LMFDB - Embedded newform 6048.2.a.bq.1.1

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:	$1$
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			1.1
Root			$3.31526$ 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)

$f(q)$	$=$	$q-2.58060 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-2.58060 q^{5} -1.00000 q^{7} -4.04992 q^{11} -0.190188 q^{13} +6.82071 q^{17} +1.34049 q^{19} +0.240108 q^{23} +1.65951 q^{25} -0.190188 q^{29} -2.46932 q^{31} +2.58060 q^{35} +10.4512 q^{37} +2.77079 q^{41} +11.4512 q^{43} +4.29003 q^{47} +1.00000 q^{49} -9.26104 q^{53} +10.4512 q^{55} -12.9710 q^{59} -13.6414 q^{61} +0.490799 q^{65} +0.871175 q^{67} -11.0608 q^{71} +12.5126 q^{73} +4.04992 q^{77} +15.0709 q^{79} -4.19019 q^{83} -17.6015 q^{85} +10.3904 q^{89} +0.190188 q^{91} -3.45928 q^{95} -10.3224 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.58060	−1.15408	−0.577040	−	0.816716i	$-0.695792\pi$
$5$	−2.58060	−1.15408	−0.577040	+	0.816716i	$0.695792\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.04992	−1.22110	−0.610548	−	0.791979i	$-0.709051\pi$
$11$	−4.04992	−1.22110	−0.610548	+	0.791979i	$0.709051\pi$
$12$	0	0
$13$	−0.190188	−0.0527486	−0.0263743	−	0.999652i	$-0.508396\pi$
$13$	−0.190188	−0.0527486	−0.0263743	+	0.999652i	$0.508396\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.82071	1.65427	0.827133	−	0.562007i	$-0.189971\pi$
$17$	6.82071	1.65427	0.827133	+	0.562007i	$0.189971\pi$
$18$	0	0
$19$	1.34049	0.307530	0.153765	−	0.988107i	$-0.450860\pi$
$19$	1.34049	0.307530	0.153765	+	0.988107i	$0.450860\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.240108	0.0500661	0.0250330	−	0.999687i	$-0.492031\pi$
$23$	0.240108	0.0500661	0.0250330	+	0.999687i	$0.492031\pi$
$24$	0	0
$25$	1.65951	0.331901
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.190188	−0.0353170	−0.0176585	−	0.999844i	$-0.505621\pi$
$29$	−0.190188	−0.0353170	−0.0176585	+	0.999844i	$0.505621\pi$
$30$	0	0
$31$	−2.46932	−0.443503	−0.221751	−	0.975103i	$-0.571177\pi$
$31$	−2.46932	−0.443503	−0.221751	+	0.975103i	$0.571177\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.58060	0.436201
$36$	0	0
$37$	10.4512	1.71817	0.859086	−	0.511831i	$-0.171033\pi$
$37$	10.4512	1.71817	0.859086	+	0.511831i	$0.171033\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.77079	0.432725	0.216362	−	0.976313i	$-0.430581\pi$
$41$	2.77079	0.432725	0.216362	+	0.976313i	$0.430581\pi$
$42$	0	0
$43$	11.4512	1.74630	0.873148	−	0.487455i	$-0.162075\pi$
$43$	11.4512	1.74630	0.873148	+	0.487455i	$0.162075\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.29003	0.625765	0.312883	−	0.949792i	$-0.398705\pi$
$47$	4.29003	0.625765	0.312883	+	0.949792i	$0.398705\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.26104	−1.27210	−0.636051	−	0.771647i	$-0.719433\pi$
$53$	−9.26104	−1.27210	−0.636051	+	0.771647i	$0.719433\pi$
$54$	0	0
$55$	10.4512	1.40924
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.9710	−1.68868	−0.844341	−	0.535806i	$-0.820008\pi$
$59$	−12.9710	−1.68868	−0.844341	+	0.535806i	$0.820008\pi$
$60$	0	0
$61$	−13.6414	−1.74660	−0.873302	−	0.487178i	$-0.838026\pi$
$61$	−13.6414	−1.74660	−0.873302	+	0.487178i	$0.838026\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.490799	0.0608761
$66$	0	0
$67$	0.871175	0.106431	0.0532155	−	0.998583i	$-0.483053\pi$
$67$	0.871175	0.106431	0.0532155	+	0.998583i	$0.483053\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.0608	−1.31268	−0.656339	−	0.754466i	$-0.727896\pi$
$71$	−11.0608	−1.31268	−0.656339	+	0.754466i	$0.727896\pi$
$72$	0	0
$73$	12.5126	1.46449	0.732244	−	0.681042i	$-0.238473\pi$
