LMFDB - Embedded newform 4680.2.a.bg.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.1016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 2 $ x^3 - x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.321637$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} -4.53982 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -4.53982 q^{7} +5.89655 q^{11} -1.00000 q^{13} -2.53982 q^{17} +1.35673 q^{19} -5.25328 q^{23} +1.00000 q^{25} -0.643274 q^{29} -2.00000 q^{31} +4.53982 q^{35} -3.89655 q^{37} +3.89655 q^{41} -3.79310 q^{43} +3.79310 q^{47} +13.6100 q^{49} -2.10345 q^{53} -5.89655 q^{55} +4.00000 q^{59} +0.103450 q^{61} +1.00000 q^{65} -5.79310 q^{67} +3.18310 q^{71} +10.4364 q^{73} -26.7693 q^{77} +10.9762 q^{79} +4.00000 q^{83} +2.53982 q^{85} -10.6100 q^{89} +4.53982 q^{91} -1.35673 q^{95} +3.46018 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} - q^{7}+O(q^{10}) $ 3 * q - 3 * q^5 - q^7	$ 3 q - 3 q^{5} - q^{7} + 5 q^{11} - 3 q^{13} + 5 q^{17} + 4 q^{19} - 3 q^{23} + 3 q^{25} - 2 q^{29} - 6 q^{31} + q^{35} + q^{37} - q^{41} + 14 q^{43} - 14 q^{47} + 28 q^{49} - 19 q^{53} - 5 q^{55} + 12 q^{59} + 13 q^{61} + 3 q^{65} + 8 q^{67} - 3 q^{71} + 6 q^{73} - 17 q^{77} - 5 q^{79} + 12 q^{83} - 5 q^{85} - 19 q^{89} + q^{91} - 4 q^{95} + 23 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 - q^7 + 5 * q^11 - 3 * q^13 + 5 * q^17 + 4 * q^19 - 3 * q^23 + 3 * q^25 - 2 * q^29 - 6 * q^31 + q^35 + q^37 - q^41 + 14 * q^43 - 14 * q^47 + 28 * q^49 - 19 * q^53 - 5 * q^55 + 12 * q^59 + 13 * q^61 + 3 * q^65 + 8 * q^67 - 3 * q^71 + 6 * q^73 - 17 * q^77 - 5 * q^79 + 12 * q^83 - 5 * q^85 - 19 * q^89 + q^91 - 4 * q^95 + 23 * 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
$7$	−4.53982	−1.71589	−0.857946	−	0.513740i	$-0.828260\pi$
$7$	−4.53982	−1.71589	−0.857946	+	0.513740i	$0.828260\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.89655	1.77788	0.888938	−	0.458027i	$-0.151444\pi$
$11$	5.89655	1.77788	0.888938	+	0.458027i	$0.151444\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.53982	−0.615998	−0.307999	−	0.951387i	$-0.599659\pi$
$17$	−2.53982	−0.615998	−0.307999	+	0.951387i	$0.599659\pi$
$18$	0	0
$19$	1.35673	0.311254	0.155627	−	0.987816i	$-0.450260\pi$
$19$	1.35673	0.311254	0.155627	+	0.987816i	$0.450260\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.25328	−1.09538	−0.547692	−	0.836680i	$-0.684493\pi$
$23$	−5.25328	−1.09538	−0.547692	+	0.836680i	$0.684493\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.643274	−0.119453	−0.0597265	−	0.998215i	$-0.519023\pi$
$29$	−0.643274	−0.119453	−0.0597265	+	0.998215i	$0.519023\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.53982	0.767370
$36$	0	0
$37$	−3.89655	−0.640589	−0.320294	−	0.947318i	$-0.603782\pi$
$37$	−3.89655	−0.640589	−0.320294	+	0.947318i	$0.603782\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.89655	0.608539	0.304269	−	0.952586i	$-0.401588\pi$
$41$	3.89655	0.608539	0.304269	+	0.952586i	$0.401588\pi$
$42$	0	0
$43$	−3.79310	−0.578442	−0.289221	−	0.957262i	$-0.593396\pi$
$43$	−3.79310	−0.578442	−0.289221	+	0.957262i	$0.593396\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.79310	0.553280	0.276640	−	0.960974i	$-0.410779\pi$
$47$	3.79310	0.553280	0.276640	+	0.960974i	$0.410779\pi$
$48$	0	0
$49$	13.6100	1.94429
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.10345	−0.288931	−0.144466	−	0.989510i	$-0.546146\pi$
$53$	−2.10345	−0.288931	−0.144466	+	0.989510i	$0.546146\pi$
$54$	0	0
$55$	−5.89655	−0.795091
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	0.103450	0.0132455	0.00662274	−	0.999978i	$-0.497892\pi$
$61$	0.103450	0.0132455	0.00662274	+	0.999978i	$0.497892\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−5.79310	−0.707740	−0.353870	−	0.935295i	$-0.615134\pi$
$67$	−5.79310	−0.707740	−0.353870	+	0.935295i	$0.615134\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.18310	0.377764	0.188882	−	0.982000i	$-0.439514\pi$
$71$	3.18310	0.377764	0.188882	+	0.982000i	$0.439514\pi$
$72$	0	0
$73$	10.4364	1.22149	0.610743	−	0.791829i	$-0.290871\pi$
$73$	10.4364	1.22149	0.610743	+	0.791829i	$0.290871\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−26.7693	−3.05064
$78$	0	0
$79$	10.9762	1.23492	0.617459	−	0.786603i	$-0.288162\pi$
$79$	10.9762	1.23492	0.617459	+	0.786603i	$0.288162\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	2.53982	0.275483
