LMFDB - Embedded newform 6400.2.a.cm.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6400,2,Mod(1,6400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6400, 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("6400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6400 = 2^{8} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6400.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:	$51.1042572936$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.517638$ of defining polynomial
Character	$\chi$	$=$	6400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{3} -2.44949 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.41421 q^{3} -2.44949 q^{7} -1.00000 q^{9} +3.46410 q^{11} +4.89898 q^{17} -3.46410 q^{19} -3.46410 q^{21} -2.44949 q^{23} -5.65685 q^{27} -4.00000 q^{31} +4.89898 q^{33} -8.48528 q^{37} +4.24264 q^{43} +7.34847 q^{47} -1.00000 q^{49} +6.92820 q^{51} -5.65685 q^{53} -4.89898 q^{57} -10.3923 q^{59} -3.46410 q^{61} +2.44949 q^{63} -4.24264 q^{67} -3.46410 q^{69} -12.0000 q^{71} +4.89898 q^{73} -8.48528 q^{77} -4.00000 q^{79} -5.00000 q^{81} +9.89949 q^{83} +6.00000 q^{89} -5.65685 q^{93} -4.89898 q^{97} -3.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} - 16 q^{31} - 4 q^{49} - 48 q^{71} - 16 q^{79} - 20 q^{81} + 24 q^{89}+O(q^{100}) $ 4 * q - 4 * q^9 - 16 * q^31 - 4 * q^49 - 48 * q^71 - 16 * q^79 - 20 * q^81 + 24 * q^89

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
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.44949	−0.925820	−0.462910	−	0.886405i	$-0.653195\pi$
$7$	−2.44949	−0.925820	−0.462910	+	0.886405i	$0.653195\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.89898	1.18818	0.594089	−	0.804400i	$-0.297513\pi$
$17$	4.89898	1.18818	0.594089	+	0.804400i	$0.297513\pi$
$18$	0	0
$19$	−3.46410	−0.794719	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	−3.46410	−0.794719	−0.397360	+	0.917663i	$0.630073\pi$
$20$	0	0
$21$	−3.46410	−0.755929
$22$	0	0
$23$	−2.44949	−0.510754	−0.255377	−	0.966842i	$-0.582200\pi$
$23$	−2.44949	−0.510754	−0.255377	+	0.966842i	$0.582200\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	4.89898	0.852803
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.48528	−1.39497	−0.697486	−	0.716599i	$-0.745698\pi$
$37$	−8.48528	−1.39497	−0.697486	+	0.716599i	$0.745698\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	4.24264	0.646997	0.323498	−	0.946229i	$-0.395141\pi$
$43$	4.24264	0.646997	0.323498	+	0.946229i	$0.395141\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.34847	1.07188	0.535942	−	0.844255i	$-0.319956\pi$
$47$	7.34847	1.07188	0.535942	+	0.844255i	$0.319956\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	6.92820	0.970143
$52$	0	0
$53$	−5.65685	−0.777029	−0.388514	−	0.921443i	$-0.627012\pi$
$53$	−5.65685	−0.777029	−0.388514	+	0.921443i	$0.627012\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.89898	−0.648886
$58$	0	0
$59$	−10.3923	−1.35296	−0.676481	−	0.736460i	$-0.736496\pi$
$59$	−10.3923	−1.35296	−0.676481	+	0.736460i	$0.736496\pi$
$60$	0	0
$61$	−3.46410	−0.443533	−0.221766	−	0.975100i	$-0.571182\pi$
$61$	−3.46410	−0.443533	−0.221766	+	0.975100i	$0.571182\pi$
$62$	0	0
$63$	2.44949	0.308607
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.24264	−0.518321	−0.259161	−	0.965834i	$-0.583446\pi$
$67$	−4.24264	−0.518321	−0.259161	+	0.965834i	$0.583446\pi$
$68$	0	0
$69$	−3.46410	−0.417029
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	4.89898	0.573382	0.286691	−	0.958023i	$-0.407445\pi$
$73$	4.89898	0.573382	0.286691	+	0.958023i	$0.407445\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.48528	−0.966988
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	9.89949	1.08661	0.543305	−	0.839535i	$-0.317173\pi$
$83$	9.89949	1.08661	0.543305	+	0.839535i	$0.317173\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−5.65685	−0.586588
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−4.89898	−0.497416	−0.248708	−	0.968579i	$-0.580006\pi$
$97$	−4.89898	−0.497416	−0.248708	+	0.968579i	$0.580006\pi$
$98$	0	0
$99$	−3.46410	−0.348155
$100$	0	0
