LMFDB - Embedded newform 6048.2.a.bu.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(1,6048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.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:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.25808.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 10x^{2} - 6x + 9 $ x^4 - 10x^2 - 6x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.707500$ of defining polynomial
Character	$\chi$	$=$	6048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.96666 q^{5} +1.00000 q^{7} +O(q^{10})$	$q+1.96666 q^{5} +1.00000 q^{7} +3.38166 q^{11} +6.48057 q^{13} -5.06557 q^{17} -4.13225 q^{19} +7.09891 q^{23} -1.13225 q^{25} +6.48057 q^{29} +6.34832 q^{31} +1.96666 q^{35} -6.65057 q^{37} -8.44723 q^{41} +5.65057 q^{43} +3.71725 q^{47} +1.00000 q^{49} +1.17000 q^{53} +6.65057 q^{55} +10.5473 q^{59} +10.1311 q^{61} +12.7451 q^{65} +0.216067 q^{67} -8.16448 q^{71} -12.3472 q^{73} +3.38166 q^{77} -11.3106 q^{79} -2.48057 q^{83} -9.96225 q^{85} +12.5139 q^{89} +6.48057 q^{91} -8.12673 q^{95} +7.86664 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{5} + 4 q^{7}+O(q^{10}) $ 4 * q + 2 * q^5 + 4 * q^7	$ 4 q + 2 q^{5} + 4 q^{7} + 2 q^{11} - 4 q^{13} + 4 q^{17} - 4 q^{19} + 10 q^{23} + 8 q^{25} - 4 q^{29} + 8 q^{31} + 2 q^{35} - 8 q^{37} + 2 q^{41} + 4 q^{43} + 8 q^{47} + 4 q^{49} + 16 q^{53} + 8 q^{55} + 24 q^{59} - 8 q^{61} - 4 q^{65} - 4 q^{67} + 10 q^{71} + 4 q^{73} + 2 q^{77} - 4 q^{79} + 20 q^{83} - 16 q^{85} + 26 q^{89} - 4 q^{91} + 34 q^{95} + 8 q^{97}+O(q^{100}) $ 4 * q + 2 * q^5 + 4 * q^7 + 2 * q^11 - 4 * q^13 + 4 * q^17 - 4 * q^19 + 10 * q^23 + 8 * q^25 - 4 * q^29 + 8 * q^31 + 2 * q^35 - 8 * q^37 + 2 * q^41 + 4 * q^43 + 8 * q^47 + 4 * q^49 + 16 * q^53 + 8 * q^55 + 24 * q^59 - 8 * q^61 - 4 * q^65 - 4 * q^67 + 10 * q^71 + 4 * q^73 + 2 * q^77 - 4 * q^79 + 20 * q^83 - 16 * q^85 + 26 * q^89 - 4 * q^91 + 34 * q^95 + 8 * 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
$5$	1.96666	0.879517	0.439758	−	0.898116i	$-0.355064\pi$
$5$	1.96666	0.879517	0.439758	+	0.898116i	$0.355064\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.38166	1.01961	0.509804	−	0.860290i	$-0.329718\pi$
$11$	3.38166	1.01961	0.509804	+	0.860290i	$0.329718\pi$
$12$	0	0
$13$	6.48057	1.79739	0.898693	−	0.438578i	$-0.144518\pi$
$13$	6.48057	1.79739	0.898693	+	0.438578i	$0.144518\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.06557	−1.22858	−0.614291	−	0.789080i	$-0.710558\pi$
$17$	−5.06557	−1.22858	−0.614291	+	0.789080i	$0.710558\pi$
$18$	0	0
$19$	−4.13225	−0.948003	−0.474002	−	0.880524i	$-0.657191\pi$
$19$	−4.13225	−0.948003	−0.474002	+	0.880524i	$0.657191\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.09891	1.48023	0.740113	−	0.672483i	$-0.234772\pi$
$23$	7.09891	1.48023	0.740113	+	0.672483i	$0.234772\pi$
$24$	0	0
$25$	−1.13225	−0.226450
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.48057	1.20341	0.601706	−	0.798718i	$-0.294488\pi$
$29$	6.48057	1.20341	0.601706	+	0.798718i	$0.294488\pi$
$30$	0	0
$31$	6.34832	1.14019	0.570096	−	0.821578i	$-0.306906\pi$
$31$	6.34832	1.14019	0.570096	+	0.821578i	$0.306906\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.96666	0.332426
$36$	0	0
$37$	−6.65057	−1.09335	−0.546674	−	0.837346i	$-0.684106\pi$
$37$	−6.65057	−1.09335	−0.546674	+	0.837346i	$0.684106\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.44723	−1.31924	−0.659618	−	0.751601i	$-0.729282\pi$
$41$	−8.44723	−1.31924	−0.659618	+	0.751601i	$0.729282\pi$
$42$	0	0
$43$	5.65057	0.861704	0.430852	−	0.902423i	$-0.358213\pi$
$43$	5.65057	0.861704	0.430852	+	0.902423i	$0.358213\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.71725	0.542217	0.271108	−	0.962549i	$-0.412610\pi$
$47$	3.71725	0.542217	0.271108	+	0.962549i	$0.412610\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.17000	0.160712	0.0803560	−	0.996766i	$-0.474394\pi$
$53$	1.17000	0.160712	0.0803560	+	0.996766i	$0.474394\pi$
$54$	0	0
$55$	6.65057	0.896763
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.5473	1.37314	0.686568	−	0.727066i	$-0.259117\pi$
$59$	10.5473	1.37314	0.686568	+	0.727066i	$0.259117\pi$
$60$	0	0
$61$	10.1311	1.29716	0.648580	−	0.761147i	$-0.275363\pi$
$61$	10.1311	1.29716	0.648580	+	0.761147i	$0.275363\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	12.7451	1.58083
$66$	0	0
$67$	0.216067	0.0263968	0.0131984	−	0.999913i	$-0.495799\pi$
$67$	0.216067	0.0263968	0.0131984	+	0.999913i	$0.495799\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.16448	−0.968946	−0.484473	−	0.874806i	$-0.660989\pi$
