LMFDB - Embedded newform 4864.2.a.bm.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:	$8$
Coefficient field:	8.8.34309996544.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 8x^{6} + 28x^{5} + 31x^{4} - 36x^{3} - 22x^{2} + 12x + 2 $ x^8 - 4x^7 - 8x^6 + 28x^5 + 31x^4 - 36x^3 - 22x^2 + 12*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 2432)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.139869$ 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)

$f(q)$	$=$	$q-2.63640 q^{3} +2.63403 q^{5} +3.88603 q^{7} +3.95063 q^{9} +O(q^{10})$	$q-2.63640 q^{3} +2.63403 q^{5} +3.88603 q^{7} +3.95063 q^{9} -5.95063 q^{11} +4.14940 q^{13} -6.94438 q^{15} +1.82843 q^{17} +1.00000 q^{19} -10.2452 q^{21} -4.14940 q^{23} +1.93813 q^{25} -2.50625 q^{27} -8.19638 q^{29} +1.67631 q^{31} +15.6883 q^{33} +10.2359 q^{35} -8.14452 q^{37} -10.9395 q^{39} -0.514201 q^{41} -10.9631 q^{43} +10.4061 q^{45} -13.3035 q^{47} +8.10124 q^{49} -4.82047 q^{51} +1.62446 q^{53} -15.6742 q^{55} -2.63640 q^{57} -12.2234 q^{59} -5.98666 q^{61} +15.3523 q^{63} +10.9297 q^{65} +4.09328 q^{67} +10.9395 q^{69} +11.8696 q^{71} -9.28531 q^{73} -5.10971 q^{75} -23.1243 q^{77} -2.42124 q^{79} -5.24441 q^{81} -0.987504 q^{83} +4.81614 q^{85} +21.6090 q^{87} -2.25389 q^{89} +16.1247 q^{91} -4.41944 q^{93} +2.63403 q^{95} +9.27281 q^{97} -23.5087 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} + 4 q^{9}+O(q^{10}) $ 8 * q - 4 * q^3 + 4 * q^9	$ 8 q - 4 q^{3} + 4 q^{9} - 20 q^{11} - 8 q^{17} + 8 q^{19} + 20 q^{25} - 4 q^{27} + 24 q^{33} - 12 q^{35} + 8 q^{41} - 28 q^{43} + 8 q^{49} - 12 q^{51} - 4 q^{57} - 36 q^{59} + 8 q^{65} - 28 q^{67} - 8 q^{73} - 68 q^{75} - 32 q^{81} - 40 q^{83} - 8 q^{89} + 12 q^{91} + 40 q^{97} - 60 q^{99}+O(q^{100}) $ 8 * q - 4 * q^3 + 4 * q^9 - 20 * q^11 - 8 * q^17 + 8 * q^19 + 20 * q^25 - 4 * q^27 + 24 * q^33 - 12 * q^35 + 8 * q^41 - 28 * q^43 + 8 * q^49 - 12 * q^51 - 4 * q^57 - 36 * q^59 + 8 * q^65 - 28 * q^67 - 8 * q^73 - 68 * q^75 - 32 * q^81 - 40 * q^83 - 8 * q^89 + 12 * q^91 + 40 * q^97 - 60 * 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
$3$	−2.63640	−1.52213	−0.761065	−	0.648676i	$-0.775323\pi$
$3$	−2.63640	−1.52213	−0.761065	+	0.648676i	$0.775323\pi$
$4$	0	0
$5$	2.63403	1.17798	0.588988	−	0.808142i	$-0.299527\pi$
$5$	2.63403	1.17798	0.588988	+	0.808142i	$0.299527\pi$
$6$	0	0
$7$	3.88603	1.46878	0.734391	−	0.678727i	$-0.237468\pi$
$7$	3.88603	1.46878	0.734391	+	0.678727i	$0.237468\pi$
$8$	0	0
$9$	3.95063	1.31688
$10$	0	0
$11$	−5.95063	−1.79418	−0.897091	−	0.441845i	$-0.854324\pi$
$11$	−5.95063	−1.79418	−0.897091	+	0.441845i	$0.854324\pi$
$12$	0	0
$13$	4.14940	1.15084	0.575418	−	0.817859i	$-0.304839\pi$
$13$	4.14940	1.15084	0.575418	+	0.817859i	$0.304839\pi$
$14$	0	0
$15$	−6.94438	−1.79303
$16$	0	0
$17$	1.82843	0.443459	0.221729	−	0.975108i	$-0.428830\pi$
$17$	1.82843	0.443459	0.221729	+	0.975108i	$0.428830\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−10.2452	−2.23568
$22$	0	0
$23$	−4.14940	−0.865210	−0.432605	−	0.901584i	$-0.642406\pi$
$23$	−4.14940	−0.865210	−0.432605	+	0.901584i	$0.642406\pi$
$24$	0	0
$25$	1.93813	0.387627
$26$	0	0
$27$	−2.50625	−0.482328
$28$	0	0
$29$	−8.19638	−1.52203	−0.761015	−	0.648735i	$-0.775298\pi$
$29$	−8.19638	−1.52203	−0.761015	+	0.648735i	$0.775298\pi$
$30$	0	0
$31$	1.67631	0.301075	0.150537	−	0.988604i	$-0.451900\pi$
$31$	1.67631	0.301075	0.150537	+	0.988604i	$0.451900\pi$
$32$	0	0
$33$	15.6883	2.73098
$34$	0	0
$35$	10.2359	1.73019
$36$	0	0
$37$	−8.14452	−1.33895	−0.669476	−	0.742834i	$-0.733481\pi$
$37$	−8.14452	−1.33895	−0.669476	+	0.742834i	$0.733481\pi$
$38$	0	0
$39$	−10.9395	−1.75172
$40$	0	0
$41$	−0.514201	−0.0803047	−0.0401524	−	0.999194i	$-0.512784\pi$
$41$	−0.514201	−0.0803047	−0.0401524	+	0.999194i	$0.512784\pi$
$42$	0	0
$43$	−10.9631	−1.67186	−0.835931	−	0.548835i	$-0.815071\pi$
$43$	−10.9631	−1.67186	−0.835931	+	0.548835i	$0.815071\pi$
$44$	0	0
$45$	10.4061	1.55125
$46$	0	0
$47$	−13.3035	−1.94051	−0.970257	−	0.242075i	$-0.922172\pi$
$47$	−13.3035	−1.94051	−0.970257	+	0.242075i	$0.922172\pi$
$48$	0	0
$49$	8.10124	1.15732
$50$	0	0
$51$	−4.82047	−0.675001
$52$	0	0
$53$	1.62446	0.223137	0.111568	−	0.993757i	$-0.464413\pi$
$53$	1.62446	0.223137	0.111568	+	0.993757i	$0.464413\pi$
$54$	0	0
$55$	−15.6742	−2.11350
$56$	0	0
$57$	−2.63640	−0.349200
$58$	0	0
$59$	−12.2234	−1.59136	−0.795678	−	0.605720i	$-0.792885\pi$
$59$	−12.2234	−1.59136	−0.795678	+	0.605720i	$0.792885\pi$
$60$	0	0
$61$	−5.98666	−0.766513	−0.383257	−	0.923642i	$-0.625197\pi$
$61$	−5.98666	−0.766513	−0.383257	+	0.923642i	$0.625197\pi$
$62$	0	0
$63$	15.3523	1.93420
$64$	0	0
$65$	10.9297	1.35566
$66$	0	0
