LMFDB - Embedded newform 4680.2.a.bd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4680,2,Mod(1,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4680.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:	$37.3699881460$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 520)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	4680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -3.46410 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -3.46410 q^{7} +0.732051 q^{11} +1.00000 q^{13} -0.535898 q^{17} +0.732051 q^{19} -2.73205 q^{23} +1.00000 q^{25} -2.53590 q^{29} +10.1962 q^{31} -3.46410 q^{35} -7.46410 q^{41} -10.7321 q^{43} -11.4641 q^{47} +5.00000 q^{49} +7.46410 q^{53} +0.732051 q^{55} +11.6603 q^{59} +8.39230 q^{61} +1.00000 q^{65} +0.928203 q^{67} -12.7321 q^{71} +10.9282 q^{73} -2.53590 q^{77} -13.4641 q^{79} -10.3923 q^{83} -0.535898 q^{85} -0.928203 q^{89} -3.46410 q^{91} +0.732051 q^{95} -11.8564 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^5	$ 2 q + 2 q^{5} - 2 q^{11} + 2 q^{13} - 8 q^{17} - 2 q^{19} - 2 q^{23} + 2 q^{25} - 12 q^{29} + 10 q^{31} - 8 q^{41} - 18 q^{43} - 16 q^{47} + 10 q^{49} + 8 q^{53} - 2 q^{55} + 6 q^{59} - 4 q^{61} + 2 q^{65} - 12 q^{67} - 22 q^{71} + 8 q^{73} - 12 q^{77} - 20 q^{79} - 8 q^{85} + 12 q^{89} - 2 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^11 + 2 * q^13 - 8 * q^17 - 2 * q^19 - 2 * q^23 + 2 * q^25 - 12 * q^29 + 10 * q^31 - 8 * q^41 - 18 * q^43 - 16 * q^47 + 10 * q^49 + 8 * q^53 - 2 * q^55 + 6 * q^59 - 4 * q^61 + 2 * q^65 - 12 * q^67 - 22 * q^71 + 8 * q^73 - 12 * q^77 - 20 * q^79 - 8 * q^85 + 12 * q^89 - 2 * q^95 + 4 * 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.732051	0.220722	0.110361	−	0.993892i	$-0.464799\pi$
$11$	0.732051	0.220722	0.110361	+	0.993892i	$0.464799\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.535898	−0.129974	−0.0649872	−	0.997886i	$-0.520701\pi$
$17$	−0.535898	−0.129974	−0.0649872	+	0.997886i	$0.520701\pi$
$18$	0	0
$19$	0.732051	0.167944	0.0839720	−	0.996468i	$-0.473239\pi$
$19$	0.732051	0.167944	0.0839720	+	0.996468i	$0.473239\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.73205	−0.569672	−0.284836	−	0.958576i	$-0.591939\pi$
$23$	−2.73205	−0.569672	−0.284836	+	0.958576i	$0.591939\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.53590	−0.470905	−0.235452	−	0.971886i	$-0.575657\pi$
$29$	−2.53590	−0.470905	−0.235452	+	0.971886i	$0.575657\pi$
$30$	0	0
$31$	10.1962	1.83128	0.915642	−	0.401996i	$-0.131683\pi$
$31$	10.1962	1.83128	0.915642	+	0.401996i	$0.131683\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.46410	−0.585540
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.46410	−1.16570	−0.582848	−	0.812581i	$-0.698062\pi$
$41$	−7.46410	−1.16570	−0.582848	+	0.812581i	$0.698062\pi$
$42$	0	0
$43$	−10.7321	−1.63662	−0.818311	−	0.574775i	$-0.805089\pi$
$43$	−10.7321	−1.63662	−0.818311	+	0.574775i	$0.805089\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.4641	−1.67221	−0.836106	−	0.548569i	$-0.815173\pi$
$47$	−11.4641	−1.67221	−0.836106	+	0.548569i	$0.815173\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.46410	1.02527	0.512637	−	0.858606i	$-0.328669\pi$
$53$	7.46410	1.02527	0.512637	+	0.858606i	$0.328669\pi$
$54$	0	0
$55$	0.732051	0.0987097
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.6603	1.51804	0.759018	−	0.651070i	$-0.225679\pi$
$59$	11.6603	1.51804	0.759018	+	0.651070i	$0.225679\pi$
$60$	0	0
$61$	8.39230	1.07452	0.537262	−	0.843415i	$-0.319459\pi$
$61$	8.39230	1.07452	0.537262	+	0.843415i	$0.319459\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	0.928203	0.113398	0.0566990	−	0.998391i	$-0.481942\pi$
$67$	0.928203	0.113398	0.0566990	+	0.998391i	$0.481942\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.7321	−1.51102	−0.755508	−	0.655139i	$-0.772610\pi$
$71$	−12.7321	−1.51102	−0.755508	+	0.655139i	$0.772610\pi$
$72$	0	0
$73$	10.9282	1.27905	0.639525	−	0.768771i	$-0.279131\pi$
$73$	10.9282	1.27905	0.639525	+	0.768771i	$0.279131\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.53590	−0.288992
$78$	0	0
$79$	−13.4641	−1.51483	−0.757415	−	0.652934i	$-0.773538\pi$
$79$	−13.4641	−1.51483	−0.757415	+	0.652934i	$0.773538\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.3923	−1.14070	−0.570352	−	0.821401i	$-0.693193\pi$
$83$	−10.3923	−1.14070	−0.570352	+	0.821401i	$0.693193\pi$