$71$	−8.16448	−0.968946	−0.484473	+	0.874806i	$0.660989\pi$
$72$	0	0
$73$	−12.3472	−1.44513	−0.722566	−	0.691302i	$-0.757037\pi$
$73$	−12.3472	−1.44513	−0.722566	+	0.691302i	$0.757037\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.38166	0.385376
$78$	0	0
$79$	−11.3106	−1.27254	−0.636269	−	0.771467i	$-0.719523\pi$
$79$	−11.3106	−1.27254	−0.636269	+	0.771467i	$0.719523\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.48057	−0.272278	−0.136139	−	0.990690i	$-0.543469\pi$
$83$	−2.48057	−0.272278	−0.136139	+	0.990690i	$0.543469\pi$
$84$	0	0
$85$	−9.96225	−1.08056
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.5139	1.32647	0.663236	−	0.748410i	$-0.269183\pi$
$89$	12.5139	1.32647	0.663236	+	0.748410i	$0.269183\pi$
$90$	0	0
$91$	6.48057	0.679348
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−8.12673	−0.833785
$96$	0	0
$97$	7.86664	0.798736	0.399368	−	0.916791i	$-0.369230\pi$
$97$	7.86664	0.798736	0.399368	+	0.916791i	$0.369230\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.19893	−0.119298	−0.0596491	−	0.998219i	$-0.518998\pi$
$101$	−1.19893	−0.119298	−0.0596491	+	0.998219i	$0.518998\pi$
$102$	0	0
$103$	19.3095	1.90262	0.951309	−	0.308240i	$-0.0997400\pi$
$103$	19.3095	1.90262	0.951309	+	0.308240i	$0.0997400\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.86775	0.663930	0.331965	−	0.943292i	$-0.392288\pi$
$107$	6.86775	0.663930	0.331965	+	0.943292i	$0.392288\pi$
$108$	0	0
$109$	−6.34832	−0.608059	−0.304029	−	0.952663i	$-0.598332\pi$
$109$	−6.34832	−0.608059	−0.304029	+	0.952663i	$0.598332\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−15.9333	−1.49888	−0.749440	−	0.662072i	$-0.769677\pi$
$113$	−15.9333	−1.49888	−0.749440	+	0.662072i	$0.769677\pi$
$114$	0	0
$115$	13.9611	1.30188
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.06557	−0.464360
$120$	0	0
$121$	0.435615	0.0396014
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.0600	−1.07868
$126$	0	0
$127$	−3.65057	−0.323936	−0.161968	−	0.986796i	$-0.551784\pi$
$127$	−3.65057	−0.323936	−0.161968	+	0.986796i	$0.551784\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.41389	−0.210902	−0.105451	−	0.994424i	$-0.533629\pi$
$131$	−2.41389	−0.210902	−0.105451	+	0.994424i	$0.533629\pi$
$132$	0	0
$133$	−4.13225	−0.358312
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.9151	−1.18884	−0.594422	−	0.804153i	$-0.702619\pi$
$137$	−13.9151	−1.18884	−0.594422	+	0.804153i	$0.702619\pi$
$138$	0	0
$139$	−6.69664	−0.568001	−0.284001	−	0.958824i	$-0.591662\pi$
$139$	−6.69664	−0.568001	−0.284001	+	0.958824i	$0.591662\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	21.9151	1.83263
$144$	0	0
$145$	12.7451	1.05842
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.8945	1.05636	0.528178	−	0.849134i	$-0.322876\pi$
$149$	12.8945	1.05636	0.528178	+	0.849134i	$0.322876\pi$
$150$	0	0
$151$	−16.1311	−1.31273	−0.656367	−	0.754442i	$-0.727908\pi$
$151$	−16.1311	−1.31273	−0.656367	+	0.754442i	$0.727908\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	12.4850	1.00282
$156$	0	0
$157$	−15.5172	−1.23841	−0.619204	−	0.785230i	$-0.712545\pi$
$157$	−15.5172	−1.23841	−0.619204	+	0.785230i	$0.712545\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.09891	0.559473
$162$	0	0
$163$	−7.04607	−0.551890	−0.275945	−	0.961173i	$-0.588991\pi$
$163$	−7.04607	−0.551890	−0.275945	+	0.961173i	$0.588991\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.6300	1.75116	0.875579	−	0.483075i	$-0.160480\pi$
$167$	22.6300	1.75116	0.875579	+	0.483075i	$0.160480\pi$
$168$	0	0
$169$	28.9978	2.23060
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−7.09891	−0.539720	−0.269860	−	0.962900i	$-0.586977\pi$
$173$	−7.09891	−0.539720	−0.269860	+	0.962900i	$0.586977\pi$
$174$	0	0
$175$	−1.13225	−0.0855901
$176$	0	0
$177$	0	0
$178$	0	0
$179$	19.2634	1.43981	0.719907	−	0.694071i	$-0.244185\pi$
$179$	19.2634	1.43981	0.719907	+	0.694071i	$0.244185\pi$
$180$	0	0
$181$	−15.5266	−1.15409	−0.577043	−	0.816714i	$-0.695793\pi$
$181$	−15.5266	−1.15409	−0.577043	+	0.816714i	$0.695793\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−13.0794	−0.961617
$186$	0	0
$187$	−17.1300	−1.25267
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.99559	0.433826	0.216913	−	0.976191i	$-0.430401\pi$
$191$	5.99559	0.433826	0.216913	+	0.976191i	$0.430401\pi$
$192$	0	0
$193$	16.6966	1.20185	0.600925	−	0.799305i	$-0.294799\pi$
$193$	16.6966	1.20185	0.600925	+	0.799305i	$0.294799\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.8945	1.20368	0.601840	−	0.798617i	$-0.294434\pi$