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.6100	−1.12466	−0.562329	−	0.826914i	$-0.690095\pi$
$89$	−10.6100	−1.12466	−0.562329	+	0.826914i	$0.690095\pi$
$90$	0	0
$91$	4.53982	0.475903
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.35673	−0.139197
$96$	0	0
$97$	3.46018	0.351328	0.175664	−	0.984450i	$-0.443793\pi$
$97$	3.46018	0.351328	0.175664	+	0.984450i	$0.443793\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.1498	−1.10945	−0.554725	−	0.832034i	$-0.687176\pi$
$101$	−11.1498	−1.10945	−0.554725	+	0.832034i	$0.687176\pi$
$102$	0	0
$103$	17.0796	1.68291	0.841454	−	0.540329i	$-0.181700\pi$
$103$	17.0796	1.68291	0.841454	+	0.540329i	$0.181700\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.9762	−1.06111	−0.530555	−	0.847651i	$-0.678016\pi$
$107$	−10.9762	−1.06111	−0.530555	+	0.847651i	$0.678016\pi$
$108$	0	0
$109$	−2.43637	−0.233362	−0.116681	−	0.993169i	$-0.537226\pi$
$109$	−2.43637	−0.233362	−0.116681	+	0.993169i	$0.537226\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.35673	0.315774	0.157887	−	0.987457i	$-0.449532\pi$
$113$	3.35673	0.315774	0.157887	+	0.987457i	$0.449532\pi$
$114$	0	0
$115$	5.25328	0.489870
$116$	0	0
$117$	0	0
$118$	0	0
$119$	11.5304	1.05699
$120$	0	0
$121$	23.7693	2.16085
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.72292	0.849496	0.424748	−	0.905312i	$-0.360363\pi$
$131$	9.72292	0.849496	0.424748	+	0.905312i	$0.360363\pi$
$132$	0	0
$133$	−6.15930	−0.534079
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.20690	−0.188548	−0.0942741	−	0.995546i	$-0.530053\pi$
$137$	−2.20690	−0.188548	−0.0942741	+	0.995546i	$0.530053\pi$
$138$	0	0
$139$	15.1831	1.28781	0.643907	−	0.765104i	$-0.277312\pi$
$139$	15.1831	1.28781	0.643907	+	0.765104i	$0.277312\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.89655	−0.493094
$144$	0	0
$145$	0.643274	0.0534210
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−20.7693	−1.70149	−0.850744	−	0.525581i	$-0.823848\pi$
$149$	−20.7693	−1.70149	−0.850744	+	0.525581i	$0.823848\pi$
$150$	0	0
$151$	21.7931	1.77350	0.886749	−	0.462252i	$-0.152958\pi$
$151$	21.7931	1.77350	0.886749	+	0.462252i	$0.152958\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	16.3662	1.30616	0.653082	−	0.757287i	$-0.273476\pi$
$157$	16.3662	1.30616	0.653082	+	0.757287i	$0.273476\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	23.8489	1.87956
$162$	0	0
$163$	23.4827	1.83931	0.919655	−	0.392726i	$-0.128468\pi$
$163$	23.4827	1.83931	0.919655	+	0.392726i	$0.128468\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.0796	−1.01213	−0.506067	−	0.862494i	$-0.668901\pi$
$167$	−13.0796	−1.01213	−0.506067	+	0.862494i	$0.668901\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.5066	1.40703	0.703513	−	0.710682i	$-0.251613\pi$
$173$	18.5066	1.40703	0.703513	+	0.710682i	$0.251613\pi$
$174$	0	0
$175$	−4.53982	−0.343178
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.6433	0.945003	0.472501	−	0.881330i	$-0.343351\pi$
$179$	12.6433	0.945003	0.472501	+	0.881330i	$0.343351\pi$
$180$	0	0
$181$	−9.32345	−0.693007	−0.346503	−	0.938049i	$-0.612631\pi$
$181$	−9.32345	−0.693007	−0.346503	+	0.938049i	$0.612631\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.89655	0.286480
$186$	0	0
$187$	−14.9762	−1.09517
$188$	0	0
$189$	0	0
$190$	0	0
$191$	23.7931	1.72161	0.860804	−	0.508936i	$-0.169961\pi$
$191$	23.7931	1.72161	0.860804	+	0.508936i	$0.169961\pi$
$192$	0	0
$193$	4.53982	0.326784	0.163392	−	0.986561i	$-0.447757\pi$
$193$	4.53982	0.326784	0.163392	+	0.986561i	$0.447757\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.50655	0.321078	0.160539	−	0.987029i	$-0.448677\pi$
$197$	4.50655	0.321078	0.160539	+	0.987029i	$0.448677\pi$
$198$	0	0
$199$	−5.42690	−0.384703	−0.192351	−	0.981326i	$-0.561611\pi$
$199$	−5.42690	−0.384703	−0.192351	+	0.981326i	$0.561611\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.92035	0.204969
$204$	0	0
$205$	−3.89655	−0.272147
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	14.7135	1.01292	0.506458	−	0.862265i	$-0.330955\pi$
$211$	14.7135	1.01292	0.506458	+	0.862265i	$0.330955\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3.79310	0.258687
$216$	0	0
$217$	9.07965	0.616367
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.53982	0.170847
$222$	0	0
$223$	20.9429	1.40244	0.701221	−	0.712944i	$-0.252639\pi$
