LMFDB - Embedded newform 4864.2.a.w.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
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			1.2
Root			$-1.73205$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.46410 q^{5} -1.73205 q^{7} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.46410 q^{5} -1.73205 q^{7} -2.00000 q^{9} -5.19615 q^{13} +3.46410 q^{15} -3.00000 q^{17} -1.00000 q^{19} -1.73205 q^{21} +1.73205 q^{23} +7.00000 q^{25} -5.00000 q^{27} -1.73205 q^{29} -3.46410 q^{31} -6.00000 q^{35} -5.19615 q^{39} +10.0000 q^{43} -6.92820 q^{45} +10.3923 q^{47} -4.00000 q^{49} -3.00000 q^{51} -5.19615 q^{53} -1.00000 q^{57} -9.00000 q^{59} -10.3923 q^{61} +3.46410 q^{63} -18.0000 q^{65} -13.0000 q^{67} +1.73205 q^{69} +3.46410 q^{71} +1.00000 q^{73} +7.00000 q^{75} +10.3923 q^{79} +1.00000 q^{81} -10.3923 q^{85} -1.73205 q^{87} -18.0000 q^{89} +9.00000 q^{91} -3.46410 q^{93} -3.46410 q^{95} -14.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^9	$ 2 q + 2 q^{3} - 4 q^{9} - 6 q^{17} - 2 q^{19} + 14 q^{25} - 10 q^{27} - 12 q^{35} + 20 q^{43} - 8 q^{49} - 6 q^{51} - 2 q^{57} - 18 q^{59} - 36 q^{65} - 26 q^{67} + 2 q^{73} + 14 q^{75} + 2 q^{81} - 36 q^{89} + 18 q^{91} - 28 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^9 - 6 * q^17 - 2 * q^19 + 14 * q^25 - 10 * q^27 - 12 * q^35 + 20 * q^43 - 8 * q^49 - 6 * q^51 - 2 * q^57 - 18 * q^59 - 36 * q^65 - 26 * q^67 + 2 * q^73 + 14 * q^75 + 2 * q^81 - 36 * q^89 + 18 * q^91 - 28 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	0	0
$7$	−1.73205	−0.654654	−0.327327	−	0.944911i	$-0.606148\pi$
$7$	−1.73205	−0.654654	−0.327327	+	0.944911i	$0.606148\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−5.19615	−1.44115	−0.720577	−	0.693375i	$-0.756123\pi$
$13$	−5.19615	−1.44115	−0.720577	+	0.693375i	$0.756123\pi$
$14$	0	0
$15$	3.46410	0.894427
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.73205	−0.377964
$22$	0	0
$23$	1.73205	0.361158	0.180579	−	0.983561i	$-0.442203\pi$
$23$	1.73205	0.361158	0.180579	+	0.983561i	$0.442203\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	−1.73205	−0.321634	−0.160817	−	0.986984i	$-0.551413\pi$
$29$	−1.73205	−0.321634	−0.160817	+	0.986984i	$0.551413\pi$
$30$	0	0
$31$	−3.46410	−0.622171	−0.311086	−	0.950382i	$-0.600693\pi$
$31$	−3.46410	−0.622171	−0.311086	+	0.950382i	$0.600693\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−6.00000	−1.01419
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	−5.19615	−0.832050
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	0	0
$45$	−6.92820	−1.03280
$46$	0	0
$47$	10.3923	1.51587	0.757937	−	0.652328i	$-0.226208\pi$
$47$	10.3923	1.51587	0.757937	+	0.652328i	$0.226208\pi$
$48$	0	0
$49$	−4.00000	−0.571429
$50$	0	0
$51$	−3.00000	−0.420084
$52$	0	0
$53$	−5.19615	−0.713746	−0.356873	−	0.934153i	$-0.616157\pi$
$53$	−5.19615	−0.713746	−0.356873	+	0.934153i	$0.616157\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	0	0
$61$	−10.3923	−1.33060	−0.665299	−	0.746577i	$-0.731696\pi$
$61$	−10.3923	−1.33060	−0.665299	+	0.746577i	$0.731696\pi$
$62$	0	0
$63$	3.46410	0.436436
$64$	0	0
$65$	−18.0000	−2.23263
$66$	0	0
$67$	−13.0000	−1.58820	−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−13.0000	−1.58820	−0.794101	+	0.607785i	$0.792058\pi$
$68$	0	0
$69$	1.73205	0.208514
$70$	0	0
$71$	3.46410	0.411113	0.205557	−	0.978645i	$-0.434100\pi$
$71$	3.46410	0.411113	0.205557	+	0.978645i	$0.434100\pi$
$72$	0	0
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	0	0
$75$	7.00000	0.808290
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.3923	1.16923	0.584613	−	0.811312i	$-0.301246\pi$
$79$	10.3923	1.16923	0.584613	+	0.811312i	$0.301246\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−10.3923	−1.12720
$86$	0	0
$87$	−1.73205	−0.185695
