LMFDB - Embedded newform 4864.2.a.r.1.2

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:	$2$
Coefficient field:	$\Q(\sqrt{11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 11 $ x^2 - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.2
Root			$3.31662$ 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)

$f(q)$	$=$	$q-2.00000 q^{3} +3.31662 q^{5} -3.31662 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-2.00000 q^{3} +3.31662 q^{5} -3.31662 q^{7} +1.00000 q^{9} +5.00000 q^{11} -6.63325 q^{15} +5.00000 q^{17} +1.00000 q^{19} +6.63325 q^{21} -6.63325 q^{23} +6.00000 q^{25} +4.00000 q^{27} +6.63325 q^{29} -10.0000 q^{33} -11.0000 q^{35} -6.63325 q^{37} +6.00000 q^{41} -1.00000 q^{43} +3.31662 q^{45} +9.94987 q^{47} +4.00000 q^{49} -10.0000 q^{51} -13.2665 q^{53} +16.5831 q^{55} -2.00000 q^{57} -6.00000 q^{59} +9.94987 q^{61} -3.31662 q^{63} -8.00000 q^{67} +13.2665 q^{69} +6.63325 q^{71} +9.00000 q^{73} -12.0000 q^{75} -16.5831 q^{77} -13.2665 q^{79} -11.0000 q^{81} +4.00000 q^{83} +16.5831 q^{85} -13.2665 q^{87} +4.00000 q^{89} +3.31662 q^{95} -12.0000 q^{97} +5.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^3 + 2 * q^9	$ 2 q - 4 q^{3} + 2 q^{9} + 10 q^{11} + 10 q^{17} + 2 q^{19} + 12 q^{25} + 8 q^{27} - 20 q^{33} - 22 q^{35} + 12 q^{41} - 2 q^{43} + 8 q^{49} - 20 q^{51} - 4 q^{57} - 12 q^{59} - 16 q^{67} + 18 q^{73} - 24 q^{75} - 22 q^{81} + 8 q^{83} + 8 q^{89} - 24 q^{97} + 10 q^{99}+O(q^{100}) $ 2 * q - 4 * q^3 + 2 * q^9 + 10 * q^11 + 10 * q^17 + 2 * q^19 + 12 * q^25 + 8 * q^27 - 20 * q^33 - 22 * q^35 + 12 * q^41 - 2 * q^43 + 8 * q^49 - 20 * q^51 - 4 * q^57 - 12 * q^59 - 16 * q^67 + 18 * q^73 - 24 * q^75 - 22 * q^81 + 8 * q^83 + 8 * q^89 - 24 * q^97 + 10 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	0	0
$5$	3.31662	1.48324	0.741620	−	0.670820i	$-0.234058\pi$
$5$	3.31662	1.48324	0.741620	+	0.670820i	$0.234058\pi$
$6$	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
$9$	1.00000	0.333333
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−6.63325	−1.71270
$16$	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
$21$	6.63325	1.44749
$22$	0	0
$23$	−6.63325	−1.38313	−0.691564	−	0.722315i	$-0.743078\pi$
$23$	−6.63325	−1.38313	−0.691564	+	0.722315i	$0.743078\pi$
$24$	0	0
$25$	6.00000	1.20000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	6.63325	1.23176	0.615882	−	0.787839i	$-0.288800\pi$
$29$	6.63325	1.23176	0.615882	+	0.787839i	$0.288800\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−10.0000	−1.74078
$34$	0	0
$35$	−11.0000	−1.85934
$36$	0	0
$37$	−6.63325	−1.09050	−0.545250	−	0.838274i	$-0.683565\pi$
$37$	−6.63325	−1.09050	−0.545250	+	0.838274i	$0.683565\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	0	0
$45$	3.31662	0.494413
$46$	0	0
$47$	9.94987	1.45134	0.725669	−	0.688044i	$-0.241530\pi$
$47$	9.94987	1.45134	0.725669	+	0.688044i	$0.241530\pi$
$48$	0	0
$49$	4.00000	0.571429
$50$	0	0
$51$	−10.0000	−1.40028
$52$	0	0
$53$	−13.2665	−1.82229	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	−13.2665	−1.82229	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	16.5831	2.23607
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	9.94987	1.27395	0.636975	−	0.770884i	$-0.280185\pi$
$61$	9.94987	1.27395	0.636975	+	0.770884i	$0.280185\pi$
$62$	0	0
$63$	−3.31662	−0.417855
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	13.2665	1.59710
$70$	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
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	0	0
$75$	−12.0000	−1.38564
$76$	0	0
$77$	−16.5831	−1.88982
$78$	0	0
$79$	−13.2665	−1.49260	−0.746299	−	0.665611i	$-0.768171\pi$
$79$	−13.2665	−1.49260	−0.746299	+	0.665611i	$0.768171\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	16.5831	1.79869
$86$	0	0
$87$	−13.2665	−1.42232
$88$	0	0
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.31662	0.340279
$96$	0	0