$197$	16.8945	1.20368	0.601840	+	0.798617i	$0.294434\pi$
$198$	0	0
$199$	−11.6117	−0.823132	−0.411566	−	0.911380i	$-0.635018\pi$
$199$	−11.6117	−0.823132	−0.411566	+	0.911380i	$0.635018\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.48057	0.454847
$204$	0	0
$205$	−16.6128	−1.16029
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−13.9739	−0.966592
$210$	0	0
$211$	−18.8278	−1.29616	−0.648079	−	0.761573i	$-0.724427\pi$
$211$	−18.8278	−1.29616	−0.648079	+	0.761573i	$0.724427\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	11.1127	0.757883
$216$	0	0
$217$	6.34832	0.430952
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−32.8278	−2.20824
$222$	0	0
$223$	−8.65057	−0.579285	−0.289643	−	0.957135i	$-0.593536\pi$
$223$	−8.65057	−0.579285	−0.289643	+	0.957135i	$0.593536\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.8945	−0.855835	−0.427918	−	0.903818i	$-0.640753\pi$
$227$	−12.8945	−0.855835	−0.427918	+	0.903818i	$0.640753\pi$
$228$	0	0
$229$	−26.1311	−1.72679	−0.863397	−	0.504525i	$-0.831668\pi$
$229$	−26.1311	−1.72679	−0.863397	+	0.504525i	$0.831668\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.49881	0.556776	0.278388	−	0.960469i	$-0.410200\pi$
$233$	8.49881	0.556776	0.278388	+	0.960469i	$0.410200\pi$
$234$	0	0
$235$	7.31057	0.476889
$236$	0	0
$237$	0	0
$238$	0	0
$239$	26.0934	1.68784	0.843921	−	0.536468i	$-0.180242\pi$
$239$	26.0934	1.68784	0.843921	+	0.536468i	$0.180242\pi$
$240$	0	0
$241$	−13.5172	−0.870720	−0.435360	−	0.900256i	$-0.643379\pi$
$241$	−13.5172	−0.870720	−0.435360	+	0.900256i	$0.643379\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.96666	0.125645
$246$	0	0
$247$	−26.7793	−1.70393
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.39787	0.151352	0.0756760	−	0.997132i	$-0.475889\pi$
$251$	2.39787	0.151352	0.0756760	+	0.997132i	$0.475889\pi$
$252$	0	0
$253$	24.0061	1.50925
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.3428	1.01944	0.509718	−	0.860342i	$-0.329750\pi$
$257$	16.3428	1.01944	0.509718	+	0.860342i	$0.329750\pi$
$258$	0	0
$259$	−6.65057	−0.413246
$260$	0	0
$261$	0	0
$262$	0	0
$263$	30.8900	1.90476	0.952381	−	0.304911i	$-0.0986266\pi$
$263$	30.8900	1.90476	0.952381	+	0.304911i	$0.0986266\pi$
$264$	0	0
$265$	2.30099	0.141349
$266$	0	0
$267$	0	0
$268$	0	0
$269$	20.1934	1.23121	0.615607	−	0.788053i	$-0.288911\pi$
$269$	20.1934	1.23121	0.615607	+	0.788053i	$0.288911\pi$
$270$	0	0
$271$	23.2256	1.41086	0.705429	−	0.708781i	$-0.250755\pi$
$271$	23.2256	1.41086	0.705429	+	0.708781i	$0.250755\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.82889	−0.230891
$276$	0	0
$277$	1.08270	0.0650534	0.0325267	−	0.999471i	$-0.489645\pi$
$277$	1.08270	0.0650534	0.0325267	+	0.999471i	$0.489645\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.58611	−0.0946195	−0.0473098	−	0.998880i	$-0.515065\pi$
$281$	−1.58611	−0.0946195	−0.0473098	+	0.998880i	$0.515065\pi$
$282$	0	0
$283$	8.96114	0.532684	0.266342	−	0.963879i	$-0.414185\pi$
$283$	8.96114	0.532684	0.266342	+	0.963879i	$0.414185\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.44723	−0.498624
$288$	0	0
$289$	8.66000	0.509412
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9.49770	0.554862	0.277431	−	0.960746i	$-0.410517\pi$
$293$	9.49770	0.554862	0.277431	+	0.960746i	$0.410517\pi$
$294$	0	0
$295$	20.7428	1.20770
$296$	0	0
$297$	0	0
$298$	0	0
$299$	46.0050	2.66054
$300$	0	0
$301$	5.65057	0.325693
$302$	0	0
$303$	0	0
$304$	0	0
$305$	19.9245	1.14087
$306$	0	0
$307$	−5.86775	−0.334890	−0.167445	−	0.985881i	$-0.553552\pi$
$307$	−5.86775	−0.334890	−0.167445	+	0.985881i	$0.553552\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	22.2138	1.25963	0.629816	−	0.776745i	$-0.283130\pi$
$311$	22.2138	1.25963	0.629816	+	0.776745i	$0.283130\pi$
$312$	0	0
$313$	13.5194	0.764163	0.382082	−	0.924129i	$-0.375207\pi$
$313$	13.5194	0.764163	0.382082	+	0.924129i	$0.375207\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.11052	−0.511698	−0.255849	−	0.966717i	$-0.582355\pi$
$317$	−9.11052	−0.511698	−0.255849	+	0.966717i	$0.582355\pi$
$318$	0	0
$319$	21.9151	1.22701
$320$	0	0
$321$	0	0
$322$	0	0
$323$	20.9322	1.16470
$324$	0	0
$325$	−7.33763	−0.407019
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.71725	0.204939
$330$	0	0
$331$	26.6966	1.46738	0.733690	−	0.679484i	$-0.237796\pi$
$331$	26.6966	1.46738	0.733690	+	0.679484i	$0.237796\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.424930	0.0232164
$336$	0	0
$337$	15.9611	0.869459	0.434729	−	0.900561i	$-0.356844\pi$