$84$	0	0
$85$	−0.535898	−0.0581263
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.928203	−0.0983893	−0.0491947	−	0.998789i	$-0.515665\pi$
$89$	−0.928203	−0.0983893	−0.0491947	+	0.998789i	$0.515665\pi$
$90$	0	0
$91$	−3.46410	−0.363137
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.732051	0.0751068
$96$	0	0
$97$	−11.8564	−1.20384	−0.601918	−	0.798558i	$-0.705597\pi$
$97$	−11.8564	−1.20384	−0.601918	+	0.798558i	$0.705597\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.92820	0.888389	0.444195	−	0.895930i	$-0.353490\pi$
$101$	8.92820	0.888389	0.444195	+	0.895930i	$0.353490\pi$
$102$	0	0
$103$	4.19615	0.413459	0.206730	−	0.978398i	$-0.433718\pi$
$103$	4.19615	0.413459	0.206730	+	0.978398i	$0.433718\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.0526	−1.74521	−0.872603	−	0.488429i	$-0.837570\pi$
$107$	−18.0526	−1.74521	−0.872603	+	0.488429i	$0.837570\pi$
$108$	0	0
$109$	−11.8564	−1.13564	−0.567819	−	0.823154i	$-0.692213\pi$
$109$	−11.8564	−1.13564	−0.567819	+	0.823154i	$0.692213\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.39230	−0.225049	−0.112525	−	0.993649i	$-0.535894\pi$
$113$	−2.39230	−0.225049	−0.112525	+	0.993649i	$0.535894\pi$
$114$	0	0
$115$	−2.73205	−0.254765
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.85641	0.170177
$120$	0	0
$121$	−10.4641	−0.951282
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−12.1962	−1.08223	−0.541117	−	0.840947i	$-0.681998\pi$
$127$	−12.1962	−1.08223	−0.541117	+	0.840947i	$0.681998\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−10.9282	−0.954802	−0.477401	−	0.878686i	$-0.658421\pi$
$131$	−10.9282	−0.954802	−0.477401	+	0.878686i	$0.658421\pi$
$132$	0	0
$133$	−2.53590	−0.219890
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.92820	0.421045	0.210522	−	0.977589i	$-0.432484\pi$
$137$	4.92820	0.421045	0.210522	+	0.977589i	$0.432484\pi$
$138$	0	0
$139$	−6.53590	−0.554368	−0.277184	−	0.960817i	$-0.589401\pi$
$139$	−6.53590	−0.554368	−0.277184	+	0.960817i	$0.589401\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.732051	0.0612172
$144$	0	0
$145$	−2.53590	−0.210595
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−19.8564	−1.62670	−0.813350	−	0.581775i	$-0.802359\pi$
$149$	−19.8564	−1.62670	−0.813350	+	0.581775i	$0.802359\pi$
$150$	0	0
$151$	22.1962	1.80630	0.903149	−	0.429328i	$-0.141250\pi$
$151$	22.1962	1.80630	0.903149	+	0.429328i	$0.141250\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	10.1962	0.818975
$156$	0	0
$157$	19.8564	1.58471	0.792357	−	0.610058i	$-0.208854\pi$
$157$	19.8564	1.58471	0.792357	+	0.610058i	$0.208854\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	9.46410	0.745876
$162$	0	0
$163$	4.92820	0.386007	0.193003	−	0.981198i	$-0.438177\pi$
$163$	4.92820	0.386007	0.193003	+	0.981198i	$0.438177\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.3205	−1.03077	−0.515386	−	0.856958i	$-0.672351\pi$
$167$	−13.3205	−1.03077	−0.515386	+	0.856958i	$0.672351\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−18.3923	−1.39834	−0.699171	−	0.714955i	$-0.746447\pi$
$173$	−18.3923	−1.39834	−0.699171	+	0.714955i	$0.746447\pi$
$174$	0	0
$175$	−3.46410	−0.261861
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−5.85641	−0.437728	−0.218864	−	0.975755i	$-0.570235\pi$
$179$	−5.85641	−0.437728	−0.218864	+	0.975755i	$0.570235\pi$
$180$	0	0
$181$	13.4641	1.00078	0.500389	−	0.865800i	$-0.333190\pi$
$181$	13.4641	1.00078	0.500389	+	0.865800i	$0.333190\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.392305	−0.0286882
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.07180	−0.366982	−0.183491	−	0.983021i	$-0.558740\pi$
$191$	−5.07180	−0.366982	−0.183491	+	0.983021i	$0.558740\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.92820	0.351120	0.175560	−	0.984469i	$-0.443826\pi$
$197$	4.92820	0.351120	0.175560	+	0.984469i	$0.443826\pi$
$198$	0	0
$199$	6.92820	0.491127	0.245564	−	0.969380i	$-0.421027\pi$
$199$	6.92820	0.491127	0.245564	+	0.969380i	$0.421027\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.78461	0.616559
$204$	0	0
$205$	−7.46410	−0.521315
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.535898	0.0370689
$210$	0	0
$211$	−10.9282	−0.752329	−0.376164	−	0.926553i	$-0.622757\pi$
$211$	−10.9282	−0.752329	−0.376164	+	0.926553i	$0.622757\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−10.7321	−0.731920
