LMFDB - Embedded newform 4864.2.a.bl.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4864,2,Mod(1,4864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4864 = 2^{8} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4864.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:	$38.8392355432$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 4 $ x^4 - 7*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1216)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.52434$ of defining polynomial
Character	$\chi$	$=$	4864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.37228 q^{3} +2.52434 q^{5} -3.31662 q^{7} +8.37228 q^{9} +O(q^{10})$	$q+3.37228 q^{3} +2.52434 q^{5} -3.31662 q^{7} +8.37228 q^{9} +2.37228 q^{11} -5.84096 q^{13} +8.51278 q^{15} +5.00000 q^{17} -1.00000 q^{19} -11.1846 q^{21} -0.792287 q^{23} +1.37228 q^{25} +18.1168 q^{27} +2.67181 q^{29} +3.46410 q^{31} +8.00000 q^{33} -8.37228 q^{35} +10.0974 q^{37} -19.6974 q^{39} -3.62772 q^{43} +21.1345 q^{45} -0.644810 q^{47} +4.00000 q^{49} +16.8614 q^{51} +6.13592 q^{53} +5.98844 q^{55} -3.37228 q^{57} +1.37228 q^{59} +14.5012 q^{61} -27.7677 q^{63} -14.7446 q^{65} -2.62772 q^{67} -2.67181 q^{69} +6.63325 q^{71} -8.48913 q^{73} +4.62772 q^{75} -7.86797 q^{77} -0.294954 q^{79} +35.9783 q^{81} +8.00000 q^{83} +12.6217 q^{85} +9.01011 q^{87} +15.4891 q^{89} +19.3723 q^{91} +11.6819 q^{93} -2.52434 q^{95} -2.74456 q^{97} +19.8614 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 22 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 + 22 * q^9	$ 4 q + 2 q^{3} + 22 q^{9} - 2 q^{11} + 20 q^{17} - 4 q^{19} - 6 q^{25} + 38 q^{27} + 32 q^{33} - 22 q^{35} - 26 q^{43} + 16 q^{49} + 10 q^{51} - 2 q^{57} - 6 q^{59} - 36 q^{65} - 22 q^{67} + 12 q^{73} + 30 q^{75} + 52 q^{81} + 32 q^{83} + 16 q^{89} + 66 q^{91} + 12 q^{97} + 22 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 22 * q^9 - 2 * q^11 + 20 * q^17 - 4 * q^19 - 6 * q^25 + 38 * q^27 + 32 * q^33 - 22 * q^35 - 26 * q^43 + 16 * q^49 + 10 * q^51 - 2 * q^57 - 6 * q^59 - 36 * q^65 - 22 * q^67 + 12 * q^73 + 30 * q^75 + 52 * q^81 + 32 * q^83 + 16 * q^89 + 66 * q^91 + 12 * q^97 + 22 * 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
$3$	3.37228	1.94699	0.973494	−	0.228714i	$-0.0734519\pi$
$3$	3.37228	1.94699	0.973494	+	0.228714i	$0.0734519\pi$
$4$	0	0
$5$	2.52434	1.12892	0.564459	−	0.825461i	$-0.309085\pi$
$5$	2.52434	1.12892	0.564459	+	0.825461i	$0.309085\pi$
$6$	0	0
$7$	−3.31662	−1.25357	−0.626783	−	0.779194i	$-0.715629\pi$
$7$	−3.31662	−1.25357	−0.626783	+	0.779194i	$0.715629\pi$
$8$	0	0
$9$	8.37228	2.79076
$10$	0	0
$11$	2.37228	0.715270	0.357635	−	0.933862i	$-0.383583\pi$
$11$	2.37228	0.715270	0.357635	+	0.933862i	$0.383583\pi$
$12$	0	0
$13$	−5.84096	−1.61999	−0.809996	−	0.586436i	$-0.800531\pi$
$13$	−5.84096	−1.61999	−0.809996	+	0.586436i	$0.800531\pi$
$14$	0	0
$15$	8.51278	2.19799
$16$	0	0
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−11.1846	−2.44068
$22$	0	0
$23$	−0.792287	−0.165203	−0.0826016	−	0.996583i	$-0.526323\pi$
$23$	−0.792287	−0.165203	−0.0826016	+	0.996583i	$0.526323\pi$
$24$	0	0
$25$	1.37228	0.274456
$26$	0	0
$27$	18.1168	3.48659
$28$	0	0
$29$	2.67181	0.496144	0.248072	−	0.968742i	$-0.420203\pi$
$29$	2.67181	0.496144	0.248072	+	0.968742i	$0.420203\pi$
$30$	0	0
$31$	3.46410	0.622171	0.311086	−	0.950382i	$-0.399307\pi$
$31$	3.46410	0.622171	0.311086	+	0.950382i	$0.399307\pi$
$32$	0	0
$33$	8.00000	1.39262
$34$	0	0
$35$	−8.37228	−1.41517
$36$	0	0
$37$	10.0974	1.65999	0.829997	−	0.557768i	$-0.188342\pi$
$37$	10.0974	1.65999	0.829997	+	0.557768i	$0.188342\pi$
$38$	0	0
$39$	−19.6974	−3.15410
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−3.62772	−0.553222	−0.276611	−	0.960982i	$-0.589211\pi$
$43$	−3.62772	−0.553222	−0.276611	+	0.960982i	$0.589211\pi$
$44$	0	0
$45$	21.1345	3.15054
$46$	0	0
$47$	−0.644810	−0.0940552	−0.0470276	−	0.998894i	$-0.514975\pi$
$47$	−0.644810	−0.0940552	−0.0470276	+	0.998894i	$0.514975\pi$
$48$	0	0
$49$	4.00000	0.571429
$50$	0	0
$51$	16.8614	2.36107
$52$	0	0
$53$	6.13592	0.842833	0.421416	−	0.906867i	$-0.361533\pi$
$53$	6.13592	0.842833	0.421416	+	0.906867i	$0.361533\pi$
$54$	0	0
$55$	5.98844	0.807481
$56$	0	0
$57$	−3.37228	−0.446670
$58$	0	0
$59$	1.37228	0.178656	0.0893279	−	0.996002i	$-0.471528\pi$
$59$	1.37228	0.178656	0.0893279	+	0.996002i	$0.471528\pi$
$60$	0	0
$61$	14.5012	1.85669	0.928345	−	0.371719i	$-0.121232\pi$
$61$	14.5012	1.85669	0.928345	+	0.371719i	$0.121232\pi$
$62$	0	0
$63$	−27.7677	−3.49840
$64$	0	0
$65$	−14.7446	−1.82884
$66$	0	0
$67$	−2.62772	−0.321027	−0.160513	−	0.987034i	$-0.551315\pi$