$337$	15.9611	0.869459	0.434729	+	0.900561i	$0.356844\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	21.4678	1.16255
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−34.2361	−1.83789	−0.918946	−	0.394383i	$-0.870958\pi$
$347$	−34.2361	−1.83789	−0.918946	+	0.394383i	$0.870958\pi$
$348$	0	0
$349$	14.9611	0.800851	0.400426	−	0.916329i	$-0.368862\pi$
$349$	14.9611	0.800851	0.400426	+	0.916329i	$0.368862\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−30.6450	−1.63107	−0.815536	−	0.578707i	$-0.803558\pi$
$353$	−30.6450	−1.63107	−0.815536	+	0.578707i	$0.803558\pi$
$354$	0	0
$355$	−16.0568	−0.852204
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.3023	0.543732	0.271866	−	0.962335i	$-0.412359\pi$
$359$	10.3023	0.543732	0.271866	+	0.962335i	$0.412359\pi$
$360$	0	0
$361$	−1.92450	−0.101289
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−24.2827	−1.27102
$366$	0	0
$367$	−4.38607	−0.228951	−0.114475	−	0.993426i	$-0.536519\pi$
$367$	−4.38607	−0.228951	−0.114475	+	0.993426i	$0.536519\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.17000	0.0607434
$372$	0	0
$373$	−12.0838	−0.625676	−0.312838	−	0.949806i	$-0.601280\pi$
$373$	−12.0838	−0.625676	−0.312838	+	0.949806i	$0.601280\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	41.9978	2.16300
$378$	0	0
$379$	−22.6117	−1.16149	−0.580743	−	0.814087i	$-0.697238\pi$
$379$	−22.6117	−1.16149	−0.580743	+	0.814087i	$0.697238\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.02782	−0.256910	−0.128455	−	0.991715i	$-0.541002\pi$
$383$	−5.02782	−0.256910	−0.128455	+	0.991715i	$0.541002\pi$
$384$	0	0
$385$	6.65057	0.338944
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7.39550	−0.374966	−0.187483	−	0.982268i	$-0.560033\pi$
$389$	−7.39550	−0.374966	−0.187483	+	0.982268i	$0.560033\pi$
$390$	0	0
$391$	−35.9600	−1.81858
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−22.2440	−1.11922
$396$	0	0
$397$	−28.5268	−1.43172	−0.715859	−	0.698245i	$-0.753965\pi$
$397$	−28.5268	−1.43172	−0.715859	+	0.698245i	$0.753965\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.8762	−1.14238	−0.571192	−	0.820817i	$-0.693519\pi$
$401$	−22.8762	−1.14238	−0.571192	+	0.820817i	$0.693519\pi$
$402$	0	0
$403$	41.1407	2.04936
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−22.4900	−1.11479
$408$	0	0
$409$	37.0096	1.83001	0.915003	−	0.403447i	$-0.132188\pi$
$409$	37.0096	1.83001	0.915003	+	0.403447i	$0.132188\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	10.5473	0.518996
$414$	0	0
$415$	−4.87843	−0.239473
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7.99118	0.390395	0.195197	−	0.980764i	$-0.437465\pi$
$419$	7.99118	0.390395	0.195197	+	0.980764i	$0.437465\pi$
$420$	0	0
$421$	−17.6139	−0.858451	−0.429225	−	0.903197i	$-0.641213\pi$
$421$	−17.6139	−0.858451	−0.429225	+	0.903197i	$0.641213\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.73550	0.278212
$426$	0	0
$427$	10.1311	0.490280
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7.09891	−0.341942	−0.170971	−	0.985276i	$-0.554691\pi$
$431$	−7.09891	−0.341942	−0.170971	+	0.985276i	$0.554691\pi$
$432$	0	0
$433$	−5.87606	−0.282386	−0.141193	−	0.989982i	$-0.545094\pi$
$433$	−5.87606	−0.282386	−0.141193	+	0.989982i	$0.545094\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−29.3345	−1.40326
$438$	0	0
$439$	−2.34000	−0.111682	−0.0558411	−	0.998440i	$-0.517784\pi$
$439$	−2.34000	−0.111682	−0.0558411	+	0.998440i	$0.517784\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.3516	0.586843	0.293421	−	0.955983i	$-0.405206\pi$
$443$	12.3516	0.586843	0.293421	+	0.955983i	$0.405206\pi$
$444$	0	0
$445$	24.6106	1.16665
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−29.9890	−1.41527	−0.707633	−	0.706580i	$-0.750237\pi$
$449$	−29.9890	−1.41527	−0.707633	+	0.706580i	$0.750237\pi$
$450$	0	0
$451$	−28.5656	−1.34510
$452$	0	0
$453$	0	0
$454$	0	0
$455$	12.7451	0.597498
$456$	0	0
$457$	−2.73327	−0.127857	−0.0639286	−	0.997954i	$-0.520363\pi$
$457$	−2.73327	−0.127857	−0.0639286	+	0.997954i	$0.520363\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	29.7867	1.38731	0.693653	−	0.720309i	$-0.256000\pi$
$461$	29.7867	1.38731	0.693653	+	0.720309i	$0.256000\pi$
$462$	0	0
$463$	−18.9611	−0.881199	−0.440599	−	0.897704i	$-0.645234\pi$
$463$	−18.9611	−0.881199	−0.440599	+	0.897704i	$0.645234\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.3384	−0.478404	−0.239202	−	0.970970i	$-0.576886\pi$
$467$	−10.3384	−0.478404	−0.239202	+	0.970970i	$0.576886\pi$
$468$	0	0
$469$	0.216067	0.00997704