$216$	0	0
$217$	−35.3205	−2.39771
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.535898	−0.0360484
$222$	0	0
$223$	−9.32051	−0.624147	−0.312074	−	0.950058i	$-0.601024\pi$
$223$	−9.32051	−0.624147	−0.312074	+	0.950058i	$0.601024\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	0	0
$229$	12.5359	0.828395	0.414198	−	0.910187i	$-0.364062\pi$
$229$	12.5359	0.828395	0.414198	+	0.910187i	$0.364062\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	−11.4641	−0.747836
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−28.7321	−1.85852	−0.929261	−	0.369424i	$-0.879555\pi$
$239$	−28.7321	−1.85852	−0.929261	+	0.369424i	$0.879555\pi$
$240$	0	0
$241$	−28.2487	−1.81966	−0.909830	−	0.414982i	$-0.863788\pi$
$241$	−28.2487	−1.81966	−0.909830	+	0.414982i	$0.863788\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5.00000	0.319438
$246$	0	0
$247$	0.732051	0.0465793
$248$	0	0
$249$	0	0
$250$	0	0
$251$	11.3205	0.714544	0.357272	−	0.934000i	$-0.383707\pi$
$251$	11.3205	0.714544	0.357272	+	0.934000i	$0.383707\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5.66025	−0.349026	−0.174513	−	0.984655i	$-0.555835\pi$
$263$	−5.66025	−0.349026	−0.174513	+	0.984655i	$0.555835\pi$
$264$	0	0
$265$	7.46410	0.458516
$266$	0	0
$267$	0	0
$268$	0	0
$269$	27.8564	1.69844	0.849218	−	0.528043i	$-0.177074\pi$
$269$	27.8564	1.69844	0.849218	+	0.528043i	$0.177074\pi$
$270$	0	0
$271$	1.12436	0.0682997	0.0341499	−	0.999417i	$-0.489128\pi$
$271$	1.12436	0.0682997	0.0341499	+	0.999417i	$0.489128\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.732051	0.0441443
$276$	0	0
$277$	13.3205	0.800352	0.400176	−	0.916438i	$-0.368949\pi$
$277$	13.3205	0.800352	0.400176	+	0.916438i	$0.368949\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	11.4641	0.683891	0.341945	−	0.939720i	$-0.388914\pi$
$281$	11.4641	0.683891	0.341945	+	0.939720i	$0.388914\pi$
$282$	0	0
$283$	14.0526	0.835338	0.417669	−	0.908599i	$-0.362847\pi$
$283$	14.0526	0.835338	0.417669	+	0.908599i	$0.362847\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	25.8564	1.52626
$288$	0	0
$289$	−16.7128	−0.983107
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.92820	0.404750	0.202375	−	0.979308i	$-0.435134\pi$
$293$	6.92820	0.404750	0.202375	+	0.979308i	$0.435134\pi$
$294$	0	0
$295$	11.6603	0.678886
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.73205	−0.157999
$300$	0	0
$301$	37.1769	2.14284
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.39230	0.480542
$306$	0	0
$307$	−23.4641	−1.33917	−0.669584	−	0.742737i	$-0.733527\pi$
$307$	−23.4641	−1.33917	−0.669584	+	0.742737i	$0.733527\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.32051	0.188289	0.0941444	−	0.995559i	$-0.469988\pi$
$311$	3.32051	0.188289	0.0941444	+	0.995559i	$0.469988\pi$
$312$	0	0
$313$	−10.3923	−0.587408	−0.293704	−	0.955896i	$-0.594888\pi$
$313$	−10.3923	−0.587408	−0.293704	+	0.955896i	$0.594888\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−25.8564	−1.45224	−0.726120	−	0.687568i	$-0.758678\pi$
$317$	−25.8564	−1.45224	−0.726120	+	0.687568i	$0.758678\pi$
$318$	0	0
$319$	−1.85641	−0.103939
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−0.392305	−0.0218284
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	39.7128	2.18944
$330$	0	0
$331$	22.5885	1.24157	0.620787	−	0.783979i	$-0.286813\pi$
$331$	22.5885	1.24157	0.620787	+	0.783979i	$0.286813\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.928203	0.0507132
$336$	0	0
$337$	1.60770	0.0875767	0.0437884	−	0.999041i	$-0.486057\pi$
$337$	1.60770	0.0875767	0.0437884	+	0.999041i	$0.486057\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7.46410	0.404204
$342$	0	0
$343$	6.92820	0.374088
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.19615	0.439993	0.219996	−	0.975501i	$-0.429395\pi$
$347$	8.19615	0.439993	0.219996	+	0.975501i	$0.429395\pi$
$348$	0	0
$349$	19.4641	1.04189	0.520945	−	0.853590i	$-0.325580\pi$
$349$	19.4641	1.04189	0.520945	+	0.853590i	$0.325580\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.92820	−0.368751	−0.184376	−	0.982856i	$-0.559026\pi$
$353$	−6.92820	−0.368751	−0.184376	+	0.982856i	$0.559026\pi$
$354$	0	0
$355$	−12.7321	−0.675747
$356$	0	0
$357$	0	0
$358$	0	0
$359$	13.1244	0.692677	0.346338	−	0.938110i	$-0.387425\pi$
$359$	13.1244	0.692677	0.346338	+	0.938110i	$0.387425\pi$
$360$	0	0
$361$	−18.4641	−0.971795