$223$	20.9429	1.40244	0.701221	+	0.712944i	$0.252639\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.28655	0.350881	0.175440	−	0.984490i	$-0.443865\pi$
$227$	5.28655	0.350881	0.175440	+	0.984490i	$0.443865\pi$
$228$	0	0
$229$	11.9298	0.788345	0.394172	−	0.919036i	$-0.371031\pi$
$229$	11.9298	0.788345	0.394172	+	0.919036i	$0.371031\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−0.173627	−0.0113747	−0.00568736	−	0.999984i	$-0.501810\pi$
$233$	−0.173627	−0.0113747	−0.00568736	+	0.999984i	$0.501810\pi$
$234$	0	0
$235$	−3.79310	−0.247434
$236$	0	0
$237$	0	0
$238$	0	0
$239$	25.4827	1.64834	0.824171	−	0.566341i	$-0.191642\pi$
$239$	25.4827	1.64834	0.824171	+	0.566341i	$0.191642\pi$
$240$	0	0
$241$	−27.7455	−1.78724	−0.893622	−	0.448820i	$-0.851844\pi$
$241$	−27.7455	−1.78724	−0.893622	+	0.448820i	$0.851844\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−13.6100	−0.869511
$246$	0	0
$247$	−1.35673	−0.0863264
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.27708	0.143728	0.0718639	−	0.997414i	$-0.477105\pi$
$251$	2.27708	0.143728	0.0718639	+	0.997414i	$0.477105\pi$
$252$	0	0
$253$	−30.9762	−1.94746
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.6433	−1.28769	−0.643846	−	0.765155i	$-0.722662\pi$
$257$	−20.6433	−1.28769	−0.643846	+	0.765155i	$0.722662\pi$
$258$	0	0
$259$	17.6896	1.09918
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−11.1498	−0.687528	−0.343764	−	0.939056i	$-0.611702\pi$
$263$	−11.1498	−0.687528	−0.343764	+	0.939056i	$0.611702\pi$
$264$	0	0
$265$	2.10345	0.129214
$266$	0	0
$267$	0	0
$268$	0	0
$269$	16.2295	0.989528	0.494764	−	0.869027i	$-0.335254\pi$
$269$	16.2295	0.989528	0.494764	+	0.869027i	$0.335254\pi$
$270$	0	0
$271$	−0.366196	−0.0222449	−0.0111224	−	0.999938i	$-0.503540\pi$
$271$	−0.366196	−0.0222449	−0.0111224	+	0.999938i	$0.503540\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.89655	0.355575
$276$	0	0
$277$	33.1724	1.99314	0.996568	−	0.0827774i	$-0.0263791\pi$
$277$	33.1724	1.99314	0.996568	+	0.0827774i	$0.0263791\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	28.1593	1.67984	0.839921	−	0.542708i	$-0.182601\pi$
$281$	28.1593	1.67984	0.839921	+	0.542708i	$0.182601\pi$
$282$	0	0
$283$	6.57310	0.390730	0.195365	−	0.980731i	$-0.437411\pi$
$283$	6.57310	0.390730	0.195365	+	0.980731i	$0.437411\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−17.6896	−1.04419
$288$	0	0
$289$	−10.5493	−0.620547
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3.28655	−0.192002	−0.0960011	−	0.995381i	$-0.530605\pi$
$293$	−3.28655	−0.192002	−0.0960011	+	0.995381i	$0.530605\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.25328	0.303805
$300$	0	0
$301$	17.2200	0.992544
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−0.103450	−0.00592356
$306$	0	0
$307$	26.6100	1.51871	0.759356	−	0.650675i	$-0.225514\pi$
$307$	26.6100	1.51871	0.759356	+	0.650675i	$0.225514\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−31.3793	−1.77936	−0.889678	−	0.456588i	$-0.849071\pi$
$311$	−31.3793	−1.77936	−0.889678	+	0.456588i	$0.849071\pi$
$312$	0	0
$313$	15.0796	0.852352	0.426176	−	0.904640i	$-0.359860\pi$
$313$	15.0796	0.852352	0.426176	+	0.904640i	$0.359860\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.5862	−1.21240	−0.606201	−	0.795312i	$-0.707307\pi$
$317$	−21.5862	−1.21240	−0.606201	+	0.795312i	$0.707307\pi$
$318$	0	0
$319$	−3.79310	−0.212373
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.44584	−0.191732
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−17.2200	−0.949369
$330$	0	0
$331$	−33.8157	−1.85868	−0.929339	−	0.369228i	$-0.879622\pi$
$331$	−33.8157	−1.85868	−0.929339	+	0.369228i	$0.879622\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.79310	0.316511
$336$	0	0
$337$	24.1593	1.31604	0.658020	−	0.753000i	$-0.271394\pi$
$337$	24.1593	1.31604	0.658020	+	0.753000i	$0.271394\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−11.7931	−0.638632
$342$	0	0
$343$	−30.0082	−1.62029
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.89655	0.101812	0.0509061	−	0.998703i	$-0.483789\pi$
$347$	1.89655	0.101812	0.0509061	+	0.998703i	$0.483789\pi$
$348$	0	0
$349$	−33.8157	−1.81011	−0.905056	−	0.425293i	$-0.860171\pi$
$349$	−33.8157	−1.81011	−0.905056	+	0.425293i	$0.860171\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	22.7324	1.20992	0.604962	−	0.796255i	$-0.293188\pi$
$353$	22.7324	1.20992	0.604962	+	0.796255i	$0.293188\pi$
$354$	0	0
$355$	−3.18310	−0.168941
