LMFDB - Embedded newform 6048.2.a.bt.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:	4.4.39528.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} - 2x + 12 $ x^4 - 9x^2 - 2x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.50948$ 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)

$f(q)$	$=$	$q+2.29751 q^{5} +1.00000 q^{7} +O(q^{10})$	$q+2.29751 q^{5} +1.00000 q^{7} +5.01897 q^{11} -4.55358 q^{13} +0.762893 q^{17} -3.01897 q^{19} -1.53461 q^{23} +0.278540 q^{25} +3.29751 q^{29} +3.29751 q^{31} +2.29751 q^{35} +4.72146 q^{37} +0.297507 q^{41} -3.83212 q^{43} +5.79069 q^{47} +1.00000 q^{49} +11.3857 q^{53} +11.5311 q^{55} -3.59501 q^{59} +2.42395 q^{61} -10.4619 q^{65} +4.12963 q^{67} -2.46539 q^{71} +16.6329 q^{73} +5.01897 q^{77} -5.36674 q^{79} +1.27854 q^{83} +1.75275 q^{85} +9.57255 q^{89} -4.55358 q^{91} -6.93610 q^{95} +14.0882 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 8 q^{13} + 2 q^{17} + 8 q^{19} + 14 q^{25} + 6 q^{29} + 6 q^{31} + 2 q^{35} + 6 q^{37} - 6 q^{41} - 2 q^{43} - 2 q^{47} + 4 q^{49} + 6 q^{53} - 12 q^{55} + 4 q^{61} + 4 q^{65} - 4 q^{67} - 16 q^{71} + 12 q^{73} - 2 q^{79} + 18 q^{83} + 22 q^{85} - 8 q^{89} + 8 q^{91} + 16 q^{95} + 24 q^{97}+O(q^{100}) $ 4 * q + 2 * q^5 + 4 * q^7 + 8 * q^13 + 2 * q^17 + 8 * q^19 + 14 * q^25 + 6 * q^29 + 6 * q^31 + 2 * q^35 + 6 * q^37 - 6 * q^41 - 2 * q^43 - 2 * q^47 + 4 * q^49 + 6 * q^53 - 12 * q^55 + 4 * q^61 + 4 * q^65 - 4 * q^67 - 16 * q^71 + 12 * q^73 - 2 * q^79 + 18 * q^83 + 22 * q^85 - 8 * q^89 + 8 * q^91 + 16 * q^95 + 24 * 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
$5$	2.29751	1.02748	0.513738	−	0.857947i	$-0.328260\pi$
$5$	2.29751	1.02748	0.513738	+	0.857947i	$0.328260\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.01897	1.51328	0.756638	−	0.653834i	$-0.226841\pi$
$11$	5.01897	1.51328	0.756638	+	0.653834i	$0.226841\pi$
$12$	0	0
$13$	−4.55358	−1.26294	−0.631468	−	0.775402i	$-0.717547\pi$
$13$	−4.55358	−1.26294	−0.631468	+	0.775402i	$0.717547\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.762893	0.185029	0.0925144	−	0.995711i	$-0.470510\pi$
$17$	0.762893	0.185029	0.0925144	+	0.995711i	$0.470510\pi$
$18$	0	0
$19$	−3.01897	−0.692599	−0.346299	−	0.938124i	$-0.612562\pi$
$19$	−3.01897	−0.692599	−0.346299	+	0.938124i	$0.612562\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.53461	−0.319989	−0.159995	−	0.987118i	$-0.551148\pi$
$23$	−1.53461	−0.319989	−0.159995	+	0.987118i	$0.551148\pi$
$24$	0	0
$25$	0.278540	0.0557081
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.29751	0.612332	0.306166	−	0.951978i	$-0.400954\pi$
$29$	3.29751	0.612332	0.306166	+	0.951978i	$0.400954\pi$
$30$	0	0
$31$	3.29751	0.592250	0.296125	−	0.955149i	$-0.404306\pi$
$31$	3.29751	0.592250	0.296125	+	0.955149i	$0.404306\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.29751	0.388350
$36$	0	0
$37$	4.72146	0.776203	0.388102	−	0.921617i	$-0.373131\pi$
$37$	4.72146	0.776203	0.388102	+	0.921617i	$0.373131\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.297507	0.0464629	0.0232314	−	0.999730i	$-0.492605\pi$
$41$	0.297507	0.0464629	0.0232314	+	0.999730i	$0.492605\pi$
$42$	0	0
$43$	−3.83212	−0.584393	−0.292197	−	0.956358i	$-0.594386\pi$
$43$	−3.83212	−0.584393	−0.292197	+	0.956358i	$0.594386\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.79069	0.844659	0.422329	−	0.906442i	$-0.361213\pi$
$47$	5.79069	0.844659	0.422329	+	0.906442i	$0.361213\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.3857	1.56395	0.781973	−	0.623312i	$-0.214213\pi$
$53$	11.3857	1.56395	0.781973	+	0.623312i	$0.214213\pi$
$54$	0	0
$55$	11.5311	1.55486
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.59501	−0.468031	−0.234016	−	0.972233i	$-0.575187\pi$
$59$	−3.59501	−0.468031	−0.234016	+	0.972233i	$0.575187\pi$
$60$	0	0
$61$	2.42395	0.310355	0.155178	−	0.987887i	$-0.450405\pi$
$61$	2.42395	0.310355	0.155178	+	0.987887i	$0.450405\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−10.4619	−1.29764
$66$	0	0
$67$	4.12963	0.504514	0.252257	−	0.967660i	$-0.418827\pi$
$67$	4.12963	0.504514	0.252257	+	0.967660i	$0.418827\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.46539	−0.292587	−0.146294	−	0.989241i	$-0.546734\pi$
$71$	−2.46539	−0.292587	−0.146294	+	0.989241i	$0.546734\pi$
$72$	0	0
$73$	16.6329	1.94674	0.973370	−	0.229241i	$-0.0736244\pi$
$73$	16.6329	1.94674	0.973370	+	0.229241i	$0.0736244\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.01897	0.571964
$78$	0	0
