LMFDB - Embedded newform 6048.2.a.br.1.2

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:	4.4.22896.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} - 4x + 3 $ x^4 - 9x^2 - 4x + 3
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			1.2
Root			$-0.896343$ 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)

$f(q)$	$=$	$q-0.314350 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-0.314350 q^{5} -1.00000 q^{7} -1.47834 q^{11} -1.79269 q^{13} -5.22915 q^{17} -6.22915 q^{19} -6.58683 q^{23} -4.90118 q^{25} +7.43647 q^{29} +4.43647 q^{31} +0.314350 q^{35} -2.79269 q^{37} +8.37952 q^{41} +0.207315 q^{43} +4.06517 q^{47} +1.00000 q^{49} +4.00000 q^{53} +0.464715 q^{55} +4.06517 q^{59} -1.67203 q^{61} +0.563530 q^{65} +13.9230 q^{67} +7.27102 q^{71} +14.6656 q^{73} +1.47834 q^{77} -1.46471 q^{79} -12.2792 q^{83} +1.64378 q^{85} +12.3795 q^{89} +1.79269 q^{91} +1.95813 q^{95} +16.1303 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 8 q^{17} + 4 q^{19} - 2 q^{23} + 8 q^{25} + 8 q^{29} - 4 q^{31} - 2 q^{35} - 4 q^{37} + 2 q^{41} + 8 q^{43} - 12 q^{47} + 4 q^{49} + 16 q^{53} - 4 q^{55} - 12 q^{59} - 8 q^{61} + 24 q^{65} - 8 q^{67} + 18 q^{71} + 8 q^{73} + 2 q^{77} - 8 q^{85} + 18 q^{89} - 10 q^{95} + 8 q^{97}+O(q^{100}) $ 4 * q + 2 * q^5 - 4 * q^7 - 2 * q^11 + 8 * q^17 + 4 * q^19 - 2 * q^23 + 8 * q^25 + 8 * q^29 - 4 * q^31 - 2 * q^35 - 4 * q^37 + 2 * q^41 + 8 * q^43 - 12 * q^47 + 4 * q^49 + 16 * q^53 - 4 * q^55 - 12 * q^59 - 8 * q^61 + 24 * q^65 - 8 * q^67 + 18 * q^71 + 8 * q^73 + 2 * q^77 - 8 * q^85 + 18 * q^89 - 10 * 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
$5$	−0.314350	−0.140582	−0.0702908	−	0.997527i	$-0.522393\pi$
$5$	−0.314350	−0.140582	−0.0702908	+	0.997527i	$0.522393\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.47834	−0.445735	−0.222867	−	0.974849i	$-0.571542\pi$
$11$	−1.47834	−0.445735	−0.222867	+	0.974849i	$0.571542\pi$
$12$	0	0
$13$	−1.79269	−0.497201	−0.248601	−	0.968606i	$-0.579971\pi$
$13$	−1.79269	−0.497201	−0.248601	+	0.968606i	$0.579971\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.22915	−1.26826	−0.634128	−	0.773228i	$-0.718641\pi$
$17$	−5.22915	−1.26826	−0.634128	+	0.773228i	$0.718641\pi$
$18$	0	0
$19$	−6.22915	−1.42907	−0.714533	−	0.699602i	$-0.753361\pi$
$19$	−6.22915	−1.42907	−0.714533	+	0.699602i	$0.753361\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.58683	−1.37345	−0.686725	−	0.726917i	$-0.740952\pi$
$23$	−6.58683	−1.37345	−0.686725	+	0.726917i	$0.740952\pi$
$24$	0	0
$25$	−4.90118	−0.980237
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.43647	1.38092	0.690459	−	0.723372i	$-0.257409\pi$
$29$	7.43647	1.38092	0.690459	+	0.723372i	$0.257409\pi$
$30$	0	0
$31$	4.43647	0.796813	0.398407	−	0.917209i	$-0.369563\pi$
$31$	4.43647	0.796813	0.398407	+	0.917209i	$0.369563\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.314350	0.0531348
$36$	0	0
$37$	−2.79269	−0.459115	−0.229557	−	0.973295i	$-0.573728\pi$
$37$	−2.79269	−0.459115	−0.229557	+	0.973295i	$0.573728\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.37952	1.30866	0.654331	−	0.756209i	$-0.272951\pi$
$41$	8.37952	1.30866	0.654331	+	0.756209i	$0.272951\pi$
$42$	0	0
$43$	0.207315	0.0316152	0.0158076	−	0.999875i	$-0.494968\pi$
$43$	0.207315	0.0316152	0.0158076	+	0.999875i	$0.494968\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.06517	0.592966	0.296483	−	0.955038i	$-0.404186\pi$
$47$	4.06517	0.592966	0.296483	+	0.955038i	$0.404186\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	0.464715	0.0626621
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.06517	0.529240	0.264620	−	0.964353i	$-0.414753\pi$
$59$	4.06517	0.529240	0.264620	+	0.964353i	$0.414753\pi$
$60$	0	0
$61$	−1.67203	−0.214081	−0.107041	−	0.994255i	$-0.534138\pi$
$61$	−1.67203	−0.214081	−0.107041	+	0.994255i	$0.534138\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.563530	0.0698973
$66$	0	0
$67$	13.9230	1.70097	0.850484	−	0.526001i	$-0.176309\pi$
$67$	13.9230	1.70097	0.850484	+	0.526001i	$0.176309\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.27102	0.862911	0.431456	−	0.902134i	$-0.358000\pi$
$71$	7.27102	0.862911	0.431456	+	0.902134i	$0.358000\pi$
$72$	0	0
$73$	14.6656	1.71648	0.858241	−	0.513247i	$-0.171558\pi$
$73$	14.6656	1.71648	0.858241	+	0.513247i	$0.171558\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.47834	0.168472
$78$	0	0
$79$	−1.46471	−0.164793	−0.0823966	−	0.996600i	$-0.526257\pi$
