LMFDB - Embedded newform 6012.2.a.e.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:	$1$
Dimension:	$5$
Coefficient field:	5.5.149169.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} - 3x^{2} + 4x + 1 $ x^5 - 6x^3 - 3x^2 + 4*x + 1
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			$-1.81853$ 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)

$f(q)$	$=$	$q-2.12560 q^{5} +0.221696 q^{7} +O(q^{10})$	$q-2.12560 q^{5} +0.221696 q^{7} +2.12560 q^{11} -2.07866 q^{13} +5.53728 q^{17} -2.18327 q^{19} -8.50926 q^{23} -0.481818 q^{25} +9.67698 q^{29} -3.18998 q^{31} -0.471237 q^{35} -5.07948 q^{37} +3.43857 q^{41} +5.53327 q^{43} +11.8218 q^{47} -6.95085 q^{49} -2.21498 q^{53} -4.51818 q^{55} -6.60234 q^{59} +9.72573 q^{61} +4.41840 q^{65} -0.214807 q^{67} -12.8354 q^{71} -12.4910 q^{73} +0.471237 q^{77} +7.45038 q^{79} -15.9723 q^{83} -11.7701 q^{85} +6.46935 q^{89} -0.460829 q^{91} +4.64076 q^{95} -0.984908 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{5} - 2 q^{7}+O(q^{10}) $ 5 * q + 3 * q^5 - 2 * q^7	$ 5 q + 3 q^{5} - 2 q^{7} - 3 q^{11} - 4 q^{13} + 7 q^{17} - 2 q^{19} - 13 q^{23} - 2 q^{25} + 3 q^{29} - 12 q^{31} - 10 q^{35} - 7 q^{37} + 16 q^{41} - q^{47} - 17 q^{49} - 3 q^{53} - 23 q^{55} - q^{59} - 22 q^{61} + 20 q^{65} + 2 q^{67} - 9 q^{71} - 28 q^{73} + 10 q^{77} - 28 q^{79} - 7 q^{83} - 11 q^{85} + 30 q^{89} - 13 q^{91} - 3 q^{95} - 33 q^{97}+O(q^{100}) $ 5 * q + 3 * q^5 - 2 * q^7 - 3 * q^11 - 4 * q^13 + 7 * q^17 - 2 * q^19 - 13 * q^23 - 2 * q^25 + 3 * q^29 - 12 * q^31 - 10 * q^35 - 7 * q^37 + 16 * q^41 - q^47 - 17 * q^49 - 3 * q^53 - 23 * q^55 - q^59 - 22 * q^61 + 20 * q^65 + 2 * q^67 - 9 * q^71 - 28 * q^73 + 10 * q^77 - 28 * q^79 - 7 * q^83 - 11 * q^85 + 30 * q^89 - 13 * q^91 - 3 * q^95 - 33 * 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
$5$	−2.12560	−0.950598	−0.475299	−	0.879824i	$-0.657660\pi$
$5$	−2.12560	−0.950598	−0.475299	+	0.879824i	$0.657660\pi$
$6$	0	0
$7$	0.221696	0.0837931	0.0418966	−	0.999122i	$-0.486660\pi$
$7$	0.221696	0.0837931	0.0418966	+	0.999122i	$0.486660\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.12560	0.640893	0.320446	−	0.947267i	$-0.396167\pi$
$11$	2.12560	0.640893	0.320446	+	0.947267i	$0.396167\pi$
$12$	0	0
$13$	−2.07866	−0.576516	−0.288258	−	0.957553i	$-0.593076\pi$
$13$	−2.07866	−0.576516	−0.288258	+	0.957553i	$0.593076\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.53728	1.34299	0.671494	−	0.741010i	$-0.265653\pi$
$17$	5.53728	1.34299	0.671494	+	0.741010i	$0.265653\pi$
$18$	0	0
$19$	−2.18327	−0.500876	−0.250438	−	0.968133i	$-0.580575\pi$
$19$	−2.18327	−0.500876	−0.250438	+	0.968133i	$0.580575\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.50926	−1.77430	−0.887152	−	0.461477i	$-0.847320\pi$
$23$	−8.50926	−1.77430	−0.887152	+	0.461477i	$0.847320\pi$
$24$	0	0
$25$	−0.481818	−0.0963636
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.67698	1.79697	0.898485	−	0.439003i	$-0.144668\pi$
$29$	9.67698	1.79697	0.898485	+	0.439003i	$0.144668\pi$
$30$	0	0
$31$	−3.18998	−0.572938	−0.286469	−	0.958089i	$-0.592482\pi$
$31$	−3.18998	−0.572938	−0.286469	+	0.958089i	$0.592482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.471237	−0.0796536
$36$	0	0
$37$	−5.07948	−0.835061	−0.417530	−	0.908663i	$-0.637104\pi$
$37$	−5.07948	−0.835061	−0.417530	+	0.908663i	$0.637104\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.43857	0.537014	0.268507	−	0.963278i	$-0.413470\pi$
$41$	3.43857	0.537014	0.268507	+	0.963278i	$0.413470\pi$
$42$	0	0
$43$	5.53327	0.843816	0.421908	−	0.906639i	$-0.361360\pi$
$43$	5.53327	0.843816	0.421908	+	0.906639i	$0.361360\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.8218	1.72439	0.862195	−	0.506576i	$-0.169089\pi$
$47$	11.8218	1.72439	0.862195	+	0.506576i	$0.169089\pi$
$48$	0	0
$49$	−6.95085	−0.992979
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.21498	−0.304251	−0.152126	−	0.988361i	$-0.548612\pi$
$53$	−2.21498	−0.304251	−0.152126	+	0.988361i	$0.548612\pi$
$54$	0	0
$55$	−4.51818	−0.609232
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.60234	−0.859551	−0.429775	−	0.902936i	$-0.641407\pi$
$59$	−6.60234	−0.859551	−0.429775	+	0.902936i	$0.641407\pi$
$60$	0	0
$61$	9.72573	1.24525	0.622626	−	0.782519i	$-0.286066\pi$
$61$	9.72573	1.24525	0.622626	+	0.782519i	$0.286066\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.41840	0.548034
$66$	0	0
$67$	−0.214807	−0.0262429	−0.0131215	−	0.999914i	$-0.504177\pi$
$67$	−0.214807	−0.0262429	−0.0131215	+	0.999914i	$0.504177\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.8354	−1.52329	−0.761643	−	0.647997i	$-0.775607\pi$