$67$	−2.62772	−0.321027	−0.160513	+	0.987034i	$0.551315\pi$
$68$	0	0
$69$	−2.67181	−0.321649
$70$	0	0
$71$	6.63325	0.787222	0.393611	−	0.919277i	$-0.371226\pi$
$71$	6.63325	0.787222	0.393611	+	0.919277i	$0.371226\pi$
$72$	0	0
$73$	−8.48913	−0.993577	−0.496788	−	0.867872i	$-0.665488\pi$
$73$	−8.48913	−0.993577	−0.496788	+	0.867872i	$0.665488\pi$
$74$	0	0
$75$	4.62772	0.534363
$76$	0	0
$77$	−7.86797	−0.896638
$78$	0	0
$79$	−0.294954	−0.0331849	−0.0165924	−	0.999862i	$-0.505282\pi$
$79$	−0.294954	−0.0331849	−0.0165924	+	0.999862i	$0.505282\pi$
$80$	0	0
$81$	35.9783	3.99758
$82$	0	0
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	12.6217	1.36901
$86$	0	0
$87$	9.01011	0.965985
$88$	0	0
$89$	15.4891	1.64184	0.820922	−	0.571040i	$-0.193460\pi$
$89$	15.4891	1.64184	0.820922	+	0.571040i	$0.193460\pi$
$90$	0	0
$91$	19.3723	2.03077
$92$	0	0
$93$	11.6819	1.21136
$94$	0	0
$95$	−2.52434	−0.258992
$96$	0	0
$97$	−2.74456	−0.278668	−0.139334	−	0.990245i	$-0.544496\pi$
$97$	−2.74456	−0.278668	−0.139334	+	0.990245i	$0.544496\pi$
$98$	0	0
$99$	19.8614	1.99615
$100$	0	0
$101$	−13.8564	−1.37876	−0.689382	−	0.724398i	$-0.742118\pi$
$101$	−13.8564	−1.37876	−0.689382	+	0.724398i	$0.742118\pi$
$102$	0	0
$103$	−11.9769	−1.18012	−0.590058	−	0.807361i	$-0.700895\pi$
$103$	−11.9769	−1.18012	−0.590058	+	0.807361i	$0.700895\pi$
$104$	0	0
$105$	−28.2337	−2.75533
$106$	0	0
$107$	−13.3723	−1.29275	−0.646374	−	0.763021i	$-0.723715\pi$
$107$	−13.3723	−1.29275	−0.646374	+	0.763021i	$0.723715\pi$
$108$	0	0
$109$	−3.96143	−0.379437	−0.189718	−	0.981839i	$-0.560757\pi$
$109$	−3.96143	−0.379437	−0.189718	+	0.981839i	$0.560757\pi$
$110$	0	0
$111$	34.0511	3.23199
$112$	0	0
$113$	−14.7446	−1.38705	−0.693526	−	0.720432i	$-0.743944\pi$
$113$	−14.7446	−1.38705	−0.693526	+	0.720432i	$0.743944\pi$
$114$	0	0
$115$	−2.00000	−0.186501
$116$	0	0
$117$	−48.9022	−4.52101
$118$	0	0
$119$	−16.5831	−1.52017
$120$	0	0
$121$	−5.37228	−0.488389
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.15759	−0.819080
$126$	0	0
$127$	12.2718	1.08895	0.544475	−	0.838777i	$-0.316729\pi$
$127$	12.2718	1.08895	0.544475	+	0.838777i	$0.316729\pi$
$128$	0	0
$129$	−12.2337	−1.07712
$130$	0	0
$131$	4.37228	0.382008	0.191004	−	0.981589i	$-0.438826\pi$
$131$	4.37228	0.382008	0.191004	+	0.981589i	$0.438826\pi$
$132$	0	0
$133$	3.31662	0.287588
$134$	0	0
$135$	45.7330	3.93607
$136$	0	0
$137$	−5.00000	−0.427179	−0.213589	−	0.976924i	$-0.568515\pi$
$137$	−5.00000	−0.427179	−0.213589	+	0.976924i	$0.568515\pi$
$138$	0	0
$139$	−3.62772	−0.307699	−0.153850	−	0.988094i	$-0.549167\pi$
$139$	−3.62772	−0.307699	−0.153850	+	0.988094i	$0.549167\pi$
$140$	0	0
$141$	−2.17448	−0.183124
$142$	0	0
$143$	−13.8564	−1.15873
$144$	0	0
$145$	6.74456	0.560105
$146$	0	0
$147$	13.4891	1.11256
$148$	0	0
$149$	4.69882	0.384942	0.192471	−	0.981303i	$-0.438350\pi$
$149$	4.69882	0.384942	0.192471	+	0.981303i	$0.438350\pi$
$150$	0	0
$151$	−15.7359	−1.28057	−0.640286	−	0.768137i	$-0.721184\pi$
$151$	−15.7359	−1.28057	−0.640286	+	0.768137i	$0.721184\pi$
$152$	0	0
$153$	41.8614	3.38429
$154$	0	0
$155$	8.74456	0.702380
$156$	0	0
$157$	18.6101	1.48525	0.742625	−	0.669708i	$-0.233581\pi$
$157$	18.6101	1.48525	0.742625	+	0.669708i	$0.233581\pi$
$158$	0	0
$159$	20.6920	1.64099
$160$	0	0
$161$	2.62772	0.207093
$162$	0	0
$163$	−10.7446	−0.841579	−0.420790	−	0.907158i	$-0.638247\pi$
$163$	−10.7446	−0.841579	−0.420790	+	0.907158i	$0.638247\pi$
$164$	0	0
$165$	20.1947	1.57216
$166$	0	0
$167$	−6.33830	−0.490472	−0.245236	−	0.969463i	$-0.578865\pi$
$167$	−6.33830	−0.490472	−0.245236	+	0.969463i	$0.578865\pi$
$168$	0	0
$169$	21.1168	1.62437
$170$	0	0
$171$	−8.37228	−0.640244
$172$	0	0
$173$	−15.1460	−1.15153	−0.575766	−	0.817615i	$-0.695296\pi$
$173$	−15.1460	−1.15153	−0.575766	+	0.817615i	$0.695296\pi$
$174$	0	0
$175$	−4.55134	−0.344049
$176$	0	0
$177$	4.62772	0.347841
$178$	0	0
$179$	−22.9783	−1.71748	−0.858738	−	0.512416i	$-0.828751\pi$
$179$	−22.9783	−1.71748	−0.858738	+	0.512416i	$0.828751\pi$
$180$	0	0
$181$	8.21782	0.610826	0.305413	−	0.952220i	$-0.401205\pi$
$181$	8.21782	0.610826	0.305413	+	0.952220i	$0.401205\pi$
$182$	0	0
$183$	48.9022	3.61495
$184$	0	0
$185$	25.4891	1.87400
$186$	0	0
$187$	11.8614	0.867392
$188$	0	0
$189$	−60.0868	−4.37067
$190$	0	0
$191$	−7.07568	−0.511978	−0.255989	−	0.966680i	$-0.582401\pi$
$191$	−7.07568	−0.511978	−0.255989	+	0.966680i	$0.582401\pi$
$192$	0	0
$193$	3.25544	0.234332	0.117166	−	0.993112i	$-0.462619\pi$
$193$	3.25544	0.234332	0.117166	+	0.993112i	$0.462619\pi$
$194$	0	0
$195$	−49.7228	−3.56072
$196$	0	0
$197$	−6.33830	−0.451585	−0.225792	−	0.974175i	$-0.572497\pi$