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	0	0
$99$	5.00000	0.502519
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	6.63325	0.653594	0.326797	−	0.945095i	$-0.394031\pi$
$103$	6.63325	0.653594	0.326797	+	0.945095i	$0.394031\pi$
$104$	0	0
$105$	22.0000	2.14698
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	13.2665	1.27070	0.635350	−	0.772224i	$-0.280856\pi$
$109$	13.2665	1.27070	0.635350	+	0.772224i	$0.280856\pi$
$110$	0	0
$111$	13.2665	1.25920
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	−22.0000	−2.05151
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−16.5831	−1.52017
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	−12.0000	−1.08200
$124$	0	0
$125$	3.31662	0.296648
$126$	0	0
$127$	6.63325	0.588606	0.294303	−	0.955712i	$-0.404913\pi$
$127$	6.63325	0.588606	0.294303	+	0.955712i	$0.404913\pi$
$128$	0	0
$129$	2.00000	0.176090
$130$	0	0
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	0	0
$133$	−3.31662	−0.287588
$134$	0	0
$135$	13.2665	1.14180
$136$	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
$139$	11.0000	0.933008	0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000	0.933008	0.466504	+	0.884519i	$0.345513\pi$
$140$	0	0
$141$	−19.8997	−1.67586
$142$	0	0
$143$	0	0
$144$	0	0
$145$	22.0000	1.82700
$146$	0	0
$147$	−8.00000	−0.659829
$148$	0	0
$149$	3.31662	0.271708	0.135854	−	0.990729i	$-0.456622\pi$
$149$	3.31662	0.271708	0.135854	+	0.990729i	$0.456622\pi$
$150$	0	0
$151$	−6.63325	−0.539806	−0.269903	−	0.962887i	$-0.586992\pi$
$151$	−6.63325	−0.539806	−0.269903	+	0.962887i	$0.586992\pi$
$152$	0	0
$153$	5.00000	0.404226
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	26.5330	2.10420
$160$	0	0
$161$	22.0000	1.73384
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	0	0
$165$	−33.1662	−2.58199
$166$	0	0
$167$	6.63325	0.513296	0.256648	−	0.966505i	$-0.417382\pi$
$167$	6.63325	0.513296	0.256648	+	0.966505i	$0.417382\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−19.8997	−1.51295	−0.756475	−	0.654023i	$-0.773080\pi$
$173$	−19.8997	−1.51295	−0.756475	+	0.654023i	$0.773080\pi$
$174$	0	0
$175$	−19.8997	−1.50428
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	0	0
$181$	19.8997	1.47914	0.739568	−	0.673081i	$-0.235030\pi$
$181$	19.8997	1.47914	0.739568	+	0.673081i	$0.235030\pi$
$182$	0	0
$183$	−19.8997	−1.47103
$184$	0	0
$185$	−22.0000	−1.61747
$186$	0	0
$187$	25.0000	1.82818
$188$	0	0
$189$	−13.2665	−0.964996
$190$	0	0
$191$	−16.5831	−1.19991	−0.599956	−	0.800033i	$-0.704815\pi$
$191$	−16.5831	−1.19991	−0.599956	+	0.800033i	$0.704815\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−26.5330	−1.89040	−0.945199	−	0.326495i	$-0.894132\pi$
$197$	−26.5330	−1.89040	−0.945199	+	0.326495i	$0.894132\pi$
$198$	0	0
$199$	16.5831	1.17555	0.587773	−	0.809026i	$-0.300005\pi$
$199$	16.5831	1.17555	0.587773	+	0.809026i	$0.300005\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	0	0
$203$	−22.0000	−1.54410
$204$	0	0
$205$	19.8997	1.38986
$206$	0	0
$207$	−6.63325	−0.461043
$208$	0	0
$209$	5.00000	0.345857
$210$	0	0
$211$	14.0000	0.963800	0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000	0.963800	0.481900	+	0.876226i	$0.339947\pi$
$212$	0	0
$213$	−13.2665	−0.909006
$214$	0	0
$215$	−3.31662	−0.226192
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−18.0000	−1.21633
$220$	0	0
$221$	0	0
$222$	0	0
$223$	19.8997	1.33259	0.666293	−	0.745690i	$-0.267880\pi$
$223$	19.8997	1.33259	0.666293	+	0.745690i	$0.267880\pi$
$224$	0	0
$225$	6.00000	0.400000
$226$	0	0
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	0	0
$229$	−23.2164	−1.53418	−0.767091	−	0.641539i	$-0.778296\pi$
$229$	−23.2164	−1.53418	−0.767091	+	0.641539i	$0.778296\pi$
$230$	0	0
$231$	33.1662	2.18218
$232$	0	0
$233$	−3.00000	−0.196537	−0.0982683	−	0.995160i	$-0.531330\pi$
$233$	−3.00000	−0.196537	−0.0982683	+	0.995160i	$0.531330\pi$
$234$	0	0
$235$	33.0000	2.15268
$236$	0	0