$71$	−12.8354	−1.52329	−0.761643	+	0.647997i	$0.775607\pi$
$72$	0	0
$73$	−12.4910	−1.46196	−0.730981	−	0.682398i	$-0.760937\pi$
$73$	−12.4910	−1.46196	−0.730981	+	0.682398i	$0.760937\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.471237	0.0537024
$78$	0	0
$79$	7.45038	0.838233	0.419116	−	0.907932i	$-0.362340\pi$
$79$	7.45038	0.838233	0.419116	+	0.907932i	$0.362340\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−15.9723	−1.75318	−0.876591	−	0.481236i	$-0.840188\pi$
$83$	−15.9723	−1.75318	−0.876591	+	0.481236i	$0.840188\pi$
$84$	0	0
$85$	−11.7701	−1.27664
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.46935	0.685750	0.342875	−	0.939381i	$-0.388599\pi$
$89$	6.46935	0.685750	0.342875	+	0.939381i	$0.388599\pi$
$90$	0	0
$91$	−0.460829	−0.0483080
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.64076	0.476132
$96$	0	0
$97$	−0.984908	−0.100002	−0.0500011	−	0.998749i	$-0.515922\pi$
$97$	−0.984908	−0.100002	−0.0500011	+	0.998749i	$0.515922\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−3.30184	−0.328545	−0.164273	−	0.986415i	$-0.552528\pi$
$101$	−3.30184	−0.328545	−0.164273	+	0.986415i	$0.552528\pi$
$102$	0	0
$103$	−11.0295	−1.08677	−0.543385	−	0.839484i	$-0.682858\pi$
$103$	−11.0295	−1.08677	−0.543385	+	0.839484i	$0.682858\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.4356	1.78223	0.891117	−	0.453775i	$-0.149923\pi$
$107$	18.4356	1.78223	0.891117	+	0.453775i	$0.149923\pi$
$108$	0	0
$109$	−1.23544	−0.118334	−0.0591669	−	0.998248i	$-0.518844\pi$
$109$	−1.23544	−0.118334	−0.0591669	+	0.998248i	$0.518844\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.2471	1.52840	0.764200	−	0.644980i	$-0.223134\pi$
$113$	16.2471	1.52840	0.764200	+	0.644980i	$0.223134\pi$
$114$	0	0
$115$	18.0873	1.68665
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.22759	0.112533
$120$	0	0
$121$	−6.48182	−0.589256
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.6522	1.04220
$126$	0	0
$127$	−4.17746	−0.370689	−0.185345	−	0.982674i	$-0.559340\pi$
$127$	−4.17746	−0.370689	−0.185345	+	0.982674i	$0.559340\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	18.7764	1.64050	0.820250	−	0.572006i	$-0.193835\pi$
$131$	18.7764	1.64050	0.820250	+	0.572006i	$0.193835\pi$
$132$	0	0
$133$	−0.484022	−0.0419700
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.69811	−0.315951	−0.157976	−	0.987443i	$-0.550497\pi$
$137$	−3.69811	−0.315951	−0.157976	+	0.987443i	$0.550497\pi$
$138$	0	0
$139$	−19.8436	−1.68312	−0.841558	−	0.540167i	$-0.818361\pi$
$139$	−19.8436	−1.68312	−0.841558	+	0.540167i	$0.818361\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.41840	−0.369485
$144$	0	0
$145$	−20.5694	−1.70820
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−15.8494	−1.29844	−0.649219	−	0.760602i	$-0.724904\pi$
$149$	−15.8494	−1.29844	−0.649219	+	0.760602i	$0.724904\pi$
$150$	0	0
$151$	−3.34230	−0.271992	−0.135996	−	0.990709i	$-0.543423\pi$
$151$	−3.34230	−0.271992	−0.135996	+	0.990709i	$0.543423\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.78064	0.544634
$156$	0	0
$157$	−1.85583	−0.148111	−0.0740555	−	0.997254i	$-0.523594\pi$
$157$	−1.85583	−0.148111	−0.0740555	+	0.997254i	$0.523594\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.88647	−0.148675
$162$	0	0
$163$	−22.6630	−1.77510	−0.887552	−	0.460707i	$-0.847596\pi$
$163$	−22.6630	−1.77510	−0.887552	+	0.460707i	$0.847596\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−8.67919	−0.667630
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.18615	−0.470324	−0.235162	−	0.971956i	$-0.575562\pi$
$173$	−6.18615	−0.470324	−0.235162	+	0.971956i	$0.575562\pi$
$174$	0	0
$175$	−0.106817	−0.00807461
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−22.1116	−1.65270	−0.826349	−	0.563158i	$-0.809586\pi$
$179$	−22.1116	−1.65270	−0.826349	+	0.563158i	$0.809586\pi$
$180$	0	0
$181$	−5.15182	−0.382931	−0.191466	−	0.981499i	$-0.561324\pi$
$181$	−5.15182	−0.382931	−0.191466	+	0.981499i	$0.561324\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.7969	0.793807
$186$	0	0
$187$	11.7701	0.860712
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.2339	0.885217	0.442608	−	0.896715i	$-0.354053\pi$
$191$	12.2339	0.885217	0.442608	+	0.896715i	$0.354053\pi$
$192$	0	0
$193$	20.6509	1.48649	0.743244	−	0.669021i	$-0.233286\pi$
$193$	20.6509	1.48649	0.743244	+	0.669021i	$0.233286\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−19.1452	−1.36404	−0.682020	−	0.731334i	$-0.738898\pi$