$79$	−5.36674	−0.603805	−0.301902	−	0.953339i	$-0.597622\pi$
$79$	−5.36674	−0.603805	−0.301902	+	0.953339i	$0.597622\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.27854	0.140338	0.0701690	−	0.997535i	$-0.477646\pi$
$83$	1.27854	0.140338	0.0701690	+	0.997535i	$0.477646\pi$
$84$	0	0
$85$	1.75275	0.190113
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.57255	1.01469	0.507344	−	0.861744i	$-0.330627\pi$
$89$	9.57255	1.01469	0.507344	+	0.861744i	$0.330627\pi$
$90$	0	0
$91$	−4.55358	−0.477345
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.93610	−0.711629
$96$	0	0
$97$	14.0882	1.43044	0.715220	−	0.698900i	$-0.246327\pi$
$97$	14.0882	1.43044	0.715220	+	0.698900i	$0.246327\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.0379	1.19782	0.598910	−	0.800817i	$-0.295601\pi$
$101$	12.0379	1.19782	0.598910	+	0.800817i	$0.295601\pi$
$102$	0	0
$103$	3.80433	0.374852	0.187426	−	0.982279i	$-0.439986\pi$
$103$	3.80433	0.374852	0.187426	+	0.982279i	$0.439986\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	10.7215	1.02693	0.513465	−	0.858111i	$-0.328362\pi$
$109$	10.7215	1.02693	0.513465	+	0.858111i	$0.328362\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−9.01897	−0.848433	−0.424217	−	0.905561i	$-0.639450\pi$
$113$	−9.01897	−0.848433	−0.424217	+	0.905561i	$0.639450\pi$
$114$	0	0
$115$	−3.52579	−0.328781
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.762893	0.0699343
$120$	0	0
$121$	14.1900	1.29000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.8476	−0.970238
$126$	0	0
$127$	−11.7404	−1.04179	−0.520897	−	0.853619i	$-0.674403\pi$
$127$	−11.7404	−1.04179	−0.520897	+	0.853619i	$0.674403\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−18.9494	−1.65562	−0.827809	−	0.561009i	$-0.810413\pi$
$131$	−18.9494	−1.65562	−0.827809	+	0.561009i	$0.810413\pi$
$132$	0	0
$133$	−3.01897	−0.261778
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−0.981033	−0.0838153	−0.0419077	−	0.999121i	$-0.513344\pi$
$137$	−0.981033	−0.0838153	−0.0419077	+	0.999121i	$0.513344\pi$
$138$	0	0
$139$	23.3162	1.97765	0.988825	−	0.149078i	$-0.0476307\pi$
$139$	23.3162	1.97765	0.988825	+	0.149078i	$0.0476307\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−22.8543	−1.91117
$144$	0	0
$145$	7.57605	0.629157
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−11.3354	−0.928636	−0.464318	−	0.885669i	$-0.653700\pi$
$149$	−11.3354	−0.928636	−0.464318	+	0.885669i	$0.653700\pi$
$150$	0	0
$151$	−7.82329	−0.636651	−0.318325	−	0.947982i	$-0.603120\pi$
$151$	−7.82329	−0.636651	−0.318325	+	0.947982i	$0.603120\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.57605	0.608523
$156$	0	0
$157$	3.36355	0.268441	0.134220	−	0.990952i	$-0.457147\pi$
$157$	3.36355	0.268441	0.134220	+	0.990952i	$0.457147\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.53461	−0.120945
$162$	0	0
$163$	−14.4651	−1.13299	−0.566496	−	0.824065i	$-0.691701\pi$
$163$	−14.4651	−1.13299	−0.566496	+	0.824065i	$0.691701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.4236	1.11613	0.558067	−	0.829796i	$-0.311543\pi$
$167$	14.4236	1.11613	0.558067	+	0.829796i	$0.311543\pi$
$168$	0	0
$169$	7.73510	0.595008
$170$	0	0
$171$	0	0
$172$	0	0
$173$	13.4429	1.02205	0.511023	−	0.859567i	$-0.329267\pi$
$173$	13.4429	1.02205	0.511023	+	0.859567i	$0.329267\pi$
$174$	0	0
$175$	0.278540	0.0210557
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.20900	0.464082	0.232041	−	0.972706i	$-0.425460\pi$
$179$	6.20900	0.464082	0.232041	+	0.972706i	$0.425460\pi$
$180$	0	0
$181$	4.93960	0.367158	0.183579	−	0.983005i	$-0.441232\pi$
$181$	4.93960	0.367158	0.183579	+	0.983005i	$0.441232\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.8476	0.797531
$186$	0	0
$187$	3.82894	0.280000
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.36705	0.0989163	0.0494582	−	0.998776i	$-0.484251\pi$
$191$	1.36705	0.0989163	0.0494582	+	0.998776i	$0.484251\pi$
$192$	0	0
$193$	−5.63295	−0.405469	−0.202734	−	0.979234i	$-0.564983\pi$
$193$	−5.63295	−0.405469	−0.202734	+	0.979234i	$0.564983\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.68321	−0.618653	−0.309327	−	0.950956i	$-0.600104\pi$
$197$	−8.68321	−0.618653	−0.309327	+	0.950956i	$0.600104\pi$
$198$	0	0
$199$	−2.75275	−0.195138	−0.0975688	−	0.995229i	$-0.531107\pi$
$199$	−2.75275	−0.195138	−0.0975688	+	0.995229i	$0.531107\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.29751	0.231440
$204$	0	0
$205$	0.683526	0.0477395
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−15.1521	−1.04809