$73$	12.5126	1.46449	0.732244	+	0.681042i	$0.238473\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.04992	0.461531
$78$	0	0
$79$	15.0709	1.69560	0.847802	−	0.530313i	$-0.177926\pi$
$79$	15.0709	1.69560	0.847802	+	0.530313i	$0.177926\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−4.19019	−0.459933	−0.229966	−	0.973199i	$-0.573862\pi$
$83$	−4.19019	−0.459933	−0.229966	+	0.973199i	$0.573862\pi$
$84$	0	0
$85$	−17.6015	−1.90915
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.3904	1.10138	0.550691	−	0.834709i	$-0.314364\pi$
$89$	10.3904	1.10138	0.550691	+	0.834709i	$0.314364\pi$
$90$	0	0
$91$	0.190188	0.0199371
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.45928	−0.354915
$96$	0	0
$97$	−10.3224	−1.04808	−0.524041	−	0.851693i	$-0.675576\pi$
$97$	−10.3224	−1.04808	−0.524041	+	0.851693i	$0.675576\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.50170	−0.746447	−0.373223	−	0.927742i	$-0.621748\pi$
$101$	−7.50170	−0.746447	−0.373223	+	0.927742i	$0.621748\pi$
$102$	0	0
$103$	−2.08894	−0.205830	−0.102915	−	0.994690i	$-0.532817\pi$
$103$	−2.08894	−0.205830	−0.102915	+	0.994690i	$0.532817\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.65951	−0.933820	−0.466910	−	0.884305i	$-0.654633\pi$
$107$	−9.65951	−0.933820	−0.466910	+	0.884305i	$0.654633\pi$
$108$	0	0
$109$	−2.46932	−0.236518	−0.118259	−	0.992983i	$-0.537731\pi$
$109$	−2.46932	−0.236518	−0.118259	+	0.992983i	$0.537731\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.83880	−0.643340	−0.321670	−	0.946852i	$-0.604244\pi$
$113$	−6.83880	−0.643340	−0.321670	+	0.946852i	$0.604244\pi$
$114$	0	0
$115$	−0.619624	−0.0577803
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.82071	−0.625253
$120$	0	0
$121$	5.40186	0.491078
$122$	0	0
$123$	0	0
$124$	0	0
$125$	8.62048	0.771040
$126$	0	0
$127$	−13.4512	−1.19360	−0.596802	−	0.802389i	$-0.703562\pi$
$127$	−13.4512	−1.19360	−0.596802	+	0.802389i	$0.703562\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−13.3514	−1.16652	−0.583258	−	0.812287i	$-0.698222\pi$
$131$	−13.3514	−1.16652	−0.583258	+	0.812287i	$0.698222\pi$
$132$	0	0
$133$	−1.34049	−0.116236
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.77025	0.749293	0.374646	−	0.927168i	$-0.377764\pi$
$137$	8.77025	0.749293	0.374646	+	0.927168i	$0.377764\pi$
$138$	0	0
$139$	−1.06136	−0.0900236	−0.0450118	−	0.998986i	$-0.514333\pi$
$139$	−1.06136	−0.0900236	−0.0450118	+	0.998986i	$0.514333\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.770246	0.0644112
$144$	0	0
$145$	0.490799	0.0407587
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−9.54158	−0.781677	−0.390838	−	0.920459i	$-0.627815\pi$
$149$	−9.54158	−0.781677	−0.390838	+	0.920459i	$0.627815\pi$
$150$	0	0
$151$	−7.64142	−0.621850	−0.310925	−	0.950434i	$-0.600639\pi$
$151$	−7.64142	−0.621850	−0.310925	+	0.950434i	$0.600639\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.37233	0.511838
$156$	0	0
$157$	19.7736	1.57811	0.789054	−	0.614324i	$-0.210571\pi$
$157$	19.7736	1.57811	0.789054	+	0.614324i	$0.210571\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.240108	−0.0189232
$162$	0	0
$163$	16.3899	1.28375	0.641877	−	0.766808i	$-0.278156\pi$
$163$	16.3899	1.28375	0.641877	+	0.766808i	$0.278156\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.77743	−0.447071	−0.223536	−	0.974696i	$-0.571760\pi$
$167$	−5.77743	−0.447071	−0.223536	+	0.974696i	$0.571760\pi$
$168$	0	0
$169$	−12.9638	−0.997218
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0.240108	0.0182551	0.00912755	−	0.999958i	$-0.497095\pi$
$173$	0.240108	0.0182551	0.00912755	+	0.999958i	$0.497095\pi$
$174$	0	0
$175$	−1.65951	−0.125447
$176$	0	0
$177$	0	0
$178$	0	0
$179$	7.30093	0.545697	0.272848	−	0.962057i	$-0.412034\pi$