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.5862	−0.822608	−0.411304	−	0.911498i	$-0.634927\pi$
$359$	−15.5862	−0.822608	−0.411304	+	0.911498i	$0.634927\pi$
$360$	0	0
$361$	−17.1593	−0.903121
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−10.4364	−0.546265
$366$	0	0
$367$	−33.7455	−1.76150	−0.880750	−	0.473581i	$-0.842961\pi$
$367$	−33.7455	−1.76150	−0.880750	+	0.473581i	$0.842961\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	9.54929	0.495775
$372$	0	0
$373$	11.4269	0.591663	0.295831	−	0.955240i	$-0.404403\pi$
$373$	11.4269	0.591663	0.295831	+	0.955240i	$0.404403\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.643274	0.0331303
$378$	0	0
$379$	−9.56363	−0.491251	−0.245625	−	0.969365i	$-0.578993\pi$
$379$	−9.56363	−0.491251	−0.245625	+	0.969365i	$0.578993\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	32.8727	1.67972	0.839859	−	0.542804i	$-0.182637\pi$
$383$	32.8727	1.67972	0.839859	+	0.542804i	$0.182637\pi$
$384$	0	0
$385$	26.7693	1.36429
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−0.502920	−0.0254991	−0.0127495	−	0.999919i	$-0.504058\pi$
$389$	−0.502920	−0.0254991	−0.0127495	+	0.999919i	$0.504058\pi$
$390$	0	0
$391$	13.3424	0.674754
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−10.9762	−0.552272
$396$	0	0
$397$	3.68965	0.185178	0.0925891	−	0.995704i	$-0.470486\pi$
$397$	3.68965	0.185178	0.0925891	+	0.995704i	$0.470486\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.2865	−0.563623	−0.281812	−	0.959470i	$-0.590935\pi$
$401$	−11.2865	−0.563623	−0.281812	+	0.959470i	$0.590935\pi$
$402$	0	0
$403$	2.00000	0.0996271
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−22.9762	−1.13889
$408$	0	0
$409$	−15.0796	−0.745641	−0.372820	−	0.927904i	$-0.621609\pi$
$409$	−15.0796	−0.745641	−0.372820	+	0.927904i	$0.621609\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−18.1593	−0.893561
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	0	0
$418$	0	0
$419$	11.7705	0.575028	0.287514	−	0.957776i	$-0.407171\pi$
$419$	11.7705	0.575028	0.287514	+	0.957776i	$0.407171\pi$
$420$	0	0
$421$	7.65638	0.373149	0.186574	−	0.982441i	$-0.440261\pi$
$421$	7.65638	0.373149	0.186574	+	0.982441i	$0.440261\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.53982	−0.123200
$426$	0	0
$427$	−0.469647	−0.0227278
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−27.4458	−1.32202	−0.661010	−	0.750377i	$-0.729872\pi$
$431$	−27.4458	−1.32202	−0.661010	+	0.750377i	$0.729872\pi$
$432$	0	0
$433$	−19.2200	−0.923654	−0.461827	−	0.886970i	$-0.652806\pi$
$433$	−19.2200	−0.923654	−0.461827	+	0.886970i	$0.652806\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.12725	−0.340943
$438$	0	0
$439$	−3.39000	−0.161796	−0.0808979	−	0.996722i	$-0.525779\pi$
$439$	−3.39000	−0.161796	−0.0808979	+	0.996722i	$0.525779\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.26275	−0.202529	−0.101265	−	0.994860i	$-0.532289\pi$
$443$	−4.26275	−0.202529	−0.101265	+	0.994860i	$0.532289\pi$
$444$	0	0
$445$	10.6100	0.502962
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9.18310	0.433377	0.216689	−	0.976241i	$-0.430474\pi$
$449$	9.18310	0.433377	0.216689	+	0.976241i	$0.430474\pi$
$450$	0	0
$451$	22.9762	1.08191
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.53982	−0.212830
$456$	0	0
$457$	−25.5529	−1.19532	−0.597658	−	0.801751i	$-0.703902\pi$
$457$	−25.5529	−1.19532	−0.597658	+	0.801751i	$0.703902\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−22.2627	−1.03688	−0.518440	−	0.855114i	$-0.673487\pi$
$461$	−22.2627	−1.03688	−0.518440	+	0.855114i	$0.673487\pi$
$462$	0	0
$463$	17.4791	0.812323	0.406162	−	0.913801i	$-0.366867\pi$
$463$	17.4791	0.812323	0.406162	+	0.913801i	$0.366867\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.7693	0.498344	0.249172	−	0.968459i	$-0.419842\pi$
$467$	10.7693	0.498344	0.249172	+	0.968459i	$0.419842\pi$
$468$	0	0
$469$	26.2996	1.21440
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−22.3662	−1.02840
$474$	0	0
$475$	1.35673	0.0622508
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−30.9097	−1.41230	−0.706149	−	0.708063i	$-0.749569\pi$
$479$	−30.9097	−1.41230	−0.706149	+	0.708063i	$0.749569\pi$
$480$	0	0
$481$	3.89655	0.177667
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.46018	−0.157119
$486$	0	0
$487$	28.2664	1.28087	0.640436	−	0.768012i	$-0.278754\pi$