$101$	13.8564	1.37876	0.689382	−	0.724398i	$-0.257882\pi$
$101$	13.8564	1.37876	0.689382	+	0.724398i	$0.257882\pi$
$102$	0	0
$103$	7.34847	0.724066	0.362033	−	0.932165i	$-0.382083\pi$
$103$	7.34847	0.724066	0.362033	+	0.932165i	$0.382083\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.41421	−0.136717	−0.0683586	−	0.997661i	$-0.521776\pi$
$107$	−1.41421	−0.136717	−0.0683586	+	0.997661i	$0.521776\pi$
$108$	0	0
$109$	−3.46410	−0.331801	−0.165900	−	0.986143i	$-0.553053\pi$
$109$	−3.46410	−0.331801	−0.165900	+	0.986143i	$0.553053\pi$
$110$	0	0
$111$	−12.0000	−1.13899
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.0000	−1.10004
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−17.1464	−1.52150	−0.760750	−	0.649045i	$-0.775169\pi$
$127$	−17.1464	−1.52150	−0.760750	+	0.649045i	$0.775169\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	3.46410	0.302660	0.151330	−	0.988483i	$-0.451644\pi$
$131$	3.46410	0.302660	0.151330	+	0.988483i	$0.451644\pi$
$132$	0	0
$133$	8.48528	0.735767
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.79796	0.837096	0.418548	−	0.908195i	$-0.362539\pi$
$137$	9.79796	0.837096	0.418548	+	0.908195i	$0.362539\pi$
$138$	0	0
$139$	10.3923	0.881464	0.440732	−	0.897639i	$-0.354719\pi$
$139$	10.3923	0.881464	0.440732	+	0.897639i	$0.354719\pi$
$140$	0	0
$141$	10.3923	0.875190
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.41421	−0.116642
$148$	0	0
$149$	17.3205	1.41895	0.709476	−	0.704730i	$-0.248932\pi$
$149$	17.3205	1.41895	0.709476	+	0.704730i	$0.248932\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	−4.89898	−0.396059
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−8.48528	−0.677199	−0.338600	−	0.940931i	$-0.609953\pi$
$157$	−8.48528	−0.677199	−0.338600	+	0.940931i	$0.609953\pi$
$158$	0	0
$159$	−8.00000	−0.634441
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	−21.2132	−1.66155	−0.830773	−	0.556611i	$-0.812101\pi$
$163$	−21.2132	−1.66155	−0.830773	+	0.556611i	$0.812101\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.2474	0.947736	0.473868	−	0.880596i	$-0.342857\pi$
$167$	12.2474	0.947736	0.473868	+	0.880596i	$0.342857\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	3.46410	0.264906
$172$	0	0
$173$	2.82843	0.215041	0.107521	−	0.994203i	$-0.465709\pi$
$173$	2.82843	0.215041	0.107521	+	0.994203i	$0.465709\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−14.6969	−1.10469
$178$	0	0
$179$	3.46410	0.258919	0.129460	−	0.991585i	$-0.458676\pi$
$179$	3.46410	0.258919	0.129460	+	0.991585i	$0.458676\pi$
$180$	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
$183$	−4.89898	−0.362143
$184$	0	0
$185$	0	0
$186$	0	0
$187$	16.9706	1.24101
$188$	0	0
$189$	13.8564	1.00791
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	−24.4949	−1.76318	−0.881591	−	0.472015i	$-0.843527\pi$
$193$	−24.4949	−1.76318	−0.881591	+	0.472015i	$0.843527\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.65685	−0.403034	−0.201517	−	0.979485i	$-0.564587\pi$
$197$	−5.65685	−0.403034	−0.201517	+	0.979485i	$0.564587\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	−6.00000	−0.423207
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.44949	0.170251
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−24.2487	−1.66935	−0.834675	−	0.550743i	$-0.814345\pi$
$211$	−24.2487	−1.66935	−0.834675	+	0.550743i	$0.814345\pi$
$212$	0	0
$213$	−16.9706	−1.16280
$214$	0	0
$215$	0	0
$216$	0	0
$217$	9.79796	0.665129
$218$	0	0
$219$	6.92820	0.468165
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−22.0454	−1.47627	−0.738135	−	0.674653i	$-0.764293\pi$
$223$	−22.0454	−1.47627	−0.738135	+	0.674653i	$0.764293\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.41421	−0.0938647	−0.0469323	−	0.998898i	$-0.514945\pi$
$227$	−1.41421	−0.0938647	−0.0469323	+	0.998898i	$0.514945\pi$
$228$	0	0
$229$	−27.7128	−1.83131	−0.915657	−	0.401960i	$-0.868329\pi$
$229$	−27.7128	−1.83131	−0.915657	+	0.401960i	$0.868329\pi$
$230$	0	0
$231$	−12.0000	−0.789542
$232$	0	0
$233$	−14.6969	−0.962828	−0.481414	−	0.876493i	$-0.659877\pi$