$179$	7.30093	0.545697	0.272848	+	0.962057i	$0.412034\pi$
$180$	0	0
$181$	−18.1997	−1.35277	−0.676386	−	0.736547i	$-0.736455\pi$
$181$	−18.1997	−1.35277	−0.676386	+	0.736547i	$0.736455\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−26.9705	−1.98291
$186$	0	0
$187$	−27.6233	−2.02002
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.1821	−1.31561	−0.657807	−	0.753187i	$-0.728516\pi$
$191$	−18.1821	−1.31561	−0.657807	+	0.753187i	$0.728516\pi$
$192$	0	0
$193$	8.93864	0.643417	0.321709	−	0.946839i	$-0.395743\pi$
$193$	8.93864	0.643417	0.321709	+	0.946839i	$0.395743\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.54158	−0.394821	−0.197411	−	0.980321i	$-0.563253\pi$
$197$	−5.54158	−0.394821	−0.197411	+	0.980321i	$0.563253\pi$
$198$	0	0
$199$	−18.8316	−1.33494	−0.667469	−	0.744638i	$-0.732622\pi$
$199$	−18.8316	−1.33494	−0.667469	+	0.744638i	$0.732622\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.190188	0.0133486
$204$	0	0
$205$	−7.15031	−0.499399
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.42889	−0.375524
$210$	0	0
$211$	−12.7028	−0.874496	−0.437248	−	0.899341i	$-0.644047\pi$
$211$	−12.7028	−0.874496	−0.437248	+	0.899341i	$0.644047\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−29.5511	−2.01537
$216$	0	0
$217$	2.46932	0.167628
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.29722	−0.0872602
$222$	0	0
$223$	−8.45123	−0.565936	−0.282968	−	0.959129i	$-0.591319\pi$
$223$	−8.45123	−0.565936	−0.282968	+	0.959129i	$0.591319\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.54158	−0.633297	−0.316648	−	0.948543i	$-0.602558\pi$
$227$	−9.54158	−0.633297	−0.316648	+	0.948543i	$0.602558\pi$
$228$	0	0
$229$	−2.35858	−0.155859	−0.0779297	−	0.996959i	$-0.524831\pi$
$229$	−2.35858	−0.155859	−0.0779297	+	0.996959i	$0.524831\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	15.4189	1.01012	0.505061	−	0.863083i	$-0.331470\pi$
$233$	15.4189	1.01012	0.505061	+	0.863083i	$0.331470\pi$
$234$	0	0
$235$	−11.0709	−0.722183
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−9.96012	−0.644266	−0.322133	−	0.946694i	$-0.604400\pi$
$239$	−9.96012	−0.644266	−0.322133	+	0.946694i	$0.604400\pi$
$240$	0	0
$241$	21.7736	1.40256	0.701282	−	0.712884i	$-0.252612\pi$
$241$	21.7736	1.40256	0.701282	+	0.712884i	$0.252612\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.58060	−0.164869
$246$	0	0
$247$	−0.254946	−0.0162218
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−15.0034	−0.947006	−0.473503	−	0.880792i	$-0.657011\pi$
$251$	−15.0034	−0.947006	−0.473503	+	0.880792i	$0.657011\pi$
$252$	0	0
$253$	−0.972420	−0.0611355
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.66954	0.228900	0.114450	−	0.993429i	$-0.463489\pi$
$257$	3.66954	0.228900	0.114450	+	0.993429i	$0.463489\pi$
$258$	0	0
$259$	−10.4512	−0.649408
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−20.6406	−1.27275	−0.636376	−	0.771379i	$-0.719567\pi$
$263$	−20.6406	−1.27275	−0.636376	+	0.771379i	$0.719567\pi$
$264$	0	0
$265$	23.8991	1.46811
$266$	0	0
$267$	0	0
$268$	0	0
$269$	17.7019	1.07931	0.539653	−	0.841888i	$-0.318556\pi$
$269$	17.7019	1.07931	0.539653	+	0.841888i	$0.318556\pi$
$270$	0	0
$271$	−4.30061	−0.261244	−0.130622	−	0.991432i	$-0.541697\pi$
$271$	−4.30061	−0.261244	−0.130622	+	0.991432i	$0.541697\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.72087	−0.405284
$276$	0	0
$277$	−18.1936	−1.09315	−0.546573	−	0.837411i	$-0.684068\pi$
$277$	−18.1936	−1.09315	−0.546573	+	0.837411i	$0.684068\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−17.3514	−1.03510	−0.517549	−	0.855654i	$-0.673155\pi$
$281$	−17.3514	−1.03510	−0.517549	+	0.855654i	$0.673155\pi$
