LMFDB - Embedded newform 6012.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(1,6012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6012, 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("6012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.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:	$48.0060616952$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 31x^{7} + 24x^{6} + 293x^{5} - 101x^{4} - 864x^{3} - 278x^{2} + 24x + 2 $ x^9 - x^8 - 31x^7 + 24x^6 + 293x^5 - 101x^4 - 864x^3 - 278x^2 + 24*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2004)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.91239$ of defining polynomial
Character	$\chi$	$=$	6012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.91239 q^{5} +1.60938 q^{7} +O(q^{10})$	$q-3.91239 q^{5} +1.60938 q^{7} +5.85136 q^{11} +0.924191 q^{13} -6.27229 q^{17} -2.72623 q^{19} -0.550033 q^{23} +10.3068 q^{25} -0.556443 q^{29} +9.36591 q^{31} -6.29652 q^{35} +6.50453 q^{37} -2.46246 q^{41} -2.43816 q^{43} -4.96474 q^{47} -4.40989 q^{49} +3.80948 q^{53} -22.8928 q^{55} +10.0411 q^{59} -9.11969 q^{61} -3.61579 q^{65} -2.78770 q^{67} -12.9838 q^{71} +15.6878 q^{73} +9.41706 q^{77} +1.12576 q^{79} -16.5458 q^{83} +24.5396 q^{85} +7.96204 q^{89} +1.48737 q^{91} +10.6661 q^{95} -4.68094 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{5} + 2 q^{7}+O(q^{10}) $ 9 * q - q^5 + 2 * q^7	$ 9 q - q^{5} + 2 q^{7} + 9 q^{11} + 10 q^{13} - 7 q^{17} - 2 q^{19} + 3 q^{23} + 18 q^{25} - 5 q^{29} + 12 q^{31} + 6 q^{35} + 15 q^{37} - 14 q^{41} + 6 q^{43} + 3 q^{47} + 27 q^{49} - 9 q^{53} + 19 q^{55} + 9 q^{59} + 30 q^{61} - 28 q^{65} + 16 q^{67} + 3 q^{71} + 32 q^{73} - 18 q^{77} + 24 q^{79} + 3 q^{83} + 37 q^{85} - 46 q^{89} + 33 q^{91} - 11 q^{95} + 43 q^{97}+O(q^{100}) $ 9 * q - q^5 + 2 * q^7 + 9 * q^11 + 10 * q^13 - 7 * q^17 - 2 * q^19 + 3 * q^23 + 18 * q^25 - 5 * q^29 + 12 * q^31 + 6 * q^35 + 15 * q^37 - 14 * q^41 + 6 * q^43 + 3 * q^47 + 27 * q^49 - 9 * q^53 + 19 * q^55 + 9 * q^59 + 30 * q^61 - 28 * q^65 + 16 * q^67 + 3 * q^71 + 32 * q^73 - 18 * q^77 + 24 * q^79 + 3 * q^83 + 37 * q^85 - 46 * q^89 + 33 * q^91 - 11 * q^95 + 43 * 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
$5$	−3.91239	−1.74967	−0.874836	−	0.484419i	$-0.839031\pi$
$5$	−3.91239	−1.74967	−0.874836	+	0.484419i	$0.839031\pi$
$6$	0	0
$7$	1.60938	0.608289	0.304144	−	0.952626i	$-0.401629\pi$
$7$	1.60938	0.608289	0.304144	+	0.952626i	$0.401629\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.85136	1.76425	0.882126	−	0.471014i	$-0.156112\pi$
$11$	5.85136	1.76425	0.882126	+	0.471014i	$0.156112\pi$
$12$	0	0
$13$	0.924191	0.256324	0.128162	−	0.991753i	$-0.459092\pi$
$13$	0.924191	0.256324	0.128162	+	0.991753i	$0.459092\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.27229	−1.52125	−0.760627	−	0.649189i	$-0.775109\pi$
$17$	−6.27229	−1.52125	−0.760627	+	0.649189i	$0.775109\pi$
$18$	0	0
$19$	−2.72623	−0.625440	−0.312720	−	0.949845i	$-0.601240\pi$
$19$	−2.72623	−0.625440	−0.312720	+	0.949845i	$0.601240\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.550033	−0.114690	−0.0573449	−	0.998354i	$-0.518263\pi$
$23$	−0.550033	−0.114690	−0.0573449	+	0.998354i	$0.518263\pi$
$24$	0	0
$25$	10.3068	2.06135
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.556443	−0.103329	−0.0516645	−	0.998665i	$-0.516453\pi$
$29$	−0.556443	−0.103329	−0.0516645	+	0.998665i	$0.516453\pi$
$30$	0	0
$31$	9.36591	1.68217	0.841084	−	0.540905i	$-0.181918\pi$
$31$	9.36591	1.68217	0.841084	+	0.540905i	$0.181918\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−6.29652	−1.06431
$36$	0	0
$37$	6.50453	1.06934	0.534669	−	0.845062i	$-0.320436\pi$
$37$	6.50453	1.06934	0.534669	+	0.845062i	$0.320436\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.46246	−0.384572	−0.192286	−	0.981339i	$-0.561590\pi$
$41$	−2.46246	−0.384572	−0.192286	+	0.981339i	$0.561590\pi$
$42$	0	0
$43$	−2.43816	−0.371816	−0.185908	−	0.982567i	$-0.559523\pi$
$43$	−2.43816	−0.371816	−0.185908	+	0.982567i	$0.559523\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.96474	−0.724181	−0.362091	−	0.932143i	$-0.617937\pi$
$47$	−4.96474	−0.724181	−0.362091	+	0.932143i	$0.617937\pi$
$48$	0	0
$49$	−4.40989	−0.629985
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.80948	0.523272	0.261636	−	0.965167i	$-0.415738\pi$
$53$	3.80948	0.523272	0.261636	+	0.965167i	$0.415738\pi$
$54$	0	0
$55$	−22.8928	−3.08686
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.0411	1.30723	0.653617	−	0.756826i	$-0.273251\pi$
$59$	10.0411	1.30723	0.653617	+	0.756826i	$0.273251\pi$
