.16539	0.243349
$868$	0	0
$869$	−15.3792	−0.521702
$870$	0	0
$871$	−56.1987	−1.90422
$872$	0	0
$873$	−8.88435	−0.300690
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	1.33821	0.0451881	0.0225940	−	0.999745i	$-0.492807\pi$
$877$	1.33821	0.0451881	0.0225940	+	0.999745i	$0.492807\pi$
$878$	0	0
$879$	−4.88895	−0.164900
$880$	0	0
$881$	6.93497	0.233645	0.116823	−	0.993153i	$-0.462729\pi$
$881$	6.93497	0.233645	0.116823	+	0.993153i	$0.462729\pi$
$882$	0	0
$883$	14.4037	0.484723	0.242362	−	0.970186i	$-0.422078\pi$
$883$	14.4037	0.484723	0.242362	+	0.970186i	$0.422078\pi$
$884$	0	0
$885$	2.38600	0.0802047
$886$	0	0
$887$	3.33360	0.111931	0.0559657	−	0.998433i	$-0.482176\pi$
$887$	3.33360	0.111931	0.0559657	+	0.998433i	$0.482176\pi$
$888$	0	0
$889$	−4.90004	−0.164342
$890$	0	0
$891$	−4.05440	−0.135828
$892$	0	0
$893$	−35.4814	−1.18734
$894$	0	0
$895$	−24.6243	−0.823098
$896$	0	0
$897$	−24.2099	−0.808346
$898$	0	0
$899$	−9.29065	−0.309861
$900$	0	0
$901$	−28.1168	−0.936707
$902$	0	0
$903$	8.14975	0.271207
$904$	0	0
$905$	0.283805	0.00943401
$906$	0	0
$907$	−24.0283	−0.797847	−0.398924	−	0.916984i	$-0.630616\pi$
$907$	−24.0283	−0.797847	−0.398924	+	0.916984i	$0.630616\pi$
$908$	0	0
$909$	−18.2795	−0.606293
$910$	0	0
$911$	−34.8808	−1.15565	−0.577826	−	0.816160i	$-0.696099\pi$
$911$	−34.8808	−1.15565	−0.577826	+	0.816160i	$0.696099\pi$
$912$	0	0
$913$	−5.14090	−0.170139
$914$	0	0
$915$	−6.57222	−0.217271
$916$	0	0
$917$	−4.00000	−0.132092
$918$	0	0
$919$	5.83302	0.192413	0.0962067	−	0.995361i	$-0.469329\pi$
$919$	5.83302	0.192413	0.0962067	+	0.995361i	$0.469329\pi$
$920$	0	0
$921$	−5.71973	−0.188471
$922$	0	0
$923$	−32.7812	−1.07901
$924$	0	0
$925$	11.0953	0.364813
$926$	0	0
$927$	−32.9655	−1.08273
$928$	0	0
$929$	−13.2168	−0.433628	−0.216814	−	0.976213i	$-0.569567\pi$
$929$	−13.2168	−0.433628	−0.216814	+	0.976213i	$0.569567\pi$
$930$	0	0
$931$	−7.21678	−0.236520
$932$	0	0
$933$	−17.8107	−0.583097
$934$	0	0
$935$	−2.76676	−0.0904826
$936$	0	0
$937$	32.1176	1.04924	0.524618	−	0.851338i	$-0.324208\pi$
$937$	32.1176	1.04924	0.524618	+	0.851338i	$0.324208\pi$
$938$	0	0
$939$	22.1011	0.721243
$940$	0	0
$941$	−27.6582	−0.901631	−0.450816	−	0.892617i	$-0.648867\pi$
$941$	−27.6582	−0.901631	−0.450816	+	0.892617i	$0.648867\pi$
$942$	0	0
$943$	−58.0317	−1.88977
$944$	0	0
$945$	4.14975	0.134991
$946$	0	0
$947$	−1.98538	−0.0645161	−0.0322580	−	0.999480i	$-0.510270\pi$
$947$	−1.98538	−0.0645161	−0.0322580	+	0.999480i	$0.510270\pi$
$948$	0	0
$949$	−23.2880	−0.755962
$950$	0	0
$951$	23.0655	0.747950
$952$	0	0
$953$	37.4767	1.21399	0.606995	−	0.794705i	$-0.292375\pi$
$953$	37.4767	1.21399	0.606995	+	0.794705i	$0.292375\pi$
$954$	0	0
$955$	−13.8330	−0.447626
$956$	0	0
$957$	1.53351	0.0495714
$958$	0	0
$959$	10.1624	0.328160
$960$	0	0
$961$	−9.42093	−0.303901
$962$	0	0
$963$	−16.4605	−0.530431
$964$	0	0
$965$	−13.6959	−0.440886
$966$	0	0
$967$	−46.4113	−1.49249	−0.746243	−	0.665674i	$-0.768144\pi$
$967$	−46.4113	−1.49249	−0.746243	+	0.665674i	$0.768144\pi$
$968$	0	0
$969$	15.3099	0.491824
$970$	0	0
$971$	−12.1869	−0.391095	−0.195547	−	0.980694i	$-0.562648\pi$
$971$	−12.1869	−0.391095	−0.195547	+	0.980694i	$0.562648\pi$
$972$	0	0
$973$	12.9000	0.413556
$974$	0	0
$975$	3.41208	0.109274
$976$	0	0
$977$	−17.7901	−0.569155	−0.284577	−	0.958653i	$-0.591853\pi$
$977$	−17.7901	−0.569155	−0.284577	+	0.958653i	$0.591853\pi$
$978$	0	0
$979$	14.5074	0.463660
$980$	0	0
$981$	−1.12520	−0.0359250
$982$	0	0
$983$	−36.0325	−1.14926	−0.574629	−	0.818414i	$-0.694854\pi$
$983$	−36.0325	−1.14926	−0.574629	+	0.818414i	$0.694854\pi$
$984$	0	0
$985$	−5.99539	−0.191029
$986$	0	0
$987$	−3.76977	−0.119993
$988$	0	0
$989$	75.4155	2.39807
$990$	0	0
$991$	36.5913	1.16236	0.581181	−	0.813774i	$-0.302591\pi$
$991$	36.5913	1.16236	0.581181	+	0.813774i	$0.302591\pi$
$992$	0	0
$993$	−4.42907	−0.140552
$994$	0	0
$995$	4.49558	0.142519
$996$	0	0
$997$	−25.7993	−0.817073	−0.408536	−	0.912742i	$-0.633961\pi$
$997$	−25.7993	−0.817073	−0.408536	+	0.912742i	$0.633961\pi$
$998$	0	0
$999$	46.0429	1.45673