$470$	0	0
$471$	0	0
$472$	0	0
$473$	19.1083	0.878601
$474$	0	0
$475$	4.67875	0.214676
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−28.3956	−1.29743	−0.648715	−	0.761031i	$-0.724693\pi$
$479$	−28.3956	−1.29743	−0.648715	+	0.761031i	$0.724693\pi$
$480$	0	0
$481$	−43.0995	−1.96517
$482$	0	0
$483$	0	0
$484$	0	0
$485$	15.4710	0.702502
$486$	0	0
$487$	−5.44171	−0.246587	−0.123294	−	0.992370i	$-0.539346\pi$
$487$	−5.44171	−0.246587	−0.123294	+	0.992370i	$0.539346\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17.4750	0.788638	0.394319	−	0.918974i	$-0.370981\pi$
$491$	17.4750	0.788638	0.394319	+	0.918974i	$0.370981\pi$
$492$	0	0
$493$	−32.8278	−1.47849
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8.16448	−0.366227
$498$	0	0
$499$	17.7084	0.792738	0.396369	−	0.918091i	$-0.370270\pi$
$499$	17.7084	0.792738	0.396369	+	0.918091i	$0.370270\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	11.5172	0.513527	0.256763	−	0.966474i	$-0.417344\pi$
$503$	11.5172	0.513527	0.256763	+	0.966474i	$0.417344\pi$
$504$	0	0
$505$	−2.35789	−0.104925
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−19.6288	−0.870033	−0.435017	−	0.900422i	$-0.643257\pi$
$509$	−19.6288	−0.870033	−0.435017	+	0.900422i	$0.643257\pi$
$510$	0	0
$511$	−12.3472	−0.546208
$512$	0	0
$513$	0	0
$514$	0	0
$515$	37.9751	1.67338
$516$	0	0
$517$	12.5705	0.552849
$518$	0	0
$519$	0	0
$520$	0	0
$521$	30.0506	1.31654	0.658271	−	0.752781i	$-0.271288\pi$
$521$	30.0506	1.31654	0.658271	+	0.752781i	$0.271288\pi$
$522$	0	0
$523$	6.56564	0.287096	0.143548	−	0.989643i	$-0.454149\pi$
$523$	6.56564	0.287096	0.143548	+	0.989643i	$0.454149\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−32.1578	−1.40082
$528$	0	0
$529$	27.3945	1.19107
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−54.7428	−2.37118
$534$	0	0
$535$	13.5065	0.583938
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.38166	0.145658
$540$	0	0
$541$	2.91507	0.125329	0.0626644	−	0.998035i	$-0.480040\pi$
$541$	2.91507	0.125329	0.0626644	+	0.998035i	$0.480040\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−12.4850	−0.534798
$546$	0	0
$547$	16.4321	0.702587	0.351294	−	0.936265i	$-0.385742\pi$
$547$	16.4321	0.702587	0.351294	+	0.936265i	$0.385742\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−26.7793	−1.14084
$552$	0	0
$553$	−11.3106	−0.480974
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11.9333	0.505631	0.252815	−	0.967515i	$-0.418643\pi$
$557$	11.9333	0.505631	0.252815	+	0.967515i	$0.418643\pi$
$558$	0	0
$559$	36.6189	1.54881
$560$	0	0
$561$	0	0
$562$	0	0
$563$	15.5751	0.656411	0.328205	−	0.944606i	$-0.393556\pi$
$563$	15.5751	0.656411	0.328205	+	0.944606i	$0.393556\pi$
$564$	0	0
$565$	−31.3354	−1.31829
$566$	0	0
$567$	0	0
$568$	0	0
$569$	42.8922	1.79814	0.899068	−	0.437809i	$-0.144245\pi$
$569$	42.8922	1.79814	0.899068	+	0.437809i	$0.144245\pi$
$570$	0	0
$571$	−23.0945	−0.966475	−0.483237	−	0.875489i	$-0.660539\pi$
$571$	−23.0945	−0.966475	−0.483237	+	0.875489i	$0.660539\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8.03775	−0.335197
$576$	0	0
$577$	−17.2256	−0.717113	−0.358556	−	0.933508i	$-0.616731\pi$
$577$	−17.2256	−0.717113	−0.358556	+	0.933508i	$0.616731\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.48057	−0.102911
$582$	0	0
$583$	3.95654	0.163863
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0.200044	0.00825671	0.00412835	−	0.999991i	$-0.498686\pi$
$587$	0.200044	0.00825671	0.00412835	+	0.999991i	$0.498686\pi$
$588$	0	0
$589$	−26.2328	−1.08091
$590$	0	0
$591$	0	0
$592$	0	0
$593$	18.4872	0.759178	0.379589	−	0.925155i	$-0.376065\pi$
$593$	18.4872	0.759178	0.379589	+	0.925155i	$0.376065\pi$
$594$	0	0
$595$	−9.96225	−0.408412
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.37223	0.219503	0.109752	−	0.993959i	$-0.464994\pi$
$599$	5.37223	0.219503	0.109752	+	0.993959i	$0.464994\pi$
$600$	0	0
$601$	8.34943	0.340580	0.170290	−	0.985394i	$-0.445529\pi$
$601$	8.34943	0.340580	0.170290	+	0.985394i	$0.445529\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0.856707	0.0348301
$606$	0	0
$607$	25.0366	1.01621	0.508103	−	0.861296i	$-0.330347\pi$
$607$	25.0366	1.01621	0.508103	+	0.861296i	$0.330347\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	24.0899	0.974573
$612$	0	0
$613$	10.6106	0.428558	0.214279	−	0.976772i	$-0.431260\pi$
$613$	10.6106	0.428558	0.214279	+	0.976772i	$0.431260\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−30.5473	−1.22979	−0.614893	−	0.788611i	$-0.710801\pi$
$617$	−30.5473	−1.22979	−0.614893	+	0.788611i	$0.710801\pi$