$67$	4.09328	0.500074	0.250037	−	0.968236i	$-0.419557\pi$
$67$	4.09328	0.500074	0.250037	+	0.968236i	$0.419557\pi$
$68$	0	0
$69$	10.9395	1.31696
$70$	0	0
$71$	11.8696	1.40866	0.704332	−	0.709871i	$-0.251247\pi$
$71$	11.8696	1.40866	0.704332	+	0.709871i	$0.251247\pi$
$72$	0	0
$73$	−9.28531	−1.08676	−0.543381	−	0.839486i	$-0.682856\pi$
$73$	−9.28531	−1.08676	−0.543381	+	0.839486i	$0.682856\pi$
$74$	0	0
$75$	−5.10971	−0.590018
$76$	0	0
$77$	−23.1243	−2.63526
$78$	0	0
$79$	−2.42124	−0.272410	−0.136205	−	0.990681i	$-0.543491\pi$
$79$	−2.42124	−0.272410	−0.136205	+	0.990681i	$0.543491\pi$
$80$	0	0
$81$	−5.24441	−0.582712
$82$	0	0
$83$	−0.987504	−0.108393	−0.0541963	−	0.998530i	$-0.517260\pi$
$83$	−0.987504	−0.108393	−0.0541963	+	0.998530i	$0.517260\pi$
$84$	0	0
$85$	4.81614	0.522384
$86$	0	0
$87$	21.6090	2.31672
$88$	0	0
$89$	−2.25389	−0.238912	−0.119456	−	0.992840i	$-0.538115\pi$
$89$	−2.25389	−0.238912	−0.119456	+	0.992840i	$0.538115\pi$
$90$	0	0
$91$	16.1247	1.69033
$92$	0	0
$93$	−4.41944	−0.458274
$94$	0	0
$95$	2.63403	0.270246
$96$	0	0
$97$	9.27281	0.941511	0.470756	−	0.882264i	$-0.343981\pi$
$97$	9.27281	0.941511	0.470756	+	0.882264i	$0.343981\pi$
$98$	0	0
$99$	−23.5087	−2.36272
$100$	0	0
$101$	−13.3620	−1.32957	−0.664785	−	0.747034i	$-0.731477\pi$
$101$	−13.3620	−1.32957	−0.664785	+	0.747034i	$0.731477\pi$
$102$	0	0
$103$	−10.0094	−0.986255	−0.493127	−	0.869957i	$-0.664146\pi$
$103$	−10.0094	−0.986255	−0.493127	+	0.869957i	$0.664146\pi$
$104$	0	0
$105$	−26.9861	−2.63357
$106$	0	0
$107$	4.85189	0.469050	0.234525	−	0.972110i	$-0.424646\pi$
$107$	4.85189	0.469050	0.234525	+	0.972110i	$0.424646\pi$
$108$	0	0
$109$	15.1198	1.44822	0.724108	−	0.689687i	$-0.242252\pi$
$109$	15.1198	1.44822	0.724108	+	0.689687i	$0.242252\pi$
$110$	0	0
$111$	21.4723	2.03806
$112$	0	0
$113$	−3.10175	−0.291789	−0.145894	−	0.989300i	$-0.546606\pi$
$113$	−3.10175	−0.291789	−0.145894	+	0.989300i	$0.546606\pi$
$114$	0	0
$115$	−10.9297	−1.01920
$116$	0	0
$117$	16.3928	1.51551
$118$	0	0
$119$	7.10532	0.651344
$120$	0	0
$121$	24.4100	2.21909
$122$	0	0
$123$	1.35564	0.122234
$124$	0	0
$125$	−8.06506	−0.721361
$126$	0	0
$127$	3.59176	0.318717	0.159358	−	0.987221i	$-0.449057\pi$
$127$	3.59176	0.318717	0.159358	+	0.987221i	$0.449057\pi$
$128$	0	0
$129$	28.9032	2.54479
$130$	0	0
$131$	9.76906	0.853527	0.426763	−	0.904363i	$-0.359654\pi$
$131$	9.76906	0.853527	0.426763	+	0.904363i	$0.359654\pi$
$132$	0	0
$133$	3.88603	0.336962
$134$	0	0
$135$	−6.60154	−0.568170
$136$	0	0
$137$	4.25690	0.363692	0.181846	−	0.983327i	$-0.441793\pi$
$137$	4.25690	0.363692	0.181846	+	0.983327i	$0.441793\pi$
$138$	0	0
$139$	9.50875	0.806521	0.403261	−	0.915085i	$-0.367877\pi$
$139$	9.50875	0.806521	0.403261	+	0.915085i	$0.367877\pi$
$140$	0	0
$141$	35.0734	2.95371
$142$	0	0
$143$	−24.6916	−2.06481
$144$	0	0
$145$	−21.5895	−1.79291
$146$	0	0
$147$	−21.3581	−1.76159
$148$	0	0
$149$	5.13803	0.420924	0.210462	−	0.977602i	$-0.432503\pi$
$149$	5.13803	0.420924	0.210462	+	0.977602i	$0.432503\pi$
$150$	0	0
$151$	5.26807	0.428709	0.214355	−	0.976756i	$-0.431235\pi$
$151$	5.26807	0.428709	0.214355	+	0.976756i	$0.431235\pi$
$152$	0	0
$153$	7.22344	0.583981
$154$	0	0
$155$	4.41546	0.354659
$156$	0	0
$157$	7.12828	0.568899	0.284449	−	0.958691i	$-0.408189\pi$
$157$	7.12828	0.568899	0.284449	+	0.958691i	$0.408189\pi$
$158$	0	0
$159$	−4.28273	−0.339643
$160$	0	0
$161$	−16.1247	−1.27080
$162$	0	0
$163$	12.4060	0.971712	0.485856	−	0.874039i	$-0.338508\pi$
$163$	12.4060	0.971712	0.485856	+	0.874039i	$0.338508\pi$
$164$	0	0
$165$	41.3234	3.21703
$166$	0	0
$167$	−8.80460	−0.681320	−0.340660	−	0.940187i	$-0.610651\pi$
$167$	−8.80460	−0.681320	−0.340660	+	0.940187i	$0.610651\pi$
$168$	0	0
$169$	4.21753	0.324426
$170$	0	0
$171$	3.95063	0.302112
$172$	0	0
$173$	1.52203	0.115718	0.0578590	−	0.998325i	$-0.481573\pi$
$173$	1.52203	0.115718	0.0578590	+	0.998325i	$0.481573\pi$
$174$	0	0
$175$	7.53165	0.569339
$176$	0	0
$177$	32.2259	2.42225
$178$	0	0
$179$	−18.6599	−1.39470	−0.697352	−	0.716729i	$-0.745639\pi$
$179$	−18.6599	−1.39470	−0.697352	+	0.716729i	$0.745639\pi$
$180$	0	0
$181$	11.3132	0.840907	0.420453	−	0.907314i	$-0.361871\pi$
$181$	11.3132	0.840907	0.420453	+	0.907314i	$0.361871\pi$
$182$	0	0
$183$	15.7833	1.16673
$184$	0	0
$185$	−21.4530	−1.57725
$186$	0	0
$187$	−10.8803	−0.795646
$188$	0	0
$189$	−9.73936	−0.708434
$190$	0	0
$191$	11.5963	0.839078	0.419539	−	0.907737i	$-0.362192\pi$
$191$	11.5963	0.839078	0.419539	+	0.907737i	$0.362192\pi$
$192$	0	0
$193$	−3.78701	−0.272595	−0.136298	−	0.990668i	$-0.543520\pi$
$193$	−3.78701	−0.272595	−0.136298	+	0.990668i	$0.543520\pi$
$194$	0	0
$195$	−28.8150	−2.06349
$196$	0	0
