LMFDB - Embedded newform 6012.2.a.f.1.3

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:	$5$
Coefficient field:	5.5.161121.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 3x^{2} + 5x + 1 $ x^5 - x^4 - 6x^3 + 3x^2 + 5*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			1.3
Root			$-0.261082$ 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)

$f(q)$	$=$	$q+1.38924 q^{5} -0.871845 q^{7} +O(q^{10})$	$q+1.38924 q^{5} -0.871845 q^{7} +5.74903 q^{11} +3.90849 q^{13} +0.477836 q^{17} +2.16161 q^{19} -3.53227 q^{23} -3.07002 q^{25} +5.05174 q^{29} +3.41763 q^{31} -1.21120 q^{35} -4.98967 q^{37} +5.50706 q^{41} -3.83237 q^{43} +13.6631 q^{47} -6.23989 q^{49} +11.6543 q^{53} +7.98676 q^{55} +0.528261 q^{59} +1.27251 q^{61} +5.42981 q^{65} +8.13778 q^{67} -4.17968 q^{71} -6.12260 q^{73} -5.01226 q^{77} -15.4461 q^{79} +1.01912 q^{83} +0.663827 q^{85} +9.40919 q^{89} -3.40760 q^{91} +3.00299 q^{95} -0.321073 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 7 q^{5} - 2 q^{7}+O(q^{10}) $ 5 * q + 7 * q^5 - 2 * q^7	$ 5 q + 7 q^{5} - 2 q^{7} + 5 q^{11} - 8 q^{13} + 7 q^{17} + 2 q^{19} + 13 q^{23} + 2 q^{25} + 11 q^{29} - 12 q^{31} + 12 q^{35} - 7 q^{37} + 12 q^{41} + 19 q^{47} - 9 q^{49} + 21 q^{53} - q^{55} + 7 q^{59} - 6 q^{61} - 14 q^{65} + 10 q^{67} + 35 q^{71} - 8 q^{73} + 6 q^{77} + 11 q^{83} + 5 q^{85} + 32 q^{89} + 5 q^{91} + 19 q^{95} + 11 q^{97}+O(q^{100}) $ 5 * q + 7 * q^5 - 2 * q^7 + 5 * q^11 - 8 * q^13 + 7 * q^17 + 2 * q^19 + 13 * q^23 + 2 * q^25 + 11 * q^29 - 12 * q^31 + 12 * q^35 - 7 * q^37 + 12 * q^41 + 19 * q^47 - 9 * q^49 + 21 * q^53 - q^55 + 7 * q^59 - 6 * q^61 - 14 * q^65 + 10 * q^67 + 35 * q^71 - 8 * q^73 + 6 * q^77 + 11 * q^83 + 5 * q^85 + 32 * q^89 + 5 * q^91 + 19 * q^95 + 11 * 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
$5$	1.38924	0.621285	0.310643	−	0.950527i	$-0.399456\pi$
$5$	1.38924	0.621285	0.310643	+	0.950527i	$0.399456\pi$
$6$	0	0
$7$	−0.871845	−0.329527	−0.164763	−	0.986333i	$-0.552686\pi$
$7$	−0.871845	−0.329527	−0.164763	+	0.986333i	$0.552686\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.74903	1.73340	0.866698	−	0.498833i	$-0.166238\pi$
$11$	5.74903	1.73340	0.866698	+	0.498833i	$0.166238\pi$
$12$	0	0
$13$	3.90849	1.08402	0.542009	−	0.840372i	$-0.317664\pi$
$13$	3.90849	1.08402	0.542009	+	0.840372i	$0.317664\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.477836	0.115892	0.0579461	−	0.998320i	$-0.481545\pi$
$17$	0.477836	0.115892	0.0579461	+	0.998320i	$0.481545\pi$
$18$	0	0
$19$	2.16161	0.495908	0.247954	−	0.968772i	$-0.420242\pi$
$19$	2.16161	0.495908	0.247954	+	0.968772i	$0.420242\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.53227	−0.736530	−0.368265	−	0.929721i	$-0.620048\pi$
$23$	−3.53227	−0.736530	−0.368265	+	0.929721i	$0.620048\pi$
$24$	0	0
$25$	−3.07002	−0.614004
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.05174	0.938085	0.469042	−	0.883176i	$-0.344599\pi$
$29$	5.05174	0.938085	0.469042	+	0.883176i	$0.344599\pi$
$30$	0	0
$31$	3.41763	0.613824	0.306912	−	0.951738i	$-0.400704\pi$
$31$	3.41763	0.613824	0.306912	+	0.951738i	$0.400704\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.21120	−0.204730
$36$	0	0
$37$	−4.98967	−0.820297	−0.410149	−	0.912019i	$-0.634523\pi$
$37$	−4.98967	−0.820297	−0.410149	+	0.912019i	$0.634523\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.50706	0.860059	0.430029	−	0.902815i	$-0.358503\pi$
$41$	5.50706	0.860059	0.430029	+	0.902815i	$0.358503\pi$
$42$	0	0
$43$	−3.83237	−0.584431	−0.292215	−	0.956353i	$-0.594392\pi$
$43$	−3.83237	−0.584431	−0.292215	+	0.956353i	$0.594392\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	13.6631	1.99297	0.996486	−	0.0837634i	$-0.0266940\pi$
$47$	13.6631	1.99297	0.996486	+	0.0837634i	$0.0266940\pi$
$48$	0	0
$49$	−6.23989	−0.891412
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.6543	1.60085	0.800423	−	0.599436i	$-0.204608\pi$
$53$	11.6543	1.60085	0.800423	+	0.599436i	$0.204608\pi$
$54$	0	0
$55$	7.98676	1.07693
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.528261	0.0687737	0.0343868	−	0.999409i	$-0.489052\pi$
$59$	0.528261	0.0687737	0.0343868	+	0.999409i	$0.489052\pi$
$60$	0	0
$61$	1.27251	0.162929	0.0814643	−	0.996676i	$-0.474040\pi$
$61$	1.27251	0.162929	0.0814643	+	0.996676i	$0.474040\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.42981	0.673485
$66$	0	0
$67$	8.13778	0.994188	0.497094	−	0.867697i	$-0.334400\pi$
$67$	8.13778	0.994188	0.497094	+	0.867697i	$0.334400\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.17968	−0.496036	−0.248018	−	0.968755i	$-0.579779\pi$