$197$	−6.33830	−0.451585	−0.225792	+	0.974175i	$0.572497\pi$
$198$	0	0
$199$	0.147477	0.0104544	0.00522718	−	0.999986i	$-0.498336\pi$
$199$	0.147477	0.0104544	0.00522718	+	0.999986i	$0.498336\pi$
$200$	0	0
$201$	−8.86141	−0.625035
$202$	0	0
$203$	−8.86141	−0.621949
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−6.63325	−0.461043
$208$	0	0
$209$	−2.37228	−0.164094
$210$	0	0
$211$	2.62772	0.180900	0.0904498	−	0.995901i	$-0.471170\pi$
$211$	2.62772	0.180900	0.0904498	+	0.995901i	$0.471170\pi$
$212$	0	0
$213$	22.3692	1.53271
$214$	0	0
$215$	−9.15759	−0.624542
$216$	0	0
$217$	−11.4891	−0.779933
$218$	0	0
$219$	−28.6277	−1.93448
$220$	0	0
$221$	−29.2048	−1.96453
$222$	0	0
$223$	−6.92820	−0.463947	−0.231973	−	0.972722i	$-0.574518\pi$
$223$	−6.92820	−0.463947	−0.231973	+	0.972722i	$0.574518\pi$
$224$	0	0
$225$	11.4891	0.765942
$226$	0	0
$227$	14.1168	0.936968	0.468484	−	0.883472i	$-0.344800\pi$
$227$	14.1168	0.936968	0.468484	+	0.883472i	$0.344800\pi$
$228$	0	0
$229$	12.6217	0.834065	0.417032	−	0.908892i	$-0.363070\pi$
$229$	12.6217	0.834065	0.417032	+	0.908892i	$0.363070\pi$
$230$	0	0
$231$	−26.5330	−1.74574
$232$	0	0
$233$	10.8832	0.712979	0.356490	−	0.934299i	$-0.383974\pi$
$233$	10.8832	0.712979	0.356490	+	0.934299i	$0.383974\pi$
$234$	0	0
$235$	−1.62772	−0.106181
$236$	0	0
$237$	−0.994667	−0.0646105
$238$	0	0
$239$	−17.4680	−1.12991	−0.564955	−	0.825122i	$-0.691106\pi$
$239$	−17.4680	−1.12991	−0.564955	+	0.825122i	$0.691106\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	66.9783	4.29666
$244$	0	0
$245$	10.0974	0.645096
$246$	0	0
$247$	5.84096	0.371652
$248$	0	0
$249$	26.9783	1.70968
$250$	0	0
$251$	−17.8614	−1.12740	−0.563701	−	0.825979i	$-0.690623\pi$
$251$	−17.8614	−1.12740	−0.563701	+	0.825979i	$0.690623\pi$
$252$	0	0
$253$	−1.87953	−0.118165
$254$	0	0
$255$	42.5639	2.66545
$256$	0	0
$257$	−16.2337	−1.01263	−0.506315	−	0.862349i	$-0.668993\pi$
$257$	−16.2337	−1.01263	−0.506315	+	0.862349i	$0.668993\pi$
$258$	0	0
$259$	−33.4891	−2.08091
$260$	0	0
$261$	22.3692	1.38462
$262$	0	0
$263$	−18.9600	−1.16912	−0.584561	−	0.811349i	$-0.698733\pi$
$263$	−18.9600	−1.16912	−0.584561	+	0.811349i	$0.698733\pi$
$264$	0	0
$265$	15.4891	0.951489
$266$	0	0
$267$	52.2337	3.19665
$268$	0	0
$269$	−1.87953	−0.114597	−0.0572984	−	0.998357i	$-0.518249\pi$
$269$	−1.87953	−0.114597	−0.0572984	+	0.998357i	$0.518249\pi$
$270$	0	0
$271$	2.37686	0.144384	0.0721920	−	0.997391i	$-0.477001\pi$
$271$	2.37686	0.144384	0.0721920	+	0.997391i	$0.477001\pi$
$272$	0	0
$273$	65.3288	3.95388
$274$	0	0
$275$	3.25544	0.196310
$276$	0	0
$277$	−16.3807	−0.984224	−0.492112	−	0.870532i	$-0.663775\pi$
$277$	−16.3807	−0.984224	−0.492112	+	0.870532i	$0.663775\pi$
$278$	0	0
$279$	29.0024	1.73633
$280$	0	0
$281$	−6.23369	−0.371871	−0.185935	−	0.982562i	$-0.559531\pi$
$281$	−6.23369	−0.371871	−0.185935	+	0.982562i	$0.559531\pi$
$282$	0	0
$283$	17.6277	1.04786	0.523930	−	0.851762i	$-0.324466\pi$
$283$	17.6277	1.04786	0.523930	+	0.851762i	$0.324466\pi$
$284$	0	0
$285$	−8.51278	−0.504253
$286$	0	0
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	−9.25544	−0.542563
$292$	0	0
$293$	4.84630	0.283124	0.141562	−	0.989929i	$-0.454788\pi$
$293$	4.84630	0.283124	0.141562	+	0.989929i	$0.454788\pi$
$294$	0	0
$295$	3.46410	0.201688
$296$	0	0
$297$	42.9783	2.49385
$298$	0	0
$299$	4.62772	0.267628
$300$	0	0
$301$	12.0318	0.693500
$302$	0	0
$303$	−46.7277	−2.68444
$304$	0	0
$305$	36.6060	2.09605
$306$	0	0
$307$	15.2554	0.870674	0.435337	−	0.900268i	$-0.356629\pi$
$307$	15.2554	0.870674	0.435337	+	0.900268i	$0.356629\pi$
$308$	0	0
$309$	−40.3894	−2.29767
$310$	0	0
$311$	−23.2164	−1.31648	−0.658240	−	0.752808i	$-0.728699\pi$
$311$	−23.2164	−1.31648	−0.658240	+	0.752808i	$0.728699\pi$
$312$	0	0
$313$	−3.88316	−0.219489	−0.109744	−	0.993960i	$-0.535003\pi$
$313$	−3.88316	−0.219489	−0.109744	+	0.993960i	$0.535003\pi$
$314$	0	0
$315$	−70.0951	−3.94941
$316$	0	0
$317$	−21.5769	−1.21188	−0.605940	−	0.795511i	$-0.707203\pi$
$317$	−21.5769	−1.21188	−0.605940	+	0.795511i	$0.707203\pi$
$318$	0	0
$319$	6.33830	0.354876
$320$	0	0
$321$	−45.0951	−2.51696
$322$	0	0
$323$	−5.00000	−0.278207
$324$	0	0
$325$	−8.01544	−0.444617
$326$	0	0
$327$	−13.3591	−0.738758
$328$	0	0
$329$	2.13859	0.117904
$330$	0	0
$331$	−10.8614	−0.596997	−0.298498	−	0.954410i	$-0.596486\pi$
$331$	−10.8614	−0.596997	−0.298498	+	0.954410i	$0.596486\pi$
$332$	0	0
$333$	84.5379	4.63265
$334$	0	0
$335$	−6.63325	−0.362413
$336$	0	0
$337$	−13.2554	−0.722070	−0.361035	−	0.932552i	$-0.617576\pi$
$337$	−13.2554	−0.722070	−0.361035	+	0.932552i	$0.617576\pi$