$282$	0	0
$283$	4.38038	0.260386	0.130193	−	0.991489i	$-0.458440\pi$
$283$	4.38038	0.260386	0.130193	+	0.991489i	$0.458440\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.77079	−0.163555
$288$	0	0
$289$	29.5221	1.73659
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4.56306	−0.266577	−0.133288	−	0.991077i	$-0.542554\pi$
$293$	−4.56306	−0.266577	−0.133288	+	0.991077i	$0.542554\pi$
$294$	0	0
$295$	33.4730	1.94888
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.0456657	−0.00264092
$300$	0	0
$301$	−11.4512	−0.660038
$302$	0	0
$303$	0	0
$304$	0	0
$305$	35.2031	2.01572
$306$	0	0
$307$	8.65951	0.494224	0.247112	−	0.968987i	$-0.420518\pi$
$307$	8.65951	0.494224	0.247112	+	0.968987i	$0.420518\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	20.8350	1.18144	0.590722	−	0.806875i	$-0.298843\pi$
$311$	20.8350	1.18144	0.590722	+	0.806875i	$0.298843\pi$
$312$	0	0
$313$	20.1902	1.14122	0.570608	−	0.821222i	$-0.306707\pi$
$313$	20.1902	1.14122	0.570608	+	0.821222i	$0.306707\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.4128	0.809501	0.404750	−	0.914427i	$-0.367358\pi$
$317$	14.4128	0.809501	0.404750	+	0.914427i	$0.367358\pi$
$318$	0	0
$319$	0.770246	0.0431255
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9.14312	0.508737
$324$	0	0
$325$	−0.315618	−0.0175073
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4.29003	−0.236517
$330$	0	0
$331$	−18.9386	−1.04096	−0.520481	−	0.853873i	$-0.674247\pi$
$331$	−18.9386	−1.04096	−0.520481	+	0.853873i	$0.674247\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.24816	−0.122830
$336$	0	0
$337$	2.61962	0.142700	0.0713500	−	0.997451i	$-0.477269\pi$
$337$	2.61962	0.142700	0.0713500	+	0.997451i	$0.477269\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	10.0005	0.541560
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.8539	−1.17318	−0.586591	−	0.809884i	$-0.699530\pi$
$347$	−21.8539	−1.17318	−0.586591	+	0.809884i	$0.699530\pi$
$348$	0	0
$349$	1.61962	0.0866965	0.0433482	−	0.999060i	$-0.486197\pi$
$349$	1.61962	0.0866965	0.0433482	+	0.999060i	$0.486197\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4.74899	−0.252763	−0.126382	−	0.991982i	$-0.540336\pi$
$353$	−4.74899	−0.252763	−0.126382	+	0.991982i	$0.540336\pi$
$354$	0	0
$355$	28.5436	1.51494
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.92055	0.154141	0.0770704	−	0.997026i	$-0.475443\pi$
$359$	2.92055	0.154141	0.0770704	+	0.997026i	$0.475443\pi$
$360$	0	0
$361$	−17.2031	−0.905425
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−32.2900	−1.69014
$366$	0	0
$367$	−7.13222	−0.372299	−0.186149	−	0.982521i	$-0.559601\pi$
$367$	−7.13222	−0.372299	−0.186149	+	0.982521i	$0.559601\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	9.26104	0.480809
$372$	0	0
$373$	−13.7883	−0.713933	−0.356966	−	0.934117i	$-0.616189\pi$
$373$	−13.7883	−0.713933	−0.356966	+	0.934117i	$0.616189\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.0361714	0.00186292
$378$	0	0
$379$	−7.83161	−0.402283	−0.201141	−	0.979562i	$-0.564465\pi$
$379$	−7.83161	−0.402283	−0.201141	+	0.979562i	$0.564465\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0.780828	0.0398985	0.0199492	−	0.999801i	$-0.493650\pi$
$383$	0.780828	0.0398985	0.0199492	+	0.999801i	$0.493650\pi$
$384$	0	0
$385$	−10.4512	−0.532644
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−33.8411	−1.71581	−0.857906	−	0.513807i	$-0.828235\pi$
$389$	−33.8411	−1.71581	−0.857906	+	0.513807i	$0.828235\pi$
$390$	0	0
$391$	1.63771	0.0828226
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−38.8919	−1.95686
$396$	0	0
$397$	24.6019	1.23473	0.617366	−	0.786676i	$-0.288200\pi$