$71$	−4.17968	−0.496036	−0.248018	+	0.968755i	$0.579779\pi$
$72$	0	0
$73$	−6.12260	−0.716596	−0.358298	−	0.933607i	$-0.616643\pi$
$73$	−6.12260	−0.716596	−0.358298	+	0.933607i	$0.616643\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.01226	−0.571200
$78$	0	0
$79$	−15.4461	−1.73782	−0.868911	−	0.494969i	$-0.835179\pi$
$79$	−15.4461	−1.73782	−0.868911	+	0.494969i	$0.835179\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.01912	0.111863	0.0559315	−	0.998435i	$-0.482187\pi$
$83$	1.01912	0.111863	0.0559315	+	0.998435i	$0.482187\pi$
$84$	0	0
$85$	0.663827	0.0720022
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.40919	0.997372	0.498686	−	0.866783i	$-0.333816\pi$
$89$	9.40919	0.997372	0.498686	+	0.866783i	$0.333816\pi$
$90$	0	0
$91$	−3.40760	−0.357213
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.00299	0.308101
$96$	0	0
$97$	−0.321073	−0.0326000	−0.0163000	−	0.999867i	$-0.505189\pi$
$97$	−0.321073	−0.0326000	−0.0163000	+	0.999867i	$0.505189\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.5077	−1.44357	−0.721784	−	0.692119i	$-0.756678\pi$
$101$	−14.5077	−1.44357	−0.721784	+	0.692119i	$0.756678\pi$
$102$	0	0
$103$	6.14201	0.605190	0.302595	−	0.953119i	$-0.402147\pi$
$103$	6.14201	0.605190	0.302595	+	0.953119i	$0.402147\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.05659	−0.392165	−0.196083	−	0.980587i	$-0.562822\pi$
$107$	−4.05659	−0.392165	−0.196083	+	0.980587i	$0.562822\pi$
$108$	0	0
$109$	−17.8310	−1.70790	−0.853951	−	0.520353i	$-0.825800\pi$
$109$	−17.8310	−1.70790	−0.853951	+	0.520353i	$0.825800\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.27411	−0.119858	−0.0599289	−	0.998203i	$-0.519087\pi$
$113$	−1.27411	−0.119858	−0.0599289	+	0.998203i	$0.519087\pi$
$114$	0	0
$115$	−4.90716	−0.457595
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.416599	−0.0381896
$120$	0	0
$121$	22.0513	2.00466
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.2112	−1.00276
$126$	0	0
$127$	−2.67900	−0.237723	−0.118862	−	0.992911i	$-0.537924\pi$
$127$	−2.67900	−0.237723	−0.118862	+	0.992911i	$0.537924\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.43675	−0.475011	−0.237505	−	0.971386i	$-0.576330\pi$
$131$	−5.43675	−0.475011	−0.237505	+	0.971386i	$0.576330\pi$
$132$	0	0
$133$	−1.88459	−0.163415
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.68828	−0.400547	−0.200273	−	0.979740i	$-0.564183\pi$
$137$	−4.68828	−0.400547	−0.200273	+	0.979740i	$0.564183\pi$
$138$	0	0
$139$	14.4563	1.22617	0.613086	−	0.790017i	$-0.289928\pi$
$139$	14.4563	1.22617	0.613086	+	0.790017i	$0.289928\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	22.4700	1.87903
$144$	0	0
$145$	7.01806	0.582818
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.97719	−0.325824	−0.162912	−	0.986641i	$-0.552089\pi$
$149$	−3.97719	−0.325824	−0.162912	+	0.986641i	$0.552089\pi$
$150$	0	0
$151$	0.552927	0.0449965	0.0224983	−	0.999747i	$-0.492838\pi$
$151$	0.552927	0.0449965	0.0224983	+	0.999747i	$0.492838\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.74789	0.381360
$156$	0	0
$157$	−8.90396	−0.710613	−0.355307	−	0.934750i	$-0.615624\pi$
$157$	−8.90396	−0.710613	−0.355307	+	0.934750i	$0.615624\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.07960	0.242706
$162$	0	0
$163$	−2.32106	−0.181800	−0.0908999	−	0.995860i	$-0.528974\pi$
$163$	−2.32106	−0.181800	−0.0908999	+	0.995860i	$0.528974\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	2.27626	0.175097
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.53893	0.497146	0.248573	−	0.968613i	$-0.420038\pi$
$173$	6.53893	0.497146	0.248573	+	0.968613i	$0.420038\pi$
$174$	0	0
$175$	2.67658	0.202331
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.03006	0.450708	0.225354	−	0.974277i	$-0.427646\pi$
$179$	6.03006	0.450708	0.225354	+	0.974277i	$0.427646\pi$
$180$	0	0
$181$	23.9128	1.77743	0.888713	−	0.458464i	$-0.151600\pi$
$181$	23.9128	1.77743	0.888713	+	0.458464i	$0.151600\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.93184	−0.509639
$186$	0	0
$187$	2.74709	0.200887
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.15759	0.373190	0.186595	−	0.982437i	$-0.440255\pi$
$191$	5.15759	0.373190	0.186595	+	0.982437i	$0.440255\pi$
$192$	0	0
$193$	−2.38508	−0.171682	−0.0858410	−	0.996309i	$-0.527358\pi$
$193$	−2.38508	−0.171682	−0.0858410	+	0.996309i	$0.527358\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.00129542	9.22949e−5 0	4.61475e−5	−	1.00000i	$-0.499985\pi$
$197$	0.00129542	9.22949e−5 0	4.61475e−5		1.00000i	$0.499985\pi$