$88$	0	0
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	9.00000	0.943456
$92$	0	0
$93$	−3.46410	−0.359211
$94$	0	0
$95$	−3.46410	−0.355409
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−13.8564	−1.36531	−0.682656	−	0.730740i	$-0.739175\pi$
$103$	−13.8564	−1.36531	−0.682656	+	0.730740i	$0.739175\pi$
$104$	0	0
$105$	−6.00000	−0.585540
$106$	0	0
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	0	0
$109$	−5.19615	−0.497701	−0.248851	−	0.968542i	$-0.580053\pi$
$109$	−5.19615	−0.497701	−0.248851	+	0.968542i	$0.580053\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	6.00000	0.559503
$116$	0	0
$117$	10.3923	0.960769
$118$	0	0
$119$	5.19615	0.476331
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	−10.3923	−0.922168	−0.461084	−	0.887357i	$-0.652539\pi$
$127$	−10.3923	−0.922168	−0.461084	+	0.887357i	$0.652539\pi$
$128$	0	0
$129$	10.0000	0.880451
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	1.73205	0.150188
$134$	0	0
$135$	−17.3205	−1.49071
$136$	0	0
$137$	−21.0000	−1.79415	−0.897076	−	0.441877i	$-0.854313\pi$
$137$	−21.0000	−1.79415	−0.897076	+	0.441877i	$0.854313\pi$
$138$	0	0
$139$	−22.0000	−1.86602	−0.933008	−	0.359856i	$-0.882826\pi$
$139$	−22.0000	−1.86602	−0.933008	+	0.359856i	$0.882826\pi$
$140$	0	0
$141$	10.3923	0.875190
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	−4.00000	−0.329914
$148$	0	0
$149$	20.7846	1.70274	0.851371	−	0.524564i	$-0.175772\pi$
$149$	20.7846	1.70274	0.851371	+	0.524564i	$0.175772\pi$
$150$	0	0
$151$	13.8564	1.12762	0.563809	−	0.825905i	$-0.309335\pi$
$151$	13.8564	1.12762	0.563809	+	0.825905i	$0.309335\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	−12.0000	−0.963863
$156$	0	0
$157$	3.46410	0.276465	0.138233	−	0.990400i	$-0.455858\pi$
$157$	3.46410	0.276465	0.138233	+	0.990400i	$0.455858\pi$
$158$	0	0
$159$	−5.19615	−0.412082
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	−14.0000	−1.09656	−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000	−1.09656	−0.548282	+	0.836293i	$0.684718\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.8564	−1.07224	−0.536120	−	0.844141i	$-0.680111\pi$
$167$	−13.8564	−1.07224	−0.536120	+	0.844141i	$0.680111\pi$
$168$	0	0
$169$	14.0000	1.07692
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	20.7846	1.58022	0.790112	−	0.612962i	$-0.210022\pi$
$173$	20.7846	1.58022	0.790112	+	0.612962i	$0.210022\pi$
$174$	0	0
$175$	−12.1244	−0.916515
$176$	0	0
$177$	−9.00000	−0.676481
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−13.8564	−1.02994	−0.514969	−	0.857209i	$-0.672197\pi$
$181$	−13.8564	−1.02994	−0.514969	+	0.857209i	$0.672197\pi$
$182$	0	0
$183$	−10.3923	−0.768221
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	8.66025	0.629941
$190$	0	0
$191$	−1.73205	−0.125327	−0.0626634	−	0.998035i	$-0.519959\pi$
$191$	−1.73205	−0.125327	−0.0626634	+	0.998035i	$0.519959\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0	0
$195$	−18.0000	−1.28901
$196$	0	0
$197$	13.8564	0.987228	0.493614	−	0.869681i	$-0.335676\pi$
$197$	13.8564	0.987228	0.493614	+	0.869681i	$0.335676\pi$
$198$	0	0
$199$	22.5167	1.59616	0.798082	−	0.602549i	$-0.205848\pi$
$199$	22.5167	1.59616	0.798082	+	0.602549i	$0.205848\pi$
$200$	0	0
$201$	−13.0000	−0.916949
$202$	0	0
$203$	3.00000	0.210559
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3.46410	−0.240772
$208$	0	0
$209$	0	0
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	0	0
$213$	3.46410	0.237356
$214$	0	0
$215$	34.6410	2.36250
$216$	0	0
$217$	6.00000	0.407307
$218$	0	0
$219$	1.00000	0.0675737
$220$	0	0
$221$	15.5885	1.04859
$222$	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
$225$	−14.0000	−0.933333
$226$	0	0
$227$	15.0000	0.995585	0.497792	−	0.867296i	$-0.334144\pi$
$227$	15.0000	0.995585	0.497792	+	0.867296i	$0.334144\pi$