$397$	24.6019	1.23473	0.617366	+	0.786676i	$0.288200\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.1506	0.656711	0.328355	−	0.944554i	$-0.393506\pi$
$401$	13.1506	0.656711	0.328355	+	0.944554i	$0.393506\pi$
$402$	0	0
$403$	0.469634	0.0233942
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−42.3267	−2.09805
$408$	0	0
$409$	19.1718	0.947984	0.473992	−	0.880529i	$-0.342813\pi$
$409$	19.1718	0.947984	0.473992	+	0.880529i	$0.342813\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	12.9710	0.638262
$414$	0	0
$415$	10.8132	0.530799
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−32.3643	−1.58110	−0.790549	−	0.612398i	$-0.790205\pi$
$419$	−32.3643	−1.58110	−0.790549	+	0.612398i	$0.790205\pi$
$420$	0	0
$421$	−29.1322	−1.41982	−0.709909	−	0.704294i	$-0.751264\pi$
$421$	−29.1322	−1.41982	−0.709909	+	0.704294i	$0.751264\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	11.3190	0.549053
$426$	0	0
$427$	13.6414	0.660155
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−0.240108	−0.0115656	−0.00578281	−	0.999983i	$-0.501841\pi$
$431$	−0.240108	−0.0115656	−0.00578281	+	0.999983i	$0.501841\pi$
$432$	0	0
$433$	−25.6509	−1.23270	−0.616352	−	0.787471i	$-0.711390\pi$
$433$	−25.6509	−1.23270	−0.616352	+	0.787471i	$0.711390\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.321864	0.0153968
$438$	0	0
$439$	−18.5221	−0.884011	−0.442006	−	0.897012i	$-0.645733\pi$
$439$	−18.5221	−0.884011	−0.442006	+	0.897012i	$0.645733\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24.6947	1.17328	0.586641	−	0.809847i	$-0.300450\pi$
$443$	24.6947	1.17328	0.586641	+	0.809847i	$0.300450\pi$
$444$	0	0
$445$	−26.8135	−1.27108
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−12.4005	−0.585214	−0.292607	−	0.956233i	$-0.594523\pi$
$449$	−12.4005	−0.585214	−0.292607	+	0.956233i	$0.594523\pi$
$450$	0	0
$451$	−11.2215	−0.528399
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.490799	−0.0230090
$456$	0	0
$457$	33.6448	1.57384	0.786919	−	0.617056i	$-0.211675\pi$
$457$	33.6448	1.57384	0.786919	+	0.617056i	$0.211675\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	39.0628	1.81934	0.909668	−	0.415336i	$-0.136336\pi$
$461$	39.0628	1.81934	0.909668	+	0.415336i	$0.136336\pi$
$462$	0	0
$463$	5.61962	0.261166	0.130583	−	0.991437i	$-0.458315\pi$
$463$	5.61962	0.261166	0.130583	+	0.991437i	$0.458315\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9.85169	0.455882	0.227941	−	0.973675i	$-0.426801\pi$
$467$	9.85169	0.455882	0.227941	+	0.973675i	$0.426801\pi$
$468$	0	0
$469$	−0.871175	−0.0402271
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−46.3766	−2.13240
$474$	0	0
$475$	2.22456	0.102070
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−0.960434	−0.0438833	−0.0219417	−	0.999759i	$-0.506985\pi$
$479$	−0.960434	−0.0438833	−0.0219417	+	0.999759i	$0.506985\pi$
$480$	0	0
$481$	−1.98770	−0.0906312
$482$	0	0
$483$	0	0
$484$	0	0
$485$	26.6380	1.20957
$486$	0	0
$487$	−14.5706	−0.660255	−0.330128	−	0.943936i	$-0.607092\pi$
$487$	−14.5706	−0.660255	−0.330128	+	0.943936i	$0.607092\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.01004	−0.0907118	−0.0453559	−	0.998971i	$-0.514442\pi$
$491$	−2.01004	−0.0907118	−0.0453559	+	0.998971i	$0.514442\pi$
$492$	0	0
$493$	−1.29722	−0.0584237
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11.0608	0.496146
$498$	0	0
$499$	−34.0743	−1.52537	−0.762687	−	0.646768i	$-0.776120\pi$
$499$	−34.0743	−1.52537	−0.762687	+	0.646768i	$0.776120\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	23.7736	1.06001	0.530007	−	0.847993i	$-0.322189\pi$
$503$	23.7736	1.06001	0.530007	+	0.847993i	$0.322189\pi$
$504$	0	0
$505$	19.3589	0.861460