$197$	−19.1452	−1.36404	−0.682020	+	0.731334i	$0.738898\pi$
$198$	0	0
$199$	12.0432	0.853720	0.426860	−	0.904318i	$-0.359620\pi$
$199$	12.0432	0.853720	0.426860	+	0.904318i	$0.359620\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.14535	0.150574
$204$	0	0
$205$	−7.30902	−0.510484
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.64076	−0.321008
$210$	0	0
$211$	18.1845	1.25187	0.625936	−	0.779874i	$-0.284717\pi$
$211$	18.1845	1.25187	0.625936	+	0.779874i	$0.284717\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−11.7615	−0.802130
$216$	0	0
$217$	−0.707206	−0.0480083
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−11.5101	−0.774254
$222$	0	0
$223$	−15.5932	−1.04420	−0.522100	−	0.852884i	$-0.674851\pi$
$223$	−15.5932	−1.04420	−0.522100	+	0.852884i	$0.674851\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−14.4872	−0.961547	−0.480774	−	0.876845i	$-0.659644\pi$
$227$	−14.4872	−0.961547	−0.480774	+	0.876845i	$0.659644\pi$
$228$	0	0
$229$	−24.9771	−1.65053	−0.825266	−	0.564744i	$-0.808975\pi$
$229$	−24.9771	−1.65053	−0.825266	+	0.564744i	$0.808975\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−19.3461	−1.26740	−0.633702	−	0.773577i	$-0.718465\pi$
$233$	−19.3461	−1.26740	−0.633702	+	0.773577i	$0.718465\pi$
$234$	0	0
$235$	−25.1285	−1.63920
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0.789530	0.0510705	0.0255352	−	0.999674i	$-0.491871\pi$
$239$	0.789530	0.0510705	0.0255352	+	0.999674i	$0.491871\pi$
$240$	0	0
$241$	−11.7550	−0.757208	−0.378604	−	0.925559i	$-0.623596\pi$
$241$	−11.7550	−0.757208	−0.378604	+	0.925559i	$0.623596\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	14.7747	0.943924
$246$	0	0
$247$	4.53827	0.288763
$248$	0	0
$249$	0	0
$250$	0	0
$251$	31.2564	1.97289	0.986443	−	0.164103i	$-0.0524731\pi$
$251$	31.2564	1.97289	0.986443	+	0.164103i	$0.0524731\pi$
$252$	0	0
$253$	−18.0873	−1.13714
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.93107	−0.307592	−0.153796	−	0.988103i	$-0.549150\pi$
$257$	−4.93107	−0.307592	−0.153796	+	0.988103i	$0.549150\pi$
$258$	0	0
$259$	−1.12610	−0.0699723
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−28.3288	−1.74683	−0.873415	−	0.486976i	$-0.838100\pi$
$263$	−28.3288	−1.74683	−0.873415	+	0.486976i	$0.838100\pi$
$264$	0	0
$265$	4.70817	0.289220
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−13.4412	−0.819526	−0.409763	−	0.912192i	$-0.634389\pi$
$269$	−13.4412	−0.819526	−0.409763	+	0.912192i	$0.634389\pi$
$270$	0	0
$271$	−9.08921	−0.552130	−0.276065	−	0.961139i	$-0.589030\pi$
$271$	−9.08921	−0.552130	−0.276065	+	0.961139i	$0.589030\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.02415	−0.0617587
$276$	0	0
$277$	−15.7966	−0.949124	−0.474562	−	0.880222i	$-0.657394\pi$
$277$	−15.7966	−0.949124	−0.474562	+	0.880222i	$0.657394\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.10473	0.244868	0.122434	−	0.992477i	$-0.460930\pi$
$281$	4.10473	0.244868	0.122434	+	0.992477i	$0.460930\pi$
$282$	0	0
$283$	23.4844	1.39600	0.698000	−	0.716098i	$-0.254073\pi$
$283$	23.4844	1.39600	0.698000	+	0.716098i	$0.254073\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.762315	0.0449981
$288$	0	0
$289$	13.6615	0.803617
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−18.3264	−1.07064	−0.535321	−	0.844649i	$-0.679809\pi$
$293$	−18.3264	−1.07064	−0.535321	+	0.844649i	$0.679809\pi$
$294$	0	0
$295$	14.0339	0.817087
$296$	0	0
$297$	0	0
$298$	0	0
$299$	17.6878	1.02291
$300$	0	0
$301$	1.22670	0.0707060
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−20.6730	−1.18373
$306$	0	0
$307$	−25.0476	−1.42954	−0.714770	−	0.699359i	$-0.753469\pi$
$307$	−25.0476	−1.42954	−0.714770	+	0.699359i	$0.753469\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−11.2135	−0.635861	−0.317931	−	0.948114i	$-0.602988\pi$
$311$	−11.2135	−0.635861	−0.317931	+	0.948114i	$0.602988\pi$
$312$	0	0
$313$	2.39275	0.135246	0.0676232	−	0.997711i	$-0.478458\pi$
$313$	2.39275	0.135246	0.0676232	+	0.997711i	$0.478458\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−32.6943	−1.83629	−0.918147	−	0.396239i	$-0.870315\pi$
$317$	−32.6943	−1.83629	−0.918147	+	0.396239i	$0.870315\pi$
$318$	0	0
$319$	20.5694	1.15167
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−12.0894	−0.672671
$324$	0	0
$325$	1.00153	0.0555551
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.62085	0.144492
$330$	0	0
$331$	−17.2684	−0.949154	−0.474577	−	0.880214i	$-0.657399\pi$
$331$	−17.2684	−0.949154	−0.474577	+	0.880214i	$0.657399\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.456595	0.0249465