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.9282	0.572008
$366$	0	0
$367$	−32.5885	−1.70110	−0.850552	−	0.525891i	$-0.823732\pi$
$367$	−32.5885	−1.70110	−0.850552	+	0.525891i	$0.823732\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−25.8564	−1.34240
$372$	0	0
$373$	19.8564	1.02813	0.514063	−	0.857753i	$-0.328140\pi$
$373$	19.8564	1.02813	0.514063	+	0.857753i	$0.328140\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.53590	−0.130605
$378$	0	0
$379$	−9.80385	−0.503590	−0.251795	−	0.967781i	$-0.581021\pi$
$379$	−9.80385	−0.503590	−0.251795	+	0.967781i	$0.581021\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	22.3923	1.14419	0.572097	−	0.820186i	$-0.306130\pi$
$383$	22.3923	1.14419	0.572097	+	0.820186i	$0.306130\pi$
$384$	0	0
$385$	−2.53590	−0.129241
$386$	0	0
$387$	0	0
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	1.46410	0.0740428
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−13.4641	−0.677452
$396$	0	0
$397$	−18.9282	−0.949979	−0.474990	−	0.879991i	$-0.657548\pi$
$397$	−18.9282	−0.949979	−0.474990	+	0.879991i	$0.657548\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−30.7846	−1.53731	−0.768655	−	0.639664i	$-0.779074\pi$
$401$	−30.7846	−1.53731	−0.768655	+	0.639664i	$0.779074\pi$
$402$	0	0
$403$	10.1962	0.507907
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−1.32051	−0.0652949	−0.0326475	−	0.999467i	$-0.510394\pi$
$409$	−1.32051	−0.0652949	−0.0326475	+	0.999467i	$0.510394\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−40.3923	−1.98758
$414$	0	0
$415$	−10.3923	−0.510138
$416$	0	0
$417$	0	0
$418$	0	0
$419$	18.5359	0.905538	0.452769	−	0.891628i	$-0.350436\pi$
$419$	18.5359	0.905538	0.452769	+	0.891628i	$0.350436\pi$
$420$	0	0
$421$	−32.9282	−1.60482	−0.802411	−	0.596772i	$-0.796450\pi$
$421$	−32.9282	−1.60482	−0.802411	+	0.596772i	$0.796450\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.535898	−0.0259949
$426$	0	0
$427$	−29.0718	−1.40688
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.0526	−0.773225	−0.386612	−	0.922242i	$-0.626355\pi$
$431$	−16.0526	−0.773225	−0.386612	+	0.922242i	$0.626355\pi$
$432$	0	0
$433$	28.9282	1.39020	0.695100	−	0.718913i	$-0.255360\pi$
$433$	28.9282	1.39020	0.695100	+	0.718913i	$0.255360\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.00000	−0.0956730
$438$	0	0
$439$	13.8564	0.661330	0.330665	−	0.943748i	$-0.392727\pi$
$439$	13.8564	0.661330	0.330665	+	0.943748i	$0.392727\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	30.4449	1.44648	0.723240	−	0.690597i	$-0.242652\pi$
$443$	30.4449	1.44648	0.723240	+	0.690597i	$0.242652\pi$
$444$	0	0
$445$	−0.928203	−0.0440011
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6.39230	−0.301672	−0.150836	−	0.988559i	$-0.548196\pi$
$449$	−6.39230	−0.301672	−0.150836	+	0.988559i	$0.548196\pi$
$450$	0	0
$451$	−5.46410	−0.257294
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.46410	−0.162400
$456$	0	0
$457$	10.7846	0.504483	0.252241	−	0.967664i	$-0.418832\pi$
$457$	10.7846	0.504483	0.252241	+	0.967664i	$0.418832\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−36.2487	−1.68827	−0.844135	−	0.536130i	$-0.819886\pi$
$461$	−36.2487	−1.68827	−0.844135	+	0.536130i	$0.819886\pi$
$462$	0	0
$463$	0.143594	0.00667336	0.00333668	−	0.999994i	$-0.498938\pi$
$463$	0.143594	0.00667336	0.00333668	+	0.999994i	$0.498938\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.5885	0.582524	0.291262	−	0.956643i	$-0.405925\pi$
$467$	12.5885	0.582524	0.291262	+	0.956643i	$0.405925\pi$
$468$	0	0
$469$	−3.21539	−0.148473
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−7.85641	−0.361238
$474$	0	0
$475$	0.732051	0.0335888
$476$	0	0
$477$	0	0
$478$	0	0
$479$	13.1244	0.599667	0.299834	−	0.953992i	$-0.403069\pi$
$479$	13.1244	0.599667	0.299834	+	0.953992i	$0.403069\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−11.8564	−0.538372
$486$	0	0
$487$	−12.1436	−0.550279	−0.275139	−	0.961404i	$-0.588724\pi$
$487$	−12.1436	−0.550279	−0.275139	+	0.961404i	$0.588724\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	26.5359	1.19755	0.598774	−	0.800918i	$-0.295655\pi$
$491$	26.5359	1.19755	0.598774	+	0.800918i	$0.295655\pi$
$492$	0	0
$493$	1.35898	0.0612056
$494$	0	0
$495$	0	0
$496$	0	0
$497$	44.1051	1.97838
$498$	0	0
$499$	13.8038	0.617945	0.308973	−	0.951071i	$-0.400015\pi$