$506$	0	0
$507$	0	0
$508$	0	0
$509$	18.2045	0.806899	0.403450	−	0.915002i	$-0.367811\pi$
$509$	18.2045	0.806899	0.403450	+	0.915002i	$0.367811\pi$
$510$	0	0
$511$	−12.5126	−0.553525
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.39073	0.237544
$516$	0	0
$517$	−17.3743	−0.764120
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−28.5938	−1.25272	−0.626359	−	0.779535i	$-0.715455\pi$
$521$	−28.5938	−1.25272	−0.626359	+	0.779535i	$0.715455\pi$
$522$	0	0
$523$	33.2215	1.45267	0.726337	−	0.687339i	$-0.241221\pi$
$523$	33.2215	1.45267	0.726337	+	0.687339i	$0.241221\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−16.8425	−0.733671
$528$	0	0
$529$	−22.9423	−0.997493
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−0.526971	−0.0228256
$534$	0	0
$535$	24.9273	1.07770
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.04992	−0.174442
$540$	0	0
$541$	−19.7702	−0.849989	−0.424995	−	0.905196i	$-0.639724\pi$
$541$	−19.7702	−0.849989	−0.424995	+	0.905196i	$0.639724\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.37233	0.272961
$546$	0	0
$547$	−14.2577	−0.609613	−0.304807	−	0.952414i	$-0.598592\pi$
$547$	−14.2577	−0.609613	−0.304807	+	0.952414i	$0.598592\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.254946	−0.0108610
$552$	0	0
$553$	−15.0709	−0.640878
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.83880	0.120284	0.0601418	−	0.998190i	$-0.480845\pi$
$557$	2.83880	0.120284	0.0601418	+	0.998190i	$0.480845\pi$
$558$	0	0
$559$	−2.17789	−0.0921147
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−13.7518	−0.579571	−0.289786	−	0.957092i	$-0.593584\pi$
$563$	−13.7518	−0.579571	−0.289786	+	0.957092i	$0.593584\pi$
$564$	0	0
$565$	17.6482	0.742466
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21.5054	−0.901554	−0.450777	−	0.892637i	$-0.648853\pi$
$569$	−21.5054	−0.901554	−0.450777	+	0.892637i	$0.648853\pi$
$570$	0	0
$571$	27.9420	1.16934	0.584669	−	0.811272i	$-0.301224\pi$
$571$	27.9420	1.16934	0.584669	+	0.811272i	$0.301224\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.398461	0.0166170
$576$	0	0
$577$	1.69939	0.0707465	0.0353732	−	0.999374i	$-0.488738\pi$
$577$	1.69939	0.0707465	0.0353732	+	0.999374i	$0.488738\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4.19019	0.173838
$582$	0	0
$583$	37.5065	1.55336
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−27.4836	−1.13437	−0.567185	−	0.823590i	$-0.691968\pi$
$587$	−27.4836	−1.13437	−0.567185	+	0.823590i	$0.691968\pi$
$588$	0	0
$589$	−3.31011	−0.136391
$590$	0	0
$591$	0	0
$592$	0	0
$593$	41.5915	1.70796	0.853979	−	0.520307i	$-0.174183\pi$
$593$	41.5915	1.70796	0.853979	+	0.520307i	$0.174183\pi$
$594$	0	0
$595$	17.6015	0.721593
$596$	0	0
$597$	0	0
$598$	0	0
$599$	31.9234	1.30435	0.652177	−	0.758066i	$-0.273856\pi$
$599$	31.9234	1.30435	0.652177	+	0.758066i	$0.273856\pi$
$600$	0	0
$601$	25.4512	1.03818	0.519089	−	0.854720i	$-0.326271\pi$
$601$	25.4512	1.03818	0.519089	+	0.854720i	$0.326271\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−13.9400	−0.566743
$606$	0	0
$607$	3.58345	0.145448	0.0727239	−	0.997352i	$-0.476831\pi$
$607$	3.58345	0.145448	0.0727239	+	0.997352i	$0.476831\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.815911	−0.0330082
$612$	0	0
$613$	−40.8135	−1.64844	−0.824221	−	0.566268i	$-0.808387\pi$
$613$	−40.8135	−1.64844	−0.824221	+	0.566268i	$0.808387\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−32.9710	−1.32736	−0.663682	−	0.748015i	$-0.731007\pi$
$617$	−32.9710	−1.32736	−0.663682	+	0.748015i	$0.731007\pi$
$618$	0	0
$619$	7.17789	0.288504	0.144252	−	0.989541i	$-0.453922\pi$