$336$	0	0
$337$	28.1993	1.53611	0.768057	−	0.640382i	$-0.221224\pi$
$337$	28.1993	1.53611	0.768057	+	0.640382i	$0.221224\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−6.78064	−0.367192
$342$	0	0
$343$	−3.09285	−0.166998
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7.97291	−0.428008	−0.214004	−	0.976833i	$-0.568651\pi$
$347$	−7.97291	−0.428008	−0.214004	+	0.976833i	$0.568651\pi$
$348$	0	0
$349$	24.6252	1.31815	0.659077	−	0.752075i	$-0.270947\pi$
$349$	24.6252	1.31815	0.659077	+	0.752075i	$0.270947\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.38038	−0.126695	−0.0633473	−	0.997992i	$-0.520178\pi$
$353$	−2.38038	−0.126695	−0.0633473	+	0.997992i	$0.520178\pi$
$354$	0	0
$355$	27.2830	1.44803
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−19.3781	−1.02274	−0.511369	−	0.859361i	$-0.670862\pi$
$359$	−19.3781	−1.02274	−0.511369	+	0.859361i	$0.670862\pi$
$360$	0	0
$361$	−14.2333	−0.749123
$362$	0	0
$363$	0	0
$364$	0	0
$365$	26.5509	1.38974
$366$	0	0
$367$	9.87368	0.515402	0.257701	−	0.966225i	$-0.417035\pi$
$367$	9.87368	0.515402	0.257701	+	0.966225i	$0.417035\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−0.491052	−0.0254941
$372$	0	0
$373$	−30.4976	−1.57911	−0.789553	−	0.613683i	$-0.789687\pi$
$373$	−30.4976	−1.57911	−0.789553	+	0.613683i	$0.789687\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−20.1151	−1.03598
$378$	0	0
$379$	15.8226	0.812752	0.406376	−	0.913706i	$-0.366792\pi$
$379$	15.8226	0.812752	0.406376	+	0.913706i	$0.366792\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10.6600	−0.544702	−0.272351	−	0.962198i	$-0.587801\pi$
$383$	−10.6600	−0.544702	−0.272351	+	0.962198i	$0.587801\pi$
$384$	0	0
$385$	−1.00166	−0.0510494
$386$	0	0
$387$	0	0
$388$	0	0
$389$	13.8665	0.703060	0.351530	−	0.936177i	$-0.385662\pi$
$389$	13.8665	0.703060	0.351530	+	0.936177i	$0.385662\pi$
$390$	0	0
$391$	−47.1182	−2.38287
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−15.8365	−0.796822
$396$	0	0
$397$	35.7439	1.79393	0.896966	−	0.442099i	$-0.145766\pi$
$397$	35.7439	1.79393	0.896966	+	0.442099i	$0.145766\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.5873	1.52745	0.763727	−	0.645539i	$-0.223367\pi$
$401$	30.5873	1.52745	0.763727	+	0.645539i	$0.223367\pi$
$402$	0	0
$403$	6.63088	0.330308
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−10.7969	−0.535184
$408$	0	0
$409$	−0.0787983	−0.00389633	−0.00194816	−	0.999998i	$-0.500620\pi$
$409$	−0.0787983	−0.00389633	−0.00194816	+	0.999998i	$0.500620\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.46371	−0.0720245
$414$	0	0
$415$	33.9506	1.66657
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−27.0551	−1.32173	−0.660865	−	0.750505i	$-0.729811\pi$
$419$	−27.0551	−1.32173	−0.660865	+	0.750505i	$0.729811\pi$
$420$	0	0
$421$	−39.3526	−1.91793	−0.958964	−	0.283529i	$-0.908495\pi$
$421$	−39.3526	−1.91793	−0.958964	+	0.283529i	$0.908495\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.66796	−0.129415
$426$	0	0
$427$	2.15615	0.104344
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.03817	0.0500068	0.0250034	−	0.999687i	$-0.492040\pi$
$431$	1.03817	0.0500068	0.0250034	+	0.999687i	$0.492040\pi$
$432$	0	0
$433$	20.6291	0.991370	0.495685	−	0.868502i	$-0.334917\pi$
$433$	20.6291	0.991370	0.495685	+	0.868502i	$0.334917\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18.5780	0.888707
$438$	0	0
$439$	−26.1283	−1.24704	−0.623518	−	0.781809i	$-0.714297\pi$
$439$	−26.1283	−1.24704	−0.623518	+	0.781809i	$0.714297\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−21.8113	−1.03629	−0.518144	−	0.855293i	$-0.673377\pi$
$443$	−21.8113	−1.03629	−0.518144	+	0.855293i	$0.673377\pi$
$444$	0	0
$445$	−13.7513	−0.651872
$446$	0	0
$447$	0	0
$448$	0	0
$449$	35.4070	1.67096	0.835479	−	0.549522i	$-0.185190\pi$
$449$	35.4070	1.67096	0.835479	+	0.549522i	$0.185190\pi$
$450$	0	0
$451$	7.30902	0.344168
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.979540	0.0459215
$456$	0	0
$457$	18.1615	0.849559	0.424780	−	0.905297i	$-0.360352\pi$
$457$	18.1615	0.849559	0.424780	+	0.905297i	$0.360352\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.47013	−0.115046	−0.0575228	−	0.998344i	$-0.518320\pi$
$461$	−2.47013	−0.115046	−0.0575228	+	0.998344i	$0.518320\pi$
$462$	0	0
$463$	−24.8116	−1.15309	−0.576545	−	0.817065i	$-0.695600\pi$
$463$	−24.8116	−1.15309	−0.576545	+	0.817065i	$0.695600\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.4827	0.994102	0.497051	−	0.867721i	$-0.334416\pi$
$467$	21.4827	0.994102	0.497051	+	0.867721i	$0.334416\pi$