$197$	−26.0803	−1.85814	−0.929071	−	0.369902i	$-0.879391\pi$
$197$	−26.0803	−1.85814	−0.929071	+	0.369902i	$0.879391\pi$
$198$	0	0
$199$	−3.53452	−0.250555	−0.125278	−	0.992122i	$-0.539982\pi$
$199$	−3.53452	−0.250555	−0.125278	+	0.992122i	$0.539982\pi$
$200$	0	0
$201$	−10.7916	−0.761178
$202$	0	0
$203$	−31.8514	−2.23553
$204$	0	0
$205$	−1.35442	−0.0945970
$206$	0	0
$207$	−16.3928	−1.13938
$208$	0	0
$209$	−5.95063	−0.411614
$210$	0	0
$211$	−2.84939	−0.196160	−0.0980802	−	0.995179i	$-0.531270\pi$
$211$	−2.84939	−0.196160	−0.0980802	+	0.995179i	$0.531270\pi$
$212$	0	0
$213$	−31.2931	−2.14417
$214$	0	0
$215$	−28.8772	−1.96941
$216$	0	0
$217$	6.51420	0.442213
$218$	0	0
$219$	24.4798	1.65419
$220$	0	0
$221$	7.58688	0.510349
$222$	0	0
$223$	−23.4383	−1.56954	−0.784772	−	0.619785i	$-0.787220\pi$
$223$	−23.4383	−1.56954	−0.784772	+	0.619785i	$0.787220\pi$
$224$	0	0
$225$	7.65685	0.510457
$226$	0	0
$227$	−7.43341	−0.493373	−0.246686	−	0.969095i	$-0.579342\pi$
$227$	−7.43341	−0.493373	−0.246686	+	0.969095i	$0.579342\pi$
$228$	0	0
$229$	−19.2316	−1.27086	−0.635431	−	0.772157i	$-0.719178\pi$
$229$	−19.2316	−1.27086	−0.635431	+	0.772157i	$0.719178\pi$
$230$	0	0
$231$	60.9651	4.01121
$232$	0	0
$233$	−29.9306	−1.96082	−0.980411	−	0.196965i	$-0.936892\pi$
$233$	−29.9306	−1.96082	−0.980411	+	0.196965i	$0.936892\pi$
$234$	0	0
$235$	−35.0419	−2.28588
$236$	0	0
$237$	6.38336	0.414644
$238$	0	0
$239$	16.0479	1.03805	0.519026	−	0.854758i	$-0.326295\pi$
$239$	16.0479	1.03805	0.519026	+	0.854758i	$0.326295\pi$
$240$	0	0
$241$	23.9168	1.54061	0.770307	−	0.637673i	$-0.220103\pi$
$241$	23.9168	1.54061	0.770307	+	0.637673i	$0.220103\pi$
$242$	0	0
$243$	21.3451	1.36929
$244$	0	0
$245$	21.3389	1.36329
$246$	0	0
$247$	4.14940	0.264020
$248$	0	0
$249$	2.60346	0.164988
$250$	0	0
$251$	−10.1950	−0.643505	−0.321753	−	0.946824i	$-0.604272\pi$
$251$	−10.1950	−0.643505	−0.321753	+	0.946824i	$0.604272\pi$
$252$	0	0
$253$	24.6916	1.55234
$254$	0	0
$255$	−12.6973	−0.795135
$256$	0	0
$257$	20.1866	1.25920	0.629602	−	0.776918i	$-0.283218\pi$
$257$	20.1866	1.25920	0.629602	+	0.776918i	$0.283218\pi$
$258$	0	0
$259$	−31.6499	−1.96663
$260$	0	0
$261$	−32.3809	−2.00432
$262$	0	0
$263$	5.32659	0.328451	0.164226	−	0.986423i	$-0.447487\pi$
$263$	5.32659	0.328451	0.164226	+	0.986423i	$0.447487\pi$
$264$	0	0
$265$	4.27888	0.262850
$266$	0	0
$267$	5.94216	0.363654
$268$	0	0
$269$	−14.0564	−0.857032	−0.428516	−	0.903534i	$-0.640963\pi$
$269$	−14.0564	−0.857032	−0.428516	+	0.903534i	$0.640963\pi$
$270$	0	0
$271$	8.56884	0.520520	0.260260	−	0.965539i	$-0.416192\pi$
$271$	8.56884	0.520520	0.260260	+	0.965539i	$0.416192\pi$
$272$	0	0
$273$	−42.5112	−2.57290
$274$	0	0
$275$	−11.5331	−0.695474
$276$	0	0
$277$	−22.7013	−1.36399	−0.681995	−	0.731357i	$-0.738887\pi$
$277$	−22.7013	−1.36399	−0.681995	+	0.731357i	$0.738887\pi$
$278$	0	0
$279$	6.62249	0.396478
$280$	0	0
$281$	8.26940	0.493311	0.246656	−	0.969103i	$-0.420668\pi$
$281$	8.26940	0.493311	0.246656	+	0.969103i	$0.420668\pi$
$282$	0	0
$283$	−21.7941	−1.29552	−0.647761	−	0.761844i	$-0.724294\pi$
$283$	−21.7941	−1.29552	−0.647761	+	0.761844i	$0.724294\pi$
$284$	0	0
$285$	−6.94438	−0.411350
$286$	0	0
$287$	−1.99820	−0.117950
$288$	0	0
$289$	−13.6569	−0.803344
$290$	0	0
$291$	−24.4469	−1.43310
$292$	0	0
$293$	8.64036	0.504775	0.252388	−	0.967626i	$-0.418784\pi$
$293$	8.64036	0.504775	0.252388	+	0.967626i	$0.418784\pi$
$294$	0	0
$295$	−32.1970	−1.87458
$296$	0	0
$297$	14.9138	0.865384
$298$	0	0
$299$	−17.2175	−0.995716
$300$	0	0
$301$	−42.6031	−2.45560
$302$	0	0
$303$	35.2277	2.02378
$304$	0	0
$305$	−15.7691	−0.902934
$306$	0	0
$307$	11.3871	0.649894	0.324947	−	0.945732i	$-0.394654\pi$
$307$	11.3871	0.649894	0.324947	+	0.945732i	$0.394654\pi$
$308$	0	0
$309$	26.3888	1.50121
$310$	0	0
$311$	−11.3199	−0.641893	−0.320947	−	0.947097i	$-0.604001\pi$
$311$	−11.3199	−0.641893	−0.320947	+	0.947097i	$0.604001\pi$
$312$	0	0
$313$	18.3052	1.03467	0.517337	−	0.855782i	$-0.326924\pi$
$313$	18.3052	1.03467	0.517337	+	0.855782i	$0.326924\pi$
$314$	0	0
$315$	40.4384	2.27845
$316$	0	0
$317$	26.0999	1.46592	0.732959	−	0.680273i	$-0.238139\pi$
$317$	26.0999	1.46592	0.732959	+	0.680273i	$0.238139\pi$
$318$	0	0
$319$	48.7736	2.73080
$320$	0	0
$321$	−12.7916	−0.713955
$322$	0	0
$323$	1.82843	0.101736
$324$	0	0
$325$	8.04210	0.446095
$326$	0	0
$327$	−39.8619	−2.20437
$328$	0	0
$329$	−51.6978	−2.85019
$330$	0	0
$331$	12.2799	0.674962	0.337481	−	0.941332i	$-0.390425\pi$
$331$	12.2799	0.674962	0.337481	+	0.941332i	$0.390425\pi$
$332$	0	0
$333$	−32.1760	−1.76323
$334$	0	0
$335$	10.7818	0.589075
$336$	0	0
$337$	15.5263	0.845772	0.422886	−	0.906183i	$-0.361017\pi$