$233$	−14.6969	−0.962828	−0.481414	+	0.876493i	$0.659877\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−5.65685	−0.367452
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	0	0
$243$	9.89949	0.635053
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	14.0000	0.887214
$250$	0	0
$251$	−10.3923	−0.655956	−0.327978	−	0.944685i	$-0.606367\pi$
$251$	−10.3923	−0.655956	−0.327978	+	0.944685i	$0.606367\pi$
$252$	0	0
$253$	−8.48528	−0.533465
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.79796	0.611180	0.305590	−	0.952163i	$-0.401146\pi$
$257$	9.79796	0.611180	0.305590	+	0.952163i	$0.401146\pi$
$258$	0	0
$259$	20.7846	1.29149
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.34847	−0.453126	−0.226563	−	0.973997i	$-0.572749\pi$
$263$	−7.34847	−0.453126	−0.226563	+	0.973997i	$0.572749\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.48528	0.519291
$268$	0	0
$269$	−10.3923	−0.633630	−0.316815	−	0.948487i	$-0.602613\pi$
$269$	−10.3923	−0.633630	−0.316815	+	0.948487i	$0.602613\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	25.4558	1.52949	0.764747	−	0.644331i	$-0.222864\pi$
$277$	25.4558	1.52949	0.764747	+	0.644331i	$0.222864\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	0	0
$283$	−21.2132	−1.26099	−0.630497	−	0.776192i	$-0.717149\pi$
$283$	−21.2132	−1.26099	−0.630497	+	0.776192i	$0.717149\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	7.00000	0.411765
$290$	0	0
$291$	−6.92820	−0.406138
$292$	0	0
$293$	19.7990	1.15667	0.578335	−	0.815800i	$-0.303703\pi$
$293$	19.7990	1.15667	0.578335	+	0.815800i	$0.303703\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−19.5959	−1.13707
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−10.3923	−0.599002
$302$	0	0
$303$	19.5959	1.12576
$304$	0	0
$305$	0	0
$306$	0	0
$307$	29.6985	1.69498	0.847491	−	0.530810i	$-0.178112\pi$
$307$	29.6985	1.69498	0.847491	+	0.530810i	$0.178112\pi$
$308$	0	0
$309$	10.3923	0.591198
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	9.79796	0.553813	0.276907	−	0.960897i	$-0.410691\pi$
$313$	9.79796	0.553813	0.276907	+	0.960897i	$0.410691\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	28.2843	1.58860	0.794301	−	0.607524i	$-0.207837\pi$
$317$	28.2843	1.58860	0.794301	+	0.607524i	$0.207837\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−2.00000	−0.111629
$322$	0	0
$323$	−16.9706	−0.944267
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−4.89898	−0.270914
$328$	0	0
$329$	−18.0000	−0.992372
$330$	0	0
$331$	−31.1769	−1.71364	−0.856819	−	0.515617i	$-0.827563\pi$
$331$	−31.1769	−1.71364	−0.856819	+	0.515617i	$0.827563\pi$
$332$	0	0
$333$	8.48528	0.464991
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−13.8564	−0.750366
$342$	0	0
$343$	19.5959	1.05808
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−15.5563	−0.835109	−0.417554	−	0.908652i	$-0.637113\pi$
$347$	−15.5563	−0.835109	−0.417554	+	0.908652i	$0.637113\pi$
$348$	0	0
$349$	−13.8564	−0.741716	−0.370858	−	0.928689i	$-0.620936\pi$
$349$	−13.8564	−0.741716	−0.370858	+	0.928689i	$0.620936\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	29.3939	1.56448	0.782239	−	0.622978i	$-0.214078\pi$
$353$	29.3939	1.56448	0.782239	+	0.622978i	$0.214078\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−16.9706	−0.898177
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	0	0
$363$	1.41421	0.0742270
$364$	0	0
$365$	0	0
$366$	0	0
$367$	12.2474	0.639312	0.319656	−	0.947534i	$-0.396433\pi$
$367$	12.2474	0.639312	0.319656	+	0.947534i	$0.396433\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	13.8564	0.719389
$372$	0	0
$373$	8.48528	0.439351	0.219676	−	0.975573i	$-0.429500\pi$
$373$	8.48528	0.439351	0.219676	+	0.975573i	$0.429500\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	24.2487	1.24557	0.622786	−	0.782392i	$-0.286001\pi$
$379$	24.2487	1.24557	0.622786	+	0.782392i	$0.286001\pi$
$380$	0	0
$381$	−24.2487	−1.24230
$382$	0	0
$383$	26.9444	1.37679	0.688397	−	0.725334i	$-0.258315\pi$
$383$	26.9444	1.37679	0.688397	+	0.725334i	$0.258315\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.24264	−0.215666
$388$	0	0