$198$	0	0
$199$	16.7180	1.18511	0.592555	−	0.805530i	$-0.298119\pi$
$199$	16.7180	1.18511	0.592555	+	0.805530i	$0.298119\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.40434	−0.309124
$204$	0	0
$205$	7.65061	0.534342
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.4272	0.859605
$210$	0	0
$211$	−3.08873	−0.212637	−0.106319	−	0.994332i	$-0.533906\pi$
$211$	−3.08873	−0.212637	−0.106319	+	0.994332i	$0.533906\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.32407	−0.363098
$216$	0	0
$217$	−2.97964	−0.202271
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.86761	0.125629
$222$	0	0
$223$	14.4949	0.970648	0.485324	−	0.874334i	$-0.338702\pi$
$223$	14.4949	0.970648	0.485324	+	0.874334i	$0.338702\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.04675	0.600454	0.300227	−	0.953868i	$-0.402938\pi$
$227$	9.04675	0.600454	0.300227	+	0.953868i	$0.402938\pi$
$228$	0	0
$229$	−15.9796	−1.05596	−0.527982	−	0.849256i	$-0.677051\pi$
$229$	−15.9796	−1.05596	−0.527982	+	0.849256i	$0.677051\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.78880	−0.117188	−0.0585941	−	0.998282i	$-0.518662\pi$
$233$	−1.78880	−0.117188	−0.0585941	+	0.998282i	$0.518662\pi$
$234$	0	0
$235$	18.9813	1.23820
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4.43119	−0.286630	−0.143315	−	0.989677i	$-0.545776\pi$
$239$	−4.43119	−0.286630	−0.143315	+	0.989677i	$0.545776\pi$
$240$	0	0
$241$	15.3349	0.987808	0.493904	−	0.869517i	$-0.335570\pi$
$241$	15.3349	0.987808	0.493904	+	0.869517i	$0.335570\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−8.66868	−0.553821
$246$	0	0
$247$	8.44863	0.537574
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−25.6393	−1.61834	−0.809168	−	0.587577i	$-0.800082\pi$
$251$	−25.6393	−1.61834	−0.809168	+	0.587577i	$0.800082\pi$
$252$	0	0
$253$	−20.3071	−1.27670
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.6727	1.41429	0.707143	−	0.707070i	$-0.249984\pi$
$257$	22.6727	1.41429	0.707143	+	0.707070i	$0.249984\pi$
$258$	0	0
$259$	4.35022	0.270310
$260$	0	0
$261$	0	0
$262$	0	0
$263$	28.4884	1.75667	0.878335	−	0.478045i	$-0.158654\pi$
$263$	28.4884	1.75667	0.878335	+	0.478045i	$0.158654\pi$
$264$	0	0
$265$	16.1906	0.994582
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−13.0734	−0.797100	−0.398550	−	0.917147i	$-0.630486\pi$
$269$	−13.0734	−0.797100	−0.398550	+	0.917147i	$0.630486\pi$
$270$	0	0
$271$	−3.04487	−0.184963	−0.0924814	−	0.995714i	$-0.529480\pi$
$271$	−3.04487	−0.184963	−0.0924814	+	0.995714i	$0.529480\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−17.6496	−1.06431
$276$	0	0
$277$	4.04405	0.242984	0.121492	−	0.992592i	$-0.461232\pi$
$277$	4.04405	0.242984	0.121492	+	0.992592i	$0.461232\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	17.2904	1.03146	0.515728	−	0.856752i	$-0.327521\pi$
$281$	17.2904	1.03146	0.515728	+	0.856752i	$0.327521\pi$
$282$	0	0
$283$	23.1428	1.37570	0.687849	−	0.725853i	$-0.258555\pi$
$283$	23.1428	1.37570	0.687849	+	0.725853i	$0.258555\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.80131	−0.283412
$288$	0	0
$289$	−16.7717	−0.986569
$290$	0	0
$291$	0	0
$292$	0	0
$293$	28.4973	1.66483	0.832416	−	0.554152i	$-0.186957\pi$
$293$	28.4973	1.66483	0.832416	+	0.554152i	$0.186957\pi$
$294$	0	0
$295$	0.733879	0.0427281
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−13.8058	−0.798412
$300$	0	0
$301$	3.34123	0.192585
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.76782	0.101225
$306$	0	0
$307$	23.4381	1.33768	0.668842	−	0.743405i	$-0.266790\pi$
$307$	23.4381	1.33768	0.668842	+	0.743405i	$0.266790\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9.33992	−0.529618	−0.264809	−	0.964301i	$-0.585309\pi$
$311$	−9.33992	−0.529618	−0.264809	+	0.964301i	$0.585309\pi$
$312$	0	0
$313$	17.9683	1.01563	0.507815	−	0.861466i	$-0.330453\pi$
$313$	17.9683	1.01563	0.507815	+	0.861466i	$0.330453\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.1619	1.35706	0.678532	−	0.734570i	$-0.262616\pi$
$317$	24.1619	1.35706	0.678532	+	0.734570i	$0.262616\pi$
$318$	0	0
$319$	29.0426	1.62607
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.03290	0.0574719
$324$	0	0
$325$	−11.9991	−0.665592
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−11.9121	−0.656737
$330$	0	0
$331$	17.6022	0.967505	0.483753	−	0.875205i	$-0.339273\pi$
$331$	17.6022	0.967505	0.483753	+	0.875205i	$0.339273\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	11.3053	0.617674
$336$	0	0
$337$	−2.14881	−0.117053	−0.0585265	−	0.998286i	$-0.518640\pi$