$337$	15.5263	0.845772	0.422886	+	0.906183i	$0.361017\pi$
$338$	0	0
$339$	8.17748	0.444140
$340$	0	0
$341$	−9.97512	−0.540183
$342$	0	0
$343$	4.27944	0.231068
$344$	0	0
$345$	28.8150	1.55135
$346$	0	0
$347$	12.5212	0.672176	0.336088	−	0.941831i	$-0.390896\pi$
$347$	12.5212	0.672176	0.336088	+	0.941831i	$0.390896\pi$
$348$	0	0
$349$	1.36237	0.0729259	0.0364630	−	0.999335i	$-0.488391\pi$
$349$	1.36237	0.0729259	0.0364630	+	0.999335i	$0.488391\pi$
$350$	0	0
$351$	−10.3994	−0.555081
$352$	0	0
$353$	−22.3247	−1.18822	−0.594111	−	0.804383i	$-0.702496\pi$
$353$	−22.3247	−1.18822	−0.594111	+	0.804383i	$0.702496\pi$
$354$	0	0
$355$	31.2650	1.65937
$356$	0	0
$357$	−18.7325	−0.991430
$358$	0	0
$359$	16.2824	0.859351	0.429676	−	0.902983i	$-0.358628\pi$
$359$	16.2824	0.859351	0.429676	+	0.902983i	$0.358628\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−64.3547	−3.37774
$364$	0	0
$365$	−24.4578	−1.28018
$366$	0	0
$367$	−29.6214	−1.54623	−0.773113	−	0.634268i	$-0.781302\pi$
$367$	−29.6214	−1.54623	−0.773113	+	0.634268i	$0.781302\pi$
$368$	0	0
$369$	−2.03142	−0.105751
$370$	0	0
$371$	6.31270	0.327739
$372$	0	0
$373$	5.18659	0.268551	0.134276	−	0.990944i	$-0.457129\pi$
$373$	5.18659	0.268551	0.134276	+	0.990944i	$0.457129\pi$
$374$	0	0
$375$	21.2628	1.09800
$376$	0	0
$377$	−34.0101	−1.75161
$378$	0	0
$379$	−11.6485	−0.598344	−0.299172	−	0.954199i	$-0.596710\pi$
$379$	−11.6485	−0.598344	−0.299172	+	0.954199i	$0.596710\pi$
$380$	0	0
$381$	−9.46932	−0.485128
$382$	0	0
$383$	33.3465	1.70393	0.851964	−	0.523600i	$-0.175411\pi$
$383$	33.3465	1.70393	0.851964	+	0.523600i	$0.175411\pi$
$384$	0	0
$385$	−60.9103	−3.10428
$386$	0	0
$387$	−43.3113	−2.20164
$388$	0	0
$389$	−1.51874	−0.0770033	−0.0385016	−	0.999259i	$-0.512258\pi$
$389$	−1.51874	−0.0770033	−0.0385016	+	0.999259i	$0.512258\pi$
$390$	0	0
$391$	−7.58688	−0.383685
$392$	0	0
$393$	−25.7552	−1.29918
$394$	0	0
$395$	−6.37762	−0.320893
$396$	0	0
$397$	29.4592	1.47852	0.739258	−	0.673422i	$-0.235176\pi$
$397$	29.4592	1.47852	0.739258	+	0.673422i	$0.235176\pi$
$398$	0	0
$399$	−10.2452	−0.512899
$400$	0	0
$401$	−32.0523	−1.60062	−0.800308	−	0.599589i	$-0.795331\pi$
$401$	−32.0523	−1.60062	−0.800308	+	0.599589i	$0.795331\pi$
$402$	0	0
$403$	6.95569	0.346488
$404$	0	0
$405$	−13.8139	−0.686421
$406$	0	0
$407$	48.4651	2.40232
$408$	0	0
$409$	−10.7427	−0.531192	−0.265596	−	0.964084i	$-0.585569\pi$
$409$	−10.7427	−0.531192	−0.265596	+	0.964084i	$0.585569\pi$
$410$	0	0
$411$	−11.2229	−0.553586
$412$	0	0
$413$	−47.5007	−2.33736
$414$	0	0
$415$	−2.60112	−0.127684
$416$	0	0
$417$	−25.0689	−1.22763
$418$	0	0
$419$	−19.9990	−0.977013	−0.488507	−	0.872560i	$-0.662458\pi$
$419$	−19.9990	−0.977013	−0.488507	+	0.872560i	$0.662458\pi$
$420$	0	0
$421$	−10.5670	−0.515006	−0.257503	−	0.966277i	$-0.582900\pi$
$421$	−10.5670	−0.515006	−0.257503	+	0.966277i	$0.582900\pi$
$422$	0	0
$423$	−52.5572	−2.55542
$424$	0	0
$425$	3.54374	0.171897
$426$	0	0
$427$	−23.2643	−1.12584
$428$	0	0
$429$	65.0969	3.14291
$430$	0	0
$431$	−3.24892	−0.156495	−0.0782474	−	0.996934i	$-0.524932\pi$
$431$	−3.24892	−0.156495	−0.0782474	+	0.996934i	$0.524932\pi$
$432$	0	0
$433$	−25.4848	−1.22472	−0.612360	−	0.790579i	$-0.709780\pi$
$433$	−25.4848	−1.22472	−0.612360	+	0.790579i	$0.709780\pi$
$434$	0	0
$435$	56.9188	2.72905
$436$	0	0
$437$	−4.14940	−0.198493
$438$	0	0
$439$	−17.9520	−0.856804	−0.428402	−	0.903588i	$-0.640923\pi$
$439$	−17.9520	−0.856804	−0.428402	+	0.903588i	$0.640923\pi$
$440$	0	0
$441$	32.0050	1.52405
$442$	0	0
$443$	1.68123	0.0798777	0.0399388	−	0.999202i	$-0.487284\pi$
$443$	1.68123	0.0798777	0.0399388	+	0.999202i	$0.487284\pi$
$444$	0	0
$445$	−5.93682	−0.281432
$446$	0	0
$447$	−13.5459	−0.640700
$448$	0	0
$449$	−21.8275	−1.03010	−0.515052	−	0.857159i	$-0.672227\pi$
$449$	−21.8275	−1.03010	−0.515052	+	0.857159i	$0.672227\pi$
$450$	0	0
$451$	3.05982	0.144081
$452$	0	0
$453$	−13.8888	−0.652551
$454$	0	0
$455$	42.4730	1.99117
$456$	0	0
$457$	8.04431	0.376297	0.188148	−	0.982141i	$-0.439751\pi$
$457$	8.04431	0.376297	0.188148	+	0.982141i	$0.439751\pi$
$458$	0	0
$459$	−4.58249	−0.213892
$460$	0	0
$461$	−6.79339	−0.316400	−0.158200	−	0.987407i	$-0.550569\pi$
$461$	−6.79339	−0.316400	−0.158200	+	0.987407i	$0.550569\pi$
$462$	0	0
$463$	−31.2479	−1.45221	−0.726107	−	0.687582i	$-0.758672\pi$
$463$	−31.2479	−1.45221	−0.726107	+	0.687582i	$0.758672\pi$
$464$	0	0
$465$	−11.6409	−0.539836
$466$	0	0
$467$	36.9113	1.70805	0.854026	−	0.520230i	$-0.174154\pi$
$467$	36.9113	1.70805	0.854026	+	0.520230i	$0.174154\pi$
$468$	0	0
$469$	15.9066	0.734500
$470$	0	0
$471$	−18.7930	−0.865938
$472$	0	0
$473$	65.2375	2.99962
$474$	0	0
$475$	1.93813	0.0889277
$476$	0	0
$477$	6.41764	0.293844