$499$	13.8038	0.617945	0.308973	+	0.951071i	$0.400015\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	16.8756	0.752448	0.376224	−	0.926529i	$-0.377222\pi$
$503$	16.8756	0.752448	0.376224	+	0.926529i	$0.377222\pi$
$504$	0	0
$505$	8.92820	0.397300
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.46410	−0.153544	−0.0767718	−	0.997049i	$-0.524461\pi$
$509$	−3.46410	−0.153544	−0.0767718	+	0.997049i	$0.524461\pi$
$510$	0	0
$511$	−37.8564	−1.67467
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.19615	0.184905
$516$	0	0
$517$	−8.39230	−0.369093
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12.3923	0.542917	0.271458	−	0.962450i	$-0.412494\pi$
$521$	12.3923	0.542917	0.271458	+	0.962450i	$0.412494\pi$
$522$	0	0
$523$	−27.5167	−1.20322	−0.601610	−	0.798790i	$-0.705474\pi$
$523$	−27.5167	−1.20322	−0.601610	+	0.798790i	$0.705474\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.46410	−0.238020
$528$	0	0
$529$	−15.5359	−0.675474
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−7.46410	−0.323306
$534$	0	0
$535$	−18.0526	−0.780480
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.66025	0.157658
$540$	0	0
$541$	17.6077	0.757014	0.378507	−	0.925598i	$-0.376438\pi$
$541$	17.6077	0.757014	0.378507	+	0.925598i	$0.376438\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−11.8564	−0.507873
$546$	0	0
$547$	−25.6603	−1.09715	−0.548577	−	0.836100i	$-0.684830\pi$
$547$	−25.6603	−1.09715	−0.548577	+	0.836100i	$0.684830\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.85641	−0.0790856
$552$	0	0
$553$	46.6410	1.98338
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.85641	0.248144	0.124072	−	0.992273i	$-0.460405\pi$
$557$	5.85641	0.248144	0.124072	+	0.992273i	$0.460405\pi$
$558$	0	0
$559$	−10.7321	−0.453917
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−37.2679	−1.57066	−0.785328	−	0.619079i	$-0.787506\pi$
$563$	−37.2679	−1.57066	−0.785328	+	0.619079i	$0.787506\pi$
$564$	0	0
$565$	−2.39230	−0.100645
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.5359	0.441688	0.220844	−	0.975309i	$-0.429119\pi$
$569$	10.5359	0.441688	0.220844	+	0.975309i	$0.429119\pi$
$570$	0	0
$571$	7.32051	0.306354	0.153177	−	0.988199i	$-0.451050\pi$
$571$	7.32051	0.306354	0.153177	+	0.988199i	$0.451050\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.73205	−0.113934
$576$	0	0
$577$	38.6410	1.60865	0.804323	−	0.594192i	$-0.202528\pi$
$577$	38.6410	1.60865	0.804323	+	0.594192i	$0.202528\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	36.0000	1.49353
$582$	0	0
$583$	5.46410	0.226300
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−42.7846	−1.76591	−0.882955	−	0.469458i	$-0.844449\pi$
$587$	−42.7846	−1.76591	−0.882955	+	0.469458i	$0.844449\pi$
$588$	0	0
$589$	7.46410	0.307553
$590$	0	0
$591$	0	0
$592$	0	0
$593$	28.6410	1.17615	0.588073	−	0.808808i	$-0.299887\pi$
$593$	28.6410	1.17615	0.588073	+	0.808808i	$0.299887\pi$
$594$	0	0
$595$	1.85641	0.0761052
$596$	0	0
$597$	0	0
$598$	0	0
$599$	26.5359	1.08423	0.542114	−	0.840305i	$-0.317624\pi$
$599$	26.5359	1.08423	0.542114	+	0.840305i	$0.317624\pi$
$600$	0	0
$601$	23.8564	0.973123	0.486562	−	0.873646i	$-0.338251\pi$
$601$	23.8564	0.973123	0.486562	+	0.873646i	$0.338251\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10.4641	−0.425426
$606$	0	0
$607$	31.5167	1.27922	0.639611	−	0.768699i	$-0.279095\pi$
$607$	31.5167	1.27922	0.639611	+	0.768699i	$0.279095\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−11.4641	−0.463788
$612$	0	0
$613$	−27.8564	−1.12511	−0.562555	−	0.826760i	$-0.690181\pi$
$613$	−27.8564	−1.12511	−0.562555	+	0.826760i	$0.690181\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9.71281	−0.391023	−0.195512	−	0.980701i	$-0.562637\pi$
$617$	−9.71281	−0.391023	−0.195512	+	0.980701i	$0.562637\pi$
$618$	0	0
$619$	−35.6603	−1.43331	−0.716653	−	0.697430i	$-0.754327\pi$
$619$	−35.6603	−1.43331	−0.716653	+	0.697430i	$0.754327\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.21539	0.128822
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−18.1962	−0.724377	−0.362189	−	0.932105i	$-0.617970\pi$
$631$	−18.1962	−0.724377	−0.362189	+	0.932105i	$0.617970\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−12.1962	−0.483990
$636$	0	0
$637$	5.00000	0.198107
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−12.9282	−0.510633	−0.255317	−	0.966857i	$-0.582180\pi$
$641$	−12.9282	−0.510633	−0.255317	+	0.966857i	$0.582180\pi$