$487$	28.2664	1.28087	0.640436	+	0.768012i	$0.278754\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	30.9429	1.39643	0.698217	−	0.715886i	$-0.253977\pi$
$491$	30.9429	1.39643	0.698217	+	0.715886i	$0.253977\pi$
$492$	0	0
$493$	1.63380	0.0735828
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−14.4507	−0.648203
$498$	0	0
$499$	−21.1498	−0.946796	−0.473398	−	0.880849i	$-0.656973\pi$
$499$	−21.1498	−0.946796	−0.473398	+	0.880849i	$0.656973\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	33.4495	1.49144	0.745719	−	0.666260i	$-0.232106\pi$
$503$	33.4495	1.49144	0.745719	+	0.666260i	$0.232106\pi$
$504$	0	0
$505$	11.1498	0.496161
$506$	0	0
$507$	0	0
$508$	0	0
$509$	18.2627	0.809482	0.404741	−	0.914431i	$-0.367362\pi$
$509$	18.2627	0.809482	0.404741	+	0.914431i	$0.367362\pi$
$510$	0	0
$511$	−47.3793	−2.09594
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−17.0796	−0.752619
$516$	0	0
$517$	22.3662	0.983664
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−17.0796	−0.748273	−0.374136	−	0.927374i	$-0.622061\pi$
$521$	−17.0796	−0.748273	−0.374136	+	0.927374i	$0.622061\pi$
$522$	0	0
$523$	−9.42690	−0.412210	−0.206105	−	0.978530i	$-0.566079\pi$
$523$	−9.42690	−0.412210	−0.206105	+	0.978530i	$0.566079\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.07965	0.221273
$528$	0	0
$529$	4.59690	0.199865
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.89655	−0.168778
$534$	0	0
$535$	10.9762	0.474542
$536$	0	0
$537$	0	0
$538$	0	0
$539$	80.2520	3.45670
$540$	0	0
$541$	20.8026	0.894372	0.447186	−	0.894441i	$-0.352426\pi$
$541$	20.8026	0.894372	0.447186	+	0.894441i	$0.352426\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.43637	0.104363
$546$	0	0
$547$	2.36620	0.101171	0.0505856	−	0.998720i	$-0.483891\pi$
$547$	2.36620	0.101171	0.0505856	+	0.998720i	$0.483891\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.872747	−0.0371803
$552$	0	0
$553$	−49.8300	−2.11899
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.159296	0.00674957	0.00337478	−	0.999994i	$-0.498926\pi$
$557$	0.159296	0.00674957	0.00337478	+	0.999994i	$0.498926\pi$
$558$	0	0
$559$	3.79310	0.160431
$560$	0	0
$561$	0	0
$562$	0	0
$563$	42.6289	1.79660	0.898298	−	0.439387i	$-0.144804\pi$
$563$	42.6289	1.79660	0.898298	+	0.439387i	$0.144804\pi$
$564$	0	0
$565$	−3.35673	−0.141219
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11.2389	−0.471161	−0.235581	−	0.971855i	$-0.575699\pi$
$569$	−11.2389	−0.471161	−0.235581	+	0.971855i	$0.575699\pi$
$570$	0	0
$571$	26.9762	1.12892	0.564459	−	0.825461i	$-0.309085\pi$
$571$	26.9762	1.12892	0.564459	+	0.825461i	$0.309085\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.25328	−0.219077
$576$	0	0
$577$	5.75983	0.239785	0.119892	−	0.992787i	$-0.461745\pi$
$577$	5.75983	0.239785	0.119892	+	0.992787i	$0.461745\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−18.1593	−0.753375
$582$	0	0
$583$	−12.4031	−0.513684
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.8727	−0.861510	−0.430755	−	0.902469i	$-0.641753\pi$
$587$	−20.8727	−0.861510	−0.430755	+	0.902469i	$0.641753\pi$
$588$	0	0
$589$	−2.71345	−0.111806
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−23.0131	−0.945035	−0.472517	−	0.881321i	$-0.656655\pi$
$593$	−23.0131	−0.945035	−0.472517	+	0.881321i	$0.656655\pi$
$594$	0	0
$595$	−11.5304	−0.472698
$596$	0	0
$597$	0	0
$598$	0	0
$599$	21.9524	0.896951	0.448475	−	0.893795i	$-0.351967\pi$
$599$	21.9524	0.896951	0.448475	+	0.893795i	$0.351967\pi$
$600$	0	0
$601$	−12.9762	−0.529310	−0.264655	−	0.964343i	$-0.585258\pi$
$601$	−12.9762	−0.529310	−0.264655	+	0.964343i	$0.585258\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−23.7693	−0.966359
$606$	0	0
$607$	39.5862	1.60675	0.803377	−	0.595471i	$-0.203034\pi$
$607$	39.5862	1.60675	0.803377	+	0.595471i	$0.203034\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.79310	−0.153452
$612$	0	0
$613$	−47.6896	−1.92617	−0.963083	−	0.269203i	$-0.913240\pi$
$613$	−47.6896	−1.92617	−0.963083	+	0.269203i	$0.913240\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9.85965	−0.396934	−0.198467	−	0.980108i	$-0.563596\pi$
$617$	−9.85965	−0.396934	−0.198467	+	0.980108i	$0.563596\pi$
$618$	0	0
$619$	−2.78363	−0.111883	−0.0559417	−	0.998434i	$-0.517816\pi$
$619$	−2.78363	−0.111883	−0.0559417	+	0.998434i	$0.517816\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	48.1675	1.92979
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9.89655	0.394601
$630$	0	0