$478$	0	0
$479$	7.26161	0.331792	0.165896	−	0.986143i	$-0.446948\pi$
$479$	7.26161	0.331792	0.165896	+	0.986143i	$0.446948\pi$
$480$	0	0
$481$	−33.7949	−1.54091
$482$	0	0
$483$	42.5112	1.93433
$484$	0	0
$485$	24.4249	1.10908
$486$	0	0
$487$	−1.00500	−0.0455411	−0.0227705	−	0.999741i	$-0.507249\pi$
$487$	−1.00500	−0.0455411	−0.0227705	+	0.999741i	$0.507249\pi$
$488$	0	0
$489$	−32.7072	−1.47907
$490$	0	0
$491$	−9.12714	−0.411902	−0.205951	−	0.978562i	$-0.566029\pi$
$491$	−9.12714	−0.411902	−0.205951	+	0.978562i	$0.566029\pi$
$492$	0	0
$493$	−14.9865	−0.674957
$494$	0	0
$495$	−61.9228	−2.78322
$496$	0	0
$497$	46.1257	2.06902
$498$	0	0
$499$	19.8928	0.890524	0.445262	−	0.895400i	$-0.353111\pi$
$499$	19.8928	0.890524	0.445262	+	0.895400i	$0.353111\pi$
$500$	0	0
$501$	23.2125	1.03706
$502$	0	0
$503$	−18.9716	−0.845904	−0.422952	−	0.906152i	$-0.639006\pi$
$503$	−18.9716	−0.845904	−0.422952	+	0.906152i	$0.639006\pi$
$504$	0	0
$505$	−35.1960	−1.56620
$506$	0	0
$507$	−11.1191	−0.493818
$508$	0	0
$509$	24.2981	1.07700	0.538498	−	0.842627i	$-0.318992\pi$
$509$	24.2981	1.07700	0.538498	+	0.842627i	$0.318992\pi$
$510$	0	0
$511$	−36.0830	−1.59622
$512$	0	0
$513$	−2.50625	−0.110654
$514$	0	0
$515$	−26.3651	−1.16178
$516$	0	0
$517$	79.1642	3.48164
$518$	0	0
$519$	−4.01270	−0.176138
$520$	0	0
$521$	−10.1930	−0.446563	−0.223282	−	0.974754i	$-0.571677\pi$
$521$	−10.1930	−0.446563	−0.223282	+	0.974754i	$0.571677\pi$
$522$	0	0
$523$	1.42683	0.0623907	0.0311954	−	0.999513i	$-0.490069\pi$
$523$	1.42683	0.0623907	0.0311954	+	0.999513i	$0.490069\pi$
$524$	0	0
$525$	−19.8565	−0.866608
$526$	0	0
$527$	3.06501	0.133514
$528$	0	0
$529$	−5.78247	−0.251412
$530$	0	0
$531$	−48.2903	−2.09562
$532$	0	0
$533$	−2.13363	−0.0924176
$534$	0	0
$535$	12.7801	0.552530
$536$	0	0
$537$	49.1950	2.12292
$538$	0	0
$539$	−48.2075	−2.07644
$540$	0	0
$541$	−28.1257	−1.20922	−0.604610	−	0.796522i	$-0.706671\pi$
$541$	−28.1257	−1.20922	−0.604610	+	0.796522i	$0.706671\pi$
$542$	0	0
$543$	−29.8263	−1.27997
$544$	0	0
$545$	39.8261	1.70596
$546$	0	0
$547$	45.2937	1.93662	0.968310	−	0.249753i	$-0.0803493\pi$
$547$	45.2937	1.93662	0.968310	+	0.249753i	$0.0803493\pi$
$548$	0	0
$549$	−23.6511	−1.00940
$550$	0	0
$551$	−8.19638	−0.349177
$552$	0	0
$553$	−9.40900	−0.400111
$554$	0	0
$555$	56.5587	2.40078
$556$	0	0
$557$	−13.9151	−0.589601	−0.294801	−	0.955559i	$-0.595253\pi$
$557$	−13.9151	−0.589601	−0.294801	+	0.955559i	$0.595253\pi$
$558$	0	0
$559$	−45.4904	−1.92404
$560$	0	0
$561$	28.6849	1.21108
$562$	0	0
$563$	10.3367	0.435641	0.217820	−	0.975989i	$-0.430105\pi$
$563$	10.3367	0.435641	0.217820	+	0.975989i	$0.430105\pi$
$564$	0	0
$565$	−8.17013	−0.343720
$566$	0	0
$567$	−20.3799	−0.855877
$568$	0	0
$569$	39.2369	1.64490	0.822448	−	0.568840i	$-0.192608\pi$
$569$	39.2369	1.64490	0.822448	+	0.568840i	$0.192608\pi$
$570$	0	0
$571$	33.4374	1.39931	0.699657	−	0.714479i	$-0.253336\pi$
$571$	33.4374	1.39931	0.699657	+	0.714479i	$0.253336\pi$
$572$	0	0
$573$	−30.5725	−1.27718
$574$	0	0
$575$	−8.04210	−0.335379
$576$	0	0
$577$	−22.3203	−0.929206	−0.464603	−	0.885519i	$-0.653803\pi$
$577$	−22.3203	−0.929206	−0.464603	+	0.885519i	$0.653803\pi$
$578$	0	0
$579$	9.98409	0.414925
$580$	0	0
$581$	−3.83747	−0.159205
$582$	0	0
$583$	−9.66656	−0.400348
$584$	0	0
$585$	43.1791	1.78523
$586$	0	0
$587$	43.6753	1.80267	0.901337	−	0.433119i	$-0.142587\pi$
$587$	43.6753	1.80267	0.901337	+	0.433119i	$0.142587\pi$
$588$	0	0
$589$	1.67631	0.0690712
$590$	0	0
$591$	68.7581	2.82833
$592$	0	0
$593$	−19.6978	−0.808890	−0.404445	−	0.914562i	$-0.632535\pi$
$593$	−19.6978	−0.808890	−0.404445	+	0.914562i	$0.632535\pi$
$594$	0	0
$595$	18.7157	0.767267
$596$	0	0
$597$	9.31842	0.381377
$598$	0	0
$599$	−34.2570	−1.39970	−0.699851	−	0.714289i	$-0.746750\pi$
$599$	−34.2570	−1.39970	−0.699851	+	0.714289i	$0.746750\pi$
$600$	0	0
$601$	−19.1801	−0.782373	−0.391186	−	0.920311i	$-0.627935\pi$
$601$	−19.1801	−0.782373	−0.391186	+	0.920311i	$0.627935\pi$
$602$	0	0
$603$	16.1711	0.658536
$604$	0	0
$605$	64.2968	2.61404
$606$	0	0
$607$	−17.1796	−0.697297	−0.348649	−	0.937253i	$-0.613359\pi$
$607$	−17.1796	−0.697297	−0.348649	+	0.937253i	$0.613359\pi$
$608$	0	0
$609$	83.9731	3.40276
$610$	0	0
$611$	−55.2016	−2.23322
$612$	0	0
$613$	13.4502	0.543247	0.271623	−	0.962404i	$-0.412439\pi$
$613$	13.4502	0.543247	0.271623	+	0.962404i	$0.412439\pi$
$614$	0	0
$615$	3.57081	0.143989
$616$	0	0
$617$	−15.2065	−0.612191	−0.306095	−	0.952001i	$-0.599023\pi$
$617$	−15.2065	−0.612191	−0.306095	+	0.952001i	$0.599023\pi$
$618$	0	0
$619$	40.1696	1.61455	0.807277	−	0.590173i	$-0.200940\pi$
$619$	40.1696	1.61455	0.807277	+	0.590173i	$0.200940\pi$
$620$	0	0