$389$	3.46410	0.175637	0.0878185	−	0.996136i	$-0.472010\pi$
$389$	3.46410	0.175637	0.0878185	+	0.996136i	$0.472010\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	4.89898	0.247121
$394$	0	0
$395$	0	0
$396$	0	0
$397$	16.9706	0.851728	0.425864	−	0.904787i	$-0.359970\pi$
$397$	16.9706	0.851728	0.425864	+	0.904787i	$0.359970\pi$
$398$	0	0
$399$	12.0000	0.600751
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−29.3939	−1.45700
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	0	0
$411$	13.8564	0.683486
$412$	0	0
$413$	25.4558	1.25260
$414$	0	0
$415$	0	0
$416$	0	0
$417$	14.6969	0.719712
$418$	0	0
$419$	−10.3923	−0.507697	−0.253849	−	0.967244i	$-0.581697\pi$
$419$	−10.3923	−0.507697	−0.253849	+	0.967244i	$0.581697\pi$
$420$	0	0
$421$	24.2487	1.18181	0.590905	−	0.806741i	$-0.298771\pi$
$421$	24.2487	1.18181	0.590905	+	0.806741i	$0.298771\pi$
$422$	0	0
$423$	−7.34847	−0.357295
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.48528	0.410632
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	4.89898	0.235430	0.117715	−	0.993047i	$-0.462443\pi$
$433$	4.89898	0.235430	0.117715	+	0.993047i	$0.462443\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.48528	0.405906
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	1.41421	0.0671913	0.0335957	−	0.999436i	$-0.489304\pi$
$443$	1.41421	0.0671913	0.0335957	+	0.999436i	$0.489304\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	24.4949	1.15857
$448$	0	0
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−22.6274	−1.06313
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.5959	−0.916658	−0.458329	−	0.888783i	$-0.651552\pi$
$457$	−19.5959	−0.916658	−0.458329	+	0.888783i	$0.651552\pi$
$458$	0	0
$459$	−27.7128	−1.29352
$460$	0	0
$461$	13.8564	0.645357	0.322679	−	0.946509i	$-0.395417\pi$
$461$	13.8564	0.645357	0.322679	+	0.946509i	$0.395417\pi$
$462$	0	0
$463$	−17.1464	−0.796862	−0.398431	−	0.917198i	$-0.630445\pi$
$463$	−17.1464	−0.796862	−0.398431	+	0.917198i	$0.630445\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7.07107	−0.327210	−0.163605	−	0.986526i	$-0.552312\pi$
$467$	−7.07107	−0.327210	−0.163605	+	0.986526i	$0.552312\pi$
$468$	0	0
$469$	10.3923	0.479872
$470$	0	0
$471$	−12.0000	−0.552931
$472$	0	0
$473$	14.6969	0.675766
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.65685	0.259010
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	8.48528	0.386094
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.34847	0.332991	0.166495	−	0.986042i	$-0.446755\pi$
$487$	7.34847	0.332991	0.166495	+	0.986042i	$0.446755\pi$
$488$	0	0
$489$	−30.0000	−1.35665
$490$	0	0
$491$	24.2487	1.09433	0.547165	−	0.837025i	$-0.315707\pi$
$491$	24.2487	1.09433	0.547165	+	0.837025i	$0.315707\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	29.3939	1.31850
$498$	0	0
$499$	17.3205	0.775372	0.387686	−	0.921791i	$-0.373274\pi$
$499$	17.3205	0.775372	0.387686	+	0.921791i	$0.373274\pi$
$500$	0	0
$501$	17.3205	0.773823
$502$	0	0
$503$	−12.2474	−0.546087	−0.273043	−	0.962002i	$-0.588030\pi$
$503$	−12.2474	−0.546087	−0.273043	+	0.962002i	$0.588030\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−18.3848	−0.816497
$508$	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
$511$	−12.0000	−0.530849
$512$	0	0
$513$	19.5959	0.865181
$514$	0	0
$515$	0	0
$516$	0	0
$517$	25.4558	1.11955
$518$	0	0
$519$	4.00000	0.175581
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	12.7279	0.556553	0.278277	−	0.960501i	$-0.410237\pi$
$523$	12.7279	0.556553	0.278277	+	0.960501i	$0.410237\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−19.5959	−0.853612
$528$	0	0
$529$	−17.0000	−0.739130
$530$	0	0
$531$	10.3923	0.450988
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.89898	0.211407
$538$	0	0
$539$	−3.46410	−0.149209
$540$	0	0
$541$	41.5692	1.78720	0.893600	−	0.448864i	$-0.148171\pi$
$541$	41.5692	1.78720	0.893600	+	0.448864i	$0.148171\pi$
$542$	0	0
$543$	−19.5959	−0.840941
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.24264	0.181402	0.0907011	−	0.995878i	$-0.471089\pi$
$547$	4.24264	0.181402	0.0907011	+	0.995878i	$0.471089\pi$