$337$	−2.14881	−0.117053	−0.0585265	+	0.998286i	$0.518640\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	19.6480	1.06400
$342$	0	0
$343$	11.5431	0.623271
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11.0393	−0.592622	−0.296311	−	0.955091i	$-0.595757\pi$
$347$	−11.0393	−0.592622	−0.296311	+	0.955091i	$0.595757\pi$
$348$	0	0
$349$	−0.603266	−0.0322921	−0.0161460	−	0.999870i	$-0.505140\pi$
$349$	−0.603266	−0.0322921	−0.0161460	+	0.999870i	$0.505140\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−28.6385	−1.52427	−0.762137	−	0.647416i	$-0.775850\pi$
$353$	−28.6385	−1.52427	−0.762137	+	0.647416i	$0.775850\pi$
$354$	0	0
$355$	−5.80656	−0.308180
$356$	0	0
$357$	0	0
$358$	0	0
$359$	28.2365	1.49026	0.745132	−	0.666918i	$-0.232387\pi$
$359$	28.2365	1.49026	0.745132	+	0.666918i	$0.232387\pi$
$360$	0	0
$361$	−14.3274	−0.754075
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.50574	−0.445211
$366$	0	0
$367$	−19.9271	−1.04019	−0.520093	−	0.854110i	$-0.674103\pi$
$367$	−19.9271	−1.04019	−0.520093	+	0.854110i	$0.674103\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−10.1608	−0.527521
$372$	0	0
$373$	−23.4724	−1.21535	−0.607677	−	0.794184i	$-0.707899\pi$
$373$	−23.4724	−1.21535	−0.607677	+	0.794184i	$0.707899\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	19.7447	1.01690
$378$	0	0
$379$	−11.2367	−0.577192	−0.288596	−	0.957451i	$-0.593188\pi$
$379$	−11.2367	−0.577192	−0.288596	+	0.957451i	$0.593188\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	34.1612	1.74555	0.872777	−	0.488119i	$-0.162317\pi$
$383$	34.1612	1.74555	0.872777	+	0.488119i	$0.162317\pi$
$384$	0	0
$385$	−6.96322	−0.354878
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−16.6706	−0.845233	−0.422617	−	0.906309i	$-0.638888\pi$
$389$	−16.6706	−0.845233	−0.422617	+	0.906309i	$0.638888\pi$
$390$	0	0
$391$	−1.68785	−0.0853581
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−21.4583	−1.07968
$396$	0	0
$397$	13.3298	0.669006	0.334503	−	0.942395i	$-0.391432\pi$
$397$	13.3298	0.669006	0.334503	+	0.942395i	$0.391432\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−24.9727	−1.24708	−0.623539	−	0.781792i	$-0.714306\pi$
$401$	−24.9727	−1.24708	−0.623539	+	0.781792i	$0.714306\pi$
$402$	0	0
$403$	13.3577	0.665397
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−28.6858	−1.42190
$408$	0	0
$409$	−22.3474	−1.10501	−0.552504	−	0.833510i	$-0.686328\pi$
$409$	−22.3474	−1.10501	−0.552504	+	0.833510i	$0.686328\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−0.460562	−0.0226628
$414$	0	0
$415$	1.41580	0.0694988
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−26.6024	−1.29961	−0.649805	−	0.760101i	$-0.725150\pi$
$419$	−26.6024	−1.29961	−0.649805	+	0.760101i	$0.725150\pi$
$420$	0	0
$421$	−23.3727	−1.13911	−0.569557	−	0.821952i	$-0.692885\pi$
$421$	−23.3727	−1.13911	−0.569557	+	0.821952i	$0.692885\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.46697	−0.0711583
$426$	0	0
$427$	−1.10944	−0.0536893
$428$	0	0
$429$	0	0
$430$	0	0
$431$	34.7703	1.67483	0.837414	−	0.546570i	$-0.184067\pi$
$431$	34.7703	1.67483	0.837414	+	0.546570i	$0.184067\pi$
$432$	0	0
$433$	−21.3677	−1.02687	−0.513433	−	0.858130i	$-0.671626\pi$
$433$	−21.3677	−1.02687	−0.513433	+	0.858130i	$0.671626\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.63541	−0.365251
$438$	0	0
$439$	10.2625	0.489802	0.244901	−	0.969548i	$-0.421245\pi$
$439$	10.2625	0.489802	0.244901	+	0.969548i	$0.421245\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.69185	−0.222917	−0.111458	−	0.993769i	$-0.535552\pi$
$443$	−4.69185	−0.222917	−0.111458	+	0.993769i	$0.535552\pi$
$444$	0	0
$445$	13.0716	0.619653
$446$	0	0
$447$	0	0
$448$	0	0
$449$	12.2687	0.578998	0.289499	−	0.957178i	$-0.406511\pi$
$449$	12.2687	0.578998	0.289499	+	0.957178i	$0.406511\pi$
$450$	0	0
$451$	31.6603	1.49082
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.73396	−0.221931
$456$	0	0
$457$	11.1775	0.522862	0.261431	−	0.965222i	$-0.415806\pi$
$457$	11.1775	0.522862	0.261431	+	0.965222i	$0.415806\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−19.2919	−0.898516	−0.449258	−	0.893402i	$-0.648312\pi$
$461$	−19.2919	−0.898516	−0.449258	+	0.893402i	$0.648312\pi$
$462$	0	0
$463$	12.1247	0.563481	0.281741	−	0.959491i	$-0.409088\pi$
$463$	12.1247	0.563481	0.281741	+	0.959491i	$0.409088\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.6560	0.585650	0.292825	−	0.956166i	$-0.405405\pi$
$467$	12.6560	0.585650	0.292825	+	0.956166i	$0.405405\pi$
$468$	0	0
$469$	−7.09488	−0.327611
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−22.0324	−1.01305