$79$	−1.46471	−0.164793	−0.0823966	+	0.996600i	$0.526257\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−12.2792	−1.34782	−0.673911	−	0.738813i	$-0.735387\pi$
$83$	−12.2792	−1.34782	−0.673911	+	0.738813i	$0.735387\pi$
$84$	0	0
$85$	1.64378	0.178293
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.3795	1.31223	0.656113	−	0.754662i	$-0.272199\pi$
$89$	12.3795	1.31223	0.656113	+	0.754662i	$0.272199\pi$
$90$	0	0
$91$	1.79269	0.187924
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.95813	0.200900
$96$	0	0
$97$	16.1303	1.63779	0.818894	−	0.573945i	$-0.194588\pi$
$97$	16.1303	1.63779	0.818894	+	0.573945i	$0.194588\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.77085	0.275709	0.137855	−	0.990452i	$-0.455979\pi$
$101$	2.77085	0.275709	0.137855	+	0.990452i	$0.455979\pi$
$102$	0	0
$103$	−18.5668	−1.82944	−0.914721	−	0.404086i	$-0.867590\pi$
$103$	−18.5668	−1.82944	−0.914721	+	0.404086i	$0.867590\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.1858	1.17805	0.589024	−	0.808115i	$-0.299512\pi$
$107$	12.1858	1.17805	0.589024	+	0.808115i	$0.299512\pi$
$108$	0	0
$109$	−10.1085	−0.968219	−0.484109	−	0.875008i	$-0.660856\pi$
$109$	−10.1085	−0.968219	−0.484109	+	0.875008i	$0.660856\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.95667	0.654429	0.327214	−	0.944950i	$-0.393890\pi$
$113$	6.95667	0.654429	0.327214	+	0.944950i	$0.393890\pi$
$114$	0	0
$115$	2.07057	0.193082
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.22915	0.479356
$120$	0	0
$121$	−8.81452	−0.801320
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.11244	0.278385
$126$	0	0
$127$	7.05008	0.625594	0.312797	−	0.949820i	$-0.398734\pi$
$127$	7.05008	0.625594	0.312797	+	0.949820i	$0.398734\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−10.8078	−0.944279	−0.472140	−	0.881524i	$-0.656518\pi$
$131$	−10.8078	−0.944279	−0.472140	+	0.881524i	$0.656518\pi$
$132$	0	0
$133$	6.22915	0.540136
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.6072	1.58972	0.794861	−	0.606792i	$-0.207544\pi$
$137$	18.6072	1.58972	0.794861	+	0.606792i	$0.207544\pi$
$138$	0	0
$139$	−8.54497	−0.724775	−0.362387	−	0.932028i	$-0.618038\pi$
$139$	−8.54497	−0.724775	−0.362387	+	0.932028i	$0.618038\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.65019	0.221620
$144$	0	0
$145$	−2.33765	−0.194132
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.08701	−0.580590	−0.290295	−	0.956937i	$-0.593754\pi$
$149$	−7.08701	−0.580590	−0.290295	+	0.956937i	$0.593754\pi$
$150$	0	0
$151$	−3.67203	−0.298826	−0.149413	−	0.988775i	$-0.547738\pi$
$151$	−3.67203	−0.298826	−0.149413	+	0.988775i	$0.547738\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.39460	−0.112017
$156$	0	0
$157$	−13.0803	−1.04392	−0.521959	−	0.852971i	$-0.674799\pi$
$157$	−13.0803	−1.04392	−0.521959	+	0.852971i	$0.674799\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.58683	0.519115
$162$	0	0
$163$	8.25099	0.646268	0.323134	−	0.946353i	$-0.395264\pi$
$163$	8.25099	0.646268	0.323134	+	0.946353i	$0.395264\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.9567	−0.847853	−0.423926	−	0.905697i	$-0.639348\pi$
$167$	−10.9567	−0.847853	−0.423926	+	0.905697i	$0.639348\pi$
$168$	0	0
$169$	−9.78628	−0.752791
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.45650	0.186764	0.0933819	−	0.995630i	$-0.470232\pi$
$173$	2.45650	0.186764	0.0933819	+	0.995630i	$0.470232\pi$
$174$	0	0
$175$	4.90118	0.370495
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.14215	0.160111	0.0800557	−	0.996790i	$-0.474490\pi$
$179$	2.14215	0.160111	0.0800557	+	0.996790i	$0.474490\pi$
$180$	0	0
$181$	−18.7863	−1.39637	−0.698187	−	0.715916i	$-0.746009\pi$
$181$	−18.7863	−1.39637	−0.698187	+	0.715916i	$0.746009\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.877880	0.0645430
$186$	0	0
$187$	7.73044	0.565306
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.4164	−1.47728	−0.738641	−	0.674099i	$-0.764532\pi$
$191$	−20.4164	−1.47728	−0.738641	+	0.674099i	$0.764532\pi$
$192$	0	0
$193$	21.7157	1.56313	0.781565	−	0.623823i	$-0.214421\pi$
$193$	21.7157	1.56313	0.781565	+	0.623823i	$0.214421\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	24.7590	1.76401	0.882004	−	0.471241i	$-0.156194\pi$
$197$	24.7590	1.76401	0.882004	+	0.471241i	$0.156194\pi$
$198$	0	0
$199$	−4.05008	−0.287103	−0.143551	−	0.989643i	$-0.545852\pi$
$199$	−4.05008	−0.287103	−0.143551	+	0.989643i	$0.545852\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.43647	−0.521938
$204$	0	0
$205$	−2.63410	−0.183974
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9.20878	0.636985