$642$	0	0
$643$	−30.3923	−1.19856	−0.599278	−	0.800541i	$-0.704545\pi$
$643$	−30.3923	−1.19856	−0.599278	+	0.800541i	$0.704545\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−17.6603	−0.694296	−0.347148	−	0.937810i	$-0.612850\pi$
$647$	−17.6603	−0.694296	−0.347148	+	0.937810i	$0.612850\pi$
$648$	0	0
$649$	8.53590	0.335063
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−21.7128	−0.849688	−0.424844	−	0.905267i	$-0.639671\pi$
$653$	−21.7128	−0.849688	−0.424844	+	0.905267i	$0.639671\pi$
$654$	0	0
$655$	−10.9282	−0.427000
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−4.67949	−0.182287	−0.0911436	−	0.995838i	$-0.529052\pi$
$659$	−4.67949	−0.182287	−0.0911436	+	0.995838i	$0.529052\pi$
$660$	0	0
$661$	7.07180	0.275061	0.137531	−	0.990498i	$-0.456083\pi$
$661$	7.07180	0.275061	0.137531	+	0.990498i	$0.456083\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.53590	−0.0983379
$666$	0	0
$667$	6.92820	0.268261
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6.14359	0.237171
$672$	0	0
$673$	35.4641	1.36704	0.683520	−	0.729931i	$-0.260448\pi$
$673$	35.4641	1.36704	0.683520	+	0.729931i	$0.260448\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−30.3923	−1.16807	−0.584036	−	0.811728i	$-0.698527\pi$
$677$	−30.3923	−1.16807	−0.584036	+	0.811728i	$0.698527\pi$
$678$	0	0
$679$	41.0718	1.57619
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−11.0718	−0.423651	−0.211825	−	0.977308i	$-0.567941\pi$
$683$	−11.0718	−0.423651	−0.211825	+	0.977308i	$0.567941\pi$
$684$	0	0
$685$	4.92820	0.188297
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7.46410	0.284360
$690$	0	0
$691$	−21.9090	−0.833456	−0.416728	−	0.909031i	$-0.636823\pi$
$691$	−21.9090	−0.833456	−0.416728	+	0.909031i	$0.636823\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.53590	−0.247921
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	0	0
$700$	0	0
$701$	25.7128	0.971160	0.485580	−	0.874192i	$-0.338609\pi$
$701$	25.7128	0.971160	0.485580	+	0.874192i	$0.338609\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−30.9282	−1.16317
$708$	0	0
$709$	26.3923	0.991184	0.495592	−	0.868556i	$-0.334951\pi$
$709$	26.3923	0.991184	0.495592	+	0.868556i	$0.334951\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−27.8564	−1.04323
$714$	0	0
$715$	0.732051	0.0273771
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−34.6410	−1.29189	−0.645946	−	0.763383i	$-0.723537\pi$
$719$	−34.6410	−1.29189	−0.645946	+	0.763383i	$0.723537\pi$
$720$	0	0
$721$	−14.5359	−0.541345
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.53590	−0.0941809
$726$	0	0
$727$	16.9808	0.629782	0.314891	−	0.949128i	$-0.398032\pi$
$727$	16.9808	0.629782	0.314891	+	0.949128i	$0.398032\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5.75129	0.212719
$732$	0	0
$733$	11.8564	0.437926	0.218963	−	0.975733i	$-0.429733\pi$
$733$	11.8564	0.437926	0.218963	+	0.975733i	$0.429733\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.679492	0.0250294
$738$	0	0
$739$	4.73205	0.174071	0.0870357	−	0.996205i	$-0.472261\pi$
$739$	4.73205	0.174071	0.0870357	+	0.996205i	$0.472261\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.32051	0.341936	0.170968	−	0.985277i	$-0.445310\pi$
$743$	9.32051	0.341936	0.170968	+	0.985277i	$0.445310\pi$
$744$	0	0
$745$	−19.8564	−0.727482
$746$	0	0
$747$	0	0
$748$	0	0
$749$	62.5359	2.28501
$750$	0	0
$751$	38.5359	1.40619	0.703097	−	0.711094i	$-0.251800\pi$
$751$	38.5359	1.40619	0.703097	+	0.711094i	$0.251800\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	22.1962	0.807801
$756$	0	0
$757$	−16.5359	−0.601007	−0.300504	−	0.953781i	$-0.597155\pi$
$757$	−16.5359	−0.601007	−0.300504	+	0.953781i	$0.597155\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	14.0000	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$762$	0	0
$763$	41.0718	1.48690
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11.6603	0.421027
$768$	0	0
$769$	28.9282	1.04318	0.521589	−	0.853197i	$-0.325340\pi$
$769$	28.9282	1.04318	0.521589	+	0.853197i	$0.325340\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−18.9282	−0.680800	−0.340400	−	0.940281i	$-0.610563\pi$
$773$	−18.9282	−0.680800	−0.340400	+	0.940281i	$0.610563\pi$
$774$	0	0
$775$	10.1962	0.366257
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.46410	−0.195772
$780$	0	0
$781$	−9.32051	−0.333514
$782$	0	0
$783$	0	0
$784$	0	0