$631$	−5.12725	−0.204113	−0.102056	−	0.994779i	$-0.532542\pi$
$631$	−5.12725	−0.204113	−0.102056	+	0.994779i	$0.532542\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−13.6100	−0.539248
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.15930	−0.0852870	−0.0426435	−	0.999090i	$-0.513578\pi$
$641$	−2.15930	−0.0852870	−0.0426435	+	0.999090i	$0.513578\pi$
$642$	0	0
$643$	38.1962	1.50631	0.753156	−	0.657843i	$-0.228531\pi$
$643$	38.1962	1.50631	0.753156	+	0.657843i	$0.228531\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.41257	0.134162	0.0670810	−	0.997748i	$-0.478631\pi$
$647$	3.41257	0.134162	0.0670810	+	0.997748i	$0.478631\pi$
$648$	0	0
$649$	23.5862	0.925839
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8.93929	0.349822	0.174911	−	0.984584i	$-0.444036\pi$
$653$	8.93929	0.349822	0.174911	+	0.984584i	$0.444036\pi$
$654$	0	0
$655$	−9.72292	−0.379906
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.5029	0.487045	0.243522	−	0.969895i	$-0.421697\pi$
$659$	12.5029	0.487045	0.243522	+	0.969895i	$0.421697\pi$
$660$	0	0
$661$	23.1688	0.901161	0.450580	−	0.892736i	$-0.351217\pi$
$661$	23.1688	0.901161	0.450580	+	0.892736i	$0.351217\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.15930	0.238847
$666$	0	0
$667$	3.37930	0.130847
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.610001	0.0235488
$672$	0	0
$673$	46.0451	1.77491	0.887455	−	0.460895i	$-0.152471\pi$
$673$	46.0451	1.77491	0.887455	+	0.460895i	$0.152471\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	40.9286	1.57301	0.786507	−	0.617582i	$-0.211887\pi$
$677$	40.9286	1.57301	0.786507	+	0.617582i	$0.211887\pi$
$678$	0	0
$679$	−15.7086	−0.602840
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−45.3317	−1.73457	−0.867284	−	0.497813i	$-0.834137\pi$
$683$	−45.3317	−1.73457	−0.867284	+	0.497813i	$0.834137\pi$
$684$	0	0
$685$	2.20690	0.0843214
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.10345	0.0801351
$690$	0	0
$691$	33.6753	1.28107	0.640535	−	0.767929i	$-0.278713\pi$
$691$	33.6753	1.28107	0.640535	+	0.767929i	$0.278713\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.1831	−0.575928
$696$	0	0
$697$	−9.89655	−0.374859
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−30.4553	−1.15028	−0.575141	−	0.818055i	$-0.695053\pi$
$701$	−30.4553	−1.15028	−0.575141	+	0.818055i	$0.695053\pi$
$702$	0	0
$703$	−5.28655	−0.199386
$704$	0	0
$705$	0	0
$706$	0	0
$707$	50.6182	1.90369
$708$	0	0
$709$	8.94292	0.335859	0.167929	−	0.985799i	$-0.446292\pi$
$709$	8.94292	0.335859	0.167929	+	0.985799i	$0.446292\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	10.5066	0.393473
$714$	0	0
$715$	5.89655	0.220518
$716$	0	0
$717$	0	0
$718$	0	0
$719$	42.2996	1.57751	0.788755	−	0.614707i	$-0.210726\pi$
$719$	42.2996	1.57751	0.788755	+	0.614707i	$0.210726\pi$
$720$	0	0
$721$	−77.5386	−2.88769
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.643274	−0.0238906
$726$	0	0
$727$	43.3793	1.60885	0.804425	−	0.594055i	$-0.202474\pi$
$727$	43.3793	1.60885	0.804425	+	0.594055i	$0.202474\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.63380	0.356319
$732$	0	0
$733$	−28.6289	−1.05743	−0.528717	−	0.848798i	$-0.677327\pi$
$733$	−28.6289	−1.05743	−0.528717	+	0.848798i	$0.677327\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−34.1593	−1.25827
$738$	0	0
$739$	−25.5636	−0.940373	−0.470187	−	0.882567i	$-0.655813\pi$
$739$	−25.5636	−0.940373	−0.470187	+	0.882567i	$0.655813\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12.7324	0.467106	0.233553	−	0.972344i	$-0.424965\pi$
$743$	12.7324	0.467106	0.233553	+	0.972344i	$0.424965\pi$
$744$	0	0
$745$	20.7693	0.760928
$746$	0	0
$747$	0	0
$748$	0	0
$749$	49.8300	1.82075
$750$	0	0
$751$	−26.5624	−0.969276	−0.484638	−	0.874715i	$-0.661049\pi$
$751$	−26.5624	−0.969276	−0.484638	+	0.874715i	$0.661049\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−21.7931	−0.793132
$756$	0	0
$757$	−30.5993	−1.11215	−0.556075	−	0.831132i	$-0.687693\pi$
$757$	−30.5993	−1.11215	−0.556075	+	0.831132i	$0.687693\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	14.1404	0.512587	0.256294	−	0.966599i	$-0.417499\pi$
$761$	14.1404	0.512587	0.256294	+	0.966599i	$0.417499\pi$
$762$	0	0
$763$	11.0607	0.400424
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.00000	−0.144432
$768$	0	0
$769$	−5.58620	−0.201443	−0.100722	−	0.994915i	$-0.532115\pi$