$60$	0	0
$61$	−9.11969	−1.16766	−0.583828	−	0.811877i	$-0.698446\pi$
$61$	−9.11969	−1.16766	−0.583828	+	0.811877i	$0.698446\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.61579	−0.448484
$66$	0	0
$67$	−2.78770	−0.340572	−0.170286	−	0.985395i	$-0.554469\pi$
$67$	−2.78770	−0.340572	−0.170286	+	0.985395i	$0.554469\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.9838	−1.54090	−0.770448	−	0.637503i	$-0.779967\pi$
$71$	−12.9838	−1.54090	−0.770448	+	0.637503i	$0.779967\pi$
$72$	0	0
$73$	15.6878	1.83612	0.918061	−	0.396439i	$-0.129754\pi$
$73$	15.6878	1.83612	0.918061	+	0.396439i	$0.129754\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9.41706	1.07317
$78$	0	0
$79$	1.12576	0.126658	0.0633289	−	0.997993i	$-0.479828\pi$
$79$	1.12576	0.126658	0.0633289	+	0.997993i	$0.479828\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−16.5458	−1.81614	−0.908069	−	0.418821i	$-0.862443\pi$
$83$	−16.5458	−1.81614	−0.908069	+	0.418821i	$0.862443\pi$
$84$	0	0
$85$	24.5396	2.66170
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.96204	0.843975	0.421987	−	0.906602i	$-0.361333\pi$
$89$	7.96204	0.843975	0.421987	+	0.906602i	$0.361333\pi$
$90$	0	0
$91$	1.48737	0.155919
$92$	0	0
$93$	0	0
$94$	0	0
$95$	10.6661	1.09431
$96$	0	0
$97$	−4.68094	−0.475277	−0.237639	−	0.971354i	$-0.576373\pi$
$97$	−4.68094	−0.475277	−0.237639	+	0.971354i	$0.576373\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.1758	1.01253	0.506266	−	0.862378i	$-0.331026\pi$
$101$	10.1758	1.01253	0.506266	+	0.862378i	$0.331026\pi$
$102$	0	0
$103$	6.78954	0.668994	0.334497	−	0.942397i	$-0.391434\pi$
$103$	6.78954	0.668994	0.334497	+	0.942397i	$0.391434\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.57700	0.442475	0.221238	−	0.975220i	$-0.428990\pi$
$107$	4.57700	0.442475	0.221238	+	0.975220i	$0.428990\pi$
$108$	0	0
$109$	−5.17032	−0.495227	−0.247613	−	0.968859i	$-0.579646\pi$
$109$	−5.17032	−0.495227	−0.247613	+	0.968859i	$0.579646\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.41595	−0.227273	−0.113637	−	0.993522i	$-0.536250\pi$
$113$	−2.41595	−0.227273	−0.113637	+	0.993522i	$0.536250\pi$
$114$	0	0
$115$	2.15194	0.200669
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−10.0945	−0.925362
$120$	0	0
$121$	23.2384	2.11258
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−20.7621	−1.85702
$126$	0	0
$127$	14.5069	1.28728	0.643638	−	0.765330i	$-0.277424\pi$
$127$	14.5069	1.28728	0.643638	+	0.765330i	$0.277424\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.2309	−1.06862	−0.534310	−	0.845289i	$-0.679428\pi$
$131$	−12.2309	−1.06862	−0.534310	+	0.845289i	$0.679428\pi$
$132$	0	0
$133$	−4.38754	−0.380448
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.60059	0.563927	0.281963	−	0.959425i	$-0.409014\pi$
$137$	6.60059	0.563927	0.281963	+	0.959425i	$0.409014\pi$
$138$	0	0
$139$	0.831721	0.0705456	0.0352728	−	0.999378i	$-0.488770\pi$
$139$	0.831721	0.0705456	0.0352728	+	0.999378i	$0.488770\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.40777	0.452221
$144$	0	0
$145$	2.17702	0.180792
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.30187	0.434346	0.217173	−	0.976133i	$-0.430316\pi$
$149$	5.30187	0.434346	0.217173	+	0.976133i	$0.430316\pi$
$150$	0	0
$151$	−11.3630	−0.924704	−0.462352	−	0.886696i	$-0.652994\pi$
$151$	−11.3630	−0.924704	−0.462352	+	0.886696i	$0.652994\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−36.6431	−2.94324
$156$	0	0
$157$	−1.23038	−0.0981953	−0.0490976	−	0.998794i	$-0.515635\pi$
$157$	−1.23038	−0.0981953	−0.0490976	+	0.998794i	$0.515635\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.885212	−0.0697645
$162$	0	0
$163$	11.6670	0.913831	0.456916	−	0.889510i	$-0.348954\pi$
$163$	11.6670	0.913831	0.456916	+	0.889510i	$0.348954\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−12.1459	−0.934298
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.05092	0.231957	0.115979	−	0.993252i	$-0.463000\pi$
$173$	3.05092	0.231957	0.115979	+	0.993252i	$0.463000\pi$
$174$	0	0
$175$	16.5875	1.25390
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−5.61550	−0.419722	−0.209861	−	0.977731i	$-0.567301\pi$
$179$	−5.61550	−0.419722	−0.209861	+	0.977731i	$0.567301\pi$
$180$	0	0
$181$	13.3833	0.994773	0.497387	−	0.867529i	$-0.334293\pi$
$181$	13.3833	0.994773	0.497387	+	0.867529i	$0.334293\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−25.4482	−1.87099
$186$	0	0
$187$	−36.7014	−2.68388
$188$	0	0
$189$	0	0
$190$	0	0
$191$	23.2079	1.67926	0.839631	−	0.543157i	$-0.182771\pi$
$191$	23.2079	1.67926	0.839631	+	0.543157i	$0.182771\pi$