$210$	0	0
$211$	−15.2009	−1.04647	−0.523237	−	0.852187i	$-0.675276\pi$
$211$	−15.2009	−1.04647	−0.523237	+	0.852187i	$0.675276\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.0651694	−0.00444452
$216$	0	0
$217$	−4.43647	−0.301167
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9.37423	0.630579
$222$	0	0
$223$	−0.792685	−0.0530821	−0.0265411	−	0.999648i	$-0.508449\pi$
$223$	−0.792685	−0.0530821	−0.0265411	+	0.999648i	$0.508449\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.8296	0.917903	0.458952	−	0.888461i	$-0.348225\pi$
$227$	13.8296	0.917903	0.458952	+	0.888461i	$0.348225\pi$
$228$	0	0
$229$	0.842770	0.0556918	0.0278459	−	0.999612i	$-0.491135\pi$
$229$	0.842770	0.0556918	0.0278459	+	0.999612i	$0.491135\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.1737	1.12508	0.562542	−	0.826769i	$-0.309823\pi$
$233$	17.1737	1.12508	0.562542	+	0.826769i	$0.309823\pi$
$234$	0	0
$235$	−1.27789	−0.0833601
$236$	0	0
$237$	0	0
$238$	0	0
$239$	19.4734	1.25963	0.629815	−	0.776746i	$-0.283131\pi$
$239$	19.4734	1.25963	0.629815	+	0.776746i	$0.283131\pi$
$240$	0	0
$241$	−0.337654	−0.0217502	−0.0108751	−	0.999941i	$-0.503462\pi$
$241$	−0.337654	−0.0217502	−0.0108751	+	0.999941i	$0.503462\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.314350	−0.0200831
$246$	0	0
$247$	11.1669	0.710534
$248$	0	0
$249$	0	0
$250$	0	0
$251$	11.2876	0.712465	0.356233	−	0.934397i	$-0.384061\pi$
$251$	11.2876	0.712465	0.356233	+	0.934397i	$0.384061\pi$
$252$	0	0
$253$	9.73755	0.612194
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.60867	−0.100346	−0.0501732	−	0.998741i	$-0.515977\pi$
$257$	−1.60867	−0.100346	−0.0501732	+	0.998741i	$0.515977\pi$
$258$	0	0
$259$	2.79269	0.173529
$260$	0	0
$261$	0	0
$262$	0	0
$263$	13.1289	0.809561	0.404781	−	0.914414i	$-0.367348\pi$
$263$	13.1289	0.809561	0.404781	+	0.914414i	$0.367348\pi$
$264$	0	0
$265$	−1.25740	−0.0772415
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8.28610	0.505213	0.252606	−	0.967569i	$-0.418712\pi$
$269$	8.28610	0.505213	0.252606	+	0.967569i	$0.418712\pi$
$270$	0	0
$271$	22.0437	1.33906	0.669529	−	0.742786i	$-0.266496\pi$
$271$	22.0437	1.33906	0.669529	+	0.742786i	$0.266496\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.24559	0.436926
$276$	0	0
$277$	−5.12066	−0.307670	−0.153835	−	0.988097i	$-0.549162\pi$
$277$	−5.12066	−0.307670	−0.153835	+	0.988097i	$0.549162\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−18.5235	−1.10502	−0.552509	−	0.833507i	$-0.686330\pi$
$281$	−18.5235	−1.10502	−0.552509	+	0.833507i	$0.686330\pi$
$282$	0	0
$283$	2.74260	0.163031	0.0815153	−	0.996672i	$-0.474024\pi$
$283$	2.74260	0.163031	0.0815153	+	0.996672i	$0.474024\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.37952	−0.494627
$288$	0	0
$289$	10.3441	0.608474
$290$	0	0
$291$	0	0
$292$	0	0
$293$	19.7440	1.15345	0.576727	−	0.816937i	$-0.304330\pi$
$293$	19.7440	1.15345	0.576727	+	0.816937i	$0.304330\pi$
$294$	0	0
$295$	−1.27789	−0.0744014
$296$	0	0
$297$	0	0
$298$	0	0
$299$	11.8081	0.682881
$300$	0	0
$301$	−0.207315	−0.0119494
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.525602	0.0300959
$306$	0	0
$307$	−11.5975	−0.661906	−0.330953	−	0.943647i	$-0.607370\pi$
$307$	−11.5975	−0.661906	−0.330953	+	0.943647i	$0.607370\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.8242	−1.40765	−0.703826	−	0.710373i	$-0.748526\pi$
$311$	−24.8242	−1.40765	−0.703826	+	0.710373i	$0.748526\pi$
$312$	0	0
$313$	24.7960	1.40155	0.700775	−	0.713382i	$-0.252838\pi$
$313$	24.7960	1.40155	0.700775	+	0.713382i	$0.252838\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	32.8514	1.84512	0.922561	−	0.385852i	$-0.126093\pi$
$317$	32.8514	1.84512	0.922561	+	0.385852i	$0.126093\pi$
$318$	0	0
$319$	−10.9936	−0.615523
$320$	0	0
$321$	0	0
$322$	0	0
$323$	32.5732	1.81242
$324$	0	0
$325$	8.78628	0.487375
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4.06517	−0.224120
$330$	0	0
$331$	−13.3877	−0.735857	−0.367928	−	0.929854i	$-0.619933\pi$
$331$	−13.3877	−0.735857	−0.367928	+	0.929854i	$0.619933\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.37670	−0.239125
$336$	0	0
$337$	16.9133	0.921328	0.460664	−	0.887575i	$-0.347611\pi$
$337$	16.9133	0.921328	0.460664	+	0.887575i	$0.347611\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−6.55859	−0.355168
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−23.8812	−1.28201	−0.641004	−	0.767538i	$-0.721482\pi$
$347$	−23.8812	−1.28201	−0.641004	+	0.767538i	$0.721482\pi$
$348$	0	0