$548$	0	0
$549$	3.46410	0.147844
$550$	0	0
$551$	0	0
$552$	0	0
$553$	9.79796	0.416652
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.1421	0.599222	0.299611	−	0.954062i	$-0.403143\pi$
$557$	14.1421	0.599222	0.299611	+	0.954062i	$0.403143\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	24.0000	1.01328
$562$	0	0
$563$	41.0122	1.72846	0.864229	−	0.503099i	$-0.167807\pi$
$563$	41.0122	1.72846	0.864229	+	0.503099i	$0.167807\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	12.2474	0.514344
$568$	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
$571$	3.46410	0.144968	0.0724841	−	0.997370i	$-0.476907\pi$
$571$	3.46410	0.144968	0.0724841	+	0.997370i	$0.476907\pi$
$572$	0	0
$573$	−33.9411	−1.41791
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−29.3939	−1.22368	−0.611842	−	0.790980i	$-0.709571\pi$
$577$	−29.3939	−1.22368	−0.611842	+	0.790980i	$0.709571\pi$
$578$	0	0
$579$	−34.6410	−1.43963
$580$	0	0
$581$	−24.2487	−1.00601
$582$	0	0
$583$	−19.5959	−0.811580
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9.89949	−0.408596	−0.204298	−	0.978909i	$-0.565491\pi$
$587$	−9.89949	−0.408596	−0.204298	+	0.978909i	$0.565491\pi$
$588$	0	0
$589$	13.8564	0.570943
$590$	0	0
$591$	−8.00000	−0.329076
$592$	0	0
$593$	−9.79796	−0.402354	−0.201177	−	0.979555i	$-0.564477\pi$
$593$	−9.79796	−0.402354	−0.201177	+	0.979555i	$0.564477\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5.65685	−0.231520
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	28.0000	1.14214	0.571072	−	0.820900i	$-0.306528\pi$
$601$	28.0000	1.14214	0.571072	+	0.820900i	$0.306528\pi$
$602$	0	0
$603$	4.24264	0.172774
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7.34847	0.298265	0.149133	−	0.988817i	$-0.452352\pi$
$607$	7.34847	0.298265	0.149133	+	0.988817i	$0.452352\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	33.9411	1.37087	0.685435	−	0.728134i	$-0.259612\pi$
$613$	33.9411	1.37087	0.685435	+	0.728134i	$0.259612\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	34.2929	1.38058	0.690289	−	0.723534i	$-0.257483\pi$
$617$	34.2929	1.38058	0.690289	+	0.723534i	$0.257483\pi$
$618$	0	0
$619$	−10.3923	−0.417702	−0.208851	−	0.977947i	$-0.566972\pi$
$619$	−10.3923	−0.417702	−0.208851	+	0.977947i	$0.566972\pi$
$620$	0	0
$621$	13.8564	0.556038
$622$	0	0
$623$	−14.6969	−0.588820
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−16.9706	−0.677739
$628$	0	0
$629$	−41.5692	−1.65747
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	−34.2929	−1.36302
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	0	0
$643$	−29.6985	−1.17119	−0.585597	−	0.810602i	$-0.699140\pi$
$643$	−29.6985	−1.17119	−0.585597	+	0.810602i	$0.699140\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.2474	−0.481497	−0.240748	−	0.970588i	$-0.577393\pi$
$647$	−12.2474	−0.481497	−0.240748	+	0.970588i	$0.577393\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	13.8564	0.543075
$652$	0	0
$653$	11.3137	0.442740	0.221370	−	0.975190i	$-0.428947\pi$
$653$	11.3137	0.442740	0.221370	+	0.975190i	$0.428947\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−4.89898	−0.191127
$658$	0	0
$659$	−24.2487	−0.944596	−0.472298	−	0.881439i	$-0.656575\pi$
$659$	−24.2487	−0.944596	−0.472298	+	0.881439i	$0.656575\pi$
$660$	0	0
$661$	−10.3923	−0.404214	−0.202107	−	0.979363i	$-0.564779\pi$
$661$	−10.3923	−0.404214	−0.202107	+	0.979363i	$0.564779\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−31.1769	−1.20537
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	−34.2929	−1.32189	−0.660946	−	0.750433i	$-0.729845\pi$
$673$	−34.2929	−1.32189	−0.660946	+	0.750433i	$0.729845\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.6274	0.869642	0.434821	−	0.900517i	$-0.356812\pi$
$677$	22.6274	0.869642	0.434821	+	0.900517i	$0.356812\pi$
$678$	0	0
$679$	12.0000	0.460518
$680$	0	0
$681$	−2.00000	−0.0766402
$682$	0	0
$683$	15.5563	0.595247	0.297624	−	0.954683i	$-0.403806\pi$
$683$	15.5563	0.595247	0.297624	+	0.954683i	$0.403806\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−39.1918	−1.49526
$688$	0	0
$689$	0	0