$192$	0	0
$193$	21.9927	1.58307	0.791533	−	0.611126i	$-0.209283\pi$
$193$	21.9927	1.58307	0.791533	+	0.611126i	$0.209283\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.7670	0.909613	0.454807	−	0.890590i	$-0.349708\pi$
$197$	12.7670	0.909613	0.454807	+	0.890590i	$0.349708\pi$
$198$	0	0
$199$	−6.01292	−0.426245	−0.213122	−	0.977026i	$-0.568363\pi$
$199$	−6.01292	−0.426245	−0.213122	+	0.977026i	$0.568363\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.895529	−0.0628538
$204$	0	0
$205$	9.63410	0.672874
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−15.9521	−1.10343
$210$	0	0
$211$	22.1000	1.52143	0.760715	−	0.649086i	$-0.224849\pi$
$211$	22.1000	1.52143	0.760715	+	0.649086i	$0.224849\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.53903	0.650556
$216$	0	0
$217$	15.0733	1.02324
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5.79680	−0.389935
$222$	0	0
$223$	1.60906	0.107751	0.0538753	−	0.998548i	$-0.482843\pi$
$223$	1.60906	0.107751	0.0538753	+	0.998548i	$0.482843\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.7798	1.18008	0.590042	−	0.807373i	$-0.299111\pi$
$227$	17.7798	1.18008	0.590042	+	0.807373i	$0.299111\pi$
$228$	0	0
$229$	7.81127	0.516183	0.258092	−	0.966120i	$-0.416906\pi$
$229$	7.81127	0.516183	0.258092	+	0.966120i	$0.416906\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.77795	−0.247501	−0.123751	−	0.992313i	$-0.539492\pi$
$233$	−3.77795	−0.247501	−0.123751	+	0.992313i	$0.539492\pi$
$234$	0	0
$235$	19.4240	1.26708
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5.11209	0.330674	0.165337	−	0.986237i	$-0.447129\pi$
$239$	5.11209	0.330674	0.165337	+	0.986237i	$0.447129\pi$
$240$	0	0
$241$	−7.42157	−0.478065	−0.239033	−	0.971012i	$-0.576830\pi$
$241$	−7.42157	−0.478065	−0.239033	+	0.971012i	$0.576830\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	17.2532	1.10227
$246$	0	0
$247$	−2.51956	−0.160316
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.26035	0.142672	0.0713359	−	0.997452i	$-0.477274\pi$
$251$	2.26035	0.142672	0.0713359	+	0.997452i	$0.477274\pi$
$252$	0	0
$253$	−3.21844	−0.202342
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.7941	1.54661	0.773307	−	0.634032i	$-0.218601\pi$
$257$	24.7941	1.54661	0.773307	+	0.634032i	$0.218601\pi$
$258$	0	0
$259$	10.4683	0.650466
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.61784	0.0997601	0.0498801	−	0.998755i	$-0.484116\pi$
$263$	1.61784	0.0997601	0.0498801	+	0.998755i	$0.484116\pi$
$264$	0	0
$265$	−14.9041	−0.915554
$266$	0	0
$267$	0	0
$268$	0	0
$269$	16.0677	0.979662	0.489831	−	0.871817i	$-0.337058\pi$
$269$	16.0677	0.979662	0.489831	+	0.871817i	$0.337058\pi$
$270$	0	0
$271$	−19.8859	−1.20798	−0.603992	−	0.796991i	$-0.706424\pi$
$271$	−19.8859	−1.20798	−0.603992	+	0.796991i	$0.706424\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	60.3085	3.63674
$276$	0	0
$277$	22.6715	1.36220	0.681099	−	0.732191i	$-0.261502\pi$
$277$	22.6715	1.36220	0.681099	+	0.732191i	$0.261502\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.06257	0.480973	0.240486	−	0.970653i	$-0.422693\pi$
$281$	8.06257	0.480973	0.240486	+	0.970653i	$0.422693\pi$
$282$	0	0
$283$	11.2170	0.666783	0.333392	−	0.942788i	$-0.391807\pi$
$283$	11.2170	0.666783	0.333392	+	0.942788i	$0.391807\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.96304	−0.233931
$288$	0	0
$289$	22.3417	1.31422
$290$	0	0
$291$	0	0
$292$	0	0
$293$	11.1981	0.654199	0.327100	−	0.944990i	$-0.393929\pi$
$293$	11.1981	0.654199	0.327100	+	0.944990i	$0.393929\pi$
$294$	0	0
$295$	−39.2845	−2.28723
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.508335	−0.0293978
$300$	0	0
$301$	−3.92393	−0.226172
$302$	0	0
$303$	0	0
$304$	0	0
$305$	35.6797	2.04302
$306$	0	0
$307$	−2.11466	−0.120690	−0.0603451	−	0.998178i	$-0.519220\pi$
$307$	−2.11466	−0.120690	−0.0603451	+	0.998178i	$0.519220\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	15.6061	0.884942	0.442471	−	0.896783i	$-0.354102\pi$
$311$	15.6061	0.884942	0.442471	+	0.896783i	$0.354102\pi$
$312$	0	0
$313$	27.4593	1.55209	0.776045	−	0.630678i	$-0.217223\pi$
$313$	27.4593	1.55209	0.776045	+	0.630678i	$0.217223\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.951285	0.0534295	0.0267147	−	0.999643i	$-0.491495\pi$
$317$	0.951285	0.0534295	0.0267147	+	0.999643i	$0.491495\pi$
$318$	0	0
$319$	−3.25595	−0.182298
$320$	0	0
$321$	0	0
$322$	0	0
$323$	17.0997	0.951453
$324$	0	0
$325$	9.52541	0.528375
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7.99015	−0.440511
$330$	0	0
$331$	20.9964	1.15407	0.577033	−	0.816721i	$-0.304210\pi$