$618$	0	0
$619$	−41.6189	−1.67281	−0.836403	−	0.548115i	$-0.815345\pi$
$619$	−41.6189	−1.67281	−0.836403	+	0.548115i	$0.815345\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.5139	0.501359
$624$	0	0
$625$	−18.0568	−0.722270
$626$	0	0
$627$	0	0
$628$	0	0
$629$	33.6889	1.34327
$630$	0	0
$631$	24.4806	0.974556	0.487278	−	0.873247i	$-0.337990\pi$
$631$	24.4806	0.974556	0.487278	+	0.873247i	$0.337990\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−7.17943	−0.284907
$636$	0	0
$637$	6.48057	0.256769
$638$	0	0
$639$	0	0
$640$	0	0
$641$	31.0256	1.22544	0.612719	−	0.790301i	$-0.290076\pi$
$641$	31.0256	1.22544	0.612719	+	0.790301i	$0.290076\pi$
$642$	0	0
$643$	41.7145	1.64506	0.822530	−	0.568722i	$-0.192562\pi$
$643$	41.7145	1.64506	0.822530	+	0.568722i	$0.192562\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.3194	0.838152	0.419076	−	0.907951i	$-0.362354\pi$
$647$	21.3194	0.838152	0.419076	+	0.907951i	$0.362354\pi$
$648$	0	0
$649$	35.6672	1.40006
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−32.5060	−1.27206	−0.636029	−	0.771665i	$-0.719424\pi$
$653$	−32.5060	−1.27206	−0.636029	+	0.771665i	$0.719424\pi$
$654$	0	0
$655$	−4.74729	−0.185492
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24.9083	0.970289	0.485145	−	0.874434i	$-0.338767\pi$
$659$	24.9083	0.970289	0.485145	+	0.874434i	$0.338767\pi$
$660$	0	0
$661$	−34.3472	−1.33595	−0.667976	−	0.744183i	$-0.732839\pi$
$661$	−34.3472	−1.33595	−0.667976	+	0.744183i	$0.732839\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.12673	−0.315141
$666$	0	0
$667$	46.0050	1.78132
$668$	0	0
$669$	0	0
$670$	0	0
$671$	34.2601	1.32259
$672$	0	0
$673$	−28.9589	−1.11628	−0.558142	−	0.829745i	$-0.688486\pi$
$673$	−28.9589	−1.11628	−0.558142	+	0.829745i	$0.688486\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18.5045	0.711185	0.355593	−	0.934641i	$-0.384279\pi$
$677$	18.5045	0.711185	0.355593	+	0.934641i	$0.384279\pi$
$678$	0	0
$679$	7.86664	0.301894
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−7.92830	−0.303368	−0.151684	−	0.988429i	$-0.548470\pi$
$683$	−7.92830	−0.303368	−0.151684	+	0.988429i	$0.548470\pi$
$684$	0	0
$685$	−27.3662	−1.04561
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7.58227	0.288862
$690$	0	0
$691$	−7.12877	−0.271191	−0.135596	−	0.990764i	$-0.543295\pi$
$691$	−7.12877	−0.271191	−0.135596	+	0.990764i	$0.543295\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−13.1700	−0.499567
$696$	0	0
$697$	42.7900	1.62079
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−43.7795	−1.65353	−0.826764	−	0.562549i	$-0.809821\pi$
$701$	−43.7795	−1.65353	−0.826764	+	0.562549i	$0.809821\pi$
$702$	0	0
$703$	27.4818	1.03650
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.19893	−0.0450905
$708$	0	0
$709$	0.877323	0.0329486	0.0164743	−	0.999864i	$-0.494756\pi$
$709$	0.877323	0.0329486	0.0164743	+	0.999864i	$0.494756\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	45.0661	1.68774
$714$	0	0
$715$	43.0995	1.61183
$716$	0	0
$717$	0	0
$718$	0	0
$719$	31.9040	1.18982	0.594910	−	0.803792i	$-0.297188\pi$
$719$	31.9040	1.18982	0.594910	+	0.803792i	$0.297188\pi$
$720$	0	0
$721$	19.3095	0.719122
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7.33763	−0.272513
$726$	0	0
$727$	−8.69886	−0.322623	−0.161311	−	0.986904i	$-0.551572\pi$
$727$	−8.69886	−0.322623	−0.161311	+	0.986904i	$0.551572\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−28.6234	−1.05867
$732$	0	0
$733$	43.5728	1.60940	0.804700	−	0.593682i	$-0.202326\pi$
$733$	43.5728	1.60940	0.804700	+	0.593682i	$0.202326\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.730664	0.0269144
$738$	0	0
$739$	−28.8300	−1.06053	−0.530264	−	0.847832i	$-0.677907\pi$
$739$	−28.8300	−1.06053	−0.530264	+	0.847832i	$0.677907\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−34.8611	−1.27893	−0.639465	−	0.768820i	$-0.720844\pi$
$743$	−34.8611	−1.27893	−0.639465	+	0.768820i	$0.720844\pi$
$744$	0	0
$745$	25.3590	0.929082
$746$	0	0
$747$	0	0
$748$	0	0
$749$	6.86775	0.250942
$750$	0	0
$751$	−38.9495	−1.42129	−0.710644	−	0.703552i	$-0.751596\pi$
$751$	−38.9495	−1.42129	−0.710644	+	0.703552i	$0.751596\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−31.7245	−1.15457
$756$	0	0
$757$	20.3544	0.739794	0.369897	−	0.929073i	$-0.379393\pi$
$757$	20.3544	0.739794	0.369897	+	0.929073i	$0.379393\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−15.9534	−0.578311	−0.289156	−	0.957282i	$-0.593375\pi$
$761$	−15.9534	−0.578311	−0.289156	+	0.957282i	$0.593375\pi$