$349$	−14.4146	−0.771597	−0.385799	−	0.922583i	$-0.626074\pi$
$349$	−14.4146	−0.771597	−0.385799	+	0.922583i	$0.626074\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	27.0384	1.43911	0.719554	−	0.694436i	$-0.244346\pi$
$353$	27.0384	1.43911	0.719554	+	0.694436i	$0.244346\pi$
$354$	0	0
$355$	−2.28564	−0.121309
$356$	0	0
$357$	0	0
$358$	0	0
$359$	33.2758	1.75623	0.878114	−	0.478452i	$-0.158802\pi$
$359$	33.2758	1.75623	0.878114	+	0.478452i	$0.158802\pi$
$360$	0	0
$361$	19.8024	1.04223
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.61014	−0.241306
$366$	0	0
$367$	−31.7658	−1.65816	−0.829080	−	0.559129i	$-0.811136\pi$
$367$	−31.7658	−1.65816	−0.829080	+	0.559129i	$0.811136\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.00000	−0.207670
$372$	0	0
$373$	−4.49296	−0.232637	−0.116318	−	0.993212i	$-0.537109\pi$
$373$	−4.49296	−0.232637	−0.116318	+	0.993212i	$0.537109\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−13.3312	−0.686594
$378$	0	0
$379$	−8.62194	−0.442880	−0.221440	−	0.975174i	$-0.571076\pi$
$379$	−8.62194	−0.442880	−0.221440	+	0.975174i	$0.571076\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−34.4311	−1.75935	−0.879673	−	0.475580i	$-0.842238\pi$
$383$	−34.4311	−1.75935	−0.879673	+	0.475580i	$0.842238\pi$
$384$	0	0
$385$	−0.464715	−0.0236840
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9.61554	−0.487527	−0.243764	−	0.969835i	$-0.578382\pi$
$389$	−9.61554	−0.487527	−0.243764	+	0.969835i	$0.578382\pi$
$390$	0	0
$391$	34.4436	1.74189
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.460433	0.0231669
$396$	0	0
$397$	−7.36837	−0.369808	−0.184904	−	0.982757i	$-0.559197\pi$
$397$	−7.36837	−0.369808	−0.184904	+	0.982757i	$0.559197\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	22.5507	1.12613	0.563065	−	0.826413i	$-0.309622\pi$
$401$	22.5507	1.12613	0.563065	+	0.826413i	$0.309622\pi$
$402$	0	0
$403$	−7.95319	−0.396177
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.12852	0.204643
$408$	0	0
$409$	4.63546	0.229209	0.114604	−	0.993411i	$-0.463440\pi$
$409$	4.63546	0.229209	0.114604	+	0.993411i	$0.463440\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.06517	−0.200034
$414$	0	0
$415$	3.85998	0.189479
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−36.7590	−1.79580	−0.897898	−	0.440203i	$-0.854906\pi$
$419$	−36.7590	−1.79580	−0.897898	+	0.440203i	$0.854906\pi$
$420$	0	0
$421$	−36.2543	−1.76693	−0.883463	−	0.468502i	$-0.844794\pi$
$421$	−36.2543	−1.76693	−0.883463	+	0.468502i	$0.844794\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	25.6290	1.24319
$426$	0	0
$427$	1.67203	0.0809152
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11.7139	0.564238	0.282119	−	0.959379i	$-0.408963\pi$
$431$	11.7139	0.564238	0.282119	+	0.959379i	$0.408963\pi$
$432$	0	0
$433$	9.24772	0.444417	0.222208	−	0.974999i	$-0.428673\pi$
$433$	9.24772	0.444417	0.222208	+	0.974999i	$0.428673\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	41.0304	1.96275
$438$	0	0
$439$	12.9166	0.616477	0.308238	−	0.951309i	$-0.400261\pi$
$439$	12.9166	0.616477	0.308238	+	0.951309i	$0.400261\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−29.2213	−1.38835	−0.694173	−	0.719809i	$-0.744230\pi$
$443$	−29.2213	−1.38835	−0.694173	+	0.719809i	$0.744230\pi$
$444$	0	0
$445$	−3.89150	−0.184475
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9.45796	0.446349	0.223174	−	0.974779i	$-0.428358\pi$
$449$	9.45796	0.446349	0.223174	+	0.974779i	$0.428358\pi$
$450$	0	0
$451$	−12.3877	−0.583316
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.563530	−0.0264187
$456$	0	0
$457$	9.84277	0.460425	0.230213	−	0.973140i	$-0.426058\pi$
$457$	9.84277	0.460425	0.230213	+	0.973140i	$0.426058\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−9.95813	−0.463797	−0.231898	−	0.972740i	$-0.574494\pi$
$461$	−9.95813	−0.463797	−0.231898	+	0.972740i	$0.574494\pi$
$462$	0	0
$463$	1.58537	0.0736784	0.0368392	−	0.999321i	$-0.488271\pi$
$463$	1.58537	0.0736784	0.0368392	+	0.999321i	$0.488271\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−19.1794	−0.887518	−0.443759	−	0.896146i	$-0.646355\pi$
$467$	−19.1794	−0.887518	−0.443759	+	0.896146i	$0.646355\pi$
$468$	0	0
$469$	−13.9230	−0.642906
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−0.306481	−0.0140920
$474$	0	0
$475$	30.5302	1.40082
$476$	0	0
$477$	0	0
$478$	0	0
$479$	16.4281	0.750621	0.375310	−	0.926899i	$-0.377536\pi$
$479$	16.4281	0.750621	0.375310	+	0.926899i	$0.377536\pi$
$480$	0	0
$481$	5.00641	0.228272