$468$	0	0
$469$	−0.0476219	−0.00219898
$470$	0	0
$471$	0	0
$472$	0	0
$473$	11.7615	0.540796
$474$	0	0
$475$	1.05194	0.0482662
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.90720	0.178524	0.0892622	−	0.996008i	$-0.471549\pi$
$479$	3.90720	0.178524	0.0892622	+	0.996008i	$0.471549\pi$
$480$	0	0
$481$	10.5585	0.481425
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.09352	0.0950619
$486$	0	0
$487$	−2.72645	−0.123547	−0.0617737	−	0.998090i	$-0.519676\pi$
$487$	−2.72645	−0.123547	−0.0617737	+	0.998090i	$0.519676\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−32.0967	−1.44850	−0.724252	−	0.689536i	$-0.757815\pi$
$491$	−32.0967	−1.44850	−0.724252	+	0.689536i	$0.757815\pi$
$492$	0	0
$493$	53.5842	2.41331
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.84556	−0.127641
$498$	0	0
$499$	−37.1880	−1.66476	−0.832381	−	0.554204i	$-0.813023\pi$
$499$	−37.1880	−1.66476	−0.832381	+	0.554204i	$0.813023\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−20.9373	−0.933550	−0.466775	−	0.884376i	$-0.654584\pi$
$503$	−20.9373	−0.933550	−0.466775	+	0.884376i	$0.654584\pi$
$504$	0	0
$505$	7.01840	0.312315
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−25.2811	−1.12056	−0.560282	−	0.828302i	$-0.689307\pi$
$509$	−25.2811	−1.12056	−0.560282	+	0.828302i	$0.689307\pi$
$510$	0	0
$511$	−2.76920	−0.122502
$512$	0	0
$513$	0	0
$514$	0	0
$515$	23.4443	1.03308
$516$	0	0
$517$	25.1285	1.10515
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.8514	0.825893	0.412947	−	0.910755i	$-0.364500\pi$
$521$	18.8514	0.825893	0.412947	+	0.910755i	$0.364500\pi$
$522$	0	0
$523$	−0.790161	−0.0345513	−0.0172757	−	0.999851i	$-0.505499\pi$
$523$	−0.790161	−0.0345513	−0.0172757	+	0.999851i	$0.505499\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−17.6638	−0.769449
$528$	0	0
$529$	49.4076	2.14815
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−7.14760	−0.309597
$534$	0	0
$535$	−39.1867	−1.69419
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−14.7747	−0.636393
$540$	0	0
$541$	−13.9752	−0.600840	−0.300420	−	0.953807i	$-0.597127\pi$
$541$	−13.9752	−0.600840	−0.300420	+	0.953807i	$0.597127\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.62606	0.112488
$546$	0	0
$547$	7.27623	0.311109	0.155555	−	0.987827i	$-0.450284\pi$
$547$	7.27623	0.311109	0.155555	+	0.987827i	$0.450284\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−21.1275	−0.900060
$552$	0	0
$553$	1.65172	0.0702382
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.84469	−0.120533	−0.0602667	−	0.998182i	$-0.519195\pi$
$557$	−2.84469	−0.120533	−0.0602667	+	0.998182i	$0.519195\pi$
$558$	0	0
$559$	−11.5018	−0.486473
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14.6772	−0.618570	−0.309285	−	0.950969i	$-0.600090\pi$
$563$	−14.6772	−0.618570	−0.309285	+	0.950969i	$0.600090\pi$
$564$	0	0
$565$	−34.5349	−1.45289
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20.9857	0.879766	0.439883	−	0.898055i	$-0.355020\pi$
$569$	20.9857	0.879766	0.439883	+	0.898055i	$0.355020\pi$
$570$	0	0
$571$	47.1954	1.97507	0.987533	−	0.157412i	$-0.0503152\pi$
$571$	47.1954	1.97507	0.987533	+	0.157412i	$0.0503152\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.09992	0.170978
$576$	0	0
$577$	8.43182	0.351021	0.175511	−	0.984478i	$-0.443842\pi$
$577$	8.43182	0.351021	0.175511	+	0.984478i	$0.443842\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.54098	−0.146905
$582$	0	0
$583$	−4.70817	−0.194992
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.17179	−0.0896395	−0.0448198	−	0.998995i	$-0.514271\pi$
$587$	−2.17179	−0.0896395	−0.0448198	+	0.998995i	$0.514271\pi$
$588$	0	0
$589$	6.96460	0.286971
$590$	0	0
$591$	0	0
$592$	0	0
$593$	17.0848	0.701587	0.350794	−	0.936453i	$-0.385912\pi$
$593$	17.0848	0.701587	0.350794	+	0.936453i	$0.385912\pi$
$594$	0	0
$595$	−2.60937	−0.106974
$596$	0	0
$597$	0	0
$598$	0	0
$599$	35.0599	1.43251	0.716255	−	0.697839i	$-0.245855\pi$
$599$	35.0599	1.43251	0.716255	+	0.697839i	$0.245855\pi$
$600$	0	0
$601$	−10.3582	−0.422519	−0.211259	−	0.977430i	$-0.567756\pi$
$601$	−10.3582	−0.422519	−0.211259	+	0.977430i	$0.567756\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	13.7778	0.560146
$606$	0	0
$607$	−10.3560	−0.420339	−0.210169	−	0.977665i	$-0.567402\pi$
$607$	−10.3560	−0.420339	−0.210169	+	0.977665i	$0.567402\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−24.5735	−0.994138
$612$	0	0
$613$	26.4256	1.06732	0.533660	−	0.845699i	$-0.320816\pi$
$613$	26.4256	1.06732	0.533660	+	0.845699i	$0.320816\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	42.3118	1.70341	0.851705	−	0.524022i	$-0.175569\pi$