$338$	0	0
$339$	−49.7228	−2.70057
$340$	0	0
$341$	8.21782	0.445020
$342$	0	0
$343$	9.94987	0.537243
$344$	0	0
$345$	−6.74456	−0.363115
$346$	0	0
$347$	−37.1168	−1.99254	−0.996268	−	0.0863104i	$-0.972492\pi$
$347$	−37.1168	−1.99254	−0.996268	+	0.0863104i	$0.972492\pi$
$348$	0	0
$349$	−13.2116	−0.707201	−0.353600	−	0.935397i	$-0.615043\pi$
$349$	−13.2116	−0.707201	−0.353600	+	0.935397i	$0.615043\pi$
$350$	0	0
$351$	−105.820	−5.64824
$352$	0	0
$353$	3.88316	0.206680	0.103340	−	0.994646i	$-0.467047\pi$
$353$	3.88316	0.206680	0.103340	+	0.994646i	$0.467047\pi$
$354$	0	0
$355$	16.7446	0.888709
$356$	0	0
$357$	−55.9230	−2.95976
$358$	0	0
$359$	16.5831	0.875224	0.437612	−	0.899164i	$-0.355824\pi$
$359$	16.5831	0.875224	0.437612	+	0.899164i	$0.355824\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−18.1168	−0.950888
$364$	0	0
$365$	−21.4294	−1.12167
$366$	0	0
$367$	4.05401	0.211618	0.105809	−	0.994386i	$-0.466257\pi$
$367$	4.05401	0.211618	0.105809	+	0.994386i	$0.466257\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−20.3505	−1.05655
$372$	0	0
$373$	14.9436	0.773753	0.386876	−	0.922132i	$-0.373554\pi$
$373$	14.9436	0.773753	0.386876	+	0.922132i	$0.373554\pi$
$374$	0	0
$375$	−30.8820	−1.59474
$376$	0	0
$377$	−15.6060	−0.803748
$378$	0	0
$379$	−36.3505	−1.86720	−0.933601	−	0.358315i	$-0.883351\pi$
$379$	−36.3505	−1.86720	−0.933601	+	0.358315i	$0.883351\pi$
$380$	0	0
$381$	41.3841	2.12017
$382$	0	0
$383$	−25.2434	−1.28988	−0.644938	−	0.764235i	$-0.723117\pi$
$383$	−25.2434	−1.28988	−0.644938	+	0.764235i	$0.723117\pi$
$384$	0	0
$385$	−19.8614	−1.01223
$386$	0	0
$387$	−30.3723	−1.54391
$388$	0	0
$389$	24.3036	1.23224	0.616121	−	0.787651i	$-0.288703\pi$
$389$	24.3036	1.23224	0.616121	+	0.787651i	$0.288703\pi$
$390$	0	0
$391$	−3.96143	−0.200338
$392$	0	0
$393$	14.7446	0.743765
$394$	0	0
$395$	−0.744563	−0.0374630
$396$	0	0
$397$	−0.644810	−0.0323621	−0.0161810	−	0.999869i	$-0.505151\pi$
$397$	−0.644810	−0.0323621	−0.0161810	+	0.999869i	$0.505151\pi$
$398$	0	0
$399$	11.1846	0.559930
$400$	0	0
$401$	35.7228	1.78391	0.891956	−	0.452122i	$-0.149333\pi$
$401$	35.7228	1.78391	0.891956	+	0.452122i	$0.149333\pi$
$402$	0	0
$403$	−20.2337	−1.00791
$404$	0	0
$405$	90.8213	4.51294
$406$	0	0
$407$	23.9538	1.18734
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	−16.8614	−0.831712
$412$	0	0
$413$	−4.55134	−0.223957
$414$	0	0
$415$	20.1947	0.991319
$416$	0	0
$417$	−12.2337	−0.599086
$418$	0	0
$419$	−0.233688	−0.0114164	−0.00570820	−	0.999984i	$-0.501817\pi$
$419$	−0.233688	−0.0114164	−0.00570820	+	0.999984i	$0.501817\pi$
$420$	0	0
$421$	−20.3971	−0.994093	−0.497046	−	0.867724i	$-0.665582\pi$
$421$	−20.3971	−0.994093	−0.497046	+	0.867724i	$0.665582\pi$
$422$	0	0
$423$	−5.39853	−0.262486
$424$	0	0
$425$	6.86141	0.332827
$426$	0	0
$427$	−48.0951	−2.32748
$428$	0	0
$429$	−46.7277	−2.25603
$430$	0	0
$431$	−21.7793	−1.04907	−0.524535	−	0.851389i	$-0.675761\pi$
$431$	−21.7793	−1.04907	−0.524535	+	0.851389i	$0.675761\pi$
$432$	0	0
$433$	−1.48913	−0.0715628	−0.0357814	−	0.999360i	$-0.511392\pi$
$433$	−1.48913	−0.0715628	−0.0357814	+	0.999360i	$0.511392\pi$
$434$	0	0
$435$	22.7446	1.09052
$436$	0	0
$437$	0.792287	0.0379002
$438$	0	0
$439$	6.92820	0.330665	0.165333	−	0.986238i	$-0.447130\pi$
$439$	6.92820	0.330665	0.165333	+	0.986238i	$0.447130\pi$
$440$	0	0
$441$	33.4891	1.59472
$442$	0	0
$443$	−41.3505	−1.96462	−0.982312	−	0.187254i	$-0.940041\pi$
$443$	−41.3505	−1.96462	−0.982312	+	0.187254i	$0.940041\pi$
$444$	0	0
$445$	39.0998	1.85351
$446$	0	0
$447$	15.8457	0.749478
$448$	0	0
$449$	37.4891	1.76922	0.884611	−	0.466330i	$-0.154424\pi$
$449$	37.4891	1.76922	0.884611	+	0.466330i	$0.154424\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−53.0660	−2.49326
$454$	0	0
$455$	48.9022	2.29257
$456$	0	0
$457$	−31.0000	−1.45012	−0.725059	−	0.688686i	$-0.758188\pi$
$457$	−31.0000	−1.45012	−0.725059	+	0.688686i	$0.758188\pi$
$458$	0	0
$459$	90.5842	4.22811
$460$	0	0
$461$	−7.86797	−0.366448	−0.183224	−	0.983071i	$-0.558653\pi$
$461$	−7.86797	−0.366448	−0.183224	+	0.983071i	$0.558653\pi$
$462$	0	0
$463$	31.5268	1.46517	0.732587	−	0.680674i	$-0.238313\pi$
$463$	31.5268	1.46517	0.732587	+	0.680674i	$0.238313\pi$
$464$	0	0
$465$	29.4891	1.36753
$466$	0	0
$467$	−10.8832	−0.503612	−0.251806	−	0.967778i	$-0.581025\pi$
$467$	−10.8832	−0.503612	−0.251806	+	0.967778i	$0.581025\pi$
$468$	0	0
$469$	8.71516	0.402429
$470$	0	0
$471$	62.7586	2.89176
$472$	0	0
$473$	−8.60597	−0.395703
$474$	0	0
$475$	−1.37228	−0.0629646
$476$	0	0
$477$	51.3716	2.35214
$478$	0	0
$479$	27.4179	1.25275	0.626377	−	0.779520i	$-0.284537\pi$