$228$	0	0
$229$	3.46410	0.228914	0.114457	−	0.993428i	$-0.463487\pi$
$229$	3.46410	0.228914	0.114457	+	0.993428i	$0.463487\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	36.0000	2.34838
$236$	0	0
$237$	10.3923	0.675053
$238$	0	0
$239$	−19.0526	−1.23241	−0.616204	−	0.787587i	$-0.711330\pi$
$239$	−19.0526	−1.23241	−0.616204	+	0.787587i	$0.711330\pi$
$240$	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
$243$	16.0000	1.02640
$244$	0	0
$245$	−13.8564	−0.885253
$246$	0	0
$247$	5.19615	0.330623
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−10.3923	−0.650791
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	3.46410	0.214423
$262$	0	0
$263$	24.2487	1.49524	0.747620	−	0.664127i	$-0.231197\pi$
$263$	24.2487	1.49524	0.747620	+	0.664127i	$0.231197\pi$
$264$	0	0
$265$	−18.0000	−1.10573
$266$	0	0
$267$	−18.0000	−1.10158
$268$	0	0
$269$	27.7128	1.68968	0.844840	−	0.535019i	$-0.179696\pi$
$269$	27.7128	1.68968	0.844840	+	0.535019i	$0.179696\pi$
$270$	0	0
$271$	22.5167	1.36779	0.683895	−	0.729581i	$-0.260285\pi$
$271$	22.5167	1.36779	0.683895	+	0.729581i	$0.260285\pi$
$272$	0	0
$273$	9.00000	0.544705
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.3923	0.624413	0.312207	−	0.950014i	$-0.398932\pi$
$277$	10.3923	0.624413	0.312207	+	0.950014i	$0.398932\pi$
$278$	0	0
$279$	6.92820	0.414781
$280$	0	0
$281$	−24.0000	−1.43172	−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000	−1.43172	−0.715860	+	0.698244i	$0.753965\pi$
$282$	0	0
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	0	0
$285$	−3.46410	−0.205196
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−14.0000	−0.820695
$292$	0	0
$293$	1.73205	0.101187	0.0505937	−	0.998719i	$-0.483889\pi$
$293$	1.73205	0.101187	0.0505937	+	0.998719i	$0.483889\pi$
$294$	0	0
$295$	−31.1769	−1.81519
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−9.00000	−0.520483
$300$	0	0
$301$	−17.3205	−0.998337
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−36.0000	−2.06135
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	−13.8564	−0.788263
$310$	0	0
$311$	1.73205	0.0982156	0.0491078	−	0.998793i	$-0.484362\pi$
$311$	1.73205	0.0982156	0.0491078	+	0.998793i	$0.484362\pi$
$312$	0	0
$313$	−19.0000	−1.07394	−0.536972	−	0.843600i	$-0.680432\pi$
$313$	−19.0000	−1.07394	−0.536972	+	0.843600i	$0.680432\pi$
$314$	0	0
$315$	12.0000	0.676123
$316$	0	0
$317$	8.66025	0.486408	0.243204	−	0.969975i	$-0.421801\pi$
$317$	8.66025	0.486408	0.243204	+	0.969975i	$0.421801\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−3.00000	−0.167444
$322$	0	0
$323$	3.00000	0.166924
$324$	0	0
$325$	−36.3731	−2.01761
$326$	0	0
$327$	−5.19615	−0.287348
$328$	0	0
$329$	−18.0000	−0.992372
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−45.0333	−2.46043
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	0	0
$343$	19.0526	1.02874
$344$	0	0
$345$	6.00000	0.323029
$346$	0	0
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	0	0
$349$	−10.3923	−0.556287	−0.278144	−	0.960539i	$-0.589719\pi$
$349$	−10.3923	−0.556287	−0.278144	+	0.960539i	$0.589719\pi$
$350$	0	0
$351$	25.9808	1.38675
$352$	0	0
$353$	27.0000	1.43706	0.718532	−	0.695493i	$-0.244814\pi$
$353$	27.0000	1.43706	0.718532	+	0.695493i	$0.244814\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	0	0
$357$	5.19615	0.275010
$358$	0	0
$359$	8.66025	0.457071	0.228535	−	0.973536i	$-0.426606\pi$
$359$	8.66025	0.457071	0.228535	+	0.973536i	$0.426606\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−11.0000	−0.577350
$364$	0	0
$365$	3.46410	0.181319
$366$	0	0
$367$	31.1769	1.62742	0.813711	−	0.581270i	$-0.197444\pi$
$367$	31.1769	1.62742	0.813711	+	0.581270i	$0.197444\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	9.00000	0.467257
$372$	0	0