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bs.1.2		4
4.3	odd	2		1540.2.a.i.1.3	✓	4
20.3	even	4		7700.2.e.t.1849.6		8
20.7	even	4		7700.2.e.t.1849.3		8
20.19	odd	2		7700.2.a.bb.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1540.2.a.i.1.3	✓	4	4.3	odd	2
6160.2.a.bs.1.2		4	1.1	even	1	trivial
7700.2.a.bb.1.2		4	20.19	odd	2
7700.2.e.t.1849.3		8	20.7	even	4
7700.2.e.t.1849.6		8	20.3	even	4

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)

\(f(q)\)	\(=\)	\(q-0.766757 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.41208 q^{9} +O(q^{10})\)	\(q-0.766757 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.41208 q^{9} -1.00000 q^{11} -4.45002 q^{13} -0.766757 q^{15} +2.76676 q^{17} -7.21678 q^{19} -0.766757 q^{21} -7.09535 q^{23} +1.00000 q^{25} +4.14975 q^{27} +2.00000 q^{29} -4.64533 q^{31} +0.766757 q^{33} +1.00000 q^{35} +11.0953 q^{37} +3.41208 q^{39} +8.17884 q^{41} -10.6289 q^{43} -2.41208 q^{45} +4.91651 q^{47} +1.00000 q^{49} -2.12143 q^{51} -10.1624 q^{53} -1.00000 q^{55} +5.53351 q^{57} -3.11181 q^{59} +8.57145 q^{61} -2.41208 q^{63} -4.45002 q^{65} +12.6289 q^{67} +5.44041 q^{69} +7.36653 q^{71} +5.23324 q^{73} -0.766757 q^{75} -1.00000 q^{77} +15.3792 q^{79} +4.05440 q^{81} +5.14090 q^{83} +2.76676 q^{85} -1.53351 q^{87} -14.5074 q^{89} -4.45002 q^{91} +3.56184 q^{93} -7.21678 q^{95} +3.68327 q^{97} +2.41208 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 4 q^{7} + 8 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 + 4 * q^7 + 8 * q^9	\( 4 q + 4 q^{5} + 4 q^{7} + 8 q^{9} - 4 q^{11} + 2 q^{13} + 8 q^{17} - 6 q^{19} + 6 q^{23} + 4 q^{25} + 6 q^{27} + 8 q^{29} - 4 q^{31} + 4 q^{35} + 10 q^{37} - 4 q^{39} + 12 q^{41} - 2 q^{43} + 8 q^{45} + 6 q^{47} + 4 q^{49} - 20 q^{51} + 6 q^{53} - 4 q^{55} + 16 q^{57} - 4 q^{59} + 26 q^{61} + 8 q^{63} + 2 q^{65} + 10 q^{67} - 18 q^{69} - 4 q^{71} + 24 q^{73} - 4 q^{77} - 8 q^{79} + 40 q^{81} + 2 q^{83} + 8 q^{85} - 6 q^{89} + 2 q^{91} - 14 q^{93} - 6 q^{95} - 2 q^{97} - 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^5 + 4 * q^7 + 8 * q^9 - 4 * q^11 + 2 * q^13 + 8 * q^17 - 6 * q^19 + 6 * q^23 + 4 * q^25 + 6 * q^27 + 8 * q^29 - 4 * q^31 + 4 * q^35 + 10 * q^37 - 4 * q^39 + 12 * q^41 - 2 * q^43 + 8 * q^45 + 6 * q^47 + 4 * q^49 - 20 * q^51 + 6 * q^53 - 4 * q^55 + 16 * q^57 - 4 * q^59 + 26 * q^61 + 8 * q^63 + 2 * q^65 + 10 * q^67 - 18 * q^69 - 4 * q^71 + 24 * q^73 - 4 * q^77 - 8 * q^79 + 40 * q^81 + 2 * q^83 + 8 * q^85 - 6 * q^89 + 2 * q^91 - 14 * q^93 - 6 * q^95 - 2 * q^97 - 8 * q^99

Introduction

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