$210$	0	0
$211$	−20.0468	−1.38008	−0.690038	−	0.723773i	$-0.742406\pi$
$211$	−20.0468	−1.38008	−0.690038	+	0.723773i	$0.742406\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.80433	−0.600450
$216$	0	0
$217$	3.29751	0.223849
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.47390	−0.233680
$222$	0	0
$223$	−23.1777	−1.55209	−0.776047	−	0.630675i	$-0.782778\pi$
$223$	−23.1777	−1.55209	−0.776047	+	0.630675i	$0.782778\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.53143	0.168017	0.0840084	−	0.996465i	$-0.473228\pi$
$227$	2.53143	0.168017	0.0840084	+	0.996465i	$0.473228\pi$
$228$	0	0
$229$	13.5258	0.893809	0.446905	−	0.894582i	$-0.352526\pi$
$229$	13.5258	0.893809	0.446905	+	0.894582i	$0.352526\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.1398	−1.12286	−0.561432	−	0.827523i	$-0.689749\pi$
$233$	−17.1398	−1.12286	−0.561432	+	0.827523i	$0.689749\pi$
$234$	0	0
$235$	13.3041	0.867867
$236$	0	0
$237$	0	0
$238$	0	0
$239$	29.9112	1.93479	0.967396	−	0.253267i	$-0.0815050\pi$
$239$	29.9112	1.93479	0.967396	+	0.253267i	$0.0815050\pi$
$240$	0	0
$241$	5.35472	0.344928	0.172464	−	0.985016i	$-0.444827\pi$
$241$	5.35472	0.344928	0.172464	+	0.985016i	$0.444827\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.29751	0.146782
$246$	0	0
$247$	13.7471	0.874708
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.53143	−0.0966630	−0.0483315	−	0.998831i	$-0.515390\pi$
$251$	−1.53143	−0.0966630	−0.0483315	+	0.998831i	$0.515390\pi$
$252$	0	0
$253$	−7.70218	−0.484232
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−25.7022	−1.60326	−0.801629	−	0.597822i	$-0.796033\pi$
$257$	−25.7022	−1.60326	−0.801629	+	0.597822i	$0.796033\pi$
$258$	0	0
$259$	4.72146	0.293377
$260$	0	0
$261$	0	0
$262$	0	0
$263$	22.6744	1.39816	0.699081	−	0.715042i	$-0.253593\pi$
$263$	22.6744	1.39816	0.699081	+	0.715042i	$0.253593\pi$
$264$	0	0
$265$	26.1587	1.60692
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−0.930456	−0.0567309	−0.0283655	−	0.999598i	$-0.509030\pi$
$269$	−0.930456	−0.0567309	−0.0283655	+	0.999598i	$0.509030\pi$
$270$	0	0
$271$	2.75275	0.167218	0.0836089	−	0.996499i	$-0.473355\pi$
$271$	2.75275	0.167218	0.0836089	+	0.996499i	$0.473355\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.39799	0.0843017
$276$	0	0
$277$	25.3544	1.52340	0.761699	−	0.647931i	$-0.224365\pi$
$277$	25.3544	1.52340	0.761699	+	0.647931i	$0.224365\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−16.6329	−0.992239	−0.496119	−	0.868254i	$-0.665242\pi$
$281$	−16.6329	−0.992239	−0.496119	+	0.868254i	$0.665242\pi$
$282$	0	0
$283$	−1.31679	−0.0782751	−0.0391375	−	0.999234i	$-0.512461\pi$
$283$	−1.31679	−0.0782751	−0.0391375	+	0.999234i	$0.512461\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.297507	0.0175613
$288$	0	0
$289$	−16.4180	−0.965764
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.24593	0.306471	0.153235	−	0.988190i	$-0.451031\pi$
$293$	5.24593	0.306471	0.153235	+	0.988190i	$0.451031\pi$
$294$	0	0
$295$	−8.25957	−0.480891
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.98799	0.404126
$300$	0	0
$301$	−3.83212	−0.220880
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.56905	0.318883
$306$	0	0
$307$	33.8230	1.93038	0.965190	−	0.261551i	$-0.0842339\pi$
$307$	33.8230	1.93038	0.965190	+	0.261551i	$0.0842339\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−26.5444	−1.50520	−0.752598	−	0.658480i	$-0.771200\pi$
$311$	−26.5444	−1.50520	−0.752598	+	0.658480i	$0.771200\pi$
$312$	0	0
$313$	6.10052	0.344822	0.172411	−	0.985025i	$-0.444844\pi$
$313$	6.10052	0.344822	0.172411	+	0.985025i	$0.444844\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.632949	−0.0355500	−0.0177750	−	0.999842i	$-0.505658\pi$
$317$	−0.632949	−0.0355500	−0.0177750	+	0.999842i	$0.505658\pi$
$318$	0	0
$319$	16.5501	0.926627
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.30315	−0.128151
$324$	0	0
$325$	−1.26836	−0.0703558
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.79069	0.319251
$330$	0	0
$331$	25.2750	1.38924	0.694621	−	0.719376i	$-0.255572\pi$
$331$	25.2750	1.38924	0.694621	+	0.719376i	$0.255572\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.48785	0.518377
$336$	0	0
$337$	20.0243	1.09079	0.545396	−	0.838178i	$-0.316379\pi$
$337$	20.0243	1.09079	0.545396	+	0.838178i	$0.316379\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	16.5501	0.896237
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	29.2333	1.56933	0.784663	−	0.619923i	$-0.212836\pi$