$237$	26.5330	1.72350
$238$	0	0
$239$	3.31662	0.214535	0.107267	−	0.994230i	$-0.465790\pi$
$239$	3.31662	0.214535	0.107267	+	0.994230i	$0.465790\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	0	0
$245$	13.2665	0.847566
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	5.00000	0.315597	0.157799	−	0.987471i	$-0.449560\pi$
$251$	5.00000	0.315597	0.157799	+	0.987471i	$0.449560\pi$
$252$	0	0
$253$	−33.1662	−2.08514
$254$	0	0
$255$	−33.1662	−2.07695
$256$	0	0
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	0	0
$259$	22.0000	1.36701
$260$	0	0
$261$	6.63325	0.410588
$262$	0	0
$263$	3.31662	0.204512	0.102256	−	0.994758i	$-0.467394\pi$
$263$	3.31662	0.204512	0.102256	+	0.994758i	$0.467394\pi$
$264$	0	0
$265$	−44.0000	−2.70290
$266$	0	0
$267$	−8.00000	−0.489592
$268$	0	0
$269$	−13.2665	−0.808873	−0.404436	−	0.914566i	$-0.632532\pi$
$269$	−13.2665	−0.808873	−0.404436	+	0.914566i	$0.632532\pi$
$270$	0	0
$271$	19.8997	1.20882	0.604412	−	0.796672i	$-0.293408\pi$
$271$	19.8997	1.20882	0.604412	+	0.796672i	$0.293408\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	30.0000	1.80907
$276$	0	0
$277$	−3.31662	−0.199277	−0.0996383	−	0.995024i	$-0.531769\pi$
$277$	−3.31662	−0.199277	−0.0996383	+	0.995024i	$0.531769\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	11.0000	0.653882	0.326941	−	0.945045i	$-0.393982\pi$
$283$	11.0000	0.653882	0.326941	+	0.945045i	$0.393982\pi$
$284$	0	0
$285$	−6.63325	−0.392920
$286$	0	0
$287$	−19.8997	−1.17465
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	24.0000	1.40690
$292$	0	0
$293$	26.5330	1.55007	0.775037	−	0.631916i	$-0.217731\pi$
$293$	26.5330	1.55007	0.775037	+	0.631916i	$0.217731\pi$
$294$	0	0
$295$	−19.8997	−1.15861
$296$	0	0
$297$	20.0000	1.16052
$298$	0	0
$299$	0	0
$300$	0	0
$301$	3.31662	0.191167
$302$	0	0
$303$	0	0
$304$	0	0
$305$	33.0000	1.88957
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	−13.2665	−0.754705
$310$	0	0
$311$	16.5831	0.940343	0.470171	−	0.882575i	$-0.344192\pi$
$311$	16.5831	0.940343	0.470171	+	0.882575i	$0.344192\pi$
$312$	0	0
$313$	34.0000	1.92179	0.960897	−	0.276907i	$-0.0893093\pi$
$313$	34.0000	1.92179	0.960897	+	0.276907i	$0.0893093\pi$
$314$	0	0
$315$	−11.0000	−0.619780
$316$	0	0
$317$	6.63325	0.372560	0.186280	−	0.982497i	$-0.440357\pi$
$317$	6.63325	0.372560	0.186280	+	0.982497i	$0.440357\pi$
$318$	0	0
$319$	33.1662	1.85695
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	5.00000	0.278207
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−26.5330	−1.46728
$328$	0	0
$329$	−33.0000	−1.81935
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	−6.63325	−0.363500
$334$	0	0
$335$	−26.5330	−1.44965
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	0	0
$339$	−36.0000	−1.95525
$340$	0	0
$341$	0	0
$342$	0	0
$343$	9.94987	0.537243
$344$	0	0
$345$	44.0000	2.36888
$346$	0	0
$347$	27.0000	1.44944	0.724718	−	0.689046i	$-0.241970\pi$
$347$	27.0000	1.44944	0.724718	+	0.689046i	$0.241970\pi$
$348$	0	0
$349$	9.94987	0.532605	0.266302	−	0.963890i	$-0.414198\pi$
$349$	9.94987	0.532605	0.266302	+	0.963890i	$0.414198\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	22.0000	1.16764
$356$	0	0
$357$	33.1662	1.75534
$358$	0	0
$359$	−23.2164	−1.22531	−0.612657	−	0.790349i	$-0.709899\pi$
$359$	−23.2164	−1.22531	−0.612657	+	0.790349i	$0.709899\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−28.0000	−1.46962
$364$	0	0
$365$	29.8496	1.56240
$366$	0	0
$367$	6.63325	0.346253	0.173126	−	0.984900i	$-0.444613\pi$
$367$	6.63325	0.346253	0.173126	+	0.984900i	$0.444613\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	44.0000	2.28437
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	−6.63325	−0.342540
$376$	0	0
$377$	0	0
$378$	0	0
$379$	6.00000	0.308199	0.154100	−	0.988055i	$-0.450752\pi$
$379$	6.00000	0.308199	0.154100	+	0.988055i	$0.450752\pi$
$380$	0	0
$381$	−13.2665	−0.679663
$382$	0	0