$479$	27.4179	1.25275	0.626377	+	0.779520i	$0.284537\pi$
$480$	0	0
$481$	−58.9783	−2.68918
$482$	0	0
$483$	8.86141	0.403208
$484$	0	0
$485$	−6.92820	−0.314594
$486$	0	0
$487$	−22.6641	−1.02701	−0.513505	−	0.858087i	$-0.671653\pi$
$487$	−22.6641	−1.02701	−0.513505	+	0.858087i	$0.671653\pi$
$488$	0	0
$489$	−36.2337	−1.63854
$490$	0	0
$491$	−30.9783	−1.39803	−0.699014	−	0.715108i	$-0.746378\pi$
$491$	−30.9783	−1.39803	−0.699014	+	0.715108i	$0.746378\pi$
$492$	0	0
$493$	13.3591	0.601662
$494$	0	0
$495$	50.1369	2.25349
$496$	0	0
$497$	−22.0000	−0.986835
$498$	0	0
$499$	4.37228	0.195730	0.0978651	−	0.995200i	$-0.468799\pi$
$499$	4.37228	0.195730	0.0978651	+	0.995200i	$0.468799\pi$
$500$	0	0
$501$	−21.3745	−0.954943
$502$	0	0
$503$	11.4795	0.511848	0.255924	−	0.966697i	$-0.417620\pi$
$503$	11.4795	0.511848	0.255924	+	0.966697i	$0.417620\pi$
$504$	0	0
$505$	−34.9783	−1.55651
$506$	0	0
$507$	71.2119	3.16263
$508$	0	0
$509$	37.8102	1.67591	0.837953	−	0.545742i	$-0.183752\pi$
$509$	37.8102	1.67591	0.837953	+	0.545742i	$0.183752\pi$
$510$	0	0
$511$	28.1552	1.24551
$512$	0	0
$513$	−18.1168	−0.799878
$514$	0	0
$515$	−30.2337	−1.33226
$516$	0	0
$517$	−1.52967	−0.0672749
$518$	0	0
$519$	−51.0767	−2.24202
$520$	0	0
$521$	5.76631	0.252627	0.126313	−	0.991990i	$-0.459686\pi$
$521$	5.76631	0.252627	0.126313	+	0.991990i	$0.459686\pi$
$522$	0	0
$523$	25.3723	1.10945	0.554726	−	0.832033i	$-0.312823\pi$
$523$	25.3723	1.10945	0.554726	+	0.832033i	$0.312823\pi$
$524$	0	0
$525$	−15.3484	−0.669859
$526$	0	0
$527$	17.3205	0.754493
$528$	0	0
$529$	−22.3723	−0.972708
$530$	0	0
$531$	11.4891	0.498586
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−33.7562	−1.45941
$536$	0	0
$537$	−77.4891	−3.34390
$538$	0	0
$539$	9.48913	0.408726
$540$	0	0
$541$	23.6039	1.01481	0.507405	−	0.861707i	$-0.330605\pi$
$541$	23.6039	1.01481	0.507405	+	0.861707i	$0.330605\pi$
$542$	0	0
$543$	27.7128	1.18927
$544$	0	0
$545$	−10.0000	−0.428353
$546$	0	0
$547$	2.51087	0.107357	0.0536786	−	0.998558i	$-0.482905\pi$
$547$	2.51087	0.107357	0.0536786	+	0.998558i	$0.482905\pi$
$548$	0	0
$549$	121.408	5.18158
$550$	0	0
$551$	−2.67181	−0.113823
$552$	0	0
$553$	0.978251	0.0415994
$554$	0	0
$555$	85.9565	3.64865
$556$	0	0
$557$	−25.1885	−1.06727	−0.533635	−	0.845715i	$-0.679174\pi$
$557$	−25.1885	−1.06727	−0.533635	+	0.845715i	$0.679174\pi$
$558$	0	0
$559$	21.1894	0.896215
$560$	0	0
$561$	40.0000	1.68880
$562$	0	0
$563$	13.4891	0.568499	0.284249	−	0.958750i	$-0.408256\pi$
$563$	13.4891	0.568499	0.284249	+	0.958750i	$0.408256\pi$
$564$	0	0
$565$	−37.2203	−1.56587
$566$	0	0
$567$	−119.326	−5.01124
$568$	0	0
$569$	26.9783	1.13099	0.565494	−	0.824753i	$-0.308686\pi$
$569$	26.9783	1.13099	0.565494	+	0.824753i	$0.308686\pi$
$570$	0	0
$571$	−42.9783	−1.79858	−0.899292	−	0.437349i	$-0.855917\pi$
$571$	−42.9783	−1.79858	−0.899292	+	0.437349i	$0.855917\pi$
$572$	0	0
$573$	−23.8612	−0.996815
$574$	0	0
$575$	−1.08724	−0.0453411
$576$	0	0
$577$	33.4674	1.39327	0.696633	−	0.717428i	$-0.254681\pi$
$577$	33.4674	1.39327	0.696633	+	0.717428i	$0.254681\pi$
$578$	0	0
$579$	10.9783	0.456241
$580$	0	0
$581$	−26.5330	−1.10077
$582$	0	0
$583$	14.5561	0.602853
$584$	0	0
$585$	−123.446	−5.10385
$586$	0	0
$587$	31.8614	1.31506	0.657530	−	0.753428i	$-0.271601\pi$
$587$	31.8614	1.31506	0.657530	+	0.753428i	$0.271601\pi$
$588$	0	0
$589$	−3.46410	−0.142736
$590$	0	0
$591$	−21.3745	−0.879230
$592$	0	0
$593$	4.97825	0.204432	0.102216	−	0.994762i	$-0.467407\pi$
$593$	4.97825	0.204432	0.102216	+	0.994762i	$0.467407\pi$
$594$	0	0
$595$	−41.8614	−1.71615
$596$	0	0
$597$	0.497333	0.0203545
$598$	0	0
$599$	39.6897	1.62168	0.810838	−	0.585270i	$-0.199012\pi$
$599$	39.6897	1.62168	0.810838	+	0.585270i	$0.199012\pi$
$600$	0	0
$601$	22.4674	0.916463	0.458232	−	0.888833i	$-0.348483\pi$
$601$	22.4674	0.916463	0.458232	+	0.888833i	$0.348483\pi$
$602$	0	0
$603$	−22.0000	−0.895909
$604$	0	0
$605$	−13.5615	−0.551351
$606$	0	0
$607$	−38.8048	−1.57504	−0.787520	−	0.616289i	$-0.788635\pi$
$607$	−38.8048	−1.57504	−0.787520	+	0.616289i	$0.788635\pi$
$608$	0	0
$609$	−29.8832	−1.21093
$610$	0	0
$611$	3.76631	0.152369
$612$	0	0
$613$	−18.5552	−0.749439	−0.374719	−	0.927138i	$-0.622261\pi$
$613$	−18.5552	−0.749439	−0.374719	+	0.927138i	$0.622261\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	40.8397	1.64414	0.822071	−	0.569384i	$-0.192818\pi$
$617$	40.8397	1.64414	0.822071	+	0.569384i	$0.192818\pi$
$618$	0	0
$619$	16.2337	0.652487	0.326244	−	0.945286i	$-0.394217\pi$
$619$	16.2337	0.652487	0.326244	+	0.945286i	$0.394217\pi$
$620$	0	0
$621$	−14.3537	−0.575996
$622$	0	0
$623$	−51.3716	−2.05816