$762$	0	0
$763$	−6.34832	−0.229825
$764$	0	0
$765$	0	0
$766$	0	0
$767$	68.3522	2.46805
$768$	0	0
$769$	48.5362	1.75026	0.875130	−	0.483887i	$-0.160776\pi$
$769$	48.5362	1.75026	0.875130	+	0.483887i	$0.160776\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−28.0223	−1.00789	−0.503946	−	0.863735i	$-0.668119\pi$
$773$	−28.0223	−1.00789	−0.503946	+	0.863735i	$0.668119\pi$
$774$	0	0
$775$	−7.18789	−0.258197
$776$	0	0
$777$	0	0
$778$	0	0
$779$	34.9061	1.25064
$780$	0	0
$781$	−27.6095	−0.987945
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−30.5171	−1.08920
$786$	0	0
$787$	12.3378	0.439794	0.219897	−	0.975523i	$-0.429428\pi$
$787$	12.3378	0.439794	0.219897	+	0.975523i	$0.429428\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−15.9333	−0.566524
$792$	0	0
$793$	65.6556	2.33150
$794$	0	0
$795$	0	0
$796$	0	0
$797$	38.1258	1.35048	0.675242	−	0.737596i	$-0.264039\pi$
$797$	38.1258	1.35048	0.675242	+	0.737596i	$0.264039\pi$
$798$	0	0
$799$	−18.8300	−0.666157
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−41.7540	−1.47347
$804$	0	0
$805$	13.9611	0.492065
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−37.4395	−1.31630	−0.658151	−	0.752886i	$-0.728661\pi$
$809$	−37.4395	−1.31630	−0.658151	+	0.752886i	$0.728661\pi$
$810$	0	0
$811$	−18.4901	−0.649277	−0.324638	−	0.945838i	$-0.605243\pi$
$811$	−18.4901	−0.649277	−0.324638	+	0.945838i	$0.605243\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−13.8572	−0.485397
$816$	0	0
$817$	−23.3496	−0.816898
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−14.8850	−0.519491	−0.259746	−	0.965677i	$-0.583639\pi$
$821$	−14.8850	−0.519491	−0.259746	+	0.965677i	$0.583639\pi$
$822$	0	0
$823$	1.47100	0.0512757	0.0256378	−	0.999671i	$-0.491838\pi$
$823$	1.47100	0.0512757	0.0256378	+	0.999671i	$0.491838\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−37.3027	−1.29714	−0.648571	−	0.761155i	$-0.724633\pi$
$827$	−37.3027	−1.29714	−0.648571	+	0.761155i	$0.724633\pi$
$828$	0	0
$829$	−8.04844	−0.279534	−0.139767	−	0.990184i	$-0.544635\pi$
$829$	−8.04844	−0.279534	−0.139767	+	0.990184i	$0.544635\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−5.06557	−0.175512
$834$	0	0
$835$	44.5054	1.54017
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−10.4806	−0.361829	−0.180915	−	0.983499i	$-0.557906\pi$
$839$	−10.4806	−0.361829	−0.180915	+	0.983499i	$0.557906\pi$
$840$	0	0
$841$	12.9978	0.448199
$842$	0	0
$843$	0	0
$844$	0	0
$845$	57.0287	1.96185
$846$	0	0
$847$	0.435615	0.0149679
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−47.2118	−1.61840
$852$	0	0
$853$	−28.3956	−0.972248	−0.486124	−	0.873890i	$-0.661590\pi$
$853$	−28.3956	−0.972248	−0.486124	+	0.873890i	$0.661590\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−26.5784	−0.907900	−0.453950	−	0.891027i	$-0.649985\pi$
$857$	−26.5784	−0.907900	−0.453950	+	0.891027i	$0.649985\pi$
$858$	0	0
$859$	−4.47448	−0.152667	−0.0763336	−	0.997082i	$-0.524321\pi$
$859$	−4.47448	−0.152667	−0.0763336	+	0.997082i	$0.524321\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−31.2954	−1.06531	−0.532655	−	0.846333i	$-0.678805\pi$
$863$	−31.2954	−1.06531	−0.532655	+	0.846333i	$0.678805\pi$
$864$	0	0
$865$	−13.9611	−0.474693
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−38.2485	−1.29749
$870$	0	0
$871$	1.40024	0.0474452
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−12.0600	−0.407704
$876$	0	0
$877$	−3.99778	−0.134995	−0.0674977	−	0.997719i	$-0.521502\pi$
$877$	−3.99778	−0.134995	−0.0674977	+	0.997719i	$0.521502\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	35.1439	1.18403	0.592013	−	0.805928i	$-0.298333\pi$
$881$	35.1439	1.18403	0.592013	+	0.805928i	$0.298333\pi$
$882$	0	0
$883$	−19.7911	−0.666025	−0.333012	−	0.942922i	$-0.608065\pi$
$883$	−19.7911	−0.666025	−0.333012	+	0.942922i	$0.608065\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−27.9128	−0.937222	−0.468611	−	0.883405i	$-0.655245\pi$
$887$	−27.9128	−0.937222	−0.468611	+	0.883405i	$0.655245\pi$
$888$	0	0
$889$	−3.65057	−0.122436
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−15.3606	−0.514023
$894$	0	0
$895$	37.8845	1.26634
$896$	0	0
$897$	0	0
$898$	0	0
$899$	41.1407	1.37212
$900$	0	0
$901$	−5.92672	−0.197448
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−30.5356	−1.01504
$906$	0	0
$907$	−48.6922	−1.61680	−0.808399	−	0.588635i	$-0.799665\pi$
$907$	−48.6922	−1.61680	−0.808399	+	0.588635i	$0.799665\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20.1667	0.668151	0.334076	−	0.942546i	$-0.391576\pi$