$373$	−12.1244	−0.627775	−0.313888	−	0.949460i	$-0.601632\pi$
$373$	−12.1244	−0.627775	−0.313888	+	0.949460i	$0.601632\pi$
$374$	0	0
$375$	6.92820	0.357771
$376$	0	0
$377$	9.00000	0.463524
$378$	0	0
$379$	1.00000	0.0513665	0.0256833	−	0.999670i	$-0.491824\pi$
$379$	1.00000	0.0513665	0.0256833	+	0.999670i	$0.491824\pi$
$380$	0	0
$381$	−10.3923	−0.532414
$382$	0	0
$383$	34.6410	1.77007	0.885037	−	0.465521i	$-0.154133\pi$
$383$	34.6410	1.77007	0.885037	+	0.465521i	$0.154133\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−20.0000	−1.01666
$388$	0	0
$389$	−27.7128	−1.40510	−0.702548	−	0.711637i	$-0.747954\pi$
$389$	−27.7128	−1.40510	−0.702548	+	0.711637i	$0.747954\pi$
$390$	0	0
$391$	−5.19615	−0.262781
$392$	0	0
$393$	18.0000	0.907980
$394$	0	0
$395$	36.0000	1.81136
$396$	0	0
$397$	−17.3205	−0.869291	−0.434646	−	0.900602i	$-0.643126\pi$
$397$	−17.3205	−0.869291	−0.434646	+	0.900602i	$0.643126\pi$
$398$	0	0
$399$	1.73205	0.0867110
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	18.0000	0.896644
$404$	0	0
$405$	3.46410	0.172133
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	−21.0000	−1.03585
$412$	0	0
$413$	15.5885	0.767058
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−22.0000	−1.07734
$418$	0	0
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	8.66025	0.422075	0.211037	−	0.977478i	$-0.432316\pi$
$421$	8.66025	0.422075	0.211037	+	0.977478i	$0.432316\pi$
$422$	0	0
$423$	−20.7846	−1.01058
$424$	0	0
$425$	−21.0000	−1.01865
$426$	0	0
$427$	18.0000	0.871081
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.2487	−1.16802	−0.584010	−	0.811747i	$-0.698517\pi$
$431$	−24.2487	−1.16802	−0.584010	+	0.811747i	$0.698517\pi$
$432$	0	0
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	0	0
$435$	−6.00000	−0.287678
$436$	0	0
$437$	−1.73205	−0.0828552
$438$	0	0
$439$	−34.6410	−1.65333	−0.826663	−	0.562698i	$-0.809764\pi$
$439$	−34.6410	−1.65333	−0.826663	+	0.562698i	$0.809764\pi$
$440$	0	0
$441$	8.00000	0.380952
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	−62.3538	−2.95585
$446$	0	0
$447$	20.7846	0.983078
$448$	0	0
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	13.8564	0.651031
$454$	0	0
$455$	31.1769	1.46160
$456$	0	0
$457$	1.00000	0.0467780	0.0233890	−	0.999726i	$-0.492554\pi$
$457$	1.00000	0.0467780	0.0233890	+	0.999726i	$0.492554\pi$
$458$	0	0
$459$	15.0000	0.700140
$460$	0	0
$461$	−13.8564	−0.645357	−0.322679	−	0.946509i	$-0.604583\pi$
$461$	−13.8564	−0.645357	−0.322679	+	0.946509i	$0.604583\pi$
$462$	0	0
$463$	10.3923	0.482971	0.241486	−	0.970404i	$-0.422365\pi$
$463$	10.3923	0.482971	0.241486	+	0.970404i	$0.422365\pi$
$464$	0	0
$465$	−12.0000	−0.556487
$466$	0	0
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	0	0
$469$	22.5167	1.03972
$470$	0	0
$471$	3.46410	0.159617
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−7.00000	−0.321182
$476$	0	0
$477$	10.3923	0.475831
$478$	0	0
$479$	−31.1769	−1.42451	−0.712255	−	0.701921i	$-0.752326\pi$
$479$	−31.1769	−1.42451	−0.712255	+	0.701921i	$0.752326\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−3.00000	−0.136505
$484$	0	0
$485$	−48.4974	−2.20215
$486$	0	0
$487$	−34.6410	−1.56973	−0.784867	−	0.619664i	$-0.787269\pi$
$487$	−34.6410	−1.56973	−0.784867	+	0.619664i	$0.787269\pi$
$488$	0	0
$489$	−14.0000	−0.633102
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	5.19615	0.234023
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.00000	−0.269137
$498$	0	0
$499$	2.00000	0.0895323	0.0447661	−	0.998997i	$-0.485746\pi$
$499$	2.00000	0.0895323	0.0447661	+	0.998997i	$0.485746\pi$
$500$	0	0
$501$	−13.8564	−0.619059
$502$	0	0
$503$	5.19615	0.231685	0.115842	−	0.993268i	$-0.463043\pi$
$503$	5.19615	0.231685	0.115842	+	0.993268i	$0.463043\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	14.0000	0.621762