$617$	42.3118	1.70341	0.851705	+	0.524022i	$0.175569\pi$
$618$	0	0
$619$	11.5350	0.463629	0.231815	−	0.972760i	$-0.425534\pi$
$619$	11.5350	0.463629	0.231815	+	0.972760i	$0.425534\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.43423	0.0574611
$624$	0	0
$625$	−22.3588	−0.894350
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−28.1265	−1.12148
$630$	0	0
$631$	20.9442	0.833776	0.416888	−	0.908958i	$-0.363121\pi$
$631$	20.9442	0.833776	0.416888	+	0.908958i	$0.363121\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	8.87961	0.352377
$636$	0	0
$637$	14.4484	0.572468
$638$	0	0
$639$	0	0
$640$	0	0
$641$	46.7978	1.84840	0.924201	−	0.381908i	$-0.124733\pi$
$641$	46.7978	1.84840	0.924201	+	0.381908i	$0.124733\pi$
$642$	0	0
$643$	−37.3967	−1.47478	−0.737390	−	0.675467i	$-0.763942\pi$
$643$	−37.3967	−1.47478	−0.737390	+	0.675467i	$0.763942\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.14478	−0.241576	−0.120788	−	0.992678i	$-0.538542\pi$
$647$	−6.14478	−0.241576	−0.120788	+	0.992678i	$0.538542\pi$
$648$	0	0
$649$	−14.0339	−0.550880
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8.28774	0.324324	0.162162	−	0.986764i	$-0.448153\pi$
$653$	8.28774	0.324324	0.162162	+	0.986764i	$0.448153\pi$
$654$	0	0
$655$	−39.9111	−1.55946
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−17.1321	−0.667371	−0.333686	−	0.942684i	$-0.608292\pi$
$659$	−17.1321	−0.667371	−0.333686	+	0.942684i	$0.608292\pi$
$660$	0	0
$661$	13.6715	0.531761	0.265880	−	0.964006i	$-0.414337\pi$
$661$	13.6715	0.531761	0.265880	+	0.964006i	$0.414337\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.02884	0.0398966
$666$	0	0
$667$	−82.3440	−3.18837
$668$	0	0
$669$	0	0
$670$	0	0
$671$	20.6730	0.798074
$672$	0	0
$673$	−29.7525	−1.14687	−0.573437	−	0.819250i	$-0.694390\pi$
$673$	−29.7525	−1.14687	−0.573437	+	0.819250i	$0.694390\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−37.2609	−1.43205	−0.716026	−	0.698074i	$-0.754041\pi$
$677$	−37.2609	−1.43205	−0.716026	+	0.698074i	$0.754041\pi$
$678$	0	0
$679$	−0.218350	−0.00837950
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−45.9067	−1.75657	−0.878286	−	0.478135i	$-0.841313\pi$
$683$	−45.9067	−1.75657	−0.878286	+	0.478135i	$0.841313\pi$
$684$	0	0
$685$	7.86072	0.300343
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.60418	0.175405
$690$	0	0
$691$	10.4030	0.395750	0.197875	−	0.980227i	$-0.436596\pi$
$691$	10.4030	0.395750	0.197875	+	0.980227i	$0.436596\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	42.1797	1.59997
$696$	0	0
$697$	19.0403	0.721203
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.09973	−0.0793057	−0.0396528	−	0.999214i	$-0.512625\pi$
$701$	−2.09973	−0.0793057	−0.0396528	+	0.999214i	$0.512625\pi$
$702$	0	0
$703$	11.0899	0.418262
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−0.732004	−0.0275299
$708$	0	0
$709$	−44.1091	−1.65655	−0.828275	−	0.560321i	$-0.810678\pi$
$709$	−44.1091	−1.65655	−0.828275	+	0.560321i	$0.810678\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	27.1444	1.01657
$714$	0	0
$715$	9.39175	0.351231
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9.03699	0.337023	0.168511	−	0.985700i	$-0.446104\pi$
$719$	9.03699	0.337023	0.168511	+	0.985700i	$0.446104\pi$
$720$	0	0
$721$	−2.44520	−0.0910638
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.66254	−0.173163
$726$	0	0
$727$	25.1230	0.931759	0.465879	−	0.884848i	$-0.345738\pi$
$727$	25.1230	0.931759	0.465879	+	0.884848i	$0.345738\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	30.6393	1.13324
$732$	0	0
$733$	29.2768	1.08137	0.540683	−	0.841227i	$-0.318166\pi$
$733$	29.2768	1.08137	0.540683	+	0.841227i	$0.318166\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.456595	−0.0168189
$738$	0	0
$739$	47.8374	1.75973	0.879863	−	0.475228i	$-0.157634\pi$
$739$	47.8374	1.75973	0.879863	+	0.475228i	$0.157634\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17.7810	0.652321	0.326160	−	0.945314i	$-0.394245\pi$
$743$	17.7810	0.652321	0.326160	+	0.945314i	$0.394245\pi$
$744$	0	0
$745$	33.6896	1.23429
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.08709	0.149339
$750$	0	0
$751$	−21.4708	−0.783480	−0.391740	−	0.920076i	$-0.628127\pi$
$751$	−21.4708	−0.783480	−0.391740	+	0.920076i	$0.628127\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.10439	0.258555
$756$	0	0
$757$	48.4916	1.76246	0.881229	−	0.472689i	$-0.156717\pi$
$757$	48.4916	1.76246	0.881229	+	0.472689i	$0.156717\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0.182652	0.00662112	0.00331056	−	0.999995i	$-0.498946\pi$