$769$	−5.58620	−0.201443	−0.100722	+	0.994915i	$0.532115\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−33.2389	−1.19552	−0.597761	−	0.801674i	$-0.703943\pi$
$773$	−33.2389	−1.19552	−0.597761	+	0.801674i	$0.703943\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.28655	0.189410
$780$	0	0
$781$	18.7693	0.671618
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−16.3662	−0.584135
$786$	0	0
$787$	−24.5066	−0.873564	−0.436782	−	0.899567i	$-0.643882\pi$
$787$	−24.5066	−0.873564	−0.436782	+	0.899567i	$0.643882\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−15.2389	−0.541834
$792$	0	0
$793$	−0.103450	−0.00367363
$794$	0	0
$795$	0	0
$796$	0	0
$797$	19.8489	0.703086	0.351543	−	0.936172i	$-0.385657\pi$
$797$	19.8489	0.703086	0.351543	+	0.936172i	$0.385657\pi$
$798$	0	0
$799$	−9.63380	−0.340819
$800$	0	0
$801$	0	0
$802$	0	0
$803$	61.5386	2.17165
$804$	0	0
$805$	−23.8489	−0.840565
$806$	0	0
$807$	0	0
$808$	0	0
$809$	29.7455	1.04580	0.522898	−	0.852395i	$-0.324851\pi$
$809$	29.7455	1.04580	0.522898	+	0.852395i	$0.324851\pi$
$810$	0	0
$811$	−0.277078	−0.00972952	−0.00486476	−	0.999988i	$-0.501549\pi$
$811$	−0.277078	−0.00972952	−0.00486476	+	0.999988i	$0.501549\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−23.4827	−0.822565
$816$	0	0
$817$	−5.14619	−0.180043
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−48.5624	−1.69484	−0.847420	−	0.530924i	$-0.821845\pi$
$821$	−48.5624	−1.69484	−0.847420	+	0.530924i	$0.821845\pi$
$822$	0	0
$823$	28.4589	0.992016	0.496008	−	0.868318i	$-0.334799\pi$
$823$	28.4589	0.992016	0.496008	+	0.868318i	$0.334799\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−23.3793	−0.812978	−0.406489	−	0.913656i	$-0.633247\pi$
$827$	−23.3793	−0.812978	−0.406489	+	0.913656i	$0.633247\pi$
$828$	0	0
$829$	24.1593	0.839087	0.419544	−	0.907735i	$-0.362190\pi$
$829$	24.1593	0.839087	0.419544	+	0.907735i	$0.362190\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−34.5670	−1.19768
$834$	0	0
$835$	13.0796	0.452640
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−26.4220	−0.912190	−0.456095	−	0.889931i	$-0.650752\pi$
$839$	−26.4220	−0.912190	−0.456095	+	0.889931i	$0.650752\pi$
$840$	0	0
$841$	−28.5862	−0.985731
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−107.908	−3.70778
$848$	0	0
$849$	0	0
$850$	0	0
$851$	20.4696	0.701690
$852$	0	0
$853$	26.0558	0.892135	0.446068	−	0.894999i	$-0.352824\pi$
$853$	26.0558	0.892135	0.446068	+	0.894999i	$0.352824\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−48.6326	−1.66126	−0.830629	−	0.556827i	$-0.812019\pi$
$857$	−48.6326	−1.66126	−0.830629	+	0.556827i	$0.812019\pi$
$858$	0	0
$859$	−32.2627	−1.10079	−0.550395	−	0.834904i	$-0.685523\pi$
$859$	−32.2627	−1.10079	−0.550395	+	0.834904i	$0.685523\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−46.4113	−1.57986	−0.789930	−	0.613197i	$-0.789883\pi$
$863$	−46.4113	−1.57986	−0.789930	+	0.613197i	$0.789883\pi$
$864$	0	0
$865$	−18.5066	−0.629242
$866$	0	0
$867$	0	0
$868$	0	0
$869$	64.7217	2.19553
$870$	0	0
$871$	5.79310	0.196292
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.53982	0.153474
$876$	0	0
$877$	−12.5731	−0.424563	−0.212282	−	0.977209i	$-0.568089\pi$
$877$	−12.5731	−0.424563	−0.212282	+	0.977209i	$0.568089\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−16.6658	−0.561487	−0.280743	−	0.959783i	$-0.590581\pi$
$881$	−16.6658	−0.561487	−0.280743	+	0.959783i	$0.590581\pi$
$882$	0	0
$883$	−41.5386	−1.39788	−0.698942	−	0.715178i	$-0.746346\pi$
$883$	−41.5386	−1.39788	−0.698942	+	0.715178i	$0.746346\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−22.1926	−0.745154	−0.372577	−	0.928001i	$-0.621526\pi$
$887$	−22.1926	−0.745154	−0.372577	+	0.928001i	$0.621526\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.14619	0.172211
$894$	0	0
$895$	−12.6433	−0.422618
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.28655	0.0429088
$900$	0	0
$901$	5.34239	0.177981
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.32345	0.309922
$906$	0	0
$907$	−49.5386	−1.64490	−0.822451	−	0.568836i	$-0.807394\pi$
$907$	−49.5386	−1.64490	−0.822451	+	0.568836i	$0.807394\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20.4589	0.677835	0.338918	−	0.940816i	$-0.389939\pi$
$911$	20.4589	0.677835	0.338918	+	0.940816i	$0.389939\pi$
$912$	0	0
$913$	23.5862	0.780589
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−44.1404	−1.45764
$918$	0	0
$919$	10.5624	0.348421	0.174211	−	0.984708i	$-0.444263\pi$