$624$	0	0
$625$	−29.9783	−1.19913
$626$	0	0
$627$	−8.00000	−0.319489
$628$	0	0
$629$	50.4868	2.01304
$630$	0	0
$631$	37.8651	1.50738	0.753692	−	0.657227i	$-0.228271\pi$
$631$	37.8651	1.50738	0.753692	+	0.657227i	$0.228271\pi$
$632$	0	0
$633$	8.86141	0.352209
$634$	0	0
$635$	30.9783	1.22933
$636$	0	0
$637$	−23.3639	−0.925709
$638$	0	0
$639$	55.5354	2.19695
$640$	0	0
$641$	−10.9783	−0.433615	−0.216807	−	0.976214i	$-0.569564\pi$
$641$	−10.9783	−0.433615	−0.216807	+	0.976214i	$0.569564\pi$
$642$	0	0
$643$	−36.3723	−1.43438	−0.717191	−	0.696876i	$-0.754573\pi$
$643$	−36.3723	−1.43438	−0.717191	+	0.696876i	$0.754573\pi$
$644$	0	0
$645$	−30.8820	−1.21598
$646$	0	0
$647$	16.5831	0.651950	0.325975	−	0.945378i	$-0.394307\pi$
$647$	16.5831	0.651950	0.325975	+	0.945378i	$0.394307\pi$
$648$	0	0
$649$	3.25544	0.127787
$650$	0	0
$651$	−38.7446	−1.51852
$652$	0	0
$653$	35.3956	1.38514	0.692569	−	0.721352i	$-0.256479\pi$
$653$	35.3956	1.38514	0.692569	+	0.721352i	$0.256479\pi$
$654$	0	0
$655$	11.0371	0.431256
$656$	0	0
$657$	−71.0733	−2.77284
$658$	0	0
$659$	46.8614	1.82546	0.912731	−	0.408562i	$-0.133970\pi$
$659$	46.8614	1.82546	0.912731	+	0.408562i	$0.133970\pi$
$660$	0	0
$661$	10.5947	0.412085	0.206043	−	0.978543i	$-0.433941\pi$
$661$	10.5947	0.412085	0.206043	+	0.978543i	$0.433941\pi$
$662$	0	0
$663$	−98.4868	−3.82491
$664$	0	0
$665$	8.37228	0.324663
$666$	0	0
$667$	−2.11684	−0.0819645
$668$	0	0
$669$	−23.3639	−0.903299
$670$	0	0
$671$	34.4010	1.32803
$672$	0	0
$673$	16.7446	0.645455	0.322728	−	0.946492i	$-0.395400\pi$
$673$	16.7446	0.645455	0.322728	+	0.946492i	$0.395400\pi$
$674$	0	0
$675$	24.8614	0.956916
$676$	0	0
$677$	−5.84096	−0.224486	−0.112243	−	0.993681i	$-0.535804\pi$
$677$	−5.84096	−0.224486	−0.112243	+	0.993681i	$0.535804\pi$
$678$	0	0
$679$	9.10268	0.349329
$680$	0	0
$681$	47.6060	1.82426
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	−12.6217	−0.482250
$686$	0	0
$687$	42.5639	1.62391
$688$	0	0
$689$	−35.8397	−1.36538
$690$	0	0
$691$	−17.8614	−0.679480	−0.339740	−	0.940519i	$-0.610339\pi$
$691$	−17.8614	−0.679480	−0.339740	+	0.940519i	$0.610339\pi$
$692$	0	0
$693$	−65.8728	−2.50230
$694$	0	0
$695$	−9.15759	−0.347367
$696$	0	0
$697$	0	0
$698$	0	0
$699$	36.7011	1.38816
$700$	0	0
$701$	−27.7128	−1.04670	−0.523349	−	0.852118i	$-0.675318\pi$
$701$	−27.7128	−1.04670	−0.523349	+	0.852118i	$0.675318\pi$
$702$	0	0
$703$	−10.0974	−0.380829
$704$	0	0
$705$	−5.48913	−0.206732
$706$	0	0
$707$	45.9565	1.72837
$708$	0	0
$709$	−45.7330	−1.71754	−0.858770	−	0.512361i	$-0.828771\pi$
$709$	−45.7330	−1.71754	−0.858770	+	0.512361i	$0.828771\pi$
$710$	0	0
$711$	−2.46943	−0.0926110
$712$	0	0
$713$	−2.74456	−0.102785
$714$	0	0
$715$	−34.9783	−1.30811
$716$	0	0
$717$	−58.9070	−2.19992
$718$	0	0
$719$	−40.5369	−1.51177	−0.755885	−	0.654704i	$-0.772793\pi$
$719$	−40.5369	−1.51177	−0.755885	+	0.654704i	$0.772793\pi$
$720$	0	0
$721$	39.7228	1.47935
$722$	0	0
$723$	26.9783	1.00333
$724$	0	0
$725$	3.66648	0.136170
$726$	0	0
$727$	14.0039	0.519375	0.259688	−	0.965693i	$-0.416380\pi$
$727$	14.0039	0.519375	0.259688	+	0.965693i	$0.416380\pi$
$728$	0	0
$729$	117.935	4.36795
$730$	0	0
$731$	−18.1386	−0.670880
$732$	0	0
$733$	21.1894	0.782647	0.391324	−	0.920253i	$-0.372017\pi$
$733$	21.1894	0.782647	0.391324	+	0.920253i	$0.372017\pi$
$734$	0	0
$735$	34.0511	1.25599
$736$	0	0
$737$	−6.23369	−0.229621
$738$	0	0
$739$	12.6060	0.463718	0.231859	−	0.972749i	$-0.425519\pi$
$739$	12.6060	0.463718	0.231859	+	0.972749i	$0.425519\pi$
$740$	0	0
$741$	19.6974	0.723601
$742$	0	0
$743$	6.63325	0.243350	0.121675	−	0.992570i	$-0.461173\pi$
$743$	6.63325	0.243350	0.121675	+	0.992570i	$0.461173\pi$
$744$	0	0
$745$	11.8614	0.434568
$746$	0	0
$747$	66.9783	2.45061
$748$	0	0
$749$	44.3508	1.62054
$750$	0	0
$751$	−9.50744	−0.346932	−0.173466	−	0.984840i	$-0.555497\pi$
$751$	−9.50744	−0.346932	−0.173466	+	0.984840i	$0.555497\pi$
$752$	0	0
$753$	−60.2337	−2.19504
$754$	0	0
$755$	−39.7228	−1.44566
$756$	0	0
$757$	18.8502	0.685121	0.342561	−	0.939496i	$-0.388706\pi$
$757$	18.8502	0.685121	0.342561	+	0.939496i	$0.388706\pi$
$758$	0	0
$759$	−6.33830	−0.230066
$760$	0	0
$761$	11.0000	0.398750	0.199375	−	0.979923i	$-0.436109\pi$
$761$	11.0000	0.398750	0.199375	+	0.979923i	$0.436109\pi$
$762$	0	0
$763$	13.1386	0.475649
$764$	0	0
$765$	105.672	3.82059
$766$	0	0
$767$	−8.01544	−0.289421
$768$	0	0
$769$	37.4674	1.35111	0.675554	−	0.737310i	$-0.263904\pi$
$769$	37.4674	1.35111	0.675554	+	0.737310i	$0.263904\pi$
$770$	0	0
$771$	−54.7446	−1.97158
$772$	0	0
$773$	−24.1561	−0.868836	−0.434418	−	0.900711i	$-0.643046\pi$