$621$	10.3994	0.417315
$622$	0	0
$623$	−8.75868	−0.350909
$624$	0	0
$625$	−30.9343	−1.23737
$626$	0	0
$627$	15.6883	0.626529
$628$	0	0
$629$	−14.8917	−0.593770
$630$	0	0
$631$	12.0318	0.478980	0.239490	−	0.970899i	$-0.423020\pi$
$631$	12.0318	0.478980	0.239490	+	0.970899i	$0.423020\pi$
$632$	0	0
$633$	7.51216	0.298581
$634$	0	0
$635$	9.46081	0.375441
$636$	0	0
$637$	33.6153	1.33189
$638$	0	0
$639$	46.8925	1.85504
$640$	0	0
$641$	28.1707	1.11267	0.556337	−	0.830957i	$-0.312206\pi$
$641$	28.1707	1.11267	0.556337	+	0.830957i	$0.312206\pi$
$642$	0	0
$643$	38.2190	1.50721	0.753605	−	0.657327i	$-0.228313\pi$
$643$	38.2190	1.50721	0.753605	+	0.657327i	$0.228313\pi$
$644$	0	0
$645$	76.1321	2.99770
$646$	0	0
$647$	7.26170	0.285487	0.142743	−	0.989760i	$-0.454408\pi$
$647$	7.26170	0.285487	0.142743	+	0.989760i	$0.454408\pi$
$648$	0	0
$649$	72.7372	2.85518
$650$	0	0
$651$	−17.1741	−0.673105
$652$	0	0
$653$	16.0972	0.629932	0.314966	−	0.949103i	$-0.398007\pi$
$653$	16.0972	0.629932	0.314966	+	0.949103i	$0.398007\pi$
$654$	0	0
$655$	25.7320	1.00543
$656$	0	0
$657$	−36.6828	−1.43113
$658$	0	0
$659$	−17.8679	−0.696035	−0.348018	−	0.937488i	$-0.613145\pi$
$659$	−17.8679	−0.696035	−0.348018	+	0.937488i	$0.613145\pi$
$660$	0	0
$661$	−21.6386	−0.841644	−0.420822	−	0.907143i	$-0.638258\pi$
$661$	−21.6386	−0.841644	−0.420822	+	0.907143i	$0.638258\pi$
$662$	0	0
$663$	−20.0021	−0.776817
$664$	0	0
$665$	10.2359	0.396933
$666$	0	0
$667$	34.0101	1.31687
$668$	0	0
$669$	61.7928	2.38905
$670$	0	0
$671$	35.6244	1.37526
$672$	0	0
$673$	−43.9881	−1.69561	−0.847807	−	0.530304i	$-0.822078\pi$
$673$	−43.9881	−1.69561	−0.847807	+	0.530304i	$0.822078\pi$
$674$	0	0
$675$	−4.85745	−0.186963
$676$	0	0
$677$	−29.6983	−1.14140	−0.570699	−	0.821160i	$-0.693328\pi$
$677$	−29.6983	−1.14140	−0.570699	+	0.821160i	$0.693328\pi$
$678$	0	0
$679$	36.0344	1.38287
$680$	0	0
$681$	19.5975	0.750977
$682$	0	0
$683$	−6.20351	−0.237371	−0.118685	−	0.992932i	$-0.537868\pi$
$683$	−6.20351	−0.237371	−0.118685	+	0.992932i	$0.537868\pi$
$684$	0	0
$685$	11.2128	0.428420
$686$	0	0
$687$	50.7024	1.93442
$688$	0	0
$689$	6.74053	0.256794
$690$	0	0
$691$	−28.1172	−1.06963	−0.534814	−	0.844970i	$-0.679618\pi$
$691$	−28.1172	−1.06963	−0.534814	+	0.844970i	$0.679618\pi$
$692$	0	0
$693$	−91.3557	−3.47032
$694$	0	0
$695$	25.0464	0.950063
$696$	0	0
$697$	−0.940179	−0.0356118
$698$	0	0
$699$	78.9093	2.98462
$700$	0	0
$701$	−8.35403	−0.315527	−0.157764	−	0.987477i	$-0.550428\pi$
$701$	−8.35403	−0.315527	−0.157764	+	0.987477i	$0.550428\pi$
$702$	0	0
$703$	−8.14452	−0.307177
$704$	0	0
$705$	92.3846	3.47940
$706$	0	0
$707$	−51.9252	−1.95285
$708$	0	0
$709$	31.7187	1.19122	0.595610	−	0.803274i	$-0.296910\pi$
$709$	31.7187	1.19122	0.595610	+	0.803274i	$0.296910\pi$
$710$	0	0
$711$	−9.56541	−0.358731
$712$	0	0
$713$	−6.95569	−0.260493
$714$	0	0
$715$	−65.0384	−2.43230
$716$	0	0
$717$	−42.3088	−1.58005
$718$	0	0
$719$	33.7972	1.26042	0.630211	−	0.776424i	$-0.282968\pi$
$719$	33.7972	1.26042	0.630211	+	0.776424i	$0.282968\pi$
$720$	0	0
$721$	−38.8968	−1.44859
$722$	0	0
$723$	−63.0543	−2.34501
$724$	0	0
$725$	−15.8857	−0.589979
$726$	0	0
$727$	32.8637	1.21885	0.609423	−	0.792845i	$-0.291401\pi$
$727$	32.8637	1.21885	0.609423	+	0.792845i	$0.291401\pi$
$728$	0	0
$729$	−40.5412	−1.50153
$730$	0	0
$731$	−20.0453	−0.741401
$732$	0	0
$733$	14.3118	0.528618	0.264309	−	0.964438i	$-0.414856\pi$
$733$	14.3118	0.528618	0.264309	+	0.964438i	$0.414856\pi$
$734$	0	0
$735$	−56.2581	−2.07511
$736$	0	0
$737$	−24.3576	−0.897225
$738$	0	0
$739$	−22.6101	−0.831726	−0.415863	−	0.909427i	$-0.636520\pi$
$739$	−22.6101	−0.831726	−0.415863	+	0.909427i	$0.636520\pi$
$740$	0	0
$741$	−10.9395	−0.401873
$742$	0	0
$743$	12.6222	0.463062	0.231531	−	0.972828i	$-0.425627\pi$
$743$	12.6222	0.463062	0.231531	+	0.972828i	$0.425627\pi$
$744$	0	0
$745$	13.5337	0.495838
$746$	0	0
$747$	−3.90126	−0.142740
$748$	0	0
$749$	18.8546	0.688932
$750$	0	0
$751$	−4.69943	−0.171485	−0.0857423	−	0.996317i	$-0.527326\pi$
$751$	−4.69943	−0.171485	−0.0857423	+	0.996317i	$0.527326\pi$
$752$	0	0
$753$	26.8783	0.979498
$754$	0	0
$755$	13.8763	0.505009
$756$	0	0
$757$	−21.8920	−0.795679	−0.397839	−	0.917455i	$-0.630240\pi$
$757$	−21.8920	−0.795679	−0.397839	+	0.917455i	$0.630240\pi$
$758$	0	0
$759$	−65.0969	−2.36287
$760$	0	0
$761$	−0.164985	−0.00598068	−0.00299034	−	0.999996i	$-0.500952\pi$
$761$	−0.164985	−0.00598068	−0.00299034	+	0.999996i	$0.500952\pi$
$762$	0	0
$763$	58.7560	2.12711
$764$	0	0
$765$	19.0268	0.687915
$766$	0	0
$767$	−50.7200	−1.83139
$768$	0	0
$769$	−8.55812	−0.308614	−0.154307	−	0.988023i	$-0.549314\pi$
$769$	−8.55812	−0.308614	−0.154307	+	0.988023i	$0.549314\pi$