$619$	7.17789	0.288504	0.144252	+	0.989541i	$0.453922\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−10.3904	−0.416283
$624$	0	0
$625$	−30.5436	−1.22174
$626$	0	0
$627$	0	0
$628$	0	0
$629$	71.2848	2.84231
$630$	0	0
$631$	−17.8098	−0.708997	−0.354499	−	0.935057i	$-0.615348\pi$
$631$	−17.8098	−0.708997	−0.354499	+	0.935057i	$0.615348\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	34.7123	1.37751
$636$	0	0
$637$	−0.190188	−0.00753552
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−15.1830	−0.599692	−0.299846	−	0.953988i	$-0.596935\pi$
$641$	−15.1830	−0.599692	−0.299846	+	0.953988i	$0.596935\pi$
$642$	0	0
$643$	−33.1018	−1.30541	−0.652704	−	0.757613i	$-0.726366\pi$
$643$	−33.1018	−1.30541	−0.652704	+	0.757613i	$0.726366\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−0.706576	−0.0277784	−0.0138892	−	0.999904i	$-0.504421\pi$
$647$	−0.706576	−0.0277784	−0.0138892	+	0.999904i	$0.504421\pi$
$648$	0	0
$649$	52.5316	2.06205
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−35.4283	−1.38642	−0.693209	−	0.720736i	$-0.743804\pi$
$653$	−35.4283	−1.38642	−0.693209	+	0.720736i	$0.743804\pi$
$654$	0	0
$655$	34.4546	1.34625
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−28.2496	−1.10045	−0.550224	−	0.835017i	$-0.685458\pi$
$659$	−28.2496	−1.10045	−0.550224	+	0.835017i	$0.685458\pi$
$660$	0	0
$661$	−9.48740	−0.369017	−0.184509	−	0.982831i	$-0.559069\pi$
$661$	−9.48740	−0.369017	−0.184509	+	0.982831i	$0.559069\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.45928	0.134145
$666$	0	0
$667$	−0.0456657	−0.00176818
$668$	0	0
$669$	0	0
$670$	0	0
$671$	55.2467	2.13277
$672$	0	0
$673$	26.3442	1.01549	0.507747	−	0.861506i	$-0.330478\pi$
$673$	26.3442	1.01549	0.507747	+	0.861506i	$0.330478\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−21.5829	−0.829499	−0.414749	−	0.909936i	$-0.636131\pi$
$677$	−21.5829	−0.829499	−0.414749	+	0.909936i	$0.636131\pi$
$678$	0	0
$679$	10.3224	0.396138
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−51.3167	−1.96358	−0.981789	−	0.189975i	$-0.939159\pi$
$683$	−51.3167	−1.96358	−0.981789	+	0.189975i	$0.939159\pi$
$684$	0	0
$685$	−22.6325	−0.864744
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.76134	0.0671017
$690$	0	0
$691$	−2.80371	−0.106658	−0.0533291	−	0.998577i	$-0.516983\pi$
$691$	−2.80371	−0.106658	−0.0533291	+	0.998577i	$0.516983\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.73896	0.103894
$696$	0	0
$697$	18.8988	0.715841
$698$	0	0
$699$	0	0
$700$	0	0
$701$	39.0565	1.47514	0.737571	−	0.675269i	$-0.235972\pi$
$701$	39.0565	1.47514	0.737571	+	0.675269i	$0.235972\pi$
$702$	0	0
$703$	14.0098	0.528390
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.50170	0.282130
$708$	0	0
$709$	−14.1687	−0.532117	−0.266058	−	0.963957i	$-0.585721\pi$
$709$	−14.1687	−0.532117	−0.266058	+	0.963957i	$0.585721\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.592904	−0.0222044
$714$	0	0
$715$	−1.98770	−0.0743357
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8.36979	0.312141	0.156070	−	0.987746i	$-0.450117\pi$
$719$	8.36979	0.312141	0.156070	+	0.987746i	$0.450117\pi$
$720$	0	0
$721$	2.08894	0.0777963
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.315618	−0.0117218
$726$	0	0
$727$	42.9025	1.59116	0.795582	−	0.605846i	$-0.207165\pi$
$727$	42.9025	1.59116	0.795582	+	0.605846i	$0.207165\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	78.1055	2.88884
$732$	0	0
$733$	−0.211984	−0.00782982	−0.00391491	−	0.999992i	$-0.501246\pi$
$733$	−0.211984	−0.00782982	−0.00391491	+	0.999992i	$0.501246\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.52819	−0.129963
$738$	0	0
$739$	39.2610	1.44424	0.722120	−	0.691767i	$-0.243168\pi$