$911$	20.1667	0.668151	0.334076	+	0.942546i	$0.391576\pi$
$912$	0	0
$913$	−8.38844	−0.277617
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.41389	−0.0797136
$918$	0	0
$919$	51.9779	1.71459	0.857297	−	0.514823i	$-0.172142\pi$
$919$	51.9779	1.71459	0.857297	+	0.514823i	$0.172142\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−52.9105	−1.74157
$924$	0	0
$925$	7.53012	0.247589
$926$	0	0
$927$	0	0
$928$	0	0
$929$	9.15992	0.300527	0.150264	−	0.988646i	$-0.451988\pi$
$929$	9.15992	0.300527	0.150264	+	0.988646i	$0.451988\pi$
$930$	0	0
$931$	−4.13225	−0.135429
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−33.6889	−1.10175
$936$	0	0
$937$	50.9224	1.66356	0.831782	−	0.555103i	$-0.187321\pi$
$937$	50.9224	1.66356	0.831782	+	0.555103i	$0.187321\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−31.6568	−1.03198	−0.515992	−	0.856593i	$-0.672577\pi$
$941$	−31.6568	−1.03198	−0.515992	+	0.856593i	$0.672577\pi$
$942$	0	0
$943$	−59.9661	−1.95277
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−11.3804	−0.369813	−0.184907	−	0.982756i	$-0.559198\pi$
$947$	−11.3804	−0.369813	−0.184907	+	0.982756i	$0.559198\pi$
$948$	0	0
$949$	−80.0169	−2.59746
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22.7428	0.736713	0.368357	−	0.929685i	$-0.379921\pi$
$953$	22.7428	0.736713	0.368357	+	0.929685i	$0.379921\pi$
$954$	0	0
$955$	11.7913	0.381557
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−13.9151	−0.449341
$960$	0	0
$961$	9.30114	0.300037
$962$	0	0
$963$	0	0
$964$	0	0
$965$	32.8366	1.05705
$966$	0	0
$967$	15.5845	0.501164	0.250582	−	0.968095i	$-0.419378\pi$
$967$	15.5845	0.501164	0.250582	+	0.968095i	$0.419378\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12.6507	0.405981	0.202990	−	0.979181i	$-0.434934\pi$
$971$	12.6507	0.405981	0.202990	+	0.979181i	$0.434934\pi$
$972$	0	0
$973$	−6.69664	−0.214684
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−15.5679	−0.498060	−0.249030	−	0.968496i	$-0.580112\pi$
$977$	−15.5679	−0.498060	−0.249030	+	0.968496i	$0.580112\pi$
$978$	0	0
$979$	42.3178	1.35248
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.45497	−0.110197	−0.0550983	−	0.998481i	$-0.517547\pi$
$983$	−3.45497	−0.110197	−0.0550983	+	0.998481i	$0.517547\pi$
$984$	0	0
$985$	33.2256	1.05866
$986$	0	0
$987$	0	0
$988$	0	0
$989$	40.1129	1.27552
$990$	0	0
$991$	9.44393	0.299996	0.149998	−	0.988686i	$-0.452073\pi$
$991$	9.44393	0.299996	0.149998	+	0.988686i	$0.452073\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−22.8363	−0.723959
$996$	0	0
$997$	18.5290	0.586819	0.293410	−	0.955987i	$-0.405210\pi$
$997$	18.5290	0.586819	0.293410	+	0.955987i	$0.405210\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.a.bu.1.3	yes	4
3.2	odd	2		6048.2.a.bn.1.2	yes	4
4.3	odd	2		6048.2.a.bq.1.3	yes	4
12.11	even	2		6048.2.a.bl.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.a.bl.1.2	✓	4	12.11	even	2
6048.2.a.bn.1.2	yes	4	3.2	odd	2
6048.2.a.bq.1.3	yes	4	4.3	odd	2
6048.2.a.bu.1.3	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.a (trivial)

\(f(q)\)	\(=\)	\(q+1.96666 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+1.96666 q^{5} +1.00000 q^{7} +3.38166 q^{11} +6.48057 q^{13} -5.06557 q^{17} -4.13225 q^{19} +7.09891 q^{23} -1.13225 q^{25} +6.48057 q^{29} +6.34832 q^{31} +1.96666 q^{35} -6.65057 q^{37} -8.44723 q^{41} +5.65057 q^{43} +3.71725 q^{47} +1.00000 q^{49} +1.17000 q^{53} +6.65057 q^{55} +10.5473 q^{59} +10.1311 q^{61} +12.7451 q^{65} +0.216067 q^{67} -8.16448 q^{71} -12.3472 q^{73} +3.38166 q^{77} -11.3106 q^{79} -2.48057 q^{83} -9.96225 q^{85} +12.5139 q^{89} +6.48057 q^{91} -8.12673 q^{95} +7.86664 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5} + 4 q^{7}+O(q^{10}) \) 4 * q + 2 * q^5 + 4 * q^7	\( 4 q + 2 q^{5} + 4 q^{7} + 2 q^{11} - 4 q^{13} + 4 q^{17} - 4 q^{19} + 10 q^{23} + 8 q^{25} - 4 q^{29} + 8 q^{31} + 2 q^{35} - 8 q^{37} + 2 q^{41} + 4 q^{43} + 8 q^{47} + 4 q^{49} + 16 q^{53} + 8 q^{55} + 24 q^{59} - 8 q^{61} - 4 q^{65} - 4 q^{67} + 10 q^{71} + 4 q^{73} + 2 q^{77} - 4 q^{79} + 20 q^{83} - 16 q^{85} + 26 q^{89} - 4 q^{91} + 34 q^{95} + 8 q^{97}+O(q^{100}) \) 4 * q + 2 * q^5 + 4 * q^7 + 2 * q^11 - 4 * q^13 + 4 * q^17 - 4 * q^19 + 10 * q^23 + 8 * q^25 - 4 * q^29 + 8 * q^31 + 2 * q^35 - 8 * q^37 + 2 * q^41 + 4 * q^43 + 8 * q^47 + 4 * q^49 + 16 * q^53 + 8 * q^55 + 24 * q^59 - 8 * q^61 - 4 * q^65 - 4 * q^67 + 10 * q^71 + 4 * q^73 + 2 * q^77 - 4 * q^79 + 20 * q^83 - 16 * q^85 + 26 * q^89 - 4 * q^91 + 34 * q^95 + 8 * q^97

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