$331$	20.9964	1.15407	0.577033	+	0.816721i	$0.304210\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	10.9065	0.595888
$336$	0	0
$337$	4.40439	0.239922	0.119961	−	0.992779i	$-0.461723\pi$
$337$	4.40439	0.239922	0.119961	+	0.992779i	$0.461723\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	54.8033	2.96777
$342$	0	0
$343$	−18.3629	−0.991501
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.2574	0.872741	0.436371	−	0.899767i	$-0.356264\pi$
$347$	16.2574	0.872741	0.436371	+	0.899767i	$0.356264\pi$
$348$	0	0
$349$	−23.8182	−1.27496	−0.637479	−	0.770468i	$-0.720023\pi$
$349$	−23.8182	−1.27496	−0.637479	+	0.770468i	$0.720023\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	26.2345	1.39632	0.698160	−	0.715942i	$-0.254003\pi$
$353$	26.2345	1.39632	0.698160	+	0.715942i	$0.254003\pi$
$354$	0	0
$355$	50.7977	2.69606
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−14.2502	−0.752096	−0.376048	−	0.926600i	$-0.622717\pi$
$359$	−14.2502	−0.752096	−0.376048	+	0.926600i	$0.622717\pi$
$360$	0	0
$361$	−11.5677	−0.608825
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−61.3769	−3.21261
$366$	0	0
$367$	−15.9009	−0.830021	−0.415011	−	0.909817i	$-0.636222\pi$
$367$	−15.9009	−0.830021	−0.415011	+	0.909817i	$0.636222\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6.13090	0.318300
$372$	0	0
$373$	−0.661408	−0.0342464	−0.0171232	−	0.999853i	$-0.505451\pi$
$373$	−0.661408	−0.0342464	−0.0171232	+	0.999853i	$0.505451\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.514260	−0.0264857
$378$	0	0
$379$	9.53053	0.489550	0.244775	−	0.969580i	$-0.421286\pi$
$379$	9.53053	0.489550	0.244775	+	0.969580i	$0.421286\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	33.7861	1.72639	0.863194	−	0.504873i	$-0.168461\pi$
$383$	33.7861	1.72639	0.863194	+	0.504873i	$0.168461\pi$
$384$	0	0
$385$	−36.8432	−1.87770
$386$	0	0
$387$	0	0
$388$	0	0
$389$	20.2311	1.02576	0.512879	−	0.858461i	$-0.328579\pi$
$389$	20.2311	1.02576	0.512879	+	0.858461i	$0.328579\pi$
$390$	0	0
$391$	3.44997	0.174472
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.40440	−0.221609
$396$	0	0
$397$	12.1945	0.612022	0.306011	−	0.952028i	$-0.401005\pi$
$397$	12.1945	0.612022	0.306011	+	0.952028i	$0.401005\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15.8930	−0.793657	−0.396829	−	0.917893i	$-0.629889\pi$
$401$	−15.8930	−0.793657	−0.396829	+	0.917893i	$0.629889\pi$
$402$	0	0
$403$	8.65589	0.431181
$404$	0	0
$405$	0	0
$406$	0	0
$407$	38.0603	1.88658
$408$	0	0
$409$	−30.7358	−1.51979	−0.759893	−	0.650048i	$-0.774749\pi$
$409$	−30.7358	−1.51979	−0.759893	+	0.650048i	$0.774749\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	16.1599	0.795176
$414$	0	0
$415$	64.7335	3.17764
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−20.9517	−1.02356	−0.511778	−	0.859118i	$-0.671013\pi$
$419$	−20.9517	−1.02356	−0.511778	+	0.859118i	$0.671013\pi$
$420$	0	0
$421$	26.9234	1.31217	0.656084	−	0.754688i	$-0.272212\pi$
$421$	26.9234	1.31217	0.656084	+	0.754688i	$0.272212\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−64.6470	−3.13584
$426$	0	0
$427$	−14.6770	−0.710272
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.58775	0.172816	0.0864080	−	0.996260i	$-0.472461\pi$
$431$	3.58775	0.172816	0.0864080	+	0.996260i	$0.472461\pi$
$432$	0	0
$433$	−9.90767	−0.476132	−0.238066	−	0.971249i	$-0.576513\pi$
$433$	−9.90767	−0.476132	−0.238066	+	0.971249i	$0.576513\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.49952	0.0717315
$438$	0	0
$439$	−41.8048	−1.99524	−0.997618	−	0.0689843i	$-0.978024\pi$
$439$	−41.8048	−1.99524	−0.997618	+	0.0689843i	$0.978024\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−26.1889	−1.24427	−0.622137	−	0.782908i	$-0.713735\pi$
$443$	−26.1889	−1.24427	−0.622137	+	0.782908i	$0.713735\pi$
$444$	0	0
$445$	−31.1506	−1.47668
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−33.2672	−1.56997	−0.784987	−	0.619512i	$-0.787331\pi$
$449$	−33.2672	−1.56997	−0.784987	+	0.619512i	$0.787331\pi$
$450$	0	0
$451$	−14.4087	−0.678481
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.81918	−0.272808
$456$	0	0
$457$	−20.3028	−0.949723	−0.474861	−	0.880061i	$-0.657502\pi$
$457$	−20.3028	−0.949723	−0.474861	+	0.880061i	$0.657502\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	31.9620	1.48862	0.744310	−	0.667834i	$-0.232778\pi$
$461$	31.9620	1.48862	0.744310	+	0.667834i	$0.232778\pi$
$462$	0	0
$463$	5.32886	0.247653	0.123826	−	0.992304i	$-0.460483\pi$
$463$	5.32886	0.247653	0.123826	+	0.992304i	$0.460483\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.8800	1.15131	0.575656	−	0.817692i	$-0.304747\pi$