$761$	0.182652	0.00662112	0.00331056	+	0.999995i	$0.498946\pi$
$762$	0	0
$763$	−0.273892	−0.00991556
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.7240	0.495544
$768$	0	0
$769$	−6.09075	−0.219638	−0.109819	−	0.993952i	$-0.535027\pi$
$769$	−6.09075	−0.219638	−0.109819	+	0.993952i	$0.535027\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	9.39591	0.337947	0.168974	−	0.985621i	$-0.445955\pi$
$773$	9.39591	0.337947	0.168974	+	0.985621i	$0.445955\pi$
$774$	0	0
$775$	1.53699	0.0552104
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.50731	−0.268977
$780$	0	0
$781$	−27.2830	−0.976263
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.94475	0.140794
$786$	0	0
$787$	−44.3464	−1.58078	−0.790389	−	0.612605i	$-0.790122\pi$
$787$	−44.3464	−1.58078	−0.790389	+	0.612605i	$0.790122\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.60192	0.128069
$792$	0	0
$793$	−20.2165	−0.717907
$794$	0	0
$795$	0	0
$796$	0	0
$797$	52.1209	1.84622	0.923109	−	0.384538i	$-0.125639\pi$
$797$	52.1209	1.84622	0.923109	+	0.384538i	$0.125639\pi$
$798$	0	0
$799$	65.4608	2.31584
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−26.5509	−0.936961
$804$	0	0
$805$	4.00988	0.141330
$806$	0	0
$807$	0	0
$808$	0	0
$809$	15.8448	0.557073	0.278537	−	0.960426i	$-0.410151\pi$
$809$	15.8448	0.557073	0.278537	+	0.960426i	$0.410151\pi$
$810$	0	0
$811$	47.0530	1.65225	0.826127	−	0.563483i	$-0.190539\pi$
$811$	47.0530	1.65225	0.826127	+	0.563483i	$0.190539\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	48.1725	1.68741
$816$	0	0
$817$	−12.0806	−0.422648
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−24.9098	−0.869356	−0.434678	−	0.900586i	$-0.643138\pi$
$821$	−24.9098	−0.869356	−0.434678	+	0.900586i	$0.643138\pi$
$822$	0	0
$823$	26.9393	0.939045	0.469523	−	0.882920i	$-0.344426\pi$
$823$	26.9393	0.939045	0.469523	+	0.882920i	$0.344426\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−6.07808	−0.211355	−0.105678	−	0.994400i	$-0.533701\pi$
$827$	−6.07808	−0.211355	−0.105678	+	0.994400i	$0.533701\pi$
$828$	0	0
$829$	−15.7278	−0.546249	−0.273124	−	0.961979i	$-0.588057\pi$
$829$	−15.7278	−0.546249	−0.273124	+	0.961979i	$0.588057\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−38.4888	−1.33356
$834$	0	0
$835$	2.12560	0.0735595
$836$	0	0
$837$	0	0
$838$	0	0
$839$	23.2845	0.803870	0.401935	−	0.915668i	$-0.368338\pi$
$839$	23.2845	0.803870	0.401935	+	0.915668i	$0.368338\pi$
$840$	0	0
$841$	64.6440	2.22910
$842$	0	0
$843$	0	0
$844$	0	0
$845$	18.4485	0.634648
$846$	0	0
$847$	−1.43699	−0.0493756
$848$	0	0
$849$	0	0
$850$	0	0
$851$	43.2226	1.48165
$852$	0	0
$853$	20.9765	0.718220	0.359110	−	0.933295i	$-0.383080\pi$
$853$	20.9765	0.718220	0.359110	+	0.933295i	$0.383080\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	17.9102	0.611800	0.305900	−	0.952064i	$-0.401043\pi$
$857$	17.9102	0.611800	0.305900	+	0.952064i	$0.401043\pi$
$858$	0	0
$859$	0.572629	0.0195379	0.00976893	−	0.999952i	$-0.496890\pi$
$859$	0.572629	0.0195379	0.00976893	+	0.999952i	$0.496890\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	20.3880	0.694017	0.347008	−	0.937862i	$-0.387198\pi$
$863$	20.3880	0.694017	0.347008	+	0.937862i	$0.387198\pi$
$864$	0	0
$865$	13.1493	0.447089
$866$	0	0
$867$	0	0
$868$	0	0
$869$	15.8365	0.537218
$870$	0	0
$871$	0.446511	0.0151294
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.58324	0.0873293
$876$	0	0
$877$	−35.5316	−1.19982	−0.599909	−	0.800069i	$-0.704796\pi$
$877$	−35.5316	−1.19982	−0.599909	+	0.800069i	$0.704796\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	23.1356	0.779458	0.389729	−	0.920930i	$-0.372569\pi$
$881$	23.1356	0.779458	0.389729	+	0.920930i	$0.372569\pi$
$882$	0	0
$883$	−29.7695	−1.00182	−0.500912	−	0.865498i	$-0.667002\pi$
$883$	−29.7695	−1.00182	−0.500912	+	0.865498i	$0.667002\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−33.5567	−1.12672	−0.563362	−	0.826210i	$-0.690492\pi$
$887$	−33.5567	−1.12672	−0.563362	+	0.826210i	$0.690492\pi$
$888$	0	0
$889$	−0.926125	−0.0310612
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−25.8102	−0.863707
$894$	0	0
$895$	47.0004	1.57105
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−30.8694	−1.02955
$900$	0	0
$901$	−12.2650	−0.408606
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10.9507	0.364014
$906$	0	0
$907$	−1.20646	−0.0400597	−0.0200298	−	0.999799i	$-0.506376\pi$
$907$	−1.20646	−0.0400597	−0.0200298	+	0.999799i	$0.506376\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−28.8600	−0.956173	−0.478086	−	0.878313i	$-0.658669\pi$