$770$	0	0
$771$	−53.2200	−1.91667
$772$	0	0
$773$	−9.49557	−0.341532	−0.170766	−	0.985312i	$-0.554624\pi$
$773$	−9.49557	−0.341532	−0.170766	+	0.985312i	$0.554624\pi$
$774$	0	0
$775$	3.24892	0.116705
$776$	0	0
$777$	83.4419	2.99346
$778$	0	0
$779$	−0.514201	−0.0184232
$780$	0	0
$781$	−70.6317	−2.52740
$782$	0	0
$783$	20.5422	0.734117
$784$	0	0
$785$	18.7761	0.670149
$786$	0	0
$787$	8.59840	0.306500	0.153250	−	0.988187i	$-0.451026\pi$
$787$	8.59840	0.306500	0.153250	+	0.988187i	$0.451026\pi$
$788$	0	0
$789$	−14.0430	−0.499945
$790$	0	0
$791$	−12.0535	−0.428574
$792$	0	0
$793$	−24.8410	−0.882131
$794$	0	0
$795$	−11.2809	−0.400091
$796$	0	0
$797$	−23.3568	−0.827340	−0.413670	−	0.910427i	$-0.635753\pi$
$797$	−23.3568	−0.827340	−0.413670	+	0.910427i	$0.635753\pi$
$798$	0	0
$799$	−24.3245	−0.860538
$800$	0	0
$801$	−8.90428	−0.314617
$802$	0	0
$803$	55.2534	1.94985
$804$	0	0
$805$	−42.4730	−1.49698
$806$	0	0
$807$	37.0583	1.30451
$808$	0	0
$809$	−11.0634	−0.388968	−0.194484	−	0.980906i	$-0.562303\pi$
$809$	−11.0634	−0.388968	−0.194484	+	0.980906i	$0.562303\pi$
$810$	0	0
$811$	−12.9630	−0.455193	−0.227596	−	0.973756i	$-0.573087\pi$
$811$	−12.9630	−0.455193	−0.227596	+	0.973756i	$0.573087\pi$
$812$	0	0
$813$	−22.5909	−0.792299
$814$	0	0
$815$	32.6778	1.14465
$816$	0	0
$817$	−10.9631	−0.383551
$818$	0	0
$819$	63.7028	2.22595
$820$	0	0
$821$	14.8255	0.517415	0.258707	−	0.965956i	$-0.416703\pi$
$821$	14.8255	0.517415	0.258707	+	0.965956i	$0.416703\pi$
$822$	0	0
$823$	4.04240	0.140909	0.0704547	−	0.997515i	$-0.477555\pi$
$823$	4.04240	0.140909	0.0704547	+	0.997515i	$0.477555\pi$
$824$	0	0
$825$	30.4060	1.05860
$826$	0	0
$827$	33.6070	1.16863	0.584314	−	0.811527i	$-0.301364\pi$
$827$	33.6070	1.16863	0.584314	+	0.811527i	$0.301364\pi$
$828$	0	0
$829$	38.3399	1.33160	0.665800	−	0.746130i	$-0.268090\pi$
$829$	38.3399	1.33160	0.665800	+	0.746130i	$0.268090\pi$
$830$	0	0
$831$	59.8498	2.07617
$832$	0	0
$833$	14.8125	0.513223
$834$	0	0
$835$	−23.1916	−0.802579
$836$	0	0
$837$	−4.20125	−0.145217
$838$	0	0
$839$	−37.6832	−1.30097	−0.650484	−	0.759520i	$-0.725434\pi$
$839$	−37.6832	−1.30097	−0.650484	+	0.759520i	$0.725434\pi$
$840$	0	0
$841$	38.1806	1.31657
$842$	0	0
$843$	−21.8015	−0.750883
$844$	0	0
$845$	11.1091	0.382165
$846$	0	0
$847$	94.8581	3.25936
$848$	0	0
$849$	57.4580	1.97195
$850$	0	0
$851$	33.7949	1.15847
$852$	0	0
$853$	7.06647	0.241951	0.120976	−	0.992655i	$-0.461398\pi$
$853$	7.06647	0.241951	0.120976	+	0.992655i	$0.461398\pi$
$854$	0	0
$855$	10.4061	0.355881
$856$	0	0
$857$	7.95910	0.271878	0.135939	−	0.990717i	$-0.456595\pi$
$857$	7.95910	0.271878	0.135939	+	0.990717i	$0.456595\pi$
$858$	0	0
$859$	−13.1466	−0.448558	−0.224279	−	0.974525i	$-0.572003\pi$
$859$	−13.1466	−0.448558	−0.224279	+	0.974525i	$0.572003\pi$
$860$	0	0
$861$	5.26807	0.179535
$862$	0	0
$863$	34.9886	1.19102	0.595512	−	0.803346i	$-0.296949\pi$
$863$	34.9886	1.19102	0.595512	+	0.803346i	$0.296949\pi$
$864$	0	0
$865$	4.00909	0.136313
$866$	0	0
$867$	36.0050	1.22279
$868$	0	0
$869$	14.4079	0.488754
$870$	0	0
$871$	16.9847	0.575504
$872$	0	0
$873$	36.6335	1.23985
$874$	0	0
$875$	−31.3411	−1.05952
$876$	0	0
$877$	49.1816	1.66075	0.830373	−	0.557208i	$-0.188127\pi$
$877$	49.1816	1.66075	0.830373	+	0.557208i	$0.188127\pi$
$878$	0	0
$879$	−22.7795	−0.768333
$880$	0	0
$881$	−7.58658	−0.255598	−0.127799	−	0.991800i	$-0.540791\pi$
$881$	−7.58658	−0.255598	−0.127799	+	0.991800i	$0.540791\pi$
$882$	0	0
$883$	−15.1875	−0.511101	−0.255551	−	0.966796i	$-0.582257\pi$
$883$	−15.1875	−0.511101	−0.255551	+	0.966796i	$0.582257\pi$
$884$	0	0
$885$	84.8842	2.85335
$886$	0	0
$887$	32.9000	1.10467	0.552337	−	0.833621i	$-0.313736\pi$
$887$	32.9000	1.10467	0.552337	+	0.833621i	$0.313736\pi$
$888$	0	0
$889$	13.9577	0.468125
$890$	0	0
$891$	31.2075	1.04549
$892$	0	0
$893$	−13.3035	−0.445185
$894$	0	0
$895$	−49.1507	−1.64293
$896$	0	0
$897$	45.3924	1.51561
$898$	0	0
$899$	−13.7397	−0.458244
$900$	0	0
$901$	2.97021	0.0989519
$902$	0	0
$903$	112.319	3.73774
$904$	0	0
$905$	29.7995	0.990568
$906$	0	0
$907$	55.9392	1.85743	0.928715	−	0.370793i	$-0.120914\pi$
$907$	55.9392	1.85743	0.928715	+	0.370793i	$0.120914\pi$
$908$	0	0
$909$	−52.7884	−1.75088
$910$	0	0
$911$	−44.6551	−1.47949	−0.739745	−	0.672887i	$-0.765054\pi$
$911$	−44.6551	−1.47949	−0.739745	+	0.672887i	$0.765054\pi$
$912$	0	0
$913$	5.87627	0.194476
$914$	0	0
$915$	41.5736	1.37438
$916$	0	0
$917$	37.9629	1.25364
$918$	0	0
$919$	7.52507	0.248229	0.124115	−	0.992268i	$-0.460391\pi$
$919$	7.52507	0.248229	0.124115	+	0.992268i	$0.460391\pi$
$920$	0	0
$921$	−30.0209	−0.989222
$922$	0	0
$923$	49.2518	1.62114
$924$	0	0
$925$	−15.7852	−0.519014
$926$	0	0