$383$	13.2665	0.677886	0.338943	−	0.940807i	$-0.389931\pi$
$383$	13.2665	0.677886	0.338943	+	0.940807i	$0.389931\pi$
$384$	0	0
$385$	−55.0000	−2.80306
$386$	0	0
$387$	−1.00000	−0.0508329
$388$	0	0
$389$	16.5831	0.840798	0.420399	−	0.907339i	$-0.361890\pi$
$389$	16.5831	0.840798	0.420399	+	0.907339i	$0.361890\pi$
$390$	0	0
$391$	−33.1662	−1.67729
$392$	0	0
$393$	−30.0000	−1.51330
$394$	0	0
$395$	−44.0000	−2.21388
$396$	0	0
$397$	−9.94987	−0.499370	−0.249685	−	0.968327i	$-0.580327\pi$
$397$	−9.94987	−0.499370	−0.249685	+	0.968327i	$0.580327\pi$
$398$	0	0
$399$	6.63325	0.332078
$400$	0	0
$401$	16.0000	0.799002	0.399501	−	0.916733i	$-0.369183\pi$
$401$	16.0000	0.799002	0.399501	+	0.916733i	$0.369183\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−36.4829	−1.81285
$406$	0	0
$407$	−33.1662	−1.64399
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	10.0000	0.493264
$412$	0	0
$413$	19.8997	0.979203
$414$	0	0
$415$	13.2665	0.651227
$416$	0	0
$417$	−22.0000	−1.07734
$418$	0	0
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	13.2665	0.646570	0.323285	−	0.946302i	$-0.395213\pi$
$421$	13.2665	0.646570	0.323285	+	0.946302i	$0.395213\pi$
$422$	0	0
$423$	9.94987	0.483779
$424$	0	0
$425$	30.0000	1.45521
$426$	0	0
$427$	−33.0000	−1.59698
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−26.5330	−1.27805	−0.639025	−	0.769186i	$-0.720662\pi$
$431$	−26.5330	−1.27805	−0.639025	+	0.769186i	$0.720662\pi$
$432$	0	0
$433$	−38.0000	−1.82616	−0.913082	−	0.407777i	$-0.866304\pi$
$433$	−38.0000	−1.82616	−0.913082	+	0.407777i	$0.866304\pi$
$434$	0	0
$435$	−44.0000	−2.10964
$436$	0	0
$437$	−6.63325	−0.317311
$438$	0	0
$439$	19.8997	0.949763	0.474882	−	0.880050i	$-0.342491\pi$
$439$	19.8997	0.949763	0.474882	+	0.880050i	$0.342491\pi$
$440$	0	0
$441$	4.00000	0.190476
$442$	0	0
$443$	−13.0000	−0.617649	−0.308824	−	0.951119i	$-0.599936\pi$
$443$	−13.0000	−0.617649	−0.308824	+	0.951119i	$0.599936\pi$
$444$	0	0
$445$	13.2665	0.628892
$446$	0	0
$447$	−6.63325	−0.313742
$448$	0	0
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	30.0000	1.41264
$452$	0	0
$453$	13.2665	0.623315
$454$	0	0
$455$	0	0
$456$	0	0
$457$	29.0000	1.35656	0.678281	−	0.734802i	$-0.262725\pi$
$457$	29.0000	1.35656	0.678281	+	0.734802i	$0.262725\pi$
$458$	0	0
$459$	20.0000	0.933520
$460$	0	0
$461$	−16.5831	−0.772353	−0.386177	−	0.922425i	$-0.626204\pi$
$461$	−16.5831	−0.772353	−0.386177	+	0.922425i	$0.626204\pi$
$462$	0	0
$463$	−23.2164	−1.07896	−0.539478	−	0.842000i	$-0.681378\pi$
$463$	−23.2164	−1.07896	−0.539478	+	0.842000i	$0.681378\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.00000	−0.138823	−0.0694117	−	0.997588i	$-0.522112\pi$
$467$	−3.00000	−0.138823	−0.0694117	+	0.997588i	$0.522112\pi$
$468$	0	0
$469$	26.5330	1.22518
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−5.00000	−0.229900
$474$	0	0
$475$	6.00000	0.275299
$476$	0	0
$477$	−13.2665	−0.607431
$478$	0	0
$479$	6.63325	0.303081	0.151540	−	0.988451i	$-0.451577\pi$
$479$	6.63325	0.303081	0.151540	+	0.988451i	$0.451577\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−44.0000	−2.00207
$484$	0	0
$485$	−39.7995	−1.80720
$486$	0	0
$487$	33.1662	1.50291	0.751453	−	0.659787i	$-0.229353\pi$
$487$	33.1662	1.50291	0.751453	+	0.659787i	$0.229353\pi$
$488$	0	0
$489$	−40.0000	−1.80886
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	33.1662	1.49373
$494$	0	0
$495$	16.5831	0.745356
$496$	0	0
$497$	−22.0000	−0.986835
$498$	0	0
$499$	3.00000	0.134298	0.0671492	−	0.997743i	$-0.478610\pi$
$499$	3.00000	0.134298	0.0671492	+	0.997743i	$0.478610\pi$
$500$	0	0
$501$	−13.2665	−0.592703
$502$	0	0
$503$	19.8997	0.887286	0.443643	−	0.896204i	$-0.353686\pi$
$503$	19.8997	0.887286	0.443643	+	0.896204i	$0.353686\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	26.0000	1.15470
$508$	0	0
$509$	−26.5330	−1.17605	−0.588027	−	0.808841i	$-0.700095\pi$
$509$	−26.5330	−1.17605	−0.588027	+	0.808841i	$0.700095\pi$