$474$	0	0
$475$	−6.63620	−0.304490
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−32.3944	−1.48014	−0.740068	−	0.672532i	$-0.765207\pi$
$479$	−32.3944	−1.48014	−0.740068	+	0.672532i	$0.765207\pi$
$480$	0	0
$481$	−19.5021	−0.889218
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.446046	−0.0202539
$486$	0	0
$487$	38.5332	1.74610	0.873052	−	0.487627i	$-0.162138\pi$
$487$	38.5332	1.74610	0.873052	+	0.487627i	$0.162138\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−17.3207	−0.781672	−0.390836	−	0.920460i	$-0.627814\pi$
$491$	−17.3207	−0.781672	−0.390836	+	0.920460i	$0.627814\pi$
$492$	0	0
$493$	2.41390	0.108717
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.64403	0.163457
$498$	0	0
$499$	−16.7924	−0.751729	−0.375865	−	0.926675i	$-0.622654\pi$
$499$	−16.7924	−0.751729	−0.375865	+	0.926675i	$0.622654\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.45950	0.154252	0.0771258	−	0.997021i	$-0.475426\pi$
$503$	3.45950	0.154252	0.0771258	+	0.997021i	$0.475426\pi$
$504$	0	0
$505$	−20.1546	−0.896868
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.8751	0.659326	0.329663	−	0.944099i	$-0.393065\pi$
$509$	14.8751	0.659326	0.329663	+	0.944099i	$0.393065\pi$
$510$	0	0
$511$	5.33796	0.236138
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.53270	0.375996
$516$	0	0
$517$	78.5497	3.45461
$518$	0	0
$519$	0	0
$520$	0	0
$521$	6.52656	0.285934	0.142967	−	0.989727i	$-0.454336\pi$
$521$	6.52656	0.285934	0.142967	+	0.989727i	$0.454336\pi$
$522$	0	0
$523$	32.7225	1.43086	0.715428	−	0.698686i	$-0.246232\pi$
$523$	32.7225	1.43086	0.715428	+	0.698686i	$0.246232\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.63306	0.0711374
$528$	0	0
$529$	−10.5231	−0.457524
$530$	0	0
$531$	0	0
$532$	0	0
$533$	21.5243	0.932320
$534$	0	0
$535$	−5.63556	−0.243647
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−35.8733	−1.54517
$540$	0	0
$541$	−35.3810	−1.52115	−0.760573	−	0.649252i	$-0.775082\pi$
$541$	−35.3810	−1.52115	−0.760573	+	0.649252i	$0.775082\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−24.7715	−1.06109
$546$	0	0
$547$	3.10909	0.132935	0.0664676	−	0.997789i	$-0.478827\pi$
$547$	3.10909	0.132935	0.0664676	+	0.997789i	$0.478827\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	10.9199	0.465204
$552$	0	0
$553$	13.4666	0.572658
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−21.3530	−0.904757	−0.452379	−	0.891826i	$-0.649424\pi$
$557$	−21.3530	−0.904757	−0.452379	+	0.891826i	$0.649424\pi$
$558$	0	0
$559$	−14.9788	−0.633534
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14.8320	−0.625093	−0.312546	−	0.949903i	$-0.601182\pi$
$563$	−14.8320	−0.625093	−0.312546	+	0.949903i	$0.601182\pi$
$564$	0	0
$565$	−1.77003	−0.0744659
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−29.6942	−1.24485	−0.622423	−	0.782681i	$-0.713852\pi$
$569$	−29.6942	−1.24485	−0.622423	+	0.782681i	$0.713852\pi$
$570$	0	0
$571$	−8.20664	−0.343437	−0.171719	−	0.985146i	$-0.554932\pi$
$571$	−8.20664	−0.343437	−0.171719	+	0.985146i	$0.554932\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	10.8442	0.452232
$576$	0	0
$577$	34.6609	1.44295	0.721477	−	0.692438i	$-0.243464\pi$
$577$	34.6609	1.44295	0.721477	+	0.692438i	$0.243464\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.888515	−0.0368618
$582$	0	0
$583$	67.0011	2.77490
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.3087	−0.920780	−0.460390	−	0.887717i	$-0.652290\pi$
$587$	−22.3087	−0.920780	−0.460390	+	0.887717i	$0.652290\pi$
$588$	0	0
$589$	7.38759	0.304400
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−48.3835	−1.98687	−0.993435	−	0.114394i	$-0.963507\pi$
$593$	−48.3835	−1.98687	−0.993435	+	0.114394i	$0.963507\pi$
$594$	0	0
$595$	−0.578755	−0.0237266
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24.7439	1.01101	0.505503	−	0.862825i	$-0.331307\pi$
$599$	24.7439	1.01101	0.505503	+	0.862825i	$0.331307\pi$
$600$	0	0
$601$	41.4466	1.69064	0.845320	−	0.534260i	$-0.179410\pi$
$601$	41.4466	1.69064	0.845320	+	0.534260i	$0.179410\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	30.6345	1.24547
$606$	0	0
$607$	8.13247	0.330087	0.165043	−	0.986286i	$-0.447224\pi$
$607$	8.13247	0.330087	0.165043	+	0.986286i	$0.447224\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	53.4021	2.16042
$612$	0	0
$613$	−13.6117	−0.549771	−0.274886	−	0.961477i	$-0.588640\pi$
$613$	−13.6117	−0.549771	−0.274886	+	0.961477i	$0.588640\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.02573	0.0815528	0.0407764	−	0.999168i	$-0.487017\pi$