$773$	−24.1561	−0.868836	−0.434418	+	0.900711i	$0.643046\pi$
$774$	0	0
$775$	4.75372	0.170759
$776$	0	0
$777$	−112.935	−4.05151
$778$	0	0
$779$	0	0
$780$	0	0
$781$	15.7359	0.563076
$782$	0	0
$783$	48.4048	1.72985
$784$	0	0
$785$	46.9783	1.67673
$786$	0	0
$787$	19.6060	0.698877	0.349439	−	0.936959i	$-0.386372\pi$
$787$	19.6060	0.698877	0.349439	+	0.936959i	$0.386372\pi$
$788$	0	0
$789$	−63.9384	−2.27627
$790$	0	0
$791$	48.9022	1.73876
$792$	0	0
$793$	−84.7011	−3.00782
$794$	0	0
$795$	52.2337	1.85254
$796$	0	0
$797$	14.6487	0.518883	0.259442	−	0.965759i	$-0.416461\pi$
$797$	14.6487	0.518883	0.259442	+	0.965759i	$0.416461\pi$
$798$	0	0
$799$	−3.22405	−0.114059
$800$	0	0
$801$	129.679	4.58199
$802$	0	0
$803$	−20.1386	−0.710676
$804$	0	0
$805$	6.63325	0.233791
$806$	0	0
$807$	−6.33830	−0.223119
$808$	0	0
$809$	37.9783	1.33524	0.667622	−	0.744500i	$-0.267312\pi$
$809$	37.9783	1.33524	0.667622	+	0.744500i	$0.267312\pi$
$810$	0	0
$811$	−28.8614	−1.01346	−0.506731	−	0.862105i	$-0.669146\pi$
$811$	−28.8614	−1.01346	−0.506731	+	0.862105i	$0.669146\pi$
$812$	0	0
$813$	8.01544	0.281114
$814$	0	0
$815$	−27.1229	−0.950074
$816$	0	0
$817$	3.62772	0.126918
$818$	0	0
$819$	162.190	5.66738
$820$	0	0
$821$	27.0680	0.944680	0.472340	−	0.881416i	$-0.343409\pi$
$821$	27.0680	0.944680	0.472340	+	0.881416i	$0.343409\pi$
$822$	0	0
$823$	−7.37063	−0.256924	−0.128462	−	0.991714i	$-0.541004\pi$
$823$	−7.37063	−0.256924	−0.128462	+	0.991714i	$0.541004\pi$
$824$	0	0
$825$	10.9783	0.382214
$826$	0	0
$827$	−40.3505	−1.40313	−0.701563	−	0.712608i	$-0.747514\pi$
$827$	−40.3505	−1.40313	−0.701563	+	0.712608i	$0.747514\pi$
$828$	0	0
$829$	−3.26172	−0.113284	−0.0566421	−	0.998395i	$-0.518039\pi$
$829$	−3.26172	−0.113284	−0.0566421	+	0.998395i	$0.518039\pi$
$830$	0	0
$831$	−55.2405	−1.91627
$832$	0	0
$833$	20.0000	0.692959
$834$	0	0
$835$	−16.0000	−0.553703
$836$	0	0
$837$	62.7586	2.16925
$838$	0	0
$839$	9.80240	0.338416	0.169208	−	0.985580i	$-0.445879\pi$
$839$	9.80240	0.338416	0.169208	+	0.985580i	$0.445879\pi$
$840$	0	0
$841$	−21.8614	−0.753842
$842$	0	0
$843$	−21.0217	−0.724028
$844$	0	0
$845$	53.3060	1.83378
$846$	0	0
$847$	17.8178	0.612228
$848$	0	0
$849$	59.4456	2.04017
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	−40.3894	−1.38291	−0.691453	−	0.722421i	$-0.743029\pi$
$853$	−40.3894	−1.38291	−0.691453	+	0.722421i	$0.743029\pi$
$854$	0	0
$855$	−21.1345	−0.722783
$856$	0	0
$857$	−22.2337	−0.759488	−0.379744	−	0.925092i	$-0.623988\pi$
$857$	−22.2337	−0.759488	−0.379744	+	0.925092i	$0.623988\pi$
$858$	0	0
$859$	8.60597	0.293632	0.146816	−	0.989164i	$-0.453097\pi$
$859$	8.60597	0.293632	0.146816	+	0.989164i	$0.453097\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.76439	0.0941009	0.0470504	−	0.998893i	$-0.485018\pi$
$863$	2.76439	0.0941009	0.0470504	+	0.998893i	$0.485018\pi$
$864$	0	0
$865$	−38.2337	−1.29998
$866$	0	0
$867$	26.9783	0.916229
$868$	0	0
$869$	−0.699713	−0.0237361
$870$	0	0
$871$	15.3484	0.520061
$872$	0	0
$873$	−22.9783	−0.777696
$874$	0	0
$875$	30.3723	1.02677
$876$	0	0
$877$	1.38219	0.0466734	0.0233367	−	0.999728i	$-0.492571\pi$
$877$	1.38219	0.0466734	0.0233367	+	0.999728i	$0.492571\pi$
$878$	0	0
$879$	16.3431	0.551238
$880$	0	0
$881$	−4.37228	−0.147306	−0.0736530	−	0.997284i	$-0.523466\pi$
$881$	−4.37228	−0.147306	−0.0736530	+	0.997284i	$0.523466\pi$
$882$	0	0
$883$	18.8832	0.635469	0.317734	−	0.948180i	$-0.397078\pi$
$883$	18.8832	0.635469	0.317734	+	0.948180i	$0.397078\pi$
$884$	0	0
$885$	11.6819	0.392684
$886$	0	0
$887$	−45.0333	−1.51207	−0.756035	−	0.654531i	$-0.772866\pi$
$887$	−45.0333	−1.51207	−0.756035	+	0.654531i	$0.772866\pi$
$888$	0	0
$889$	−40.7011	−1.36507
$890$	0	0
$891$	85.3505	2.85935
$892$	0	0
$893$	0.644810	0.0215777
$894$	0	0
$895$	−58.0049	−1.93889
$896$	0	0
$897$	15.6060	0.521068
$898$	0	0
$899$	9.25544	0.308686
$900$	0	0
$901$	30.6796	1.02209
$902$	0	0
$903$	40.5746	1.35024
$904$	0	0
$905$	20.7446	0.689573
$906$	0	0
$907$	46.8614	1.55601	0.778004	−	0.628260i	$-0.216232\pi$
$907$	46.8614	1.55601	0.778004	+	0.628260i	$0.216232\pi$
$908$	0	0
$909$	−116.010	−3.84780
$910$	0	0
$911$	47.6126	1.57747	0.788737	−	0.614730i	$-0.210735\pi$
$911$	47.6126	1.57747	0.788737	+	0.614730i	$0.210735\pi$
$912$	0	0
$913$	18.9783	0.628088
$914$	0	0
$915$	123.446	4.08099
$916$	0	0
$917$	−14.5012	−0.478872
$918$	0	0
$919$	8.71516	0.287486	0.143743	−	0.989615i	$-0.454086\pi$
$919$	8.71516	0.287486	0.143743	+	0.989615i	$0.454086\pi$
$920$	0	0
$921$	51.4456	1.69519
$922$	0	0
$923$	−38.7446	−1.27529
$924$	0	0
$925$	13.8564	0.455596
$926$	0	0
$927$	−100.274	−3.29342
$928$	0	0