$911$	−28.8600	−0.956173	−0.478086	+	0.878313i	$0.658669\pi$
$912$	0	0
$913$	−33.9506	−1.12360
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.16264	0.137463
$918$	0	0
$919$	33.3352	1.09963	0.549814	−	0.835287i	$-0.314699\pi$
$919$	33.3352	1.09963	0.549814	+	0.835287i	$0.314699\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	26.6804	0.878198
$924$	0	0
$925$	2.44738	0.0804694
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−36.7007	−1.20411	−0.602056	−	0.798454i	$-0.705652\pi$
$929$	−36.7007	−1.20411	−0.602056	+	0.798454i	$0.705652\pi$
$930$	0	0
$931$	15.1756	0.497360
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−25.0184	−0.818191
$936$	0	0
$937$	7.42782	0.242656	0.121328	−	0.992612i	$-0.461285\pi$
$937$	7.42782	0.242656	0.121328	+	0.992612i	$0.461285\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	6.94417	0.226374	0.113187	−	0.993574i	$-0.463894\pi$
$941$	6.94417	0.226374	0.113187	+	0.993574i	$0.463894\pi$
$942$	0	0
$943$	−29.2597	−0.952826
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.8796	0.516019	0.258009	−	0.966142i	$-0.416933\pi$
$947$	15.8796	0.516019	0.258009	+	0.966142i	$0.416933\pi$
$948$	0	0
$949$	25.9645	0.842844
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−59.2310	−1.91868	−0.959340	−	0.282252i	$-0.908919\pi$
$953$	−59.2310	−1.91868	−0.959340	+	0.282252i	$0.908919\pi$
$954$	0	0
$955$	−26.0045	−0.841485
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−0.819856	−0.0264745
$960$	0	0
$961$	−20.8240	−0.671742
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−43.8957	−1.41305
$966$	0	0
$967$	−30.8083	−0.990730	−0.495365	−	0.868685i	$-0.664966\pi$
$967$	−30.8083	−0.990730	−0.495365	+	0.868685i	$0.664966\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−42.6220	−1.36780	−0.683902	−	0.729573i	$-0.739719\pi$
$971$	−42.6220	−1.36780	−0.683902	+	0.729573i	$0.739719\pi$
$972$	0	0
$973$	−4.39925	−0.141033
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.13049	0.196132	0.0980659	−	0.995180i	$-0.468734\pi$
$977$	6.13049	0.196132	0.0980659	+	0.995180i	$0.468734\pi$
$978$	0	0
$979$	13.7513	0.439492
$980$	0	0
$981$	0	0
$982$	0	0
$983$	33.9342	1.08233	0.541167	−	0.840915i	$-0.317983\pi$
$983$	33.9342	1.08233	0.541167	+	0.840915i	$0.317983\pi$
$984$	0	0
$985$	40.6951	1.29665
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−47.0841	−1.49719
$990$	0	0
$991$	−14.1117	−0.448274	−0.224137	−	0.974558i	$-0.571956\pi$
$991$	−14.1117	−0.448274	−0.224137	+	0.974558i	$0.571956\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−25.5991	−0.811545
$996$	0	0
$997$	16.7835	0.531537	0.265769	−	0.964037i	$-0.414374\pi$
$997$	16.7835	0.531537	0.265769	+	0.964037i	$0.414374\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.e.1.1		5
3.2	odd	2		2004.2.a.a.1.5	✓	5
12.11	even	2		8016.2.a.t.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.a.1.5	✓	5	3.2	odd	2
6012.2.a.e.1.1		5	1.1	even	1	trivial
8016.2.a.t.1.5		5	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)

\(f(q)\)	\(=\)	\(q-2.12560 q^{5} +0.221696 q^{7} +O(q^{10})\)	\(q-2.12560 q^{5} +0.221696 q^{7} +2.12560 q^{11} -2.07866 q^{13} +5.53728 q^{17} -2.18327 q^{19} -8.50926 q^{23} -0.481818 q^{25} +9.67698 q^{29} -3.18998 q^{31} -0.471237 q^{35} -5.07948 q^{37} +3.43857 q^{41} +5.53327 q^{43} +11.8218 q^{47} -6.95085 q^{49} -2.21498 q^{53} -4.51818 q^{55} -6.60234 q^{59} +9.72573 q^{61} +4.41840 q^{65} -0.214807 q^{67} -12.8354 q^{71} -12.4910 q^{73} +0.471237 q^{77} +7.45038 q^{79} -15.9723 q^{83} -11.7701 q^{85} +6.46935 q^{89} -0.460829 q^{91} +4.64076 q^{95} -0.984908 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{5} - 2 q^{7}+O(q^{10}) \) 5 * q + 3 * q^5 - 2 * q^7	\( 5 q + 3 q^{5} - 2 q^{7} - 3 q^{11} - 4 q^{13} + 7 q^{17} - 2 q^{19} - 13 q^{23} - 2 q^{25} + 3 q^{29} - 12 q^{31} - 10 q^{35} - 7 q^{37} + 16 q^{41} - q^{47} - 17 q^{49} - 3 q^{53} - 23 q^{55} - q^{59} - 22 q^{61} + 20 q^{65} + 2 q^{67} - 9 q^{71} - 28 q^{73} + 10 q^{77} - 28 q^{79} - 7 q^{83} - 11 q^{85} + 30 q^{89} - 13 q^{91} - 3 q^{95} - 33 q^{97}+O(q^{100}) \) 5 * q + 3 * q^5 - 2 * q^7 - 3 * q^11 - 4 * q^13 + 7 * q^17 - 2 * q^19 - 13 * q^23 - 2 * q^25 + 3 * q^29 - 12 * q^31 - 10 * q^35 - 7 * q^37 + 16 * q^41 - q^47 - 17 * q^49 - 3 * q^53 - 23 * q^55 - q^59 - 22 * q^61 + 20 * q^65 + 2 * q^67 - 9 * q^71 - 28 * q^73 + 10 * q^77 - 28 * q^79 - 7 * q^83 - 11 * q^85 + 30 * q^89 - 13 * q^91 - 3 * q^95 - 33 * q^97

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