$508$	0	0
$509$	27.7128	1.22835	0.614174	−	0.789170i	$-0.289489\pi$
$509$	27.7128	1.22835	0.614174	+	0.789170i	$0.289489\pi$
$510$	0	0
$511$	−1.73205	−0.0766214
$512$	0	0
$513$	5.00000	0.220755
$514$	0	0
$515$	−48.0000	−2.11513
$516$	0	0
$517$	0	0
$518$	0	0
$519$	20.7846	0.912343
$520$	0	0
$521$	36.0000	1.57719	0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000	1.57719	0.788594	+	0.614914i	$0.210809\pi$
$522$	0	0
$523$	7.00000	0.306089	0.153044	−	0.988219i	$-0.451092\pi$
$523$	7.00000	0.306089	0.153044	+	0.988219i	$0.451092\pi$
$524$	0	0
$525$	−12.1244	−0.529150
$526$	0	0
$527$	10.3923	0.452696
$528$	0	0
$529$	−20.0000	−0.869565
$530$	0	0
$531$	18.0000	0.781133
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−10.3923	−0.449299
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	−13.8564	−0.594635
$544$	0	0
$545$	−18.0000	−0.771035
$546$	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
$549$	20.7846	0.887066
$550$	0	0
$551$	1.73205	0.0737878
$552$	0	0
$553$	−18.0000	−0.765438
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−24.2487	−1.02745	−0.513725	−	0.857955i	$-0.671735\pi$
$557$	−24.2487	−1.02745	−0.513725	+	0.857955i	$0.671735\pi$
$558$	0	0
$559$	−51.9615	−2.19774
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	20.7846	0.874415
$566$	0	0
$567$	−1.73205	−0.0727393
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	0	0
$573$	−1.73205	−0.0723575
$574$	0	0
$575$	12.1244	0.505621
$576$	0	0
$577$	−35.0000	−1.45707	−0.728535	−	0.685009i	$-0.759798\pi$
$577$	−35.0000	−1.45707	−0.728535	+	0.685009i	$0.759798\pi$
$578$	0	0
$579$	−16.0000	−0.664937
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	36.0000	1.48842
$586$	0	0
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	3.46410	0.142736
$590$	0	0
$591$	13.8564	0.569976
$592$	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
$595$	18.0000	0.737928
$596$	0	0
$597$	22.5167	0.921546
$598$	0	0
$599$	−27.7128	−1.13231	−0.566157	−	0.824297i	$-0.691571\pi$
$599$	−27.7128	−1.13231	−0.566157	+	0.824297i	$0.691571\pi$
$600$	0	0
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	0	0
$603$	26.0000	1.05880
$604$	0	0
$605$	−38.1051	−1.54919
$606$	0	0
$607$	−3.46410	−0.140604	−0.0703018	−	0.997526i	$-0.522396\pi$
$607$	−3.46410	−0.140604	−0.0703018	+	0.997526i	$0.522396\pi$
$608$	0	0
$609$	3.00000	0.121566
$610$	0	0
$611$	−54.0000	−2.18461
$612$	0	0
$613$	6.92820	0.279827	0.139914	−	0.990164i	$-0.455317\pi$
$613$	6.92820	0.279827	0.139914	+	0.990164i	$0.455317\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	0	0
$621$	−8.66025	−0.347524
$622$	0	0
$623$	31.1769	1.24908
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−31.1769	−1.24113	−0.620567	−	0.784154i	$-0.713097\pi$
$631$	−31.1769	−1.24113	−0.620567	+	0.784154i	$0.713097\pi$
$632$	0	0
$633$	5.00000	0.198732
$634$	0	0
$635$	−36.0000	−1.42862
$636$	0	0
$637$	20.7846	0.823516
$638$	0	0
$639$	−6.92820	−0.274075
$640$	0	0
$641$	−48.0000	−1.89589	−0.947943	−	0.318440i	$-0.896841\pi$
$641$	−48.0000	−1.89589	−0.947943	+	0.318440i	$0.896841\pi$
$642$	0	0
$643$	22.0000	0.867595	0.433798	−	0.901010i	$-0.357173\pi$
$643$	22.0000	0.867595	0.433798	+	0.901010i	$0.357173\pi$
$644$	0	0
$645$	34.6410	1.36399
$646$	0	0
$647$	22.5167	0.885221	0.442611	−	0.896714i	$-0.354052\pi$
$647$	22.5167	0.885221	0.442611	+	0.896714i	$0.354052\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	6.00000	0.235159
$652$	0	0
$653$	3.46410	0.135561	0.0677804	−	0.997700i	$-0.478408\pi$
$653$	3.46410	0.135561	0.0677804	+	0.997700i	$0.478408\pi$
$654$	0	0
$655$	62.3538	2.43637
$656$	0	0
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	51.0000	1.98668	0.993339	−	0.115229i	$-0.0367601\pi$
$659$	51.0000	1.98668	0.993339	+	0.115229i	$0.0367601\pi$
$660$	0	0