$510$	0	0
$511$	−29.8496	−1.32047
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	22.0000	0.969436
$516$	0	0
$517$	49.7494	2.18797
$518$	0	0
$519$	39.7995	1.74700
$520$	0	0
$521$	−16.0000	−0.700973	−0.350486	−	0.936568i	$-0.613984\pi$
$521$	−16.0000	−0.700973	−0.350486	+	0.936568i	$0.613984\pi$
$522$	0	0
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	0	0
$525$	39.7995	1.73699
$526$	0	0
$527$	0	0
$528$	0	0
$529$	21.0000	0.913043
$530$	0	0
$531$	−6.00000	−0.260378
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−19.8997	−0.860341
$536$	0	0
$537$	36.0000	1.55351
$538$	0	0
$539$	20.0000	0.861461
$540$	0	0
$541$	9.94987	0.427779	0.213889	−	0.976858i	$-0.431387\pi$
$541$	9.94987	0.427779	0.213889	+	0.976858i	$0.431387\pi$
$542$	0	0
$543$	−39.7995	−1.70796
$544$	0	0
$545$	44.0000	1.88475
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	9.94987	0.424650
$550$	0	0
$551$	6.63325	0.282586
$552$	0	0
$553$	44.0000	1.87107
$554$	0	0
$555$	44.0000	1.86770
$556$	0	0
$557$	3.31662	0.140530	0.0702650	−	0.997528i	$-0.477616\pi$
$557$	3.31662	0.140530	0.0702650	+	0.997528i	$0.477616\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−50.0000	−2.11100
$562$	0	0
$563$	−14.0000	−0.590030	−0.295015	−	0.955493i	$-0.595325\pi$
$563$	−14.0000	−0.590030	−0.295015	+	0.955493i	$0.595325\pi$
$564$	0	0
$565$	59.6992	2.51157
$566$	0	0
$567$	36.4829	1.53214
$568$	0	0
$569$	−20.0000	−0.838444	−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000	−0.838444	−0.419222	+	0.907884i	$0.637697\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	0	0
$573$	33.1662	1.38554
$574$	0	0
$575$	−39.7995	−1.65975
$576$	0	0
$577$	−43.0000	−1.79011	−0.895057	−	0.445952i	$-0.852865\pi$
$577$	−43.0000	−1.79011	−0.895057	+	0.445952i	$0.852865\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−13.2665	−0.550387
$582$	0	0
$583$	−66.3325	−2.74721
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.00000	0.206372	0.103186	−	0.994662i	$-0.467096\pi$
$587$	5.00000	0.206372	0.103186	+	0.994662i	$0.467096\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	53.0660	2.18284
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	−55.0000	−2.25478
$596$	0	0
$597$	−33.1662	−1.35740
$598$	0	0
$599$	13.2665	0.542054	0.271027	−	0.962572i	$-0.412637\pi$
$599$	13.2665	0.542054	0.271027	+	0.962572i	$0.412637\pi$
$600$	0	0
$601$	−42.0000	−1.71322	−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000	−1.71322	−0.856608	+	0.515968i	$0.827432\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	0	0
$605$	46.4327	1.88776
$606$	0	0
$607$	26.5330	1.07694	0.538471	−	0.842644i	$-0.319002\pi$
$607$	26.5330	1.07694	0.538471	+	0.842644i	$0.319002\pi$
$608$	0	0
$609$	44.0000	1.78297
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−3.31662	−0.133957	−0.0669786	−	0.997754i	$-0.521336\pi$
$613$	−3.31662	−0.133957	−0.0669786	+	0.997754i	$0.521336\pi$
$614$	0	0
$615$	−39.7995	−1.60487
$616$	0	0
$617$	41.0000	1.65060	0.825299	−	0.564696i	$-0.191007\pi$
$617$	41.0000	1.65060	0.825299	+	0.564696i	$0.191007\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	−26.5330	−1.06473
$622$	0	0
$623$	−13.2665	−0.531511
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−10.0000	−0.399362
$628$	0	0
$629$	−33.1662	−1.32242
$630$	0	0
$631$	−9.94987	−0.396098	−0.198049	−	0.980192i	$-0.563461\pi$
$631$	−9.94987	−0.396098	−0.198049	+	0.980192i	$0.563461\pi$
$632$	0	0
$633$	−28.0000	−1.11290
$634$	0	0
$635$	22.0000	0.873043
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6.63325	0.262407
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	5.00000	0.197181	0.0985904	−	0.995128i	$-0.468567\pi$
$643$	5.00000	0.197181	0.0985904	+	0.995128i	$0.468567\pi$
$644$	0	0
$645$	6.63325	0.261184
$646$	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
$649$	−30.0000	−1.17760
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−36.4829	−1.42769	−0.713843	−	0.700306i	$-0.753047\pi$