$617$	2.02573	0.0815528	0.0407764	+	0.999168i	$0.487017\pi$
$618$	0	0
$619$	−13.8882	−0.558213	−0.279106	−	0.960260i	$-0.590038\pi$
$619$	−13.8882	−0.558213	−0.279106	+	0.960260i	$0.590038\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−8.20336	−0.328661
$624$	0	0
$625$	−0.224859	−0.00899436
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2.38424	−0.0950661
$630$	0	0
$631$	0.724370	0.0288367	0.0144184	−	0.999896i	$-0.495410\pi$
$631$	0.724370	0.0288367	0.0144184	+	0.999896i	$0.495410\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.72177	−0.147694
$636$	0	0
$637$	−24.3885	−0.966308
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18.4957	−0.730537	−0.365269	−	0.930902i	$-0.619023\pi$
$641$	−18.4957	−0.730537	−0.365269	+	0.930902i	$0.619023\pi$
$642$	0	0
$643$	−7.03941	−0.277607	−0.138804	−	0.990320i	$-0.544326\pi$
$643$	−7.03941	−0.277607	−0.138804	+	0.990320i	$0.544326\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	46.7368	1.83741	0.918706	−	0.394942i	$-0.129235\pi$
$647$	46.7368	1.83741	0.918706	+	0.394942i	$0.129235\pi$
$648$	0	0
$649$	3.03698	0.119212
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.06283	−0.0807247	−0.0403624	−	0.999185i	$-0.512851\pi$
$653$	−2.06283	−0.0807247	−0.0403624	+	0.999185i	$0.512851\pi$
$654$	0	0
$655$	−7.55293	−0.295117
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−26.3610	−1.02688	−0.513440	−	0.858126i	$-0.671629\pi$
$659$	−26.3610	−1.02688	−0.513440	+	0.858126i	$0.671629\pi$
$660$	0	0
$661$	−25.1696	−0.978982	−0.489491	−	0.872008i	$-0.662817\pi$
$661$	−25.1696	−0.978982	−0.489491	+	0.872008i	$0.662817\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.61815	−0.101527
$666$	0	0
$667$	−17.8441	−0.690927
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7.31572	0.282420
$672$	0	0
$673$	14.4885	0.558491	0.279245	−	0.960220i	$-0.409916\pi$
$673$	14.4885	0.558491	0.279245	+	0.960220i	$0.409916\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	43.9862	1.69053	0.845263	−	0.534350i	$-0.179444\pi$
$677$	43.9862	1.69053	0.845263	+	0.534350i	$0.179444\pi$
$678$	0	0
$679$	0.279926	0.0107426
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−11.4959	−0.439878	−0.219939	−	0.975514i	$-0.570586\pi$
$683$	−11.4959	−0.439878	−0.219939	+	0.975514i	$0.570586\pi$
$684$	0	0
$685$	−6.51313	−0.248854
$686$	0	0
$687$	0	0
$688$	0	0
$689$	45.5508	1.73535
$690$	0	0
$691$	47.9678	1.82478	0.912390	−	0.409323i	$-0.134235\pi$
$691$	47.9678	1.82478	0.912390	+	0.409323i	$0.134235\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	20.0833	0.761802
$696$	0	0
$697$	2.63147	0.0996741
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−18.4427	−0.696571	−0.348285	−	0.937389i	$-0.613236\pi$
$701$	−18.4427	−0.696571	−0.348285	+	0.937389i	$0.613236\pi$
$702$	0	0
$703$	−10.7857	−0.406792
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.6485	0.475694
$708$	0	0
$709$	37.1171	1.39396	0.696981	−	0.717090i	$-0.254526\pi$
$709$	37.1171	1.39396	0.696981	+	0.717090i	$0.254526\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−12.0720	−0.452099
$714$	0	0
$715$	31.2161	1.16742
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8.72522	0.325396	0.162698	−	0.986676i	$-0.447980\pi$
$719$	8.72522	0.325396	0.162698	+	0.986676i	$0.447980\pi$
$720$	0	0
$721$	−5.35488	−0.199426
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−15.5090	−0.575988
$726$	0	0
$727$	−38.9456	−1.44441	−0.722206	−	0.691678i	$-0.756872\pi$
$727$	−38.9456	−1.44441	−0.722206	+	0.691678i	$0.756872\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.83124	−0.0677310
$732$	0	0
$733$	−11.0832	−0.409367	−0.204684	−	0.978828i	$-0.565617\pi$
$733$	−11.0832	−0.409367	−0.204684	+	0.978828i	$0.565617\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	46.7843	1.72332
$738$	0	0
$739$	−39.0330	−1.43585	−0.717927	−	0.696119i	$-0.754909\pi$
$739$	−39.0330	−1.43585	−0.717927	+	0.696119i	$0.754909\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8.69917	−0.319142	−0.159571	−	0.987186i	$-0.551011\pi$
$743$	−8.69917	−0.319142	−0.159571	+	0.987186i	$0.551011\pi$
$744$	0	0
$745$	−5.52526	−0.202430
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.53672	0.129229
$750$	0	0
$751$	−30.1732	−1.10104	−0.550518	−	0.834823i	$-0.685570\pi$
$751$	−30.1732	−1.10104	−0.550518	+	0.834823i	$0.685570\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.768146	0.0279557
$756$	0	0
$757$	−1.93772	−0.0704277	−0.0352138	−	0.999380i	$-0.511211\pi$