$467$	24.8800	1.15131	0.575656	+	0.817692i	$0.304747\pi$
$468$	0	0
$469$	−4.48647	−0.207166
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−14.2666	−0.655977
$474$	0	0
$475$	−28.0986	−1.28925
$476$	0	0
$477$	0	0
$478$	0	0
$479$	32.2912	1.47542	0.737712	−	0.675115i	$-0.235906\pi$
$479$	32.2912	1.47542	0.737712	+	0.675115i	$0.235906\pi$
$480$	0	0
$481$	6.01143	0.274097
$482$	0	0
$483$	0	0
$484$	0	0
$485$	18.3136	0.831580
$486$	0	0
$487$	30.9257	1.40138	0.700688	−	0.713468i	$-0.252877\pi$
$487$	30.9257	1.40138	0.700688	+	0.713468i	$0.252877\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	22.5371	1.01708	0.508541	−	0.861037i	$-0.330185\pi$
$491$	22.5371	1.01708	0.508541	+	0.861037i	$0.330185\pi$
$492$	0	0
$493$	3.49018	0.157190
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−20.8959	−0.937310
$498$	0	0
$499$	−30.4499	−1.36312	−0.681562	−	0.731760i	$-0.738699\pi$
$499$	−30.4499	−1.36312	−0.681562	+	0.731760i	$0.738699\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	17.3410	0.773197	0.386599	−	0.922248i	$-0.373650\pi$
$503$	17.3410	0.773197	0.386599	+	0.922248i	$0.373650\pi$
$504$	0	0
$505$	−39.8117	−1.77160
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−36.0147	−1.59632	−0.798162	−	0.602444i	$-0.794194\pi$
$509$	−36.0147	−1.59632	−0.798162	+	0.602444i	$0.794194\pi$
$510$	0	0
$511$	25.2477	1.11689
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−26.5633	−1.17052
$516$	0	0
$517$	−29.0505	−1.27764
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−41.0989	−1.80057	−0.900287	−	0.435296i	$-0.856644\pi$
$521$	−41.0989	−1.80057	−0.900287	+	0.435296i	$0.856644\pi$
$522$	0	0
$523$	15.0726	0.659078	0.329539	−	0.944142i	$-0.393107\pi$
$523$	15.0726	0.659078	0.329539	+	0.944142i	$0.393107\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−58.7458	−2.55901
$528$	0	0
$529$	−22.6975	−0.986846
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.27578	−0.0985752
$534$	0	0
$535$	−17.9070	−0.774186
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−25.8039	−1.11145
$540$	0	0
$541$	7.94096	0.341409	0.170704	−	0.985322i	$-0.445396\pi$
$541$	7.94096	0.341409	0.170704	+	0.985322i	$0.445396\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	20.2283	0.866484
$546$	0	0
$547$	−26.6998	−1.14160	−0.570800	−	0.821089i	$-0.693367\pi$
$547$	−26.6998	−1.14160	−0.570800	+	0.821089i	$0.693367\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.51699	0.0646260
$552$	0	0
$553$	1.81177	0.0770445
$554$	0	0
$555$	0	0
$556$	0	0
$557$	39.9545	1.69293	0.846463	−	0.532447i	$-0.178728\pi$
$557$	39.9545	1.69293	0.846463	+	0.532447i	$0.178728\pi$
$558$	0	0
$559$	−2.25333	−0.0953056
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9.40412	−0.396336	−0.198168	−	0.980168i	$-0.563499\pi$
$563$	−9.40412	−0.396336	−0.198168	+	0.980168i	$0.563499\pi$
$564$	0	0
$565$	9.45212	0.397654
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20.7509	−0.869925	−0.434962	−	0.900449i	$-0.643238\pi$
$569$	−20.7509	−0.869925	−0.434962	+	0.900449i	$0.643238\pi$
$570$	0	0
$571$	40.2120	1.68282	0.841410	−	0.540398i	$-0.181726\pi$
$571$	40.2120	1.68282	0.841410	+	0.540398i	$0.181726\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.66905	−0.236416
$576$	0	0
$577$	−25.9250	−1.07927	−0.539636	−	0.841898i	$-0.681438\pi$
$577$	−25.9250	−1.07927	−0.539636	+	0.841898i	$0.681438\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−26.6285	−1.10474
$582$	0	0
$583$	22.2906	0.923183
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−35.3422	−1.45873	−0.729364	−	0.684126i	$-0.760184\pi$
$587$	−35.3422	−1.45873	−0.729364	+	0.684126i	$0.760184\pi$
$588$	0	0
$589$	−25.5336	−1.05209
$590$	0	0
$591$	0	0
$592$	0	0
$593$	41.2205	1.69272	0.846362	−	0.532608i	$-0.178788\pi$
$593$	41.2205	1.69272	0.846362	+	0.532608i	$0.178788\pi$
$594$	0	0
$595$	39.4936	1.61908
$596$	0	0
$597$	0	0
$598$	0	0
$599$	35.2080	1.43856	0.719280	−	0.694721i	$-0.244472\pi$
$599$	35.2080	1.43856	0.719280	+	0.694721i	$0.244472\pi$
$600$	0	0
$601$	8.10874	0.330762	0.165381	−	0.986230i	$-0.447115\pi$
$601$	8.10874	0.330762	0.165381	+	0.986230i	$0.447115\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−90.9176	−3.69633
$606$	0	0
$607$	−9.15153	−0.371449	−0.185724	−	0.982602i	$-0.559463\pi$
$607$	−9.15153	−0.371449	−0.185724	+	0.982602i	$0.559463\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.58837	−0.185625
$612$	0	0
$613$	−22.5460	−0.910626	−0.455313	−	0.890331i	$-0.650473\pi$