$653$	−36.4829	−1.42769	−0.713843	+	0.700306i	$0.753047\pi$
$654$	0	0
$655$	49.7494	1.94387
$656$	0	0
$657$	9.00000	0.351123
$658$	0	0
$659$	26.0000	1.01282	0.506408	−	0.862294i	$-0.330973\pi$
$659$	26.0000	1.01282	0.506408	+	0.862294i	$0.330973\pi$
$660$	0	0
$661$	−39.7995	−1.54802	−0.774011	−	0.633173i	$-0.781752\pi$
$661$	−39.7995	−1.54802	−0.774011	+	0.633173i	$0.781752\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−11.0000	−0.426562
$666$	0	0
$667$	−44.0000	−1.70369
$668$	0	0
$669$	−39.7995	−1.53874
$670$	0	0
$671$	49.7494	1.92055
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	0	0
$675$	24.0000	0.923760
$676$	0	0
$677$	19.8997	0.764809	0.382405	−	0.923995i	$-0.375096\pi$
$677$	19.8997	0.764809	0.382405	+	0.923995i	$0.375096\pi$
$678$	0	0
$679$	39.7995	1.52736
$680$	0	0
$681$	−16.0000	−0.613121
$682$	0	0
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	0	0
$685$	−16.5831	−0.633609
$686$	0	0
$687$	46.4327	1.77152
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−31.0000	−1.17930	−0.589648	−	0.807661i	$-0.700733\pi$
$691$	−31.0000	−1.17930	−0.589648	+	0.807661i	$0.700733\pi$
$692$	0	0
$693$	−16.5831	−0.629941
$694$	0	0
$695$	36.4829	1.38387
$696$	0	0
$697$	30.0000	1.13633
$698$	0	0
$699$	6.00000	0.226941
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	−6.63325	−0.250178
$704$	0	0
$705$	−66.0000	−2.48570
$706$	0	0
$707$	0	0
$708$	0	0
$709$	13.2665	0.498234	0.249117	−	0.968473i	$-0.419860\pi$
$709$	13.2665	0.498234	0.249117	+	0.968473i	$0.419860\pi$
$710$	0	0
$711$	−13.2665	−0.497533
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6.63325	−0.247723
$718$	0	0
$719$	−3.31662	−0.123689	−0.0618446	−	0.998086i	$-0.519698\pi$
$719$	−3.31662	−0.123689	−0.0618446	+	0.998086i	$0.519698\pi$
$720$	0	0
$721$	−22.0000	−0.819323
$722$	0	0
$723$	−52.0000	−1.93390
$724$	0	0
$725$	39.7995	1.47812
$726$	0	0
$727$	16.5831	0.615034	0.307517	−	0.951543i	$-0.400502\pi$
$727$	16.5831	0.615034	0.307517	+	0.951543i	$0.400502\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−5.00000	−0.184932
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	−26.5330	−0.978684
$736$	0	0
$737$	−40.0000	−1.47342
$738$	0	0
$739$	−45.0000	−1.65535	−0.827676	−	0.561206i	$-0.810337\pi$
$739$	−45.0000	−1.65535	−0.827676	+	0.561206i	$0.810337\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−53.0660	−1.94680	−0.973401	−	0.229107i	$-0.926420\pi$
$743$	−53.0660	−1.94680	−0.973401	+	0.229107i	$0.926420\pi$
$744$	0	0
$745$	11.0000	0.403009
$746$	0	0
$747$	4.00000	0.146352
$748$	0	0
$749$	19.8997	0.727121
$750$	0	0
$751$	−39.7995	−1.45230	−0.726152	−	0.687534i	$-0.758693\pi$
$751$	−39.7995	−1.45230	−0.726152	+	0.687534i	$0.758693\pi$
$752$	0	0
$753$	−10.0000	−0.364420
$754$	0	0
$755$	−22.0000	−0.800662
$756$	0	0
$757$	9.94987	0.361634	0.180817	−	0.983517i	$-0.442126\pi$
$757$	9.94987	0.361634	0.180817	+	0.983517i	$0.442126\pi$
$758$	0	0
$759$	66.3325	2.40772
$760$	0	0
$761$	−1.00000	−0.0362500	−0.0181250	−	0.999836i	$-0.505770\pi$
$761$	−1.00000	−0.0362500	−0.0181250	+	0.999836i	$0.505770\pi$
$762$	0	0
$763$	−44.0000	−1.59291
$764$	0	0
$765$	16.5831	0.599564
$766$	0	0
$767$	0	0
$768$	0	0
$769$	33.0000	1.19001	0.595005	−	0.803722i	$-0.297150\pi$
$769$	33.0000	1.19001	0.595005	+	0.803722i	$0.297150\pi$
$770$	0	0
$771$	16.0000	0.576226
$772$	0	0
$773$	6.63325	0.238581	0.119291	−	0.992859i	$-0.461938\pi$
$773$	6.63325	0.238581	0.119291	+	0.992859i	$0.461938\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−44.0000	−1.57849
$778$	0	0
$779$	6.00000	0.214972
$780$	0	0
$781$	33.1662	1.18678
$782$	0	0
$783$	26.5330	0.948212
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	0	0
$789$	−6.63325	−0.236150
$790$	0	0
$791$	−59.6992	−2.12266
$792$	0	0
$793$	0	0
$794$	0	0
$795$	88.0000	3.12104
$796$	0	0
$797$	−26.5330	−0.939847	−0.469924	−	0.882707i	$-0.655719\pi$