$757$	−1.93772	−0.0704277	−0.0352138	+	0.999380i	$0.511211\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	32.4526	1.17641	0.588203	−	0.808713i	$-0.299836\pi$
$761$	32.4526	1.17641	0.588203	+	0.808713i	$0.299836\pi$
$762$	0	0
$763$	15.5459	0.562799
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.06470	0.0745519
$768$	0	0
$769$	−13.5848	−0.489882	−0.244941	−	0.969538i	$-0.578769\pi$
$769$	−13.5848	−0.489882	−0.244941	+	0.969538i	$0.578769\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	5.72819	0.206029	0.103014	−	0.994680i	$-0.467151\pi$
$773$	5.72819	0.206029	0.103014	+	0.994680i	$0.467151\pi$
$774$	0	0
$775$	−10.4922	−0.376890
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.9041	0.426510
$780$	0	0
$781$	−24.0291	−0.859828
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−12.3697	−0.441494
$786$	0	0
$787$	−49.9123	−1.77918	−0.889591	−	0.456758i	$-0.849010\pi$
$787$	−49.9123	−1.77918	−0.889591	+	0.456758i	$0.849010\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.11082	0.0394963
$792$	0	0
$793$	4.97360	0.176618
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−32.1371	−1.13836	−0.569178	−	0.822214i	$-0.692738\pi$
$797$	−32.1371	−1.13836	−0.569178	+	0.822214i	$0.692738\pi$
$798$	0	0
$799$	6.52873	0.230970
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−35.1990	−1.24215
$804$	0	0
$805$	4.27829	0.150790
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−6.77641	−0.238246	−0.119123	−	0.992880i	$-0.538008\pi$
$809$	−6.77641	−0.238246	−0.119123	+	0.992880i	$0.538008\pi$
$810$	0	0
$811$	32.4544	1.13963	0.569814	−	0.821774i	$-0.307015\pi$
$811$	32.4544	1.13963	0.569814	+	0.821774i	$0.307015\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.22451	−0.112950
$816$	0	0
$817$	−8.28410	−0.289824
$818$	0	0
$819$	0	0
$820$	0	0
$821$	24.8054	0.865716	0.432858	−	0.901462i	$-0.357505\pi$
$821$	24.8054	0.865716	0.432858	+	0.901462i	$0.357505\pi$
$822$	0	0
$823$	−15.2673	−0.532184	−0.266092	−	0.963948i	$-0.585732\pi$
$823$	−15.2673	−0.532184	−0.266092	+	0.963948i	$0.585732\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.3232	0.463292	0.231646	−	0.972800i	$-0.425589\pi$
$827$	13.3232	0.463292	0.231646	+	0.972800i	$0.425589\pi$
$828$	0	0
$829$	−49.9301	−1.73414	−0.867072	−	0.498183i	$-0.834001\pi$
$829$	−49.9301	−1.73414	−0.867072	+	0.498183i	$0.834001\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.98164	−0.103308
$834$	0	0
$835$	1.38924	0.0480765
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−19.8275	−0.684520	−0.342260	−	0.939605i	$-0.611192\pi$
$839$	−19.8275	−0.684520	−0.342260	+	0.939605i	$0.611192\pi$
$840$	0	0
$841$	−3.47992	−0.119997
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.16226	0.108785
$846$	0	0
$847$	−19.2253	−0.660590
$848$	0	0
$849$	0	0
$850$	0	0
$851$	17.6249	0.604173
$852$	0	0
$853$	−17.1965	−0.588798	−0.294399	−	0.955683i	$-0.595119\pi$
$853$	−17.1965	−0.588798	−0.294399	+	0.955683i	$0.595119\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.21284	0.0414299	0.0207149	−	0.999785i	$-0.493406\pi$
$857$	1.21284	0.0414299	0.0207149	+	0.999785i	$0.493406\pi$
$858$	0	0
$859$	22.6809	0.773863	0.386932	−	0.922108i	$-0.373535\pi$
$859$	22.6809	0.773863	0.386932	+	0.922108i	$0.373535\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−9.21615	−0.313721	−0.156861	−	0.987621i	$-0.550137\pi$
$863$	−9.21615	−0.313721	−0.156861	+	0.987621i	$0.550137\pi$
$864$	0	0
$865$	9.08412	0.308869
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−88.8000	−3.01233
$870$	0	0
$871$	31.8064	1.07772
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9.77441	0.330435
$876$	0	0
$877$	−33.5985	−1.13454	−0.567270	−	0.823532i	$-0.692000\pi$
$877$	−33.5985	−1.13454	−0.567270	+	0.823532i	$0.692000\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	49.4124	1.66475	0.832373	−	0.554216i	$-0.186982\pi$
$881$	49.4124	1.66475	0.832373	+	0.554216i	$0.186982\pi$
$882$	0	0
$883$	27.9558	0.940786	0.470393	−	0.882457i	$-0.344112\pi$
$883$	27.9558	0.940786	0.470393	+	0.882457i	$0.344112\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.31976	0.0778899	0.0389449	−	0.999241i	$-0.487600\pi$
$887$	2.31976	0.0778899	0.0389449	+	0.999241i	$0.487600\pi$
$888$	0	0
$889$	2.33568	0.0783361
$890$	0	0
$891$	0	0
$892$	0	0
$893$	29.5344	0.988331
$894$	0	0
$895$	8.37718	0.280018
$896$	0	0
$897$	0	0
$898$	0	0
$899$	17.2650	0.575819
$900$	0	0
$901$	5.56886	0.185526
$902$	0	0
$903$	0	0
$904$	0	0
$905$	33.2206	1.10429
$906$	0	0
$907$	−20.0556	−0.665936	−0.332968	−	0.942938i	$-0.608050\pi$
$907$	−20.0556	−0.665936	−0.332968	+	0.942938i	$0.608050\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	18.8720	0.625258	0.312629	−	0.949875i	$-0.398790\pi$