$661$	−19.0526	−0.741059	−0.370529	−	0.928821i	$-0.620824\pi$
$661$	−19.0526	−0.741059	−0.370529	+	0.928821i	$0.620824\pi$
$662$	0	0
$663$	15.5885	0.605406
$664$	0	0
$665$	6.00000	0.232670
$666$	0	0
$667$	−3.00000	−0.116160
$668$	0	0
$669$	−6.92820	−0.267860
$670$	0	0
$671$	0	0
$672$	0	0
$673$	28.0000	1.07932	0.539660	−	0.841883i	$-0.318553\pi$
$673$	28.0000	1.07932	0.539660	+	0.841883i	$0.318553\pi$
$674$	0	0
$675$	−35.0000	−1.34715
$676$	0	0
$677$	36.3731	1.39793	0.698965	−	0.715156i	$-0.253644\pi$
$677$	36.3731	1.39793	0.698965	+	0.715156i	$0.253644\pi$
$678$	0	0
$679$	24.2487	0.930580
$680$	0	0
$681$	15.0000	0.574801
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	−72.7461	−2.77949
$686$	0	0
$687$	3.46410	0.132164
$688$	0	0
$689$	27.0000	1.02862
$690$	0	0
$691$	34.0000	1.29342	0.646710	−	0.762736i	$-0.276144\pi$
$691$	34.0000	1.29342	0.646710	+	0.762736i	$0.276144\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−76.2102	−2.89082
$696$	0	0
$697$	0	0
$698$	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
$703$	0	0
$704$	0	0
$705$	36.0000	1.35584
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−24.2487	−0.910679	−0.455340	−	0.890318i	$-0.650482\pi$
$709$	−24.2487	−0.910679	−0.455340	+	0.890318i	$0.650482\pi$
$710$	0	0
$711$	−20.7846	−0.779484
$712$	0	0
$713$	−6.00000	−0.224702
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−19.0526	−0.711531
$718$	0	0
$719$	−8.66025	−0.322973	−0.161486	−	0.986875i	$-0.551629\pi$
$719$	−8.66025	−0.322973	−0.161486	+	0.986875i	$0.551629\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	0	0
$723$	8.00000	0.297523
$724$	0	0
$725$	−12.1244	−0.450287
$726$	0	0
$727$	−19.0526	−0.706620	−0.353310	−	0.935506i	$-0.614944\pi$
$727$	−19.0526	−0.706620	−0.353310	+	0.935506i	$0.614944\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−30.0000	−1.10959
$732$	0	0
$733$	31.1769	1.15155	0.575773	−	0.817610i	$-0.304701\pi$
$733$	31.1769	1.15155	0.575773	+	0.817610i	$0.304701\pi$
$734$	0	0
$735$	−13.8564	−0.511101
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−28.0000	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$740$	0	0
$741$	5.19615	0.190885
$742$	0	0
$743$	17.3205	0.635428	0.317714	−	0.948187i	$-0.397085\pi$
$743$	17.3205	0.635428	0.317714	+	0.948187i	$0.397085\pi$
$744$	0	0
$745$	72.0000	2.63788
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5.19615	0.189863
$750$	0	0
$751$	20.7846	0.758441	0.379221	−	0.925306i	$-0.376192\pi$
$751$	20.7846	0.758441	0.379221	+	0.925306i	$0.376192\pi$
$752$	0	0
$753$	18.0000	0.655956
$754$	0	0
$755$	48.0000	1.74690
$756$	0	0
$757$	51.9615	1.88857	0.944287	−	0.329124i	$-0.106753\pi$
$757$	51.9615	1.88857	0.944287	+	0.329124i	$0.106753\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−21.0000	−0.761249	−0.380625	−	0.924730i	$-0.624291\pi$
$761$	−21.0000	−0.761249	−0.380625	+	0.924730i	$0.624291\pi$
$762$	0	0
$763$	9.00000	0.325822
$764$	0	0
$765$	20.7846	0.751469
$766$	0	0
$767$	46.7654	1.68860
$768$	0	0
$769$	41.0000	1.47850	0.739249	−	0.673432i	$-0.235181\pi$
$769$	41.0000	1.47850	0.739249	+	0.673432i	$0.235181\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	0	0
$773$	−32.9090	−1.18365	−0.591827	−	0.806065i	$-0.701593\pi$
$773$	−32.9090	−1.18365	−0.591827	+	0.806065i	$0.701593\pi$
$774$	0	0
$775$	−24.2487	−0.871039
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	8.66025	0.309492
$784$	0	0
$785$	12.0000	0.428298
$786$	0	0
$787$	−53.0000	−1.88925	−0.944623	−	0.328158i	$-0.893572\pi$
$787$	−53.0000	−1.88925	−0.944623	+	0.328158i	$0.893572\pi$
$788$	0	0
$789$	24.2487	0.863277
$790$	0	0
$791$	−10.3923	−0.369508
$792$	0	0
$793$	54.0000	1.91760
$794$	0	0
$795$	−18.0000	−0.638394
$796$	0	0
$797$	−29.4449	−1.04299	−0.521495	−	0.853254i	$-0.674626\pi$