$785$	19.8564	0.708706
$786$	0	0
$787$	−2.39230	−0.0852765	−0.0426382	−	0.999091i	$-0.513576\pi$
$787$	−2.39230	−0.0852765	−0.0426382	+	0.999091i	$0.513576\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8.28719	0.294658
$792$	0	0
$793$	8.39230	0.298019
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5.21539	−0.184739	−0.0923693	−	0.995725i	$-0.529444\pi$
$797$	−5.21539	−0.184739	−0.0923693	+	0.995725i	$0.529444\pi$
$798$	0	0
$799$	6.14359	0.217345
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.00000	0.282314
$804$	0	0
$805$	9.46410	0.333566
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−43.3205	−1.52307	−0.761534	−	0.648125i	$-0.775553\pi$
$809$	−43.3205	−1.52307	−0.761534	+	0.648125i	$0.775553\pi$
$810$	0	0
$811$	−29.9090	−1.05025	−0.525123	−	0.851026i	$-0.675981\pi$
$811$	−29.9090	−1.05025	−0.525123	+	0.851026i	$0.675981\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.92820	0.172627
$816$	0	0
$817$	−7.85641	−0.274861
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−0.928203	−0.0323945	−0.0161973	−	0.999869i	$-0.505156\pi$
$821$	−0.928203	−0.0323945	−0.0161973	+	0.999869i	$0.505156\pi$
$822$	0	0
$823$	−4.58846	−0.159944	−0.0799718	−	0.996797i	$-0.525483\pi$
$823$	−4.58846	−0.159944	−0.0799718	+	0.996797i	$0.525483\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.46410	0.120459	0.0602293	−	0.998185i	$-0.480817\pi$
$827$	3.46410	0.120459	0.0602293	+	0.998185i	$0.480817\pi$
$828$	0	0
$829$	23.6077	0.819929	0.409965	−	0.912101i	$-0.365541\pi$
$829$	23.6077	0.819929	0.409965	+	0.912101i	$0.365541\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.67949	−0.0928389
$834$	0	0
$835$	−13.3205	−0.460975
$836$	0	0
$837$	0	0
$838$	0	0
$839$	24.4449	0.843930	0.421965	−	0.906612i	$-0.361340\pi$
$839$	24.4449	0.843930	0.421965	+	0.906612i	$0.361340\pi$
$840$	0	0
$841$	−22.5692	−0.778249
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	36.2487	1.24552
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	47.7128	1.63366	0.816828	−	0.576882i	$-0.195731\pi$
$853$	47.7128	1.63366	0.816828	+	0.576882i	$0.195731\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31.8564	1.08819	0.544097	−	0.839022i	$-0.316872\pi$
$857$	31.8564	1.08819	0.544097	+	0.839022i	$0.316872\pi$
$858$	0	0
$859$	8.39230	0.286342	0.143171	−	0.989698i	$-0.454270\pi$
$859$	8.39230	0.286342	0.143171	+	0.989698i	$0.454270\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	54.3923	1.85154	0.925768	−	0.378093i	$-0.123420\pi$
$863$	54.3923	1.85154	0.925768	+	0.378093i	$0.123420\pi$
$864$	0	0
$865$	−18.3923	−0.625357
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−9.85641	−0.334356
$870$	0	0
$871$	0.928203	0.0314510
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.46410	−0.117108
$876$	0	0
$877$	13.7128	0.463049	0.231524	−	0.972829i	$-0.425629\pi$
$877$	13.7128	0.463049	0.231524	+	0.972829i	$0.425629\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.46410	0.0493268	0.0246634	−	0.999696i	$-0.492149\pi$
$881$	1.46410	0.0493268	0.0246634	+	0.999696i	$0.492149\pi$
$882$	0	0
$883$	−8.58846	−0.289025	−0.144512	−	0.989503i	$-0.546161\pi$
$883$	−8.58846	−0.289025	−0.144512	+	0.989503i	$0.546161\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	46.4449	1.55947	0.779733	−	0.626112i	$-0.215355\pi$
$887$	46.4449	1.55947	0.779733	+	0.626112i	$0.215355\pi$
$888$	0	0
$889$	42.2487	1.41698
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.39230	−0.280838
$894$	0	0
$895$	−5.85641	−0.195758
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−25.8564	−0.862359
$900$	0	0
$901$	−4.00000	−0.133259
$902$	0	0
$903$	0	0
$904$	0	0
$905$	13.4641	0.447562
$906$	0	0
$907$	42.8372	1.42238	0.711192	−	0.702997i	$-0.248155\pi$
$907$	42.8372	1.42238	0.711192	+	0.702997i	$0.248155\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−20.7846	−0.688625	−0.344312	−	0.938855i	$-0.611888\pi$
$911$	−20.7846	−0.688625	−0.344312	+	0.938855i	$0.611888\pi$
$912$	0	0
$913$	−7.60770	−0.251778
$914$	0	0
$915$	0	0
$916$	0	0
$917$	37.8564	1.25013
$918$	0	0
$919$	4.67949	0.154362	0.0771811	−	0.997017i	$-0.475408\pi$
$919$	4.67949	0.154362	0.0771811	+	0.997017i	$0.475408\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−12.7321	−0.419081
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	19.1769	0.629174	0.314587	−	0.949229i	$-0.398134\pi$
$929$	19.1769	0.629174	0.314587	+	0.949229i	$0.398134\pi$