$347$	29.2333	1.56933	0.784663	+	0.619923i	$0.212836\pi$
$348$	0	0
$349$	−13.6105	−0.728552	−0.364276	−	0.931291i	$-0.618684\pi$
$349$	−13.6105	−0.728552	−0.364276	+	0.931291i	$0.618684\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9.35791	−0.498071	−0.249036	−	0.968494i	$-0.580114\pi$
$353$	−9.35791	−0.498071	−0.249036	+	0.968494i	$0.580114\pi$
$354$	0	0
$355$	−5.66424	−0.300627
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.4268	−1.50031	−0.750155	−	0.661262i	$-0.770021\pi$
$359$	−28.4268	−1.50031	−0.750155	+	0.661262i	$0.770021\pi$
$360$	0	0
$361$	−9.88584	−0.520307
$362$	0	0
$363$	0	0
$364$	0	0
$365$	38.2143	2.00023
$366$	0	0
$367$	−23.3857	−1.22072	−0.610362	−	0.792123i	$-0.708976\pi$
$367$	−23.3857	−1.22072	−0.610362	+	0.792123i	$0.708976\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	11.3857	0.591116
$372$	0	0
$373$	36.0702	1.86765	0.933823	−	0.357736i	$-0.116451\pi$
$373$	36.0702	1.86765	0.933823	+	0.357736i	$0.116451\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−15.0155	−0.773336
$378$	0	0
$379$	−32.4562	−1.66717	−0.833583	−	0.552395i	$-0.813714\pi$
$379$	−32.4562	−1.66717	−0.833583	+	0.552395i	$0.813714\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	20.4616	1.04554	0.522769	−	0.852475i	$-0.324899\pi$
$383$	20.4616	1.04554	0.522769	+	0.852475i	$0.324899\pi$
$384$	0	0
$385$	11.5311	0.587680
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−13.2782	−0.673233	−0.336616	−	0.941642i	$-0.609283\pi$
$389$	−13.2782	−0.673233	−0.336616	+	0.941642i	$0.609283\pi$
$390$	0	0
$391$	−1.17075	−0.0592072
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−12.3301	−0.620395
$396$	0	0
$397$	36.3351	1.82361	0.911804	−	0.410626i	$-0.134690\pi$
$397$	36.3351	1.82361	0.911804	+	0.410626i	$0.134690\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−30.1764	−1.50694	−0.753468	−	0.657484i	$-0.771621\pi$
$401$	−30.1764	−1.50694	−0.753468	+	0.657484i	$0.771621\pi$
$402$	0	0
$403$	−15.0155	−0.747974
$404$	0	0
$405$	0	0
$406$	0	0
$407$	23.6968	1.17461
$408$	0	0
$409$	−4.15874	−0.205636	−0.102818	−	0.994700i	$-0.532786\pi$
$409$	−4.15874	−0.205636	−0.102818	+	0.994700i	$0.532786\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.59501	−0.176899
$414$	0	0
$415$	2.93746	0.144194
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−23.1764	−1.13224	−0.566120	−	0.824323i	$-0.691556\pi$
$419$	−23.1764	−1.13224	−0.566120	+	0.824323i	$0.691556\pi$
$420$	0	0
$421$	32.9301	1.60492	0.802458	−	0.596708i	$-0.203525\pi$
$421$	32.9301	1.60492	0.802458	+	0.596708i	$0.203525\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.212497	0.0103076
$426$	0	0
$427$	2.42395	0.117303
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−22.0379	−1.06153	−0.530765	−	0.847519i	$-0.678095\pi$
$431$	−22.0379	−1.06153	−0.530765	+	0.847519i	$0.678095\pi$
$432$	0	0
$433$	−19.8732	−0.955047	−0.477523	−	0.878619i	$-0.658465\pi$
$433$	−19.8732	−0.955047	−0.477523	+	0.878619i	$0.658465\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.63295	0.221624
$438$	0	0
$439$	28.4752	1.35905	0.679524	−	0.733653i	$-0.262186\pi$
$439$	28.4752	1.35905	0.679524	+	0.733653i	$0.262186\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−24.2849	−1.15381	−0.576904	−	0.816812i	$-0.695739\pi$
$443$	−24.2849	−1.15381	−0.576904	+	0.816812i	$0.695739\pi$
$444$	0	0
$445$	21.9930	1.04257
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−16.4240	−0.775094	−0.387547	−	0.921850i	$-0.626678\pi$
$449$	−16.4240	−0.775094	−0.387547	+	0.921850i	$0.626678\pi$
$450$	0	0
$451$	1.49318	0.0703111
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−10.4619	−0.490461
$456$	0	0
$457$	21.8978	1.02434	0.512169	−	0.858885i	$-0.328842\pi$
$457$	21.8978	1.02434	0.512169	+	0.858885i	$0.328842\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	27.9997	1.30408	0.652038	−	0.758187i	$-0.273914\pi$
$461$	27.9997	1.30408	0.652038	+	0.758187i	$0.273914\pi$
$462$	0	0
$463$	6.55677	0.304719	0.152359	−	0.988325i	$-0.451313\pi$
$463$	6.55677	0.304719	0.152359	+	0.988325i	$0.451313\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.9864	−0.508388	−0.254194	−	0.967153i	$-0.581810\pi$
$467$	−10.9864	−0.508388	−0.254194	+	0.967153i	$0.581810\pi$
$468$	0	0
$469$	4.12963	0.190689
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−19.2333	−0.884348
$474$	0	0
$475$	−0.840904	−0.0385833
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7.44856	0.340333	0.170167	−	0.985415i	$-0.445569\pi$