$690$	0	0
$691$	3.46410	0.131781	0.0658903	−	0.997827i	$-0.479011\pi$
$691$	3.46410	0.131781	0.0658903	+	0.997827i	$0.479011\pi$
$692$	0	0
$693$	8.48528	0.322329
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−20.7846	−0.786146
$700$	0	0
$701$	−24.2487	−0.915861	−0.457931	−	0.888988i	$-0.651409\pi$
$701$	−24.2487	−0.915861	−0.457931	+	0.888988i	$0.651409\pi$
$702$	0	0
$703$	29.3939	1.10861
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−33.9411	−1.27649
$708$	0	0
$709$	41.5692	1.56116	0.780582	−	0.625053i	$-0.214923\pi$
$709$	41.5692	1.56116	0.780582	+	0.625053i	$0.214923\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	9.79796	0.366936
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−16.9706	−0.633777
$718$	0	0
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	−18.0000	−0.670355
$722$	0	0
$723$	5.65685	0.210381
$724$	0	0
$725$	0	0
$726$	0	0
$727$	46.5403	1.72608	0.863042	−	0.505132i	$-0.168556\pi$
$727$	46.5403	1.72608	0.863042	+	0.505132i	$0.168556\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	20.7846	0.768747
$732$	0	0
$733$	42.4264	1.56706	0.783528	−	0.621357i	$-0.213418\pi$
$733$	42.4264	1.56706	0.783528	+	0.621357i	$0.213418\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−14.6969	−0.541369
$738$	0	0
$739$	3.46410	0.127429	0.0637145	−	0.997968i	$-0.479705\pi$
$739$	3.46410	0.127429	0.0637145	+	0.997968i	$0.479705\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	51.4393	1.88712	0.943562	−	0.331195i	$-0.107452\pi$
$743$	51.4393	1.88712	0.943562	+	0.331195i	$0.107452\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−9.89949	−0.362204
$748$	0	0
$749$	3.46410	0.126576
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	−14.6969	−0.535586
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−25.4558	−0.925208	−0.462604	−	0.886565i	$-0.653085\pi$
$757$	−25.4558	−0.925208	−0.462604	+	0.886565i	$0.653085\pi$
$758$	0	0
$759$	−12.0000	−0.435572
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	8.48528	0.307188
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	13.8564	0.499026
$772$	0	0
$773$	−28.2843	−1.01731	−0.508657	−	0.860969i	$-0.669858\pi$
$773$	−28.2843	−1.01731	−0.508657	+	0.860969i	$0.669858\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	29.3939	1.05450
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−41.5692	−1.48746
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−21.2132	−0.756169	−0.378085	−	0.925771i	$-0.623417\pi$
$787$	−21.2132	−0.756169	−0.378085	+	0.925771i	$0.623417\pi$
$788$	0	0
$789$	−10.3923	−0.369976
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−39.5980	−1.40263	−0.701316	−	0.712850i	$-0.747404\pi$
$797$	−39.5980	−1.40263	−0.701316	+	0.712850i	$0.747404\pi$
$798$	0	0
$799$	36.0000	1.27359
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	16.9706	0.598878
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−14.6969	−0.517357
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−10.3923	−0.364923	−0.182462	−	0.983213i	$-0.558407\pi$
$811$	−10.3923	−0.364923	−0.182462	+	0.983213i	$0.558407\pi$
$812$	0	0
$813$	−28.2843	−0.991973
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−14.6969	−0.514181
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.1769	1.08808	0.544041	−	0.839059i	$-0.316894\pi$
$821$	31.1769	1.08808	0.544041	+	0.839059i	$0.316894\pi$
$822$	0	0
$823$	26.9444	0.939222	0.469611	−	0.882873i	$-0.344394\pi$
$823$	26.9444	0.939222	0.469611	+	0.882873i	$0.344394\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	35.3553	1.22943	0.614713	−	0.788751i	$-0.289272\pi$
$827$	35.3553	1.22943	0.614713	+	0.788751i	$0.289272\pi$
$828$	0	0
$829$	−10.3923	−0.360940	−0.180470	−	0.983581i	$-0.557762\pi$
$829$	−10.3923	−0.360940	−0.180470	+	0.983581i	$0.557762\pi$
$830$	0	0
$831$	36.0000	1.24883
$832$	0	0
$833$	−4.89898	−0.169740
$834$	0	0
$835$	0	0
$836$	0	0
$837$	22.6274	0.782118
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	16.9706	0.584497
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.44949	−0.0841655
$848$	0	0
$849$	−30.0000	−1.02960