$613$	−22.5460	−0.910626	−0.455313	+	0.890331i	$0.650473\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	36.2434	1.45910	0.729552	−	0.683925i	$-0.239729\pi$
$617$	36.2434	1.45910	0.729552	+	0.683925i	$0.239729\pi$
$618$	0	0
$619$	8.66166	0.348142	0.174071	−	0.984733i	$-0.444308\pi$
$619$	8.66166	0.348142	0.174071	+	0.984733i	$0.444308\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.8140	0.513380
$624$	0	0
$625$	29.6955	1.18782
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−40.7983	−1.62674
$630$	0	0
$631$	36.6406	1.45864	0.729320	−	0.684173i	$-0.239837\pi$
$631$	36.6406	1.45864	0.729320	+	0.684173i	$0.239837\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−56.7564	−2.25231
$636$	0	0
$637$	−4.07558	−0.161481
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−24.2511	−0.957861	−0.478930	−	0.877853i	$-0.658975\pi$
$641$	−24.2511	−0.957861	−0.478930	+	0.877853i	$0.658975\pi$
$642$	0	0
$643$	−36.3735	−1.43443	−0.717216	−	0.696851i	$-0.754584\pi$
$643$	−36.3735	−1.43443	−0.717216	+	0.696851i	$0.754584\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−26.2114	−1.03048	−0.515239	−	0.857047i	$-0.672297\pi$
$647$	−26.2114	−1.03048	−0.515239	+	0.857047i	$0.672297\pi$
$648$	0	0
$649$	58.7538	2.30629
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.01675	0.0397886	0.0198943	−	0.999802i	$-0.493667\pi$
$653$	1.01675	0.0397886	0.0198943	+	0.999802i	$0.493667\pi$
$654$	0	0
$655$	47.8520	1.86973
$656$	0	0
$657$	0	0
$658$	0	0
$659$	37.0705	1.44406	0.722030	−	0.691861i	$-0.243209\pi$
$659$	37.0705	1.44406	0.722030	+	0.691861i	$0.243209\pi$
$660$	0	0
$661$	27.8954	1.08500	0.542502	−	0.840055i	$-0.317477\pi$
$661$	27.8954	1.08500	0.542502	+	0.840055i	$0.317477\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	17.1657	0.665659
$666$	0	0
$667$	0.306062	0.0118508
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−53.3626	−2.06004
$672$	0	0
$673$	−38.9003	−1.49950	−0.749748	−	0.661723i	$-0.769826\pi$
$673$	−38.9003	−1.49950	−0.749748	+	0.661723i	$0.769826\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.3217	0.857893	0.428947	−	0.903330i	$-0.358885\pi$
$677$	22.3217	0.857893	0.428947	+	0.903330i	$0.358885\pi$
$678$	0	0
$679$	−7.53341	−0.289106
$680$	0	0
$681$	0	0
$682$	0	0
$683$	48.0011	1.83671	0.918356	−	0.395755i	$-0.129517\pi$
$683$	48.0011	1.83671	0.918356	+	0.395755i	$0.129517\pi$
$684$	0	0
$685$	−25.8241	−0.986687
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.52068	0.134127
$690$	0	0
$691$	−34.6561	−1.31838	−0.659189	−	0.751977i	$-0.729100\pi$
$691$	−34.6561	−1.31838	−0.659189	+	0.751977i	$0.729100\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.25401	−0.123432
$696$	0	0
$697$	15.4453	0.585032
$698$	0	0
$699$	0	0
$700$	0	0
$701$	43.2115	1.63207	0.816037	−	0.577999i	$-0.196166\pi$
$701$	43.2115	1.63207	0.816037	+	0.577999i	$0.196166\pi$
$702$	0	0
$703$	−17.7328	−0.668807
$704$	0	0
$705$	0	0
$706$	0	0
$707$	16.3768	0.615911
$708$	0	0
$709$	−23.4464	−0.880548	−0.440274	−	0.897863i	$-0.645119\pi$
$709$	−23.4464	−0.880548	−0.440274	+	0.897863i	$0.645119\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5.15156	−0.192927
$714$	0	0
$715$	−21.1573	−0.791238
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−41.9480	−1.56440	−0.782199	−	0.623029i	$-0.785902\pi$
$719$	−41.9480	−1.56440	−0.782199	+	0.623029i	$0.785902\pi$
$720$	0	0
$721$	10.9270	0.406941
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5.73513	−0.212997
$726$	0	0
$727$	−1.78676	−0.0662673	−0.0331336	−	0.999451i	$-0.510549\pi$
$727$	−1.78676	−0.0662673	−0.0331336	+	0.999451i	$0.510549\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	15.2929	0.565627
$732$	0	0
$733$	47.3649	1.74946	0.874731	−	0.484609i	$-0.161038\pi$
$733$	47.3649	1.74946	0.874731	+	0.484609i	$0.161038\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.3118	−0.600854
$738$	0	0
$739$	−18.1010	−0.665855	−0.332927	−	0.942952i	$-0.608036\pi$
$739$	−18.1010	−0.665855	−0.332927	+	0.942952i	$0.608036\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	38.7719	1.42240	0.711201	−	0.702989i	$-0.248152\pi$
$743$	38.7719	1.42240	0.711201	+	0.702989i	$0.248152\pi$
$744$	0	0
$745$	−20.7430	−0.759963
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7.36613	0.269153
$750$	0	0
$751$	44.9158	1.63900	0.819500	−	0.573079i	$-0.194251\pi$
$751$	44.9158	1.63900	0.819500	+	0.573079i	$0.194251\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	44.4563	1.61793
$756$	0	0
$757$	49.9053	1.81384	0.906919	−	0.421305i	$-0.138428\pi$
$757$	49.9053	1.81384	0.906919	+	0.421305i	$0.138428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−49.4064	−1.79098	−0.895491	−	0.445080i	$-0.853175\pi$