$479$	7.44856	0.340333	0.170167	+	0.985415i	$0.445569\pi$
$480$	0	0
$481$	−21.4995	−0.980295
$482$	0	0
$483$	0	0
$484$	0	0
$485$	32.3677	1.46974
$486$	0	0
$487$	−25.8230	−1.17015	−0.585075	−	0.810979i	$-0.698935\pi$
$487$	−25.8230	−1.17015	−0.585075	+	0.810979i	$0.698935\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−25.4482	−1.14846	−0.574232	−	0.818693i	$-0.694699\pi$
$491$	−25.4482	−1.14846	−0.574232	+	0.818693i	$0.694699\pi$
$492$	0	0
$493$	2.51565	0.113299
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.46539	−0.110588
$498$	0	0
$499$	−29.6463	−1.32715	−0.663575	−	0.748110i	$-0.730962\pi$
$499$	−29.6463	−1.32715	−0.663575	+	0.748110i	$0.730962\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3.61430	−0.161154	−0.0805768	−	0.996748i	$-0.525676\pi$
$503$	−3.61430	−0.161154	−0.0805768	+	0.996748i	$0.525676\pi$
$504$	0	0
$505$	27.6572	1.23073
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8.71613	0.386336	0.193168	−	0.981166i	$-0.438124\pi$
$509$	8.71613	0.386336	0.193168	+	0.981166i	$0.438124\pi$
$510$	0	0
$511$	16.6329	0.735798
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.74047	0.385151
$516$	0	0
$517$	29.0633	1.27820
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−20.2614	−0.887668	−0.443834	−	0.896109i	$-0.646382\pi$
$521$	−20.2614	−0.887668	−0.443834	+	0.896109i	$0.646382\pi$
$522$	0	0
$523$	19.1574	0.837696	0.418848	−	0.908056i	$-0.362434\pi$
$523$	19.1574	0.837696	0.418848	+	0.908056i	$0.362434\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.51565	0.109583
$528$	0	0
$529$	−20.6450	−0.897607
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.35472	−0.0586796
$534$	0	0
$535$	−9.19003	−0.397320
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.01897	0.216182
$540$	0	0
$541$	−34.6652	−1.49038	−0.745188	−	0.666855i	$-0.767640\pi$
$541$	−34.6652	−1.49038	−0.745188	+	0.666855i	$0.767640\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	24.6326	1.05515
$546$	0	0
$547$	17.4805	0.747414	0.373707	−	0.927547i	$-0.378087\pi$
$547$	17.4805	0.747414	0.373707	+	0.927547i	$0.378087\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9.95507	−0.424100
$552$	0	0
$553$	−5.36674	−0.228217
$554$	0	0
$555$	0	0
$556$	0	0
$557$	44.7901	1.89782	0.948908	−	0.315553i	$-0.102190\pi$
$557$	44.7901	1.89782	0.948908	+	0.315553i	$0.102190\pi$
$558$	0	0
$559$	17.4499	0.738051
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−16.6965	−0.703675	−0.351838	−	0.936061i	$-0.614443\pi$
$563$	−16.6965	−0.703675	−0.351838	+	0.936061i	$0.614443\pi$
$564$	0	0
$565$	−20.7211	−0.871745
$566$	0	0
$567$	0	0
$568$	0	0
$569$	33.6396	1.41024	0.705122	−	0.709086i	$-0.250892\pi$
$569$	33.6396	1.41024	0.705122	+	0.709086i	$0.250892\pi$
$570$	0	0
$571$	−35.8251	−1.49923	−0.749617	−	0.661871i	$-0.769762\pi$
$571$	−35.8251	−1.49923	−0.749617	+	0.661871i	$0.769762\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.427452	−0.0178260
$576$	0	0
$577$	21.7022	0.903473	0.451737	−	0.892151i	$-0.350805\pi$
$577$	21.7022	0.903473	0.451737	+	0.892151i	$0.350805\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.27854	0.0530428
$582$	0	0
$583$	57.1445	2.36668
$584$	0	0
$585$	0	0
$586$	0	0
$587$	41.9358	1.73088	0.865438	−	0.501017i	$-0.167041\pi$
$587$	41.9358	1.73088	0.865438	+	0.501017i	$0.167041\pi$
$588$	0	0
$589$	−9.95507	−0.410191
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−20.2975	−0.833519	−0.416759	−	0.909017i	$-0.636834\pi$
$593$	−20.2975	−0.833519	−0.416759	+	0.909017i	$0.636834\pi$
$594$	0	0
$595$	1.75275	0.0718559
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−15.0101	−0.613298	−0.306649	−	0.951823i	$-0.599208\pi$
$599$	−15.0101	−0.613298	−0.306649	+	0.951823i	$0.599208\pi$
$600$	0	0
$601$	12.4998	0.509878	0.254939	−	0.966957i	$-0.417945\pi$
$601$	12.4998	0.509878	0.254939	+	0.966957i	$0.417945\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	32.6017	1.32545
$606$	0	0
$607$	35.7268	1.45011	0.725053	−	0.688693i	$-0.241815\pi$
$607$	35.7268	1.45011	0.725053	+	0.688693i	$0.241815\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−26.3684	−1.06675
$612$	0	0
$613$	29.9438	1.20942	0.604709	−	0.796447i	$-0.293289\pi$
$613$	29.9438	1.20942	0.604709	+	0.796447i	$0.293289\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.6709	1.23476	0.617382	−	0.786664i	$-0.288193\pi$
$617$	30.6709	1.23476	0.617382	+	0.786664i	$0.288193\pi$
$618$	0	0
$619$	−6.47954	−0.260435	−0.130217	−	0.991485i	$-0.541568\pi$