$850$	0	0
$851$	20.7846	0.712487
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−39.1918	−1.33877	−0.669384	−	0.742917i	$-0.733442\pi$
$857$	−39.1918	−1.33877	−0.669384	+	0.742917i	$0.733442\pi$
$858$	0	0
$859$	3.46410	0.118194	0.0590968	−	0.998252i	$-0.481178\pi$
$859$	3.46410	0.118194	0.0590968	+	0.998252i	$0.481178\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.44949	0.0833816	0.0416908	−	0.999131i	$-0.486726\pi$
$863$	2.44949	0.0833816	0.0416908	+	0.999131i	$0.486726\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9.89949	0.336204
$868$	0	0
$869$	−13.8564	−0.470046
$870$	0	0
$871$	0	0
$872$	0	0
$873$	4.89898	0.165805
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.48528	0.286528	0.143264	−	0.989685i	$-0.454240\pi$
$877$	8.48528	0.286528	0.143264	+	0.989685i	$0.454240\pi$
$878$	0	0
$879$	28.0000	0.944417
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	−4.24264	−0.142776	−0.0713881	−	0.997449i	$-0.522743\pi$
$883$	−4.24264	−0.142776	−0.0713881	+	0.997449i	$0.522743\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26.9444	0.904704	0.452352	−	0.891839i	$-0.350585\pi$
$887$	26.9444	0.904704	0.452352	+	0.891839i	$0.350585\pi$
$888$	0	0
$889$	42.0000	1.40863
$890$	0	0
$891$	−17.3205	−0.580259
$892$	0	0
$893$	−25.4558	−0.851847
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−27.7128	−0.923248
$902$	0	0
$903$	−14.6969	−0.489083
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.24264	0.140875	0.0704373	−	0.997516i	$-0.477561\pi$
$907$	4.24264	0.140875	0.0704373	+	0.997516i	$0.477561\pi$
$908$	0	0
$909$	−13.8564	−0.459588
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	34.2929	1.13493
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.48528	−0.280209
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	42.0000	1.38395
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−7.34847	−0.241355
$928$	0	0
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	3.46410	0.113531
$932$	0	0
$933$	−33.9411	−1.11118
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.89898	0.160043	0.0800213	−	0.996793i	$-0.474501\pi$
$937$	4.89898	0.160043	0.0800213	+	0.996793i	$0.474501\pi$
$938$	0	0
$939$	13.8564	0.452187
$940$	0	0
$941$	−13.8564	−0.451706	−0.225853	−	0.974161i	$-0.572517\pi$
$941$	−13.8564	−0.451706	−0.225853	+	0.974161i	$0.572517\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	41.0122	1.33272	0.666359	−	0.745631i	$-0.267852\pi$
$947$	41.0122	1.33272	0.666359	+	0.745631i	$0.267852\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	40.0000	1.29709
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−24.0000	−0.775000
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	1.41421	0.0455724
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−17.1464	−0.551392	−0.275696	−	0.961245i	$-0.588908\pi$
$967$	−17.1464	−0.551392	−0.275696	+	0.961245i	$0.588908\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	−3.46410	−0.111168	−0.0555842	−	0.998454i	$-0.517702\pi$
$971$	−3.46410	−0.111168	−0.0555842	+	0.998454i	$0.517702\pi$
$972$	0	0
$973$	−25.4558	−0.816077
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−44.0908	−1.41059	−0.705295	−	0.708914i	$-0.749185\pi$
$977$	−44.0908	−1.41059	−0.705295	+	0.708914i	$0.749185\pi$
$978$	0	0
$979$	20.7846	0.664279
$980$	0	0
$981$	3.46410	0.110600
$982$	0	0
$983$	−56.3383	−1.79691	−0.898456	−	0.439064i	$-0.855310\pi$
$983$	−56.3383	−1.79691	−0.898456	+	0.439064i	$0.855310\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−25.4558	−0.810268
$988$	0	0
$989$	−10.3923	−0.330456
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	−44.0908	−1.39918
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−50.9117	−1.61239	−0.806195	−	0.591650i	$-0.798477\pi$
$997$	−50.9117	−1.61239	−0.806195	+	0.591650i	$0.798477\pi$
$998$	0	0
$999$	48.0000	1.51865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6400.2.a.cm.1.3		4
4.3	odd	2		6400.2.a.co.1.2		4
5.2	odd	4		1280.2.c.k.769.2		4
5.3	odd	4		1280.2.c.k.769.4		4
5.4	even	2	inner	6400.2.a.cm.1.2		4
8.3	odd	2		6400.2.a.co.1.4		4
8.5	even	2	inner	6400.2.a.cm.1.1		4