$761$	−49.4064	−1.79098	−0.895491	+	0.445080i	$0.853175\pi$
$762$	0	0
$763$	−8.32101	−0.301241
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.27985	0.335076
$768$	0	0
$769$	−29.9419	−1.07973	−0.539866	−	0.841751i	$-0.681525\pi$
$769$	−29.9419	−1.07973	−0.539866	+	0.841751i	$0.681525\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−49.6906	−1.78725	−0.893624	−	0.448816i	$-0.851846\pi$
$773$	−49.6906	−1.78725	−0.893624	+	0.448816i	$0.851846\pi$
$774$	0	0
$775$	96.5322	3.46754
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.71323	0.240527
$780$	0	0
$781$	−75.9730	−2.71853
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.81373	0.171809
$786$	0	0
$787$	−31.1813	−1.11149	−0.555746	−	0.831352i	$-0.687567\pi$
$787$	−31.1813	−1.11149	−0.555746	+	0.831352i	$0.687567\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.88818	−0.138248
$792$	0	0
$793$	−8.42833	−0.299299
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31.4265	1.11318	0.556591	−	0.830786i	$-0.312109\pi$
$797$	31.4265	1.11318	0.556591	+	0.830786i	$0.312109\pi$
$798$	0	0
$799$	31.1403	1.10166
$800$	0	0
$801$	0	0
$802$	0	0
$803$	91.7952	3.23938
$804$	0	0
$805$	3.46329	0.122065
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−51.0207	−1.79379	−0.896896	−	0.442242i	$-0.854183\pi$
$809$	−51.0207	−1.79379	−0.896896	+	0.442242i	$0.854183\pi$
$810$	0	0
$811$	−19.3841	−0.680669	−0.340334	−	0.940304i	$-0.610540\pi$
$811$	−19.3841	−0.680669	−0.340334	+	0.940304i	$0.610540\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−45.6459	−1.59891
$816$	0	0
$817$	6.64699	0.232549
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.19967	0.146570	0.0732848	−	0.997311i	$-0.476652\pi$
$821$	4.19967	0.146570	0.0732848	+	0.997311i	$0.476652\pi$
$822$	0	0
$823$	−37.7371	−1.31543	−0.657717	−	0.753265i	$-0.728478\pi$
$823$	−37.7371	−1.31543	−0.657717	+	0.753265i	$0.728478\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.27697	−0.113951	−0.0569757	−	0.998376i	$-0.518146\pi$
$827$	−3.27697	−0.113951	−0.0569757	+	0.998376i	$0.518146\pi$
$828$	0	0
$829$	36.2047	1.25744	0.628720	−	0.777632i	$-0.283579\pi$
$829$	36.2047	1.25744	0.628720	+	0.777632i	$0.283579\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	27.6602	0.958368
$834$	0	0
$835$	−3.91239	−0.135394
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12.1609	−0.419839	−0.209920	−	0.977719i	$-0.567320\pi$
$839$	−12.1609	−0.419839	−0.209920	+	0.977719i	$0.567320\pi$
$840$	0	0
$841$	−28.6904	−0.989323
$842$	0	0
$843$	0	0
$844$	0	0
$845$	47.5193	1.63471
$846$	0	0
$847$	37.3994	1.28506
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3.57770	−0.122642
$852$	0	0
$853$	−40.9721	−1.40286	−0.701430	−	0.712739i	$-0.747455\pi$
$853$	−40.9721	−1.40286	−0.701430	+	0.712739i	$0.747455\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25.4349	−0.868839	−0.434419	−	0.900711i	$-0.643046\pi$
$857$	−25.4349	−0.868839	−0.434419	+	0.900711i	$0.643046\pi$
$858$	0	0
$859$	−2.34657	−0.0800639	−0.0400320	−	0.999198i	$-0.512746\pi$
$859$	−2.34657	−0.0800639	−0.0400320	+	0.999198i	$0.512746\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	30.9243	1.05268	0.526338	−	0.850276i	$-0.323565\pi$
$863$	30.9243	1.05268	0.526338	+	0.850276i	$0.323565\pi$
$864$	0	0
$865$	−11.9364	−0.405849
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6.58722	0.223456
$870$	0	0
$871$	−2.57637	−0.0872968
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−33.4141	−1.12960
$876$	0	0
$877$	−21.1946	−0.715692	−0.357846	−	0.933781i	$-0.616489\pi$
$877$	−21.1946	−0.715692	−0.357846	+	0.933781i	$0.616489\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−18.4441	−0.621398	−0.310699	−	0.950508i	$-0.600563\pi$
$881$	−18.4441	−0.621398	−0.310699	+	0.950508i	$0.600563\pi$
$882$	0	0
$883$	−1.44488	−0.0486240	−0.0243120	−	0.999704i	$-0.507740\pi$
$883$	−1.44488	−0.0486240	−0.0243120	+	0.999704i	$0.507740\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−31.9537	−1.07290	−0.536450	−	0.843932i	$-0.680235\pi$
$887$	−31.9537	−1.07290	−0.536450	+	0.843932i	$0.680235\pi$
$888$	0	0
$889$	23.3471	0.783035
$890$	0	0
$891$	0	0
$892$	0	0
$893$	13.5350	0.452932
$894$	0	0
$895$	21.9700	0.734376
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−5.21160	−0.173817
$900$	0	0
$901$	−23.8942	−0.796030
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−52.3606	−1.74053
$906$	0	0
$907$	−8.49466	−0.282061	−0.141030	−	0.990005i	$-0.545041\pi$
$907$	−8.49466	−0.282061	−0.141030	+	0.990005i	$0.545041\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	16.6975	0.553214	0.276607	−	0.960983i	$-0.410790\pi$