$619$	−6.47954	−0.260435	−0.130217	+	0.991485i	$0.541568\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	9.57255	0.383516
$624$	0	0
$625$	−26.3151	−1.05260
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.60197	0.143620
$630$	0	0
$631$	−20.5125	−0.816588	−0.408294	−	0.912850i	$-0.633876\pi$
$631$	−20.5125	−0.816588	−0.408294	+	0.912850i	$0.633876\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−26.9737	−1.07042
$636$	0	0
$637$	−4.55358	−0.180419
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−13.1511	−0.519435	−0.259718	−	0.965685i	$-0.583630\pi$
$641$	−13.1511	−0.519435	−0.259718	+	0.965685i	$0.583630\pi$
$642$	0	0
$643$	−14.0935	−0.555794	−0.277897	−	0.960611i	$-0.589637\pi$
$643$	−14.0935	−0.555794	−0.277897	+	0.960611i	$0.589637\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−22.5950	−0.888302	−0.444151	−	0.895952i	$-0.646495\pi$
$647$	−22.5950	−0.888302	−0.444151	+	0.895952i	$0.646495\pi$
$648$	0	0
$649$	−18.0433	−0.708260
$650$	0	0
$651$	0	0
$652$	0	0
$653$	39.5118	1.54622	0.773109	−	0.634274i	$-0.218701\pi$
$653$	39.5118	1.54622	0.773109	+	0.634274i	$0.218701\pi$
$654$	0	0
$655$	−43.5364	−1.70111
$656$	0	0
$657$	0	0
$658$	0	0
$659$	48.8473	1.90282	0.951410	−	0.307928i	$-0.0996355\pi$
$659$	48.8473	1.90282	0.951410	+	0.307928i	$0.0996355\pi$
$660$	0	0
$661$	−34.8419	−1.35519	−0.677597	−	0.735433i	$-0.736979\pi$
$661$	−34.8419	−1.35519	−0.677597	+	0.735433i	$0.736979\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−6.93610	−0.268970
$666$	0	0
$667$	−5.06040	−0.195940
$668$	0	0
$669$	0	0
$670$	0	0
$671$	12.1657	0.469653
$672$	0	0
$673$	−40.5687	−1.56381	−0.781905	−	0.623398i	$-0.785752\pi$
$673$	−40.5687	−1.56381	−0.781905	+	0.623398i	$0.785752\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−37.2210	−1.43052	−0.715259	−	0.698860i	$-0.753691\pi$
$677$	−37.2210	−1.43052	−0.715259	+	0.698860i	$0.753691\pi$
$678$	0	0
$679$	14.0882	0.540655
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−4.42395	−0.169278	−0.0846389	−	0.996412i	$-0.526974\pi$
$683$	−4.42395	−0.169278	−0.0846389	+	0.996412i	$0.526974\pi$
$684$	0	0
$685$	−2.25393	−0.0861183
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−51.8457	−1.97516
$690$	0	0
$691$	34.2090	1.30137	0.650686	−	0.759347i	$-0.274482\pi$
$691$	34.2090	1.30137	0.650686	+	0.759347i	$0.274482\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	53.5690	2.03199
$696$	0	0
$697$	0.226966	0.00859697
$698$	0	0
$699$	0	0
$700$	0	0
$701$	7.91181	0.298825	0.149412	−	0.988775i	$-0.452262\pi$
$701$	7.91181	0.298825	0.149412	+	0.988775i	$0.452262\pi$
$702$	0	0
$703$	−14.2539	−0.537597
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.0379	0.452733
$708$	0	0
$709$	18.5894	0.698139	0.349069	−	0.937097i	$-0.386498\pi$
$709$	18.5894	0.698139	0.349069	+	0.937097i	$0.386498\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5.06040	−0.189514
$714$	0	0
$715$	−52.5079	−1.96368
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−31.4170	−1.17166	−0.585828	−	0.810435i	$-0.699231\pi$
$719$	−31.4170	−1.17166	−0.585828	+	0.810435i	$0.699231\pi$
$720$	0	0
$721$	3.80433	0.141681
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0.918489	0.0341118
$726$	0	0
$727$	−49.3727	−1.83113	−0.915567	−	0.402166i	$-0.868258\pi$
$727$	−49.3727	−1.83113	−0.915567	+	0.402166i	$0.868258\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2.92350	−0.108130
$732$	0	0
$733$	−19.2191	−0.709875	−0.354938	−	0.934890i	$-0.615498\pi$
$733$	−19.2191	−0.709875	−0.354938	+	0.934890i	$0.615498\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	20.7265	0.763469
$738$	0	0
$739$	40.5944	1.49329	0.746644	−	0.665224i	$-0.231664\pi$
$739$	40.5944	1.49329	0.746644	+	0.665224i	$0.231664\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−6.77490	−0.248547	−0.124274	−	0.992248i	$-0.539660\pi$
$743$	−6.77490	−0.248547	−0.124274	+	0.992248i	$0.539660\pi$
$744$	0	0
$745$	−26.0433	−0.954151
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	46.0309	1.67969	0.839846	−	0.542824i	$-0.182645\pi$
$751$	46.0309	1.67969	0.839846	+	0.542824i	$0.182645\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−17.9741	−0.654144
$756$	0	0
$757$	−33.1571	−1.20512	−0.602558	−	0.798075i	$-0.705852\pi$
$757$	−33.1571	−1.20512	−0.602558	+	0.798075i	$0.705852\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6.22347	0.225600	0.112800	−	0.993618i	$-0.464018\pi$
$761$	6.22347	0.225600	0.112800	+	0.993618i	$0.464018\pi$
$762$	0	0
$763$	10.7215	0.388143