$929$	−27.8832	−0.914817	−0.457408	−	0.889257i	$-0.651222\pi$
$929$	−27.8832	−0.914817	−0.457408	+	0.889257i	$0.651222\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	0	0
$933$	−78.2921	−2.56317
$934$	0	0
$935$	29.9422	0.979215
$936$	0	0
$937$	21.9783	0.717998	0.358999	−	0.933338i	$-0.383118\pi$
$937$	21.9783	0.717998	0.358999	+	0.933338i	$0.383118\pi$
$938$	0	0
$939$	−13.0951	−0.427342
$940$	0	0
$941$	−10.8896	−0.354992	−0.177496	−	0.984122i	$-0.556800\pi$
$941$	−10.8896	−0.354992	−0.177496	+	0.984122i	$0.556800\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−151.679	−4.93413
$946$	0	0
$947$	43.2119	1.40420	0.702100	−	0.712079i	$-0.252246\pi$
$947$	43.2119	1.40420	0.702100	+	0.712079i	$0.252246\pi$
$948$	0	0
$949$	49.5847	1.60959
$950$	0	0
$951$	−72.7634	−2.35951
$952$	0	0
$953$	−26.2337	−0.849793	−0.424896	−	0.905242i	$-0.639689\pi$
$953$	−26.2337	−0.849793	−0.424896	+	0.905242i	$0.639689\pi$
$954$	0	0
$955$	−17.8614	−0.577982
$956$	0	0
$957$	21.3745	0.690940
$958$	0	0
$959$	16.5831	0.535497
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	−111.957	−3.60775
$964$	0	0
$965$	8.21782	0.264541
$966$	0	0
$967$	55.1307	1.77288	0.886441	−	0.462841i	$-0.153170\pi$
$967$	55.1307	1.77288	0.886441	+	0.462841i	$0.153170\pi$
$968$	0	0
$969$	−16.8614	−0.541666
$970$	0	0
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	0	0
$973$	12.0318	0.385721
$974$	0	0
$975$	−27.0303	−0.865663
$976$	0	0
$977$	27.4891	0.879455	0.439728	−	0.898131i	$-0.355075\pi$
$977$	27.4891	0.879455	0.439728	+	0.898131i	$0.355075\pi$
$978$	0	0
$979$	36.7446	1.17436
$980$	0	0
$981$	−33.1662	−1.05892
$982$	0	0
$983$	52.6612	1.67963	0.839816	−	0.542871i	$-0.182663\pi$
$983$	52.6612	1.67963	0.839816	+	0.542871i	$0.182663\pi$
$984$	0	0
$985$	−16.0000	−0.509802
$986$	0	0
$987$	7.21194	0.229559
$988$	0	0
$989$	2.87419	0.0913941
$990$	0	0
$991$	7.92287	0.251678	0.125839	−	0.992051i	$-0.459838\pi$
$991$	7.92287	0.251678	0.125839	+	0.992051i	$0.459838\pi$
$992$	0	0
$993$	−36.6277	−1.16235
$994$	0	0
$995$	0.372281	0.0118021
$996$	0	0
$997$	−32.8164	−1.03931	−0.519653	−	0.854378i	$-0.673939\pi$
$997$	−32.8164	−1.03931	−0.519653	+	0.854378i	$0.673939\pi$
$998$	0	0
$999$	182.932	5.78772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.bl.1.4		4
4.3	odd	2		4864.2.a.bi.1.2		4
8.3	odd	2	inner	4864.2.a.bl.1.3		4
8.5	even	2		4864.2.a.bi.1.1		4
16.3	odd	4		1216.2.c.h.609.1	✓	8
16.5	even	4		1216.2.c.h.609.2	yes	8
16.11	odd	4		1216.2.c.h.609.8	yes	8
16.13	even	4		1216.2.c.h.609.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.c.h.609.1	✓	8	16.3	odd	4
1216.2.c.h.609.2	yes	8	16.5	even	4
1216.2.c.h.609.7	yes	8	16.13	even	4
1216.2.c.h.609.8	yes	8	16.11	odd	4
4864.2.a.bi.1.1		4	8.5	even	2
4864.2.a.bi.1.2		4	4.3	odd	2
4864.2.a.bl.1.3		4	8.3	odd	2	inner
4864.2.a.bl.1.4		4	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:	\( N \)	\(=\)	\( 4864 = 2^{8} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4864.a (trivial)

\(f(q)\)	\(=\)	\(q+3.37228 q^{3} +2.52434 q^{5} -3.31662 q^{7} +8.37228 q^{9} +O(q^{10})\)	\(q+3.37228 q^{3} +2.52434 q^{5} -3.31662 q^{7} +8.37228 q^{9} +2.37228 q^{11} -5.84096 q^{13} +8.51278 q^{15} +5.00000 q^{17} -1.00000 q^{19} -11.1846 q^{21} -0.792287 q^{23} +1.37228 q^{25} +18.1168 q^{27} +2.67181 q^{29} +3.46410 q^{31} +8.00000 q^{33} -8.37228 q^{35} +10.0974 q^{37} -19.6974 q^{39} -3.62772 q^{43} +21.1345 q^{45} -0.644810 q^{47} +4.00000 q^{49} +16.8614 q^{51} +6.13592 q^{53} +5.98844 q^{55} -3.37228 q^{57} +1.37228 q^{59} +14.5012 q^{61} -27.7677 q^{63} -14.7446 q^{65} -2.62772 q^{67} -2.67181 q^{69} +6.63325 q^{71} -8.48913 q^{73} +4.62772 q^{75} -7.86797 q^{77} -0.294954 q^{79} +35.9783 q^{81} +8.00000 q^{83} +12.6217 q^{85} +9.01011 q^{87} +15.4891 q^{89} +19.3723 q^{91} +11.6819 q^{93} -2.52434 q^{95} -2.74456 q^{97} +19.8614 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 22 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 + 22 * q^9	\( 4 q + 2 q^{3} + 22 q^{9} - 2 q^{11} + 20 q^{17} - 4 q^{19} - 6 q^{25} + 38 q^{27} + 32 q^{33} - 22 q^{35} - 26 q^{43} + 16 q^{49} + 10 q^{51} - 2 q^{57} - 6 q^{59} - 36 q^{65} - 22 q^{67} + 12 q^{73} + 30 q^{75} + 52 q^{81} + 32 q^{83} + 16 q^{89} + 66 q^{91} + 12 q^{97} + 22 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 22 * q^9 - 2 * q^11 + 20 * q^17 - 4 * q^19 - 6 * q^25 + 38 * q^27 + 32 * q^33 - 22 * q^35 - 26 * q^43 + 16 * q^49 + 10 * q^51 - 2 * q^57 - 6 * q^59 - 36 * q^65 - 22 * q^67 + 12 * q^73 + 30 * q^75 + 52 * q^81 + 32 * q^83 + 16 * q^89 + 66 * q^91 + 12 * q^97 + 22 * q^99

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