$927$	−39.5434	−1.29878
$928$	0	0
$929$	−10.4360	−0.342395	−0.171198	−	0.985237i	$-0.554764\pi$
$929$	−10.4360	−0.342395	−0.171198	+	0.985237i	$0.554764\pi$
$930$	0	0
$931$	8.10124	0.265507
$932$	0	0
$933$	29.8439	0.977044
$934$	0	0
$935$	−28.6591	−0.937252
$936$	0	0
$937$	−19.1073	−0.624207	−0.312104	−	0.950048i	$-0.601034\pi$
$937$	−19.1073	−0.624207	−0.312104	+	0.950048i	$0.601034\pi$
$938$	0	0
$939$	−48.2600	−1.57491
$940$	0	0
$941$	43.2146	1.40875	0.704377	−	0.709826i	$-0.251226\pi$
$941$	43.2146	1.40875	0.704377	+	0.709826i	$0.251226\pi$
$942$	0	0
$943$	2.13363	0.0694804
$944$	0	0
$945$	−25.6538	−0.834518
$946$	0	0
$947$	−1.25385	−0.0407446	−0.0203723	−	0.999792i	$-0.506485\pi$
$947$	−1.25385	−0.0407446	−0.0203723	+	0.999792i	$0.506485\pi$
$948$	0	0
$949$	−38.5285	−1.25069
$950$	0	0
$951$	−68.8100	−2.23132
$952$	0	0
$953$	−35.0739	−1.13616	−0.568078	−	0.822975i	$-0.692313\pi$
$953$	−35.0739	−1.13616	−0.568078	+	0.822975i	$0.692313\pi$
$954$	0	0
$955$	30.5450	0.988413
$956$	0	0
$957$	−128.587	−4.15663
$958$	0	0
$959$	16.5425	0.534184
$960$	0	0
$961$	−28.1900	−0.909354
$962$	0	0
$963$	19.1680	0.617681
$964$	0	0
$965$	−9.97512	−0.321110
$966$	0	0
$967$	1.66764	0.0536275	0.0268138	−	0.999640i	$-0.491464\pi$
$967$	1.66764	0.0536275	0.0268138	+	0.999640i	$0.491464\pi$
$968$	0	0
$969$	−4.82047	−0.154856
$970$	0	0
$971$	−16.1136	−0.517110	−0.258555	−	0.965996i	$-0.583246\pi$
$971$	−16.1136	−0.517110	−0.258555	+	0.965996i	$0.583246\pi$
$972$	0	0
$973$	36.9513	1.18460
$974$	0	0
$975$	−21.2022	−0.679015
$976$	0	0
$977$	33.8857	1.08410	0.542051	−	0.840346i	$-0.317648\pi$
$977$	33.8857	1.08410	0.542051	+	0.840346i	$0.317648\pi$
$978$	0	0
$979$	13.4121	0.428651
$980$	0	0
$981$	59.7328	1.90712
$982$	0	0
$983$	9.89492	0.315599	0.157799	−	0.987471i	$-0.449560\pi$
$983$	9.89492	0.315599	0.157799	+	0.987471i	$0.449560\pi$
$984$	0	0
$985$	−68.6963	−2.18885
$986$	0	0
$987$	136.296	4.33836
$988$	0	0
$989$	45.4904	1.44651
$990$	0	0
$991$	−31.3693	−0.996478	−0.498239	−	0.867040i	$-0.666020\pi$
$991$	−31.3693	−0.996478	−0.498239	+	0.867040i	$0.666020\pi$
$992$	0	0
$993$	−32.3747	−1.02738
$994$	0	0
$995$	−9.31004	−0.295148
$996$	0	0
$997$	4.55606	0.144292	0.0721459	−	0.997394i	$-0.477015\pi$
$997$	4.55606	0.144292	0.0721459	+	0.997394i	$0.477015\pi$
$998$	0	0
$999$	20.4122	0.645813

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.bm.1.2		8
4.3	odd	2		4864.2.a.br.1.8		8
8.3	odd	2	inner	4864.2.a.bm.1.1		8
8.5	even	2		4864.2.a.br.1.7		8
16.3	odd	4		2432.2.c.i.1217.15	yes	16
16.5	even	4		2432.2.c.i.1217.16	yes	16
16.11	odd	4		2432.2.c.i.1217.2	yes	16
16.13	even	4		2432.2.c.i.1217.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.i.1217.1	✓	16	16.13	even	4
2432.2.c.i.1217.2	yes	16	16.11	odd	4
2432.2.c.i.1217.15	yes	16	16.3	odd	4
2432.2.c.i.1217.16	yes	16	16.5	even	4
4864.2.a.bm.1.1		8	8.3	odd	2	inner
4864.2.a.bm.1.2		8	1.1	even	1	trivial
4864.2.a.br.1.7		8	8.5	even	2
4864.2.a.br.1.8		8	4.3	odd	2

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)

\(f(q)\)	\(=\)	\(q-2.63640 q^{3} +2.63403 q^{5} +3.88603 q^{7} +3.95063 q^{9} +O(q^{10})\)	\(q-2.63640 q^{3} +2.63403 q^{5} +3.88603 q^{7} +3.95063 q^{9} -5.95063 q^{11} +4.14940 q^{13} -6.94438 q^{15} +1.82843 q^{17} +1.00000 q^{19} -10.2452 q^{21} -4.14940 q^{23} +1.93813 q^{25} -2.50625 q^{27} -8.19638 q^{29} +1.67631 q^{31} +15.6883 q^{33} +10.2359 q^{35} -8.14452 q^{37} -10.9395 q^{39} -0.514201 q^{41} -10.9631 q^{43} +10.4061 q^{45} -13.3035 q^{47} +8.10124 q^{49} -4.82047 q^{51} +1.62446 q^{53} -15.6742 q^{55} -2.63640 q^{57} -12.2234 q^{59} -5.98666 q^{61} +15.3523 q^{63} +10.9297 q^{65} +4.09328 q^{67} +10.9395 q^{69} +11.8696 q^{71} -9.28531 q^{73} -5.10971 q^{75} -23.1243 q^{77} -2.42124 q^{79} -5.24441 q^{81} -0.987504 q^{83} +4.81614 q^{85} +21.6090 q^{87} -2.25389 q^{89} +16.1247 q^{91} -4.41944 q^{93} +2.63403 q^{95} +9.27281 q^{97} -23.5087 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} + 4 q^{9}+O(q^{10}) \) 8 * q - 4 * q^3 + 4 * q^9	\( 8 q - 4 q^{3} + 4 q^{9} - 20 q^{11} - 8 q^{17} + 8 q^{19} + 20 q^{25} - 4 q^{27} + 24 q^{33} - 12 q^{35} + 8 q^{41} - 28 q^{43} + 8 q^{49} - 12 q^{51} - 4 q^{57} - 36 q^{59} + 8 q^{65} - 28 q^{67} - 8 q^{73} - 68 q^{75} - 32 q^{81} - 40 q^{83} - 8 q^{89} + 12 q^{91} + 40 q^{97} - 60 q^{99}+O(q^{100}) \) 8 * q - 4 * q^3 + 4 * q^9 - 20 * q^11 - 8 * q^17 + 8 * q^19 + 20 * q^25 - 4 * q^27 + 24 * q^33 - 12 * q^35 + 8 * q^41 - 28 * q^43 + 8 * q^49 - 12 * q^51 - 4 * q^57 - 36 * q^59 + 8 * q^65 - 28 * q^67 - 8 * q^73 - 68 * q^75 - 32 * q^81 - 40 * q^83 - 8 * q^89 + 12 * q^91 + 40 * q^97 - 60 * q^99

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