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.07057	−0.230243
$486$	0	0
$487$	−4.44863	−0.201586	−0.100793	−	0.994907i	$-0.532138\pi$
$487$	−4.44863	−0.201586	−0.100793	+	0.994907i	$0.532138\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1.23602	−0.0557808	−0.0278904	−	0.999611i	$-0.508879\pi$
$491$	−1.23602	−0.0557808	−0.0278904	+	0.999611i	$0.508879\pi$
$492$	0	0
$493$	−38.8864	−1.75136
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.27102	−0.326150
$498$	0	0
$499$	−18.1207	−0.811192	−0.405596	−	0.914052i	$-0.632936\pi$
$499$	−18.1207	−0.811192	−0.405596	+	0.914052i	$0.632936\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	19.4500	0.867232	0.433616	−	0.901098i	$-0.357237\pi$
$503$	19.4500	0.867232	0.433616	+	0.901098i	$0.357237\pi$
$504$	0	0
$505$	−0.871015	−0.0387597
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.1888	0.628905	0.314453	−	0.949273i	$-0.398179\pi$
$509$	14.1888	0.628905	0.314453	+	0.949273i	$0.398179\pi$
$510$	0	0
$511$	−14.6656	−0.648769
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.83647	0.257186
$516$	0	0
$517$	−6.00968	−0.264306
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−0.0633554	−0.00277565	−0.00138782	−	0.999999i	$-0.500442\pi$
$521$	−0.0633554	−0.00277565	−0.00138782	+	0.999999i	$0.500442\pi$
$522$	0	0
$523$	6.20091	0.271147	0.135573	−	0.990767i	$-0.456712\pi$
$523$	6.20091	0.271147	0.135573	+	0.990767i	$0.456712\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−23.1990	−1.01056
$528$	0	0
$529$	20.3864	0.886365
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−15.0218	−0.650668
$534$	0	0
$535$	−3.83061	−0.165612
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.47834	−0.0636764
$540$	0	0
$541$	−23.3377	−1.00336	−0.501682	−	0.865052i	$-0.667285\pi$
$541$	−23.3377	−1.00336	−0.501682	+	0.865052i	$0.667285\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.17761	0.136114
$546$	0	0
$547$	35.8460	1.53267	0.766333	−	0.642443i	$-0.222079\pi$
$547$	35.8460	1.53267	0.766333	+	0.642443i	$0.222079\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−46.3229	−1.97342
$552$	0	0
$553$	1.46471	0.0622860
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.2302	0.560580	0.280290	−	0.959915i	$-0.409569\pi$
$557$	13.2302	0.560580	0.280290	+	0.959915i	$0.409569\pi$
$558$	0	0
$559$	−0.371650	−0.0157191
$560$	0	0
$561$	0	0
$562$	0	0
$563$	21.2388	0.895110	0.447555	−	0.894256i	$-0.352295\pi$
$563$	21.2388	0.895110	0.447555	+	0.894256i	$0.352295\pi$
$564$	0	0
$565$	−2.18683	−0.0920006
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−17.9599	−0.752920	−0.376460	−	0.926433i	$-0.622859\pi$
$569$	−17.9599	−0.752920	−0.376460	+	0.926433i	$0.622859\pi$
$570$	0	0
$571$	22.3717	0.936224	0.468112	−	0.883669i	$-0.344934\pi$
$571$	22.3717	0.936224	0.468112	+	0.883669i	$0.344934\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	32.2833	1.34631
$576$	0	0
$577$	−4.04368	−0.168341	−0.0841703	−	0.996451i	$-0.526824\pi$
$577$	−4.04368	−0.168341	−0.0841703	+	0.996451i	$0.526824\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12.2792	0.509429
$582$	0	0
$583$	−5.91334	−0.244906
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.12999	−0.129188	−0.0645942	−	0.997912i	$-0.520575\pi$
$587$	−3.12999	−0.129188	−0.0645942	+	0.997912i	$0.520575\pi$
$588$	0	0
$589$	−27.6355	−1.13870
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.42185	0.0583882	0.0291941	−	0.999574i	$-0.490706\pi$
$593$	1.42185	0.0583882	0.0291941	+	0.999574i	$0.490706\pi$
$594$	0	0
$595$	−1.64378	−0.0673886
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−0.570396	−0.0233057	−0.0116529	−	0.999932i	$-0.503709\pi$
$599$	−0.570396	−0.0233057	−0.0116529	+	0.999932i	$0.503709\pi$
$600$	0	0
$601$	5.79269	0.236289	0.118144	−	0.992996i	$-0.462305\pi$
$601$	5.79269	0.236289	0.118144	+	0.992996i	$0.462305\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.77085	0.112651
$606$	0	0
$607$	7.71571	0.313171	0.156585	−	0.987664i	$-0.449951\pi$
$607$	7.71571	0.313171	0.156585	+	0.987664i	$0.449951\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7.28757	−0.294823
$612$	0	0
$613$	4.43647	0.179187	0.0895937	−	0.995978i	$-0.471443\pi$
$613$	4.43647	0.179187	0.0895937	+	0.995978i	$0.471443\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.6101	0.829733	0.414866	−	0.909882i	$-0.363828\pi$
$617$	20.6101	0.829733	0.414866	+	0.909882i	$0.363828\pi$
$618$	0	0
$619$	36.3877	1.46255	0.731273	−	0.682085i	$-0.238926\pi$