$797$	−29.4449	−1.04299	−0.521495	+	0.853254i	$0.674626\pi$
$798$	0	0
$799$	−31.1769	−1.10296
$800$	0	0
$801$	36.0000	1.27200
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−10.3923	−0.366281
$806$	0	0
$807$	27.7128	0.975537
$808$	0	0
$809$	27.0000	0.949269	0.474635	−	0.880183i	$-0.342580\pi$
$809$	27.0000	0.949269	0.474635	+	0.880183i	$0.342580\pi$
$810$	0	0
$811$	31.0000	1.08856	0.544279	−	0.838905i	$-0.316803\pi$
$811$	31.0000	1.08856	0.544279	+	0.838905i	$0.316803\pi$
$812$	0	0
$813$	22.5167	0.789694
$814$	0	0
$815$	−48.4974	−1.69879
$816$	0	0
$817$	−10.0000	−0.349856
$818$	0	0
$819$	−18.0000	−0.628971
$820$	0	0
$821$	10.3923	0.362694	0.181347	−	0.983419i	$-0.441954\pi$
$821$	10.3923	0.362694	0.181347	+	0.983419i	$0.441954\pi$
$822$	0	0
$823$	8.66025	0.301877	0.150939	−	0.988543i	$-0.451770\pi$
$823$	8.66025	0.301877	0.150939	+	0.988543i	$0.451770\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	45.0000	1.56480	0.782402	−	0.622774i	$-0.213994\pi$
$827$	45.0000	1.56480	0.782402	+	0.622774i	$0.213994\pi$
$828$	0	0
$829$	22.5167	0.782036	0.391018	−	0.920383i	$-0.372123\pi$
$829$	22.5167	0.782036	0.391018	+	0.920383i	$0.372123\pi$
$830$	0	0
$831$	10.3923	0.360505
$832$	0	0
$833$	12.0000	0.415775
$834$	0	0
$835$	−48.0000	−1.66111
$836$	0	0
$837$	17.3205	0.598684
$838$	0	0
$839$	38.1051	1.31553	0.657767	−	0.753221i	$-0.271501\pi$
$839$	38.1051	1.31553	0.657767	+	0.753221i	$0.271501\pi$
$840$	0	0
$841$	−26.0000	−0.896552
$842$	0	0
$843$	−24.0000	−0.826604
$844$	0	0
$845$	48.4974	1.66836
$846$	0	0
$847$	19.0526	0.654654
$848$	0	0
$849$	−28.0000	−0.960958
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−13.8564	−0.474434	−0.237217	−	0.971457i	$-0.576235\pi$
$853$	−13.8564	−0.474434	−0.237217	+	0.971457i	$0.576235\pi$
$854$	0	0
$855$	6.92820	0.236940
$856$	0	0
$857$	−24.0000	−0.819824	−0.409912	−	0.912125i	$-0.634441\pi$
$857$	−24.0000	−0.819824	−0.409912	+	0.912125i	$0.634441\pi$
$858$	0	0
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−45.0333	−1.53295	−0.766476	−	0.642273i	$-0.777992\pi$
$863$	−45.0333	−1.53295	−0.766476	+	0.642273i	$0.777992\pi$
$864$	0	0
$865$	72.0000	2.44807
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	0	0
$870$	0	0
$871$	67.5500	2.28884
$872$	0	0
$873$	28.0000	0.947656
$874$	0	0
$875$	−12.0000	−0.405674
$876$	0	0
$877$	5.19615	0.175462	0.0877308	−	0.996144i	$-0.472038\pi$
$877$	5.19615	0.175462	0.0877308	+	0.996144i	$0.472038\pi$
$878$	0	0
$879$	1.73205	0.0584206
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	0	0
$883$	−14.0000	−0.471138	−0.235569	−	0.971858i	$-0.575695\pi$
$883$	−14.0000	−0.471138	−0.235569	+	0.971858i	$0.575695\pi$
$884$	0	0
$885$	−31.1769	−1.04800
$886$	0	0
$887$	−51.9615	−1.74470	−0.872349	−	0.488884i	$-0.837404\pi$
$887$	−51.9615	−1.74470	−0.872349	+	0.488884i	$0.837404\pi$
$888$	0	0
$889$	18.0000	0.603701
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−10.3923	−0.347765
$894$	0	0
$895$	41.5692	1.38951
$896$	0	0
$897$	−9.00000	−0.300501
$898$	0	0
$899$	6.00000	0.200111
$900$	0	0
$901$	15.5885	0.519327
$902$	0	0
$903$	−17.3205	−0.576390
$904$	0	0
$905$	−48.0000	−1.59557
$906$	0	0
$907$	19.0000	0.630885	0.315442	−	0.948945i	$-0.397847\pi$
$907$	19.0000	0.630885	0.315442	+	0.948945i	$0.397847\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−31.1769	−1.03294	−0.516469	−	0.856306i	$-0.672754\pi$
$911$	−31.1769	−1.03294	−0.516469	+	0.856306i	$0.672754\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−36.0000	−1.19012
$916$	0	0
$917$	−31.1769	−1.02955
$918$	0	0
$919$	−32.9090	−1.08557	−0.542783	−	0.839873i	$-0.682630\pi$
$919$	−32.9090	−1.08557	−0.542783	+	0.839873i	$0.682630\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	0	0
$923$	−18.0000	−0.592477
$924$	0	0
$925$	0	0
$926$	0	0
$927$	27.7128	0.910208
$928$	0	0