$797$	−26.5330	−0.939847	−0.469924	+	0.882707i	$0.655719\pi$
$798$	0	0
$799$	49.7494	1.76001
$800$	0	0
$801$	4.00000	0.141333
$802$	0	0
$803$	45.0000	1.58802
$804$	0	0
$805$	72.9657	2.57170
$806$	0	0
$807$	26.5330	0.934006
$808$	0	0
$809$	39.0000	1.37117	0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000	1.37117	0.685583	+	0.727994i	$0.259547\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	−39.7995	−1.39583
$814$	0	0
$815$	66.3325	2.32353
$816$	0	0
$817$	−1.00000	−0.0349856
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−9.94987	−0.347253	−0.173627	−	0.984812i	$-0.555549\pi$
$821$	−9.94987	−0.347253	−0.173627	+	0.984812i	$0.555549\pi$
$822$	0	0
$823$	29.8496	1.04049	0.520246	−	0.854016i	$-0.325840\pi$
$823$	29.8496	1.04049	0.520246	+	0.854016i	$0.325840\pi$
$824$	0	0
$825$	−60.0000	−2.08893
$826$	0	0
$827$	−32.0000	−1.11275	−0.556375	−	0.830932i	$-0.687808\pi$
$827$	−32.0000	−1.11275	−0.556375	+	0.830932i	$0.687808\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	6.63325	0.230105
$832$	0	0
$833$	20.0000	0.692959
$834$	0	0
$835$	22.0000	0.761341
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−46.4327	−1.60304	−0.801518	−	0.597970i	$-0.795974\pi$
$839$	−46.4327	−1.60304	−0.801518	+	0.597970i	$0.795974\pi$
$840$	0	0
$841$	15.0000	0.517241
$842$	0	0
$843$	−52.0000	−1.79098
$844$	0	0
$845$	−43.1161	−1.48324
$846$	0	0
$847$	−46.4327	−1.59545
$848$	0	0
$849$	−22.0000	−0.755038
$850$	0	0
$851$	44.0000	1.50830
$852$	0	0
$853$	−53.0660	−1.81695	−0.908473	−	0.417945i	$-0.862751\pi$
$853$	−53.0660	−1.81695	−0.908473	+	0.417945i	$0.862751\pi$
$854$	0	0
$855$	3.31662	0.113426
$856$	0	0
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	0	0
$859$	−41.0000	−1.39890	−0.699451	−	0.714681i	$-0.746572\pi$
$859$	−41.0000	−1.39890	−0.699451	+	0.714681i	$0.746572\pi$
$860$	0	0
$861$	39.7995	1.35636
$862$	0	0
$863$	−33.1662	−1.12899	−0.564496	−	0.825436i	$-0.690929\pi$
$863$	−33.1662	−1.12899	−0.564496	+	0.825436i	$0.690929\pi$
$864$	0	0
$865$	−66.0000	−2.24407
$866$	0	0
$867$	−16.0000	−0.543388
$868$	0	0
$869$	−66.3325	−2.25018
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−12.0000	−0.406138
$874$	0	0
$875$	−11.0000	−0.371868
$876$	0	0
$877$	6.63325	0.223989	0.111994	−	0.993709i	$-0.464276\pi$
$877$	6.63325	0.223989	0.111994	+	0.993709i	$0.464276\pi$
$878$	0	0
$879$	−53.0660	−1.78987
$880$	0	0
$881$	51.0000	1.71823	0.859117	−	0.511780i	$-0.171014\pi$
$881$	51.0000	1.71823	0.859117	+	0.511780i	$0.171014\pi$
$882$	0	0
$883$	31.0000	1.04323	0.521617	−	0.853180i	$-0.325329\pi$
$883$	31.0000	1.04323	0.521617	+	0.853180i	$0.325329\pi$
$884$	0	0
$885$	39.7995	1.33785
$886$	0	0
$887$	−19.8997	−0.668168	−0.334084	−	0.942543i	$-0.608427\pi$
$887$	−19.8997	−0.668168	−0.334084	+	0.942543i	$0.608427\pi$
$888$	0	0
$889$	−22.0000	−0.737856
$890$	0	0
$891$	−55.0000	−1.84257
$892$	0	0
$893$	9.94987	0.332960
$894$	0	0
$895$	−59.6992	−1.99553
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−66.3325	−2.20986
$902$	0	0
$903$	−6.63325	−0.220741
$904$	0	0
$905$	66.0000	2.19391
$906$	0	0
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−19.8997	−0.659308	−0.329654	−	0.944102i	$-0.606932\pi$
$911$	−19.8997	−0.659308	−0.329654	+	0.944102i	$0.606932\pi$
$912$	0	0
$913$	20.0000	0.661903
$914$	0	0
$915$	−66.0000	−2.18189
$916$	0	0
$917$	−49.7494	−1.64287
$918$	0	0
$919$	33.1662	1.09405	0.547027	−	0.837115i	$-0.315760\pi$
$919$	33.1662	1.09405	0.547027	+	0.837115i	$0.315760\pi$
$920$	0	0
$921$	24.0000	0.790827
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−39.7995	−1.30860
$926$	0	0
$927$	6.63325	0.217865
$928$	0	0
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	0	0
$933$	−33.1662	−1.08581
$934$	0	0
$935$	82.9156	2.71163
$936$	0	0
$937$	−1.00000	−0.0326686	−0.0163343	−	0.999867i	$-0.505200\pi$
$937$	−1.00000	−0.0326686	−0.0163343	+	0.999867i	$0.505200\pi$
$938$	0	0
$939$	−68.0000	−2.21910