16.3	odd	4		200.2.d.e.101.3		4
16.5	even	4		800.2.d.f.401.2		4
16.11	odd	4		200.2.d.e.101.4		4
16.13	even	4		800.2.d.f.401.4		4
20.3	even	4		1280.2.c.i.769.2		4
20.7	even	4		1280.2.c.i.769.4		4
20.19	odd	2		6400.2.a.co.1.3		4
40.3	even	4		1280.2.c.i.769.3		4
40.13	odd	4		1280.2.c.k.769.1		4
40.19	odd	2		6400.2.a.co.1.1		4
40.27	even	4		1280.2.c.i.769.1		4
40.29	even	2	inner	6400.2.a.cm.1.4		4
40.37	odd	4		1280.2.c.k.769.3		4
48.5	odd	4		7200.2.k.l.3601.3		4
48.11	even	4		1800.2.k.m.901.1		4
48.29	odd	4		7200.2.k.l.3601.4		4
48.35	even	4		1800.2.k.m.901.2		4
80.3	even	4		40.2.f.a.29.1	✓	4
80.13	odd	4		160.2.f.a.49.1		4
80.19	odd	4		200.2.d.e.101.2		4
80.27	even	4		40.2.f.a.29.2	yes	4
80.29	even	4		800.2.d.f.401.1		4
80.37	odd	4		160.2.f.a.49.2		4
80.43	even	4		40.2.f.a.29.3	yes	4
80.53	odd	4		160.2.f.a.49.4		4
80.59	odd	4		200.2.d.e.101.1		4
80.67	even	4		40.2.f.a.29.4	yes	4
80.69	even	4		800.2.d.f.401.3		4
80.77	odd	4		160.2.f.a.49.3		4
240.29	odd	4		7200.2.k.l.3601.2		4
240.53	even	4		1440.2.d.c.1009.1		4
240.59	even	4		1800.2.k.m.901.4		4
240.77	even	4		1440.2.d.c.1009.2		4
240.83	odd	4		360.2.d.b.109.4		4
240.107	odd	4		360.2.d.b.109.3		4
240.149	odd	4		7200.2.k.l.3601.1		4
240.173	even	4		1440.2.d.c.1009.4		4
240.179	even	4		1800.2.k.m.901.3		4
240.197	even	4		1440.2.d.c.1009.3		4
240.203	odd	4		360.2.d.b.109.2		4
240.227	odd	4		360.2.d.b.109.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.2.f.a.29.1	✓	4	80.3	even	4
40.2.f.a.29.2	yes	4	80.27	even	4
40.2.f.a.29.3	yes	4	80.43	even	4
40.2.f.a.29.4	yes	4	80.67	even	4
160.2.f.a.49.1		4	80.13	odd	4
160.2.f.a.49.2		4	80.37	odd	4
160.2.f.a.49.3		4	80.77	odd	4
160.2.f.a.49.4		4	80.53	odd	4
200.2.d.e.101.1		4	80.59	odd	4
200.2.d.e.101.2		4	80.19	odd	4
200.2.d.e.101.3		4	16.3	odd	4
200.2.d.e.101.4		4	16.11	odd	4
360.2.d.b.109.1		4	240.227	odd	4
360.2.d.b.109.2		4	240.203	odd	4
360.2.d.b.109.3		4	240.107	odd	4
360.2.d.b.109.4		4	240.83	odd	4
800.2.d.f.401.1		4	80.29	even	4
800.2.d.f.401.2		4	16.5	even	4
800.2.d.f.401.3		4	80.69	even	4
800.2.d.f.401.4		4	16.13	even	4
1280.2.c.i.769.1		4	40.27	even	4
1280.2.c.i.769.2		4	20.3	even	4
1280.2.c.i.769.3		4	40.3	even	4
1280.2.c.i.769.4		4	20.7	even	4
1280.2.c.k.769.1		4	40.13	odd	4
1280.2.c.k.769.2		4	5.2	odd	4
1280.2.c.k.769.3		4	40.37	odd	4
1280.2.c.k.769.4		4	5.3	odd	4
1440.2.d.c.1009.1		4	240.53	even	4
1440.2.d.c.1009.2		4	240.77	even	4
1440.2.d.c.1009.3		4	240.197	even	4
1440.2.d.c.1009.4		4	240.173	even	4
1800.2.k.m.901.1		4	48.11	even	4
1800.2.k.m.901.2		4	48.35	even	4
1800.2.k.m.901.3		4	240.179	even	4
1800.2.k.m.901.4		4	240.59	even	4
6400.2.a.cm.1.1		4	8.5	even	2	inner
6400.2.a.cm.1.2		4	5.4	even	2	inner
6400.2.a.cm.1.3		4	1.1	even	1	trivial
6400.2.a.cm.1.4		4	40.29	even	2	inner
6400.2.a.co.1.1		4	40.19	odd	2
6400.2.a.co.1.2		4	4.3	odd	2
6400.2.a.co.1.3		4	20.19	odd	2
6400.2.a.co.1.4		4	8.3	odd	2
7200.2.k.l.3601.1		4	240.149	odd	4
7200.2.k.l.3601.2		4	240.29	odd	4
7200.2.k.l.3601.3		4	48.5	odd	4
7200.2.k.l.3601.4		4	48.29	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6400 = 2^{8} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6400.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{3} -2.44949 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{3} -2.44949 q^{7} -1.00000 q^{9} +3.46410 q^{11} +4.89898 q^{17} -3.46410 q^{19} -3.46410 q^{21} -2.44949 q^{23} -5.65685 q^{27} -4.00000 q^{31} +4.89898 q^{33} -8.48528 q^{37} +4.24264 q^{43} +7.34847 q^{47} -1.00000 q^{49} +6.92820 q^{51} -5.65685 q^{53} -4.89898 q^{57} -10.3923 q^{59} -3.46410 q^{61} +2.44949 q^{63} -4.24264 q^{67} -3.46410 q^{69} -12.0000 q^{71} +4.89898 q^{73} -8.48528 q^{77} -4.00000 q^{79} -5.00000 q^{81} +9.89949 q^{83} +6.00000 q^{89} -5.65685 q^{93} -4.89898 q^{97} -3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} - 16 q^{31} - 4 q^{49} - 48 q^{71} - 16 q^{79} - 20 q^{81} + 24 q^{89}+O(q^{100}) \) 4 * q - 4 * q^9 - 16 * q^31 - 4 * q^49 - 48 * q^71 - 16 * q^79 - 20 * q^81 + 24 * q^89

Introduction

L-functions

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