$619$	36.3877	1.46255	0.731273	+	0.682085i	$0.238926\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12.3795	−0.495975
$624$	0	0
$625$	23.5275	0.941101
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14.6034	0.582275
$630$	0	0
$631$	16.8633	0.671316	0.335658	−	0.941984i	$-0.391041\pi$
$631$	16.8633	0.671316	0.335658	+	0.941984i	$0.391041\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.21619	−0.0879470
$636$	0	0
$637$	−1.79269	−0.0710288
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−14.7154	−0.581222	−0.290611	−	0.956841i	$-0.593859\pi$
$641$	−14.7154	−0.581222	−0.290611	+	0.956841i	$0.593859\pi$
$642$	0	0
$643$	−10.0988	−0.398258	−0.199129	−	0.979973i	$-0.563811\pi$
$643$	−10.0988	−0.398258	−0.199129	+	0.979973i	$0.563811\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	49.4529	1.94419	0.972097	−	0.234578i	$-0.0753709\pi$
$647$	49.4529	1.94419	0.972097	+	0.234578i	$0.0753709\pi$
$648$	0	0
$649$	−6.00968	−0.235901
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20.8514	−0.815980	−0.407990	−	0.912986i	$-0.633770\pi$
$653$	−20.8514	−0.815980	−0.407990	+	0.912986i	$0.633770\pi$
$654$	0	0
$655$	3.39742	0.132748
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24.1372	0.940252	0.470126	−	0.882599i	$-0.344208\pi$
$659$	24.1372	0.940252	0.470126	+	0.882599i	$0.344208\pi$
$660$	0	0
$661$	31.0130	1.20626	0.603132	−	0.797641i	$-0.293919\pi$
$661$	31.0130	1.20626	0.603132	+	0.797641i	$0.293919\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.95813	−0.0759332
$666$	0	0
$667$	−48.9828	−1.89662
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.47182	0.0954236
$672$	0	0
$673$	25.7157	0.991268	0.495634	−	0.868532i	$-0.334936\pi$
$673$	25.7157	0.991268	0.495634	+	0.868532i	$0.334936\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	20.0301	0.769818	0.384909	−	0.922955i	$-0.374233\pi$
$677$	20.0301	0.769818	0.384909	+	0.922955i	$0.374233\pi$
$678$	0	0
$679$	−16.1303	−0.619026
$680$	0	0
$681$	0	0
$682$	0	0
$683$	45.6087	1.74517	0.872584	−	0.488465i	$-0.162443\pi$
$683$	45.6087	1.74517	0.872584	+	0.488465i	$0.162443\pi$
$684$	0	0
$685$	−5.84918	−0.223485
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−7.17074	−0.273183
$690$	0	0
$691$	11.0404	0.419997	0.209998	−	0.977702i	$-0.432654\pi$
$691$	11.0404	0.419997	0.209998	+	0.977702i	$0.432654\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.68611	0.101890
$696$	0	0
$697$	−43.8178	−1.65972
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−36.9545	−1.39575	−0.697877	−	0.716218i	$-0.745872\pi$
$701$	−36.9545	−1.39575	−0.697877	+	0.716218i	$0.745872\pi$
$702$	0	0
$703$	17.3961	0.656105
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.77085	−0.104208
$708$	0	0
$709$	46.7107	1.75426	0.877128	−	0.480257i	$-0.159457\pi$
$709$	46.7107	1.75426	0.877128	+	0.480257i	$0.159457\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−29.2223	−1.09438
$714$	0	0
$715$	−0.833087	−0.0311557
$716$	0	0
$717$	0	0
$718$	0	0
$719$	20.4368	0.762165	0.381082	−	0.924541i	$-0.375551\pi$
$719$	20.4368	0.762165	0.381082	+	0.924541i	$0.375551\pi$
$720$	0	0
$721$	18.5668	0.691464
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−36.4475	−1.35363
$726$	0	0
$727$	−28.4885	−1.05658	−0.528290	−	0.849064i	$-0.677166\pi$
$727$	−28.4885	−1.05658	−0.528290	+	0.849064i	$0.677166\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.08408	−0.0400962
$732$	0	0
$733$	−18.5353	−0.684616	−0.342308	−	0.939588i	$-0.611209\pi$
$733$	−18.5353	−0.684616	−0.342308	+	0.939588i	$0.611209\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−20.5829	−0.758181
$738$	0	0
$739$	−30.6888	−1.12891	−0.564453	−	0.825465i	$-0.690913\pi$
$739$	−30.6888	−1.12891	−0.564453	+	0.825465i	$0.690913\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−9.23097	−0.338651	−0.169326	−	0.985560i	$-0.554159\pi$
$743$	−9.23097	−0.338651	−0.169326	+	0.985560i	$0.554159\pi$
$744$	0	0
$745$	2.22780	0.0816203
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−12.1858	−0.445260
$750$	0	0
$751$	3.66235	0.133641	0.0668205	−	0.997765i	$-0.478715\pi$
$751$	3.66235	0.133641	0.0668205	+	0.997765i	$0.478715\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.15430	0.0420094
$756$	0	0
$757$	20.7561	0.754394	0.377197	−	0.926133i	$-0.376888\pi$
$757$	20.7561	0.754394	0.377197	+	0.926133i	$0.376888\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−23.1425	−0.838915	−0.419457	−	0.907775i	$-0.637780\pi$
$761$	−23.1425	−0.838915	−0.419457	+	0.907775i	$0.637780\pi$
$762$	0	0
$763$	10.1085	0.365952