$940$	0	0
$941$	−46.4327	−1.51366	−0.756832	−	0.653609i	$-0.773254\pi$
$941$	−46.4327	−1.51366	−0.756832	+	0.653609i	$0.773254\pi$
$942$	0	0
$943$	−39.7995	−1.29605
$944$	0	0
$945$	−44.0000	−1.43132
$946$	0	0
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−13.2665	−0.430196
$952$	0	0
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	0	0
$955$	−55.0000	−1.77976
$956$	0	0
$957$	−66.3325	−2.14423
$958$	0	0
$959$	16.5831	0.535497
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−6.00000	−0.193347
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−33.1662	−1.06655	−0.533277	−	0.845940i	$-0.679040\pi$
$967$	−33.1662	−1.06655	−0.533277	+	0.845940i	$0.679040\pi$
$968$	0	0
$969$	−10.0000	−0.321246
$970$	0	0
$971$	−32.0000	−1.02693	−0.513464	−	0.858111i	$-0.671638\pi$
$971$	−32.0000	−1.02693	−0.513464	+	0.858111i	$0.671638\pi$
$972$	0	0
$973$	−36.4829	−1.16959
$974$	0	0
$975$	0	0
$976$	0	0
$977$	16.0000	0.511885	0.255943	−	0.966692i	$-0.417614\pi$
$977$	16.0000	0.511885	0.255943	+	0.966692i	$0.417614\pi$
$978$	0	0
$979$	20.0000	0.639203
$980$	0	0
$981$	13.2665	0.423567
$982$	0	0
$983$	13.2665	0.423136	0.211568	−	0.977363i	$-0.432143\pi$
$983$	13.2665	0.423136	0.211568	+	0.977363i	$0.432143\pi$
$984$	0	0
$985$	−88.0000	−2.80391
$986$	0	0
$987$	66.0000	2.10080
$988$	0	0
$989$	6.63325	0.210925
$990$	0	0
$991$	−33.1662	−1.05356	−0.526780	−	0.850001i	$-0.676601\pi$
$991$	−33.1662	−1.05356	−0.526780	+	0.850001i	$0.676601\pi$
$992$	0	0
$993$	−8.00000	−0.253872
$994$	0	0
$995$	55.0000	1.74362
$996$	0	0
$997$	−43.1161	−1.36550	−0.682751	−	0.730651i	$-0.739216\pi$
$997$	−43.1161	−1.36550	−0.682751	+	0.730651i	$0.739216\pi$
$998$	0	0
$999$	−26.5330	−0.839467

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.r.1.2		2
4.3	odd	2		4864.2.a.y.1.2		2
8.3	odd	2	inner	4864.2.a.r.1.1		2
8.5	even	2		4864.2.a.y.1.1		2
16.3	odd	4		1216.2.c.f.609.3	yes	4
16.5	even	4		1216.2.c.f.609.4	yes	4
16.11	odd	4		1216.2.c.f.609.2	yes	4
16.13	even	4		1216.2.c.f.609.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.c.f.609.1	✓	4	16.13	even	4
1216.2.c.f.609.2	yes	4	16.11	odd	4
1216.2.c.f.609.3	yes	4	16.3	odd	4
1216.2.c.f.609.4	yes	4	16.5	even	4
4864.2.a.r.1.1		2	8.3	odd	2	inner
4864.2.a.r.1.2		2	1.1	even	1	trivial
4864.2.a.y.1.1		2	8.5	even	2
4864.2.a.y.1.2		2	4.3	odd	2

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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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{3} +3.31662 q^{5} -3.31662 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{3} +3.31662 q^{5} -3.31662 q^{7} +1.00000 q^{9} +5.00000 q^{11} -6.63325 q^{15} +5.00000 q^{17} +1.00000 q^{19} +6.63325 q^{21} -6.63325 q^{23} +6.00000 q^{25} +4.00000 q^{27} +6.63325 q^{29} -10.0000 q^{33} -11.0000 q^{35} -6.63325 q^{37} +6.00000 q^{41} -1.00000 q^{43} +3.31662 q^{45} +9.94987 q^{47} +4.00000 q^{49} -10.0000 q^{51} -13.2665 q^{53} +16.5831 q^{55} -2.00000 q^{57} -6.00000 q^{59} +9.94987 q^{61} -3.31662 q^{63} -8.00000 q^{67} +13.2665 q^{69} +6.63325 q^{71} +9.00000 q^{73} -12.0000 q^{75} -16.5831 q^{77} -13.2665 q^{79} -11.0000 q^{81} +4.00000 q^{83} +16.5831 q^{85} -13.2665 q^{87} +4.00000 q^{89} +3.31662 q^{95} -12.0000 q^{97} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^3 + 2 * q^9	\( 2 q - 4 q^{3} + 2 q^{9} + 10 q^{11} + 10 q^{17} + 2 q^{19} + 12 q^{25} + 8 q^{27} - 20 q^{33} - 22 q^{35} + 12 q^{41} - 2 q^{43} + 8 q^{49} - 20 q^{51} - 4 q^{57} - 12 q^{59} - 16 q^{67} + 18 q^{73} - 24 q^{75} - 22 q^{81} + 8 q^{83} + 8 q^{89} - 24 q^{97} + 10 q^{99}+O(q^{100}) \) 2 * q - 4 * q^3 + 2 * q^9 + 10 * q^11 + 10 * q^17 + 2 * q^19 + 12 * q^25 + 8 * q^27 - 20 * q^33 - 22 * q^35 + 12 * q^41 - 2 * q^43 + 8 * q^49 - 20 * q^51 - 4 * q^57 - 12 * q^59 - 16 * q^67 + 18 * q^73 - 24 * q^75 - 22 * q^81 + 8 * q^83 + 8 * q^89 - 24 * q^97 + 10 * q^99

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