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.3702	0.591093
$768$	0	0
$769$	1.06923	0.0385573	0.0192787	−	0.999814i	$-0.493863\pi$
$769$	1.06923	0.0385573	0.0192787	+	0.999814i	$0.493863\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−11.4875	−0.413178	−0.206589	−	0.978428i	$-0.566236\pi$
$773$	−11.4875	−0.413178	−0.206589	+	0.978428i	$0.566236\pi$
$774$	0	0
$775$	0.918489	0.0329931
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.898165	−0.0321801
$780$	0	0
$781$	−12.3737	−0.442765
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7.72778	0.275816
$786$	0	0
$787$	−32.7335	−1.16682	−0.583411	−	0.812177i	$-0.698282\pi$
$787$	−32.7335	−1.16682	−0.583411	+	0.812177i	$0.698282\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−9.01897	−0.320678
$792$	0	0
$793$	−11.0377	−0.391959
$794$	0	0
$795$	0	0
$796$	0	0
$797$	34.4496	1.22027	0.610133	−	0.792299i	$-0.291116\pi$
$797$	34.4496	1.22027	0.610133	+	0.792299i	$0.291116\pi$
$798$	0	0
$799$	4.41768	0.156286
$800$	0	0
$801$	0	0
$802$	0	0
$803$	83.4802	2.94595
$804$	0	0
$805$	−3.52579	−0.124268
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−21.6845	−0.762387	−0.381194	−	0.924495i	$-0.624487\pi$
$809$	−21.6845	−0.762387	−0.381194	+	0.924495i	$0.624487\pi$
$810$	0	0
$811$	21.9497	0.770760	0.385380	−	0.922758i	$-0.374070\pi$
$811$	21.9497	0.770760	0.385380	+	0.922758i	$0.374070\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−33.2336	−1.16412
$816$	0	0
$817$	11.5690	0.404750
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21.6765	0.756516	0.378258	−	0.925700i	$-0.376523\pi$
$821$	21.6765	0.756516	0.378258	+	0.925700i	$0.376523\pi$
$822$	0	0
$823$	−30.8540	−1.07550	−0.537751	−	0.843104i	$-0.680726\pi$
$823$	−30.8540	−1.07550	−0.537751	+	0.843104i	$0.680726\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−22.0503	−0.766763	−0.383381	−	0.923590i	$-0.625240\pi$
$827$	−22.0503	−0.766763	−0.383381	+	0.923590i	$0.625240\pi$
$828$	0	0
$829$	22.1464	0.769177	0.384588	−	0.923088i	$-0.374343\pi$
$829$	22.1464	0.769177	0.384588	+	0.923088i	$0.374343\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.762893	0.0264327
$834$	0	0
$835$	33.1384	1.14680
$836$	0	0
$837$	0	0
$838$	0	0
$839$	36.6450	1.26512	0.632562	−	0.774510i	$-0.282003\pi$
$839$	36.6450	1.26512	0.632562	+	0.774510i	$0.282003\pi$
$840$	0	0
$841$	−18.1264	−0.625050
$842$	0	0
$843$	0	0
$844$	0	0
$845$	17.7714	0.611356
$846$	0	0
$847$	14.1900	0.487575
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−7.24562	−0.248377
$852$	0	0
$853$	−14.1296	−0.483789	−0.241895	−	0.970303i	$-0.577769\pi$
$853$	−14.1296	−0.483789	−0.241895	+	0.970303i	$0.577769\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	24.9329	0.851692	0.425846	−	0.904796i	$-0.359977\pi$
$857$	24.9329	0.851692	0.425846	+	0.904796i	$0.359977\pi$
$858$	0	0
$859$	−23.1654	−0.790392	−0.395196	−	0.918597i	$-0.629323\pi$
$859$	−23.1654	−0.790392	−0.395196	+	0.918597i	$0.629323\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	40.3440	1.37332	0.686662	−	0.726977i	$-0.259075\pi$
$863$	40.3440	1.37332	0.686662	+	0.726977i	$0.259075\pi$
$864$	0	0
$865$	30.8852	1.05013
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−26.9355	−0.913723
$870$	0	0
$871$	−18.8046	−0.637170
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−10.8476	−0.366715
$876$	0	0
$877$	17.7971	0.600964	0.300482	−	0.953788i	$-0.402852\pi$
$877$	17.7971	0.600964	0.300482	+	0.953788i	$0.402852\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−19.5346	−0.658138	−0.329069	−	0.944306i	$-0.606735\pi$
$881$	−19.5346	−0.658138	−0.329069	+	0.944306i	$0.606735\pi$
$882$	0	0
$883$	−46.3822	−1.56089	−0.780443	−	0.625227i	$-0.785006\pi$
$883$	−46.3822	−1.56089	−0.780443	+	0.625227i	$0.785006\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.65887	0.290736	0.145368	−	0.989378i	$-0.453563\pi$
$887$	8.65887	0.290736	0.145368	+	0.989378i	$0.453563\pi$
$888$	0	0
$889$	−11.7404	−0.393761
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−17.4819	−0.585009
$894$	0	0
$895$	14.2652	0.476834
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.8736	0.362653
$900$	0	0
$901$	8.68608	0.289375
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11.3488	0.377246
$906$	0	0
$907$	23.9618	0.795637	0.397818	−	0.917464i	$-0.369767\pi$
$907$	23.9618	0.795637	0.397818	+	0.917464i	$0.369767\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−8.62062	−0.285614	−0.142807	−	0.989751i	$-0.545613\pi$
$911$	−8.62062	−0.285614	−0.142807	+	0.989751i	$0.545613\pi$