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7.28757	−0.263139
$768$	0	0
$769$	9.90951	0.357346	0.178673	−	0.983908i	$-0.442820\pi$
$769$	9.90951	0.357346	0.178673	+	0.983908i	$0.442820\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−35.8160	−1.28821	−0.644106	−	0.764936i	$-0.722770\pi$
$773$	−35.8160	−1.28821	−0.644106	+	0.764936i	$0.722770\pi$
$774$	0	0
$775$	−21.7440	−0.781066
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−52.1973	−1.87016
$780$	0	0
$781$	−10.7490	−0.384630
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.11178	0.146756
$786$	0	0
$787$	−7.30108	−0.260255	−0.130128	−	0.991497i	$-0.541539\pi$
$787$	−7.30108	−0.260255	−0.130128	+	0.991497i	$0.541539\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.95667	−0.247351
$792$	0	0
$793$	2.99742	0.106442
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−12.7756	−0.452535	−0.226267	−	0.974065i	$-0.572652\pi$
$797$	−12.7756	−0.452535	−0.226267	+	0.974065i	$0.572652\pi$
$798$	0	0
$799$	−21.2574	−0.752033
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−21.6807	−0.765096
$804$	0	0
$805$	−2.07057	−0.0729780
$806$	0	0
$807$	0	0
$808$	0	0
$809$	21.8948	0.769779	0.384890	−	0.922963i	$-0.374239\pi$
$809$	21.8948	0.769779	0.384890	+	0.922963i	$0.374239\pi$
$810$	0	0
$811$	−21.4146	−0.751969	−0.375985	−	0.926626i	$-0.622695\pi$
$811$	−21.4146	−0.751969	−0.375985	+	0.926626i	$0.622695\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.59370	−0.0908533
$816$	0	0
$817$	−1.29140	−0.0451802
$818$	0	0
$819$	0	0
$820$	0	0
$821$	20.7405	0.723848	0.361924	−	0.932208i	$-0.382120\pi$
$821$	20.7405	0.723848	0.361924	+	0.932208i	$0.382120\pi$
$822$	0	0
$823$	−19.7459	−0.688298	−0.344149	−	0.938915i	$-0.611833\pi$
$823$	−19.7459	−0.688298	−0.344149	+	0.938915i	$0.611833\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	53.5532	1.86223	0.931113	−	0.364730i	$-0.118839\pi$
$827$	53.5532	1.86223	0.931113	+	0.364730i	$0.118839\pi$
$828$	0	0
$829$	−37.5386	−1.30377	−0.651884	−	0.758319i	$-0.726021\pi$
$829$	−37.5386	−1.30377	−0.651884	+	0.758319i	$0.726021\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−5.22915	−0.181179
$834$	0	0
$835$	3.44423	0.119192
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−44.7677	−1.54555	−0.772777	−	0.634678i	$-0.781133\pi$
$839$	−44.7677	−1.54555	−0.772777	+	0.634678i	$0.781133\pi$
$840$	0	0
$841$	26.3011	0.906934
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.07632	0.105829
$846$	0	0
$847$	8.81452	0.302871
$848$	0	0
$849$	0	0
$850$	0	0
$851$	18.3950	0.630571
$852$	0	0
$853$	15.3011	0.523899	0.261949	−	0.965082i	$-0.415635\pi$
$853$	15.3011	0.523899	0.261949	+	0.965082i	$0.415635\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.8346	0.677535	0.338768	−	0.940870i	$-0.389990\pi$
$857$	19.8346	0.677535	0.338768	+	0.940870i	$0.389990\pi$
$858$	0	0
$859$	21.2452	0.724878	0.362439	−	0.932007i	$-0.381944\pi$
$859$	21.2452	0.724878	0.362439	+	0.932007i	$0.381944\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−17.7039	−0.602648	−0.301324	−	0.953522i	$-0.597429\pi$
$863$	−17.7039	−0.602648	−0.301324	+	0.953522i	$0.597429\pi$
$864$	0	0
$865$	−0.772199	−0.0262556
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.16534	0.0734541
$870$	0	0
$871$	−24.9596	−0.845724
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.11244	−0.105220
$876$	0	0
$877$	−52.9058	−1.78650	−0.893251	−	0.449558i	$-0.851582\pi$
$877$	−52.9058	−1.78650	−0.893251	+	0.449558i	$0.851582\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	24.8543	0.837362	0.418681	−	0.908133i	$-0.362493\pi$
$881$	24.8543	0.837362	0.418681	+	0.908133i	$0.362493\pi$
$882$	0	0
$883$	26.5886	0.894779	0.447390	−	0.894339i	$-0.352354\pi$
$883$	26.5886	0.894779	0.447390	+	0.894339i	$0.352354\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−10.3465	−0.347403	−0.173701	−	0.984798i	$-0.555573\pi$
$887$	−10.3465	−0.347403	−0.173701	+	0.984798i	$0.555573\pi$
$888$	0	0
$889$	−7.05008	−0.236452
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−25.3226	−0.847387
$894$	0	0
$895$	−0.673383	−0.0225087
$896$	0	0
$897$	0	0
$898$	0	0
$899$	32.9917	1.10033
$900$	0	0
$901$	−20.9166	−0.696834
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.90547	0.196304
$906$	0	0
$907$	11.9563	0.397003	0.198502	−	0.980101i	$-0.436393\pi$
$907$	11.9563	0.397003	0.198502	+	0.980101i	$0.436393\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	56.4330	1.86971	0.934854	−	0.355032i	$-0.115530\pi$
$911$	56.4330	1.86971	0.934854	+	0.355032i	$0.115530\pi$