$929$	21.0000	0.688988	0.344494	−	0.938789i	$-0.388051\pi$
$929$	21.0000	0.688988	0.344494	+	0.938789i	$0.388051\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	0	0
$933$	1.73205	0.0567048
$934$	0	0
$935$	0	0
$936$	0	0
$937$	59.0000	1.92745	0.963723	−	0.266904i	$-0.0860008\pi$
$937$	59.0000	1.92745	0.963723	+	0.266904i	$0.0860008\pi$
$938$	0	0
$939$	−19.0000	−0.620042
$940$	0	0
$941$	43.3013	1.41158	0.705791	−	0.708421i	$-0.250592\pi$
$941$	43.3013	1.41158	0.705791	+	0.708421i	$0.250592\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	30.0000	0.975900
$946$	0	0
$947$	18.0000	0.584921	0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000	0.584921	0.292461	+	0.956278i	$0.405526\pi$
$948$	0	0
$949$	−5.19615	−0.168674
$950$	0	0
$951$	8.66025	0.280828
$952$	0	0
$953$	−36.0000	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$954$	0	0
$955$	−6.00000	−0.194155
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36.3731	1.17455
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	6.00000	0.193347
$964$	0	0
$965$	−55.4256	−1.78421
$966$	0	0
$967$	24.2487	0.779786	0.389893	−	0.920860i	$-0.372512\pi$
$967$	24.2487	0.779786	0.389893	+	0.920860i	$0.372512\pi$
$968$	0	0
$969$	3.00000	0.0963739
$970$	0	0
$971$	60.0000	1.92549	0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000	1.92549	0.962746	+	0.270408i	$0.0871586\pi$
$972$	0	0
$973$	38.1051	1.22159
$974$	0	0
$975$	−36.3731	−1.16487
$976$	0	0
$977$	−6.00000	−0.191957	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000	−0.191957	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	10.3923	0.331801
$982$	0	0
$983$	45.0333	1.43634	0.718170	−	0.695868i	$-0.244980\pi$
$983$	45.0333	1.43634	0.718170	+	0.695868i	$0.244980\pi$
$984$	0	0
$985$	48.0000	1.52941
$986$	0	0
$987$	−18.0000	−0.572946
$988$	0	0
$989$	17.3205	0.550760
$990$	0	0
$991$	−31.1769	−0.990367	−0.495184	−	0.868788i	$-0.664899\pi$
$991$	−31.1769	−0.990367	−0.495184	+	0.868788i	$0.664899\pi$
$992$	0	0
$993$	−7.00000	−0.222138
$994$	0	0
$995$	78.0000	2.47277
$996$	0	0
$997$	38.1051	1.20680	0.603401	−	0.797438i	$-0.293812\pi$
$997$	38.1051	1.20680	0.603401	+	0.797438i	$0.293812\pi$
$998$	0	0
$999$	0	0

Twists

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

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

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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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.46410 q^{5} -1.73205 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.46410 q^{5} -1.73205 q^{7} -2.00000 q^{9} -5.19615 q^{13} +3.46410 q^{15} -3.00000 q^{17} -1.00000 q^{19} -1.73205 q^{21} +1.73205 q^{23} +7.00000 q^{25} -5.00000 q^{27} -1.73205 q^{29} -3.46410 q^{31} -6.00000 q^{35} -5.19615 q^{39} +10.0000 q^{43} -6.92820 q^{45} +10.3923 q^{47} -4.00000 q^{49} -3.00000 q^{51} -5.19615 q^{53} -1.00000 q^{57} -9.00000 q^{59} -10.3923 q^{61} +3.46410 q^{63} -18.0000 q^{65} -13.0000 q^{67} +1.73205 q^{69} +3.46410 q^{71} +1.00000 q^{73} +7.00000 q^{75} +10.3923 q^{79} +1.00000 q^{81} -10.3923 q^{85} -1.73205 q^{87} -18.0000 q^{89} +9.00000 q^{91} -3.46410 q^{93} -3.46410 q^{95} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^9	\( 2 q + 2 q^{3} - 4 q^{9} - 6 q^{17} - 2 q^{19} + 14 q^{25} - 10 q^{27} - 12 q^{35} + 20 q^{43} - 8 q^{49} - 6 q^{51} - 2 q^{57} - 18 q^{59} - 36 q^{65} - 26 q^{67} + 2 q^{73} + 14 q^{75} + 2 q^{81} - 36 q^{89} + 18 q^{91} - 28 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^9 - 6 * q^17 - 2 * q^19 + 14 * q^25 - 10 * q^27 - 12 * q^35 + 20 * q^43 - 8 * q^49 - 6 * q^51 - 2 * q^57 - 18 * q^59 - 36 * q^65 - 26 * q^67 + 2 * q^73 + 14 * q^75 + 2 * q^81 - 36 * q^89 + 18 * q^91 - 28 * q^97

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