$911$	18.8720	0.625258	0.312629	+	0.949875i	$0.398790\pi$
$912$	0	0
$913$	5.85895	0.193903
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.74000	0.156529
$918$	0	0
$919$	53.4602	1.76349	0.881745	−	0.471727i	$-0.156369\pi$
$919$	53.4602	1.76349	0.881745	+	0.471727i	$0.156369\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−16.3362	−0.537713
$924$	0	0
$925$	15.3184	0.503666
$926$	0	0
$927$	0	0
$928$	0	0
$929$	30.7913	1.01023	0.505114	−	0.863052i	$-0.331450\pi$
$929$	30.7913	1.01023	0.505114	+	0.863052i	$0.331450\pi$
$930$	0	0
$931$	−13.4882	−0.442059
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3.81636	0.124808
$936$	0	0
$937$	−26.5516	−0.867405	−0.433702	−	0.901056i	$-0.642793\pi$
$937$	−26.5516	−0.867405	−0.433702	+	0.901056i	$0.642793\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	46.2200	1.50673	0.753364	−	0.657603i	$-0.228430\pi$
$941$	46.2200	1.50673	0.753364	+	0.657603i	$0.228430\pi$
$942$	0	0
$943$	−19.4525	−0.633459
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−38.8101	−1.26116	−0.630579	−	0.776125i	$-0.717183\pi$
$947$	−38.8101	−1.26116	−0.630579	+	0.776125i	$0.717183\pi$
$948$	0	0
$949$	−23.9301	−0.776804
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−14.1074	−0.456982	−0.228491	−	0.973546i	$-0.573379\pi$
$953$	−14.1074	−0.456982	−0.228491	+	0.973546i	$0.573379\pi$
$954$	0	0
$955$	7.16512	0.231858
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.08746	0.131991
$960$	0	0
$961$	−19.3198	−0.623220
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−3.31344	−0.106664
$966$	0	0
$967$	−9.47705	−0.304761	−0.152381	−	0.988322i	$-0.548694\pi$
$967$	−9.47705	−0.304761	−0.152381	+	0.988322i	$0.548694\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	6.76536	0.217111	0.108555	−	0.994090i	$-0.465378\pi$
$971$	6.76536	0.217111	0.108555	+	0.994090i	$0.465378\pi$
$972$	0	0
$973$	−12.6037	−0.404056
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.48932	−0.143626	−0.0718131	−	0.997418i	$-0.522879\pi$
$977$	−4.48932	−0.143626	−0.0718131	+	0.997418i	$0.522879\pi$
$978$	0	0
$979$	54.0937	1.72884
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−6.64779	−0.212032	−0.106016	−	0.994364i	$-0.533809\pi$
$983$	−6.64779	−0.212032	−0.106016	+	0.994364i	$0.533809\pi$
$984$	0	0
$985$	0.00179965	5.73415e−5 0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	13.5370	0.430450
$990$	0	0
$991$	−27.9862	−0.889011	−0.444505	−	0.895776i	$-0.646621\pi$
$991$	−27.9862	−0.889011	−0.444505	+	0.895776i	$0.646621\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	23.2253	0.736292
$996$	0	0
$997$	9.59766	0.303961	0.151981	−	0.988383i	$-0.451435\pi$
$997$	9.59766	0.303961	0.151981	+	0.988383i	$0.451435\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.f.1.3		5
3.2	odd	2		2004.2.a.b.1.3	✓	5
12.11	even	2		8016.2.a.q.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.b.1.3	✓	5	3.2	odd	2
6012.2.a.f.1.3		5	1.1	even	1	trivial
8016.2.a.q.1.3		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)

\(f(q)\)	\(=\)	\(q+1.38924 q^{5} -0.871845 q^{7} +O(q^{10})\)	\(q+1.38924 q^{5} -0.871845 q^{7} +5.74903 q^{11} +3.90849 q^{13} +0.477836 q^{17} +2.16161 q^{19} -3.53227 q^{23} -3.07002 q^{25} +5.05174 q^{29} +3.41763 q^{31} -1.21120 q^{35} -4.98967 q^{37} +5.50706 q^{41} -3.83237 q^{43} +13.6631 q^{47} -6.23989 q^{49} +11.6543 q^{53} +7.98676 q^{55} +0.528261 q^{59} +1.27251 q^{61} +5.42981 q^{65} +8.13778 q^{67} -4.17968 q^{71} -6.12260 q^{73} -5.01226 q^{77} -15.4461 q^{79} +1.01912 q^{83} +0.663827 q^{85} +9.40919 q^{89} -3.40760 q^{91} +3.00299 q^{95} -0.321073 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 7 q^{5} - 2 q^{7}+O(q^{10}) \) 5 * q + 7 * q^5 - 2 * q^7	\( 5 q + 7 q^{5} - 2 q^{7} + 5 q^{11} - 8 q^{13} + 7 q^{17} + 2 q^{19} + 13 q^{23} + 2 q^{25} + 11 q^{29} - 12 q^{31} + 12 q^{35} - 7 q^{37} + 12 q^{41} + 19 q^{47} - 9 q^{49} + 21 q^{53} - q^{55} + 7 q^{59} - 6 q^{61} - 14 q^{65} + 10 q^{67} + 35 q^{71} - 8 q^{73} + 6 q^{77} + 11 q^{83} + 5 q^{85} + 32 q^{89} + 5 q^{91} + 19 q^{95} + 11 q^{97}+O(q^{100}) \) 5 * q + 7 * q^5 - 2 * q^7 + 5 * q^11 - 8 * q^13 + 7 * q^17 + 2 * q^19 + 13 * q^23 + 2 * q^25 + 11 * q^29 - 12 * q^31 + 12 * q^35 - 7 * q^37 + 12 * q^41 + 19 * q^47 - 9 * q^49 + 21 * q^53 - q^55 + 7 * q^59 - 6 * q^61 - 14 * q^65 + 10 * q^67 + 35 * q^71 - 8 * q^73 + 6 * q^77 + 11 * q^83 + 5 * q^85 + 32 * q^89 + 5 * q^91 + 19 * q^95 + 11 * q^97

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