$912$	0	0
$913$	18.1528	0.600771
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.8078	0.356904
$918$	0	0
$919$	56.3910	1.86017	0.930084	−	0.367347i	$-0.119734\pi$
$919$	56.3910	1.86017	0.930084	+	0.367347i	$0.119734\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−13.0347	−0.429041
$924$	0	0
$925$	13.6875	0.450041
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−33.4898	−1.09877	−0.549383	−	0.835571i	$-0.685137\pi$
$929$	−33.4898	−1.09877	−0.549383	+	0.835571i	$0.685137\pi$
$930$	0	0
$931$	−6.22915	−0.204152
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.43006	−0.0794716
$936$	0	0
$937$	4.18683	0.136778	0.0683889	−	0.997659i	$-0.478214\pi$
$937$	4.18683	0.136778	0.0683889	+	0.997659i	$0.478214\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−51.3176	−1.67291	−0.836453	−	0.548038i	$-0.815375\pi$
$941$	−51.3176	−1.67291	−0.836453	+	0.548038i	$0.815375\pi$
$942$	0	0
$943$	−55.1945	−1.79738
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19.5367	−0.634859	−0.317430	−	0.948282i	$-0.602820\pi$
$947$	−19.5367	−0.634859	−0.317430	+	0.948282i	$0.602820\pi$
$948$	0	0
$949$	−26.2908	−0.853437
$950$	0	0
$951$	0	0
$952$	0	0
$953$	19.4500	0.630047	0.315023	−	0.949084i	$-0.397988\pi$
$953$	19.4500	0.630047	0.315023	+	0.949084i	$0.397988\pi$
$954$	0	0
$955$	6.41791	0.207679
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−18.6072	−0.600858
$960$	0	0
$961$	−11.3177	−0.365088
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6.82633	−0.219747
$966$	0	0
$967$	48.8923	1.57227	0.786135	−	0.618054i	$-0.212079\pi$
$967$	48.8923	1.57227	0.786135	+	0.618054i	$0.212079\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	9.05201	0.290493	0.145246	−	0.989396i	$-0.453603\pi$
$971$	9.05201	0.290493	0.145246	+	0.989396i	$0.453603\pi$
$972$	0	0
$973$	8.54497	0.273939
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−12.3581	−0.395372	−0.197686	−	0.980265i	$-0.563343\pi$
$977$	−12.3581	−0.395372	−0.197686	+	0.980265i	$0.563343\pi$
$978$	0	0
$979$	−18.3011	−0.584905
$980$	0	0
$981$	0	0
$982$	0	0
$983$	14.7513	0.470493	0.235246	−	0.971936i	$-0.424410\pi$
$983$	14.7513	0.470493	0.235246	+	0.971936i	$0.424410\pi$
$984$	0	0
$985$	−7.78300	−0.247987
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.36555	−0.0434219
$990$	0	0
$991$	45.9969	1.46114	0.730569	−	0.682838i	$-0.239255\pi$
$991$	45.9969	1.46114	0.730569	+	0.682838i	$0.239255\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.27314	0.0403614
$996$	0	0
$997$	38.9166	1.23250	0.616251	−	0.787550i	$-0.288651\pi$
$997$	38.9166	1.23250	0.616251	+	0.787550i	$0.288651\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.a.br.1.2	yes	4
3.2	odd	2		6048.2.a.bm.1.3	✓	4
4.3	odd	2		6048.2.a.bv.1.2	yes	4
12.11	even	2		6048.2.a.bo.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.a.bm.1.3	✓	4	3.2	odd	2
6048.2.a.bo.1.3	yes	4	12.11	even	2
6048.2.a.br.1.2	yes	4	1.1	even	1	trivial
6048.2.a.bv.1.2	yes	4	4.3	odd	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:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.a (trivial)

\(f(q)\)	\(=\)	\(q-0.314350 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-0.314350 q^{5} -1.00000 q^{7} -1.47834 q^{11} -1.79269 q^{13} -5.22915 q^{17} -6.22915 q^{19} -6.58683 q^{23} -4.90118 q^{25} +7.43647 q^{29} +4.43647 q^{31} +0.314350 q^{35} -2.79269 q^{37} +8.37952 q^{41} +0.207315 q^{43} +4.06517 q^{47} +1.00000 q^{49} +4.00000 q^{53} +0.464715 q^{55} +4.06517 q^{59} -1.67203 q^{61} +0.563530 q^{65} +13.9230 q^{67} +7.27102 q^{71} +14.6656 q^{73} +1.47834 q^{77} -1.46471 q^{79} -12.2792 q^{83} +1.64378 q^{85} +12.3795 q^{89} +1.79269 q^{91} +1.95813 q^{95} +16.1303 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5} - 4 q^{7}+O(q^{10}) \) 4 * q + 2 * q^5 - 4 * q^7	\( 4 q + 2 q^{5} - 4 q^{7} - 2 q^{11} + 8 q^{17} + 4 q^{19} - 2 q^{23} + 8 q^{25} + 8 q^{29} - 4 q^{31} - 2 q^{35} - 4 q^{37} + 2 q^{41} + 8 q^{43} - 12 q^{47} + 4 q^{49} + 16 q^{53} - 4 q^{55} - 12 q^{59} - 8 q^{61} + 24 q^{65} - 8 q^{67} + 18 q^{71} + 8 q^{73} + 2 q^{77} - 8 q^{85} + 18 q^{89} - 10 q^{95} + 8 q^{97}+O(q^{100}) \) 4 * q + 2 * q^5 - 4 * q^7 - 2 * q^11 + 8 * q^17 + 4 * q^19 - 2 * q^23 + 8 * q^25 + 8 * q^29 - 4 * q^31 - 2 * q^35 - 4 * q^37 + 2 * q^41 + 8 * q^43 - 12 * q^47 + 4 * q^49 + 16 * q^53 - 4 * q^55 - 12 * q^59 - 8 * q^61 + 24 * q^65 - 8 * q^67 + 18 * q^71 + 8 * q^73 + 2 * q^77 - 8 * q^85 + 18 * q^89 - 10 * q^95 + 8 * q^97

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