$930$	0	0
$931$	3.66025	0.119960
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−0.392305	−0.0128297
$936$	0	0
$937$	−40.9282	−1.33707	−0.668533	−	0.743682i	$-0.733077\pi$
$937$	−40.9282	−1.33707	−0.668533	+	0.743682i	$0.733077\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.60770	−0.0524094	−0.0262047	−	0.999657i	$-0.508342\pi$
$941$	−1.60770	−0.0524094	−0.0262047	+	0.999657i	$0.508342\pi$
$942$	0	0
$943$	20.3923	0.664065
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.60770	−0.0522431	−0.0261215	−	0.999659i	$-0.508316\pi$
$947$	−1.60770	−0.0522431	−0.0261215	+	0.999659i	$0.508316\pi$
$948$	0	0
$949$	10.9282	0.354744
$950$	0	0
$951$	0	0
$952$	0	0
$953$	42.7846	1.38593	0.692965	−	0.720971i	$-0.256304\pi$
$953$	42.7846	1.38593	0.692965	+	0.720971i	$0.256304\pi$
$954$	0	0
$955$	−5.07180	−0.164119
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−17.0718	−0.551277
$960$	0	0
$961$	72.9615	2.35360
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	−18.0000	−0.578841	−0.289420	−	0.957202i	$-0.593463\pi$
$967$	−18.0000	−0.578841	−0.289420	+	0.957202i	$0.593463\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	11.7128	0.375882	0.187941	−	0.982180i	$-0.439819\pi$
$971$	11.7128	0.375882	0.187941	+	0.982180i	$0.439819\pi$
$972$	0	0
$973$	22.6410	0.725838
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.9282	−0.349624	−0.174812	−	0.984602i	$-0.555932\pi$
$977$	−10.9282	−0.349624	−0.174812	+	0.984602i	$0.555932\pi$
$978$	0	0
$979$	−0.679492	−0.0217167
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−11.8564	−0.378161	−0.189080	−	0.981962i	$-0.560551\pi$
$983$	−11.8564	−0.378161	−0.189080	+	0.981962i	$0.560551\pi$
$984$	0	0
$985$	4.92820	0.157026
$986$	0	0
$987$	0	0
$988$	0	0
$989$	29.3205	0.932338
$990$	0	0
$991$	−26.1436	−0.830479	−0.415239	−	0.909712i	$-0.636302\pi$
$991$	−26.1436	−0.830479	−0.415239	+	0.909712i	$0.636302\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.92820	0.219639
$996$	0	0
$997$	25.3205	0.801909	0.400954	−	0.916098i	$-0.368679\pi$
$997$	25.3205	0.801909	0.400954	+	0.916098i	$0.368679\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.a.bd.1.1		2
3.2	odd	2		520.2.a.e.1.1	✓	2
4.3	odd	2		9360.2.a.cr.1.2		2
12.11	even	2		1040.2.a.l.1.2		2
15.2	even	4		2600.2.d.m.1249.4		4
15.8	even	4		2600.2.d.m.1249.1		4
15.14	odd	2		2600.2.a.v.1.2		2
24.5	odd	2		4160.2.a.bk.1.2		2
24.11	even	2		4160.2.a.w.1.1		2
39.38	odd	2		6760.2.a.o.1.1		2
60.59	even	2		5200.2.a.bn.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.a.e.1.1	✓	2	3.2	odd	2
1040.2.a.l.1.2		2	12.11	even	2
2600.2.a.v.1.2		2	15.14	odd	2
2600.2.d.m.1249.1		4	15.8	even	4
2600.2.d.m.1249.4		4	15.2	even	4
4160.2.a.w.1.1		2	24.11	even	2
4160.2.a.bk.1.2		2	24.5	odd	2
4680.2.a.bd.1.1		2	1.1	even	1	trivial
5200.2.a.bn.1.1		2	60.59	even	2
6760.2.a.o.1.1		2	39.38	odd	2
9360.2.a.cr.1.2		2	4.3	odd	2

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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4680.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -3.46410 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -3.46410 q^{7} +0.732051 q^{11} +1.00000 q^{13} -0.535898 q^{17} +0.732051 q^{19} -2.73205 q^{23} +1.00000 q^{25} -2.53590 q^{29} +10.1962 q^{31} -3.46410 q^{35} -7.46410 q^{41} -10.7321 q^{43} -11.4641 q^{47} +5.00000 q^{49} +7.46410 q^{53} +0.732051 q^{55} +11.6603 q^{59} +8.39230 q^{61} +1.00000 q^{65} +0.928203 q^{67} -12.7321 q^{71} +10.9282 q^{73} -2.53590 q^{77} -13.4641 q^{79} -10.3923 q^{83} -0.535898 q^{85} -0.928203 q^{89} -3.46410 q^{91} +0.732051 q^{95} -11.8564 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^5	\( 2 q + 2 q^{5} - 2 q^{11} + 2 q^{13} - 8 q^{17} - 2 q^{19} - 2 q^{23} + 2 q^{25} - 12 q^{29} + 10 q^{31} - 8 q^{41} - 18 q^{43} - 16 q^{47} + 10 q^{49} + 8 q^{53} - 2 q^{55} + 6 q^{59} - 4 q^{61} + 2 q^{65} - 12 q^{67} - 22 q^{71} + 8 q^{73} - 12 q^{77} - 20 q^{79} - 8 q^{85} + 12 q^{89} - 2 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^11 + 2 * q^13 - 8 * q^17 - 2 * q^19 - 2 * q^23 + 2 * q^25 - 12 * q^29 + 10 * q^31 - 8 * q^41 - 18 * q^43 - 16 * q^47 + 10 * q^49 + 8 * q^53 - 2 * q^55 + 6 * q^59 - 4 * q^61 + 2 * q^65 - 12 * q^67 - 22 * q^71 + 8 * q^73 - 12 * q^77 - 20 * q^79 - 8 * q^85 + 12 * q^89 - 2 * q^95 + 4 * q^97

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