$911$	16.6975	0.553214	0.276607	+	0.960983i	$0.410790\pi$
$912$	0	0
$913$	−96.8154	−3.20412
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−19.6842	−0.650029
$918$	0	0
$919$	−13.0683	−0.431083	−0.215541	−	0.976495i	$-0.569152\pi$
$919$	−13.0683	−0.431083	−0.215541	+	0.976495i	$0.569152\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−11.9995	−0.394969
$924$	0	0
$925$	67.0406	2.20428
$926$	0	0
$927$	0	0
$928$	0	0
$929$	32.6412	1.07092	0.535462	−	0.844559i	$-0.320137\pi$
$929$	32.6412	1.07092	0.535462	+	0.844559i	$0.320137\pi$
$930$	0	0
$931$	12.0224	0.394018
$932$	0	0
$933$	0	0
$934$	0	0
$935$	143.590	4.69590
$936$	0	0
$937$	−11.7444	−0.383674	−0.191837	−	0.981427i	$-0.561445\pi$
$937$	−11.7444	−0.383674	−0.191837	+	0.981427i	$0.561445\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−53.8901	−1.75677	−0.878384	−	0.477956i	$-0.841378\pi$
$941$	−53.8901	−1.75677	−0.878384	+	0.477956i	$0.841378\pi$
$942$	0	0
$943$	1.35443	0.0441064
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−58.6945	−1.90732	−0.953658	−	0.300894i	$-0.902715\pi$
$947$	−58.6945	−1.90732	−0.953658	+	0.300894i	$0.902715\pi$
$948$	0	0
$949$	14.4986	0.470643
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.28484	−0.106406	−0.0532032	−	0.998584i	$-0.516943\pi$
$953$	−3.28484	−0.106406	−0.0532032	+	0.998584i	$0.516943\pi$
$954$	0	0
$955$	−90.7981	−2.93816
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10.6229	0.343030
$960$	0	0
$961$	56.7204	1.82969
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−86.0438	−2.76985
$966$	0	0
$967$	7.53672	0.242365	0.121182	−	0.992630i	$-0.461331\pi$
$967$	7.53672	0.242365	0.121182	+	0.992630i	$0.461331\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	53.1183	1.70465	0.852323	−	0.523015i	$-0.175193\pi$
$971$	53.1183	1.70465	0.852323	+	0.523015i	$0.175193\pi$
$972$	0	0
$973$	1.33856	0.0429121
$974$	0	0
$975$	0	0
$976$	0	0
$977$	23.5765	0.754278	0.377139	−	0.926157i	$-0.376908\pi$
$977$	23.5765	0.754278	0.377139	+	0.926157i	$0.376908\pi$
$978$	0	0
$979$	46.5888	1.48898
$980$	0	0
$981$	0	0
$982$	0	0
$983$	51.2141	1.63348	0.816738	−	0.577009i	$-0.195780\pi$
$983$	51.2141	1.63348	0.816738	+	0.577009i	$0.195780\pi$
$984$	0	0
$985$	−49.9495	−1.59152
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.34107	0.0426435
$990$	0	0
$991$	−46.5577	−1.47895	−0.739477	−	0.673182i	$-0.764927\pi$
$991$	−46.5577	−1.47895	−0.739477	+	0.673182i	$0.764927\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	23.5249	0.745788
$996$	0	0
$997$	−24.6427	−0.780442	−0.390221	−	0.920721i	$-0.627601\pi$
$997$	−24.6427	−0.780442	−0.390221	+	0.920721i	$0.627601\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.i.1.1		9
3.2	odd	2		2004.2.a.c.1.9	✓	9
12.11	even	2		8016.2.a.bc.1.9		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.c.1.9	✓	9	3.2	odd	2
6012.2.a.i.1.1		9	1.1	even	1	trivial
8016.2.a.bc.1.9		9	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:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.a (trivial)

\(f(q)\)	\(=\)	\(q-3.91239 q^{5} +1.60938 q^{7} +O(q^{10})\)	\(q-3.91239 q^{5} +1.60938 q^{7} +5.85136 q^{11} +0.924191 q^{13} -6.27229 q^{17} -2.72623 q^{19} -0.550033 q^{23} +10.3068 q^{25} -0.556443 q^{29} +9.36591 q^{31} -6.29652 q^{35} +6.50453 q^{37} -2.46246 q^{41} -2.43816 q^{43} -4.96474 q^{47} -4.40989 q^{49} +3.80948 q^{53} -22.8928 q^{55} +10.0411 q^{59} -9.11969 q^{61} -3.61579 q^{65} -2.78770 q^{67} -12.9838 q^{71} +15.6878 q^{73} +9.41706 q^{77} +1.12576 q^{79} -16.5458 q^{83} +24.5396 q^{85} +7.96204 q^{89} +1.48737 q^{91} +10.6661 q^{95} -4.68094 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{5} + 2 q^{7}+O(q^{10}) \) 9 * q - q^5 + 2 * q^7	\( 9 q - q^{5} + 2 q^{7} + 9 q^{11} + 10 q^{13} - 7 q^{17} - 2 q^{19} + 3 q^{23} + 18 q^{25} - 5 q^{29} + 12 q^{31} + 6 q^{35} + 15 q^{37} - 14 q^{41} + 6 q^{43} + 3 q^{47} + 27 q^{49} - 9 q^{53} + 19 q^{55} + 9 q^{59} + 30 q^{61} - 28 q^{65} + 16 q^{67} + 3 q^{71} + 32 q^{73} - 18 q^{77} + 24 q^{79} + 3 q^{83} + 37 q^{85} - 46 q^{89} + 33 q^{91} - 11 q^{95} + 43 q^{97}+O(q^{100}) \) 9 * q - q^5 + 2 * q^7 + 9 * q^11 + 10 * q^13 - 7 * q^17 - 2 * q^19 + 3 * q^23 + 18 * q^25 - 5 * q^29 + 12 * q^31 + 6 * q^35 + 15 * q^37 - 14 * q^41 + 6 * q^43 + 3 * q^47 + 27 * q^49 - 9 * q^53 + 19 * q^55 + 9 * q^59 + 30 * q^61 - 28 * q^65 + 16 * q^67 + 3 * q^71 + 32 * q^73 - 18 * q^77 + 24 * q^79 + 3 * q^83 + 37 * q^85 - 46 * q^89 + 33 * q^91 - 11 * q^95 + 43 * q^97

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