$919$	10.5624	0.348421	0.174211	+	0.984708i	$0.444263\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.18310	−0.104773
$924$	0	0
$925$	−3.89655	−0.128118
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.18310	−0.0388162	−0.0194081	−	0.999812i	$-0.506178\pi$
$929$	−1.18310	−0.0388162	−0.0194081	+	0.999812i	$0.506178\pi$
$930$	0	0
$931$	18.4650	0.605167
$932$	0	0
$933$	0	0
$934$	0	0
$935$	14.9762	0.489774
$936$	0	0
$937$	−39.0796	−1.27668	−0.638338	−	0.769756i	$-0.720378\pi$
$937$	−39.0796	−1.27668	−0.638338	+	0.769756i	$0.720378\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−22.7504	−0.741640	−0.370820	−	0.928705i	$-0.620923\pi$
$941$	−22.7504	−0.741640	−0.370820	+	0.928705i	$0.620923\pi$
$942$	0	0
$943$	−20.4696	−0.666583
$944$	0	0
$945$	0	0
$946$	0	0
$947$	17.9524	0.583374	0.291687	−	0.956514i	$-0.405783\pi$
$947$	17.9524	0.583374	0.291687	+	0.956514i	$0.405783\pi$
$948$	0	0
$949$	−10.4364	−0.338779
$950$	0	0
$951$	0	0
$952$	0	0
$953$	21.7788	0.705483	0.352742	−	0.935721i	$-0.385249\pi$
$953$	21.7788	0.705483	0.352742	+	0.935721i	$0.385249\pi$
$954$	0	0
$955$	−23.7931	−0.769927
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10.0189	0.323528
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4.53982	−0.146142
$966$	0	0
$967$	1.21637	0.0391159	0.0195579	−	0.999809i	$-0.493774\pi$
$967$	1.21637	0.0391159	0.0195579	+	0.999809i	$0.493774\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.85018	0.155650	0.0778248	−	0.996967i	$-0.475203\pi$
$971$	4.85018	0.155650	0.0778248	+	0.996967i	$0.475203\pi$
$972$	0	0
$973$	−68.9286	−2.20975
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−31.0131	−0.992197	−0.496098	−	0.868266i	$-0.665235\pi$
$977$	−31.0131	−0.992197	−0.496098	+	0.868266i	$0.665235\pi$
$978$	0	0
$979$	−62.5624	−1.99950
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−36.7324	−1.17158	−0.585791	−	0.810462i	$-0.699216\pi$
$983$	−36.7324	−1.17158	−0.585791	+	0.810462i	$0.699216\pi$
$984$	0	0
$985$	−4.50655	−0.143591
$986$	0	0
$987$	0	0
$988$	0	0
$989$	19.9262	0.633616
$990$	0	0
$991$	55.7824	1.77199	0.885993	−	0.463698i	$-0.153478\pi$
$991$	55.7824	1.77199	0.885993	+	0.463698i	$0.153478\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.42690	0.172044
$996$	0	0
$997$	3.84070	0.121636	0.0608182	−	0.998149i	$-0.480629\pi$
$997$	3.84070	0.121636	0.0608182	+	0.998149i	$0.480629\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.a.bg.1.1	✓	3
3.2	odd	2		4680.2.a.bj.1.1	yes	3
4.3	odd	2		9360.2.a.cz.1.3		3
12.11	even	2		9360.2.a.de.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4680.2.a.bg.1.1	✓	3	1.1	even	1	trivial
4680.2.a.bj.1.1	yes	3	3.2	odd	2
9360.2.a.cz.1.3		3	4.3	odd	2
9360.2.a.de.1.3		3	12.11	even	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:	\( N \)	\(=\)	\( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4680.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -4.53982 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -4.53982 q^{7} +5.89655 q^{11} -1.00000 q^{13} -2.53982 q^{17} +1.35673 q^{19} -5.25328 q^{23} +1.00000 q^{25} -0.643274 q^{29} -2.00000 q^{31} +4.53982 q^{35} -3.89655 q^{37} +3.89655 q^{41} -3.79310 q^{43} +3.79310 q^{47} +13.6100 q^{49} -2.10345 q^{53} -5.89655 q^{55} +4.00000 q^{59} +0.103450 q^{61} +1.00000 q^{65} -5.79310 q^{67} +3.18310 q^{71} +10.4364 q^{73} -26.7693 q^{77} +10.9762 q^{79} +4.00000 q^{83} +2.53982 q^{85} -10.6100 q^{89} +4.53982 q^{91} -1.35673 q^{95} +3.46018 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} - q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 - q^7	\( 3 q - 3 q^{5} - q^{7} + 5 q^{11} - 3 q^{13} + 5 q^{17} + 4 q^{19} - 3 q^{23} + 3 q^{25} - 2 q^{29} - 6 q^{31} + q^{35} + q^{37} - q^{41} + 14 q^{43} - 14 q^{47} + 28 q^{49} - 19 q^{53} - 5 q^{55} + 12 q^{59} + 13 q^{61} + 3 q^{65} + 8 q^{67} - 3 q^{71} + 6 q^{73} - 17 q^{77} - 5 q^{79} + 12 q^{83} - 5 q^{85} - 19 q^{89} + q^{91} - 4 q^{95} + 23 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 - q^7 + 5 * q^11 - 3 * q^13 + 5 * q^17 + 4 * q^19 - 3 * q^23 + 3 * q^25 - 2 * q^29 - 6 * q^31 + q^35 + q^37 - q^41 + 14 * q^43 - 14 * q^47 + 28 * q^49 - 19 * q^53 - 5 * q^55 + 12 * q^59 + 13 * q^61 + 3 * q^65 + 8 * q^67 - 3 * q^71 + 6 * q^73 - 17 * q^77 - 5 * q^79 + 12 * q^83 - 5 * q^85 - 19 * q^89 + q^91 - 4 * q^95 + 23 * q^97

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