$912$	0	0
$913$	6.41695	0.212370
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−18.9494	−0.625765
$918$	0	0
$919$	−4.67120	−0.154089	−0.0770443	−	0.997028i	$-0.524548\pi$
$919$	−4.67120	−0.154089	−0.0770443	+	0.997028i	$0.524548\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	11.2263	0.369519
$924$	0	0
$925$	1.31512	0.0432408
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.32180	0.0433669	0.0216835	−	0.999765i	$-0.493097\pi$
$929$	1.32180	0.0433669	0.0216835	+	0.999765i	$0.493097\pi$
$930$	0	0
$931$	−3.01897	−0.0989426
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8.79701	0.287693
$936$	0	0
$937$	−17.6140	−0.575424	−0.287712	−	0.957717i	$-0.592895\pi$
$937$	−17.6140	−0.575424	−0.287712	+	0.957717i	$0.592895\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−25.4875	−0.830870	−0.415435	−	0.909623i	$-0.636371\pi$
$941$	−25.4875	−0.830870	−0.415435	+	0.909623i	$0.636371\pi$
$942$	0	0
$943$	−0.456559	−0.0148676
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.50551	−0.0489224	−0.0244612	−	0.999701i	$-0.507787\pi$
$947$	−1.50551	−0.0489224	−0.0244612	+	0.999701i	$0.507787\pi$
$948$	0	0
$949$	−75.7395	−2.45861
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−4.55008	−0.147392	−0.0736958	−	0.997281i	$-0.523479\pi$
$953$	−4.55008	−0.147392	−0.0736958	+	0.997281i	$0.523479\pi$
$954$	0	0
$955$	3.14081	0.101634
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−0.981033	−0.0316792
$960$	0	0
$961$	−20.1264	−0.649240
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−12.9417	−0.416609
$966$	0	0
$967$	−45.7401	−1.47090	−0.735451	−	0.677577i	$-0.763030\pi$
$967$	−45.7401	−1.47090	−0.735451	+	0.677577i	$0.763030\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	17.5677	0.563775	0.281888	−	0.959447i	$-0.409039\pi$
$971$	17.5677	0.563775	0.281888	+	0.959447i	$0.409039\pi$
$972$	0	0
$973$	23.3162	0.747482
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−20.7538	−0.663971	−0.331986	−	0.943284i	$-0.607719\pi$
$977$	−20.7538	−0.663971	−0.331986	+	0.943284i	$0.607719\pi$
$978$	0	0
$979$	48.0443	1.53550
$980$	0	0
$981$	0	0
$982$	0	0
$983$	15.5378	0.495579	0.247789	−	0.968814i	$-0.420296\pi$
$983$	15.5378	0.495579	0.247789	+	0.968814i	$0.420296\pi$
$984$	0	0
$985$	−19.9497	−0.635652
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.88083	0.186999
$990$	0	0
$991$	−7.87888	−0.250281	−0.125140	−	0.992139i	$-0.539938\pi$
$991$	−7.87888	−0.250281	−0.125140	+	0.992139i	$0.539938\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−6.32447	−0.200499
$996$	0	0
$997$	15.6987	0.497182	0.248591	−	0.968609i	$-0.420032\pi$
$997$	15.6987	0.497182	0.248591	+	0.968609i	$0.420032\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.a.bt.1.3	yes	4
3.2	odd	2		6048.2.a.bp.1.2	yes	4
4.3	odd	2		6048.2.a.bs.1.3	yes	4
12.11	even	2		6048.2.a.bk.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.a.bk.1.2	✓	4	12.11	even	2
6048.2.a.bp.1.2	yes	4	3.2	odd	2
6048.2.a.bs.1.3	yes	4	4.3	odd	2
6048.2.a.bt.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)

\(f(q)\)	\(=\)	\(q+2.29751 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+2.29751 q^{5} +1.00000 q^{7} +5.01897 q^{11} -4.55358 q^{13} +0.762893 q^{17} -3.01897 q^{19} -1.53461 q^{23} +0.278540 q^{25} +3.29751 q^{29} +3.29751 q^{31} +2.29751 q^{35} +4.72146 q^{37} +0.297507 q^{41} -3.83212 q^{43} +5.79069 q^{47} +1.00000 q^{49} +11.3857 q^{53} +11.5311 q^{55} -3.59501 q^{59} +2.42395 q^{61} -10.4619 q^{65} +4.12963 q^{67} -2.46539 q^{71} +16.6329 q^{73} +5.01897 q^{77} -5.36674 q^{79} +1.27854 q^{83} +1.75275 q^{85} +9.57255 q^{89} -4.55358 q^{91} -6.93610 q^{95} +14.0882 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5} + 4 q^{7}+O(q^{10}) \) 4 * q + 2 * q^5 + 4 * q^7	\( 4 q + 2 q^{5} + 4 q^{7} + 8 q^{13} + 2 q^{17} + 8 q^{19} + 14 q^{25} + 6 q^{29} + 6 q^{31} + 2 q^{35} + 6 q^{37} - 6 q^{41} - 2 q^{43} - 2 q^{47} + 4 q^{49} + 6 q^{53} - 12 q^{55} + 4 q^{61} + 4 q^{65} - 4 q^{67} - 16 q^{71} + 12 q^{73} - 2 q^{79} + 18 q^{83} + 22 q^{85} - 8 q^{89} + 8 q^{91} + 16 q^{95} + 24 q^{97}+O(q^{100}) \) 4 * q + 2 * q^5 + 4 * q^7 + 8 * q^13 + 2 * q^17 + 8 * q^19 + 14 * q^25 + 6 * q^29 + 6 * q^31 + 2 * q^35 + 6 * q^37 - 6 * q^41 - 2 * q^43 - 2 * q^47 + 4 * q^49 + 6 * q^53 - 12 * q^55 + 4 * q^61 + 4 * q^65 - 4 * q^67 - 16 * q^71 + 12 * q^73 - 2 * q^79 + 18 * q^83 + 22 * q^85 - 8 * q^89 + 8 * q^91 + 16 * q^95 + 24 * q^97

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