$739$	39.2610	1.44424	0.722120	+	0.691767i	$0.243168\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7.87782	0.289009	0.144505	−	0.989504i	$-0.453841\pi$
$743$	7.87782	0.289009	0.144505	+	0.989504i	$0.453841\pi$
$744$	0	0
$745$	24.6230	0.902118
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.65951	0.352951
$750$	0	0
$751$	−54.3175	−1.98207	−0.991037	−	0.133585i	$-0.957351\pi$
$751$	−54.3175	−1.98207	−0.991037	+	0.133585i	$0.957351\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	19.7195	0.717665
$756$	0	0
$757$	−8.50310	−0.309050	−0.154525	−	0.987989i	$-0.549385\pi$
$757$	−8.50310	−0.309050	−0.154525	+	0.987989i	$0.549385\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−47.9658	−1.73876	−0.869380	−	0.494144i	$-0.835481\pi$
$761$	−47.9658	−1.73876	−0.869380	+	0.494144i	$0.835481\pi$
$762$	0	0
$763$	2.46932	0.0893953
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.46693	0.0890757
$768$	0	0
$769$	33.3715	1.20341	0.601703	−	0.798720i	$-0.294489\pi$
$769$	33.3715	1.20341	0.601703	+	0.798720i	$0.294489\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−14.9811	−0.538831	−0.269416	−	0.963024i	$-0.586831\pi$
$773$	−14.9811	−0.538831	−0.269416	+	0.963024i	$0.586831\pi$
$774$	0	0
$775$	−4.09785	−0.147199
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.71423	0.133076
$780$	0	0
$781$	44.7954	1.60291
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−51.0279	−1.82126
$786$	0	0
$787$	50.4859	1.79963	0.899814	−	0.436273i	$-0.143702\pi$
$787$	50.4859	1.79963	0.899814	+	0.436273i	$0.143702\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	6.83880	0.243160
$792$	0	0
$793$	2.59443	0.0921310
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−50.2427	−1.77969	−0.889844	−	0.456264i	$-0.849187\pi$
$797$	−50.2427	−1.77969	−0.889844	+	0.456264i	$0.849187\pi$
$798$	0	0
$799$	29.2610	1.03518
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−50.6750	−1.78828
$804$	0	0
$805$	0.619624	0.0218389
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24.5344	0.862583	0.431292	−	0.902213i	$-0.358058\pi$
$809$	24.5344	0.862583	0.431292	+	0.902213i	$0.358058\pi$
$810$	0	0
$811$	−6.01840	−0.211335	−0.105667	−	0.994402i	$-0.533698\pi$
$811$	−6.01840	−0.211335	−0.105667	+	0.994402i	$0.533698\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−42.2957	−1.48155
$816$	0	0
$817$	15.3503	0.537039
$818$	0	0
$819$	0	0
$820$	0	0
$821$	45.5149	1.58848	0.794241	−	0.607603i	$-0.207869\pi$
$821$	45.5149	1.58848	0.794241	+	0.607603i	$0.207869\pi$
$822$	0	0
$823$	−12.6380	−0.440534	−0.220267	−	0.975440i	$-0.570693\pi$
$823$	−12.6380	−0.440534	−0.220267	+	0.975440i	$0.570693\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	46.1088	1.60336	0.801680	−	0.597754i	$-0.203940\pi$
$827$	46.1088	1.60336	0.801680	+	0.597754i	$0.203940\pi$
$828$	0	0
$829$	−3.55216	−0.123372	−0.0616858	−	0.998096i	$-0.519648\pi$
$829$	−3.55216	−0.123372	−0.0616858	+	0.998096i	$0.519648\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.82071	0.236324
$834$	0	0
$835$	14.9093	0.515956
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.80981	0.131529	0.0657647	−	0.997835i	$-0.479051\pi$
$839$	3.80981	0.131529	0.0657647	+	0.997835i	$0.479051\pi$
$840$	0	0
$841$	−28.9638	−0.998753
$842$	0	0
$843$	0	0
$844$	0	0
$845$	33.4545	1.15087
$846$	0	0
$847$	−5.40186	−0.185610
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.50943	0.0860221
$852$	0	0
$853$	0.960434	0.0328846	0.0164423	−	0.999865i	$-0.494766\pi$
$853$	0.960434	0.0328846	0.0164423	+	0.999865i	$0.494766\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8.41221	0.287356	0.143678	−	0.989625i	$-0.454107\pi$
$857$	8.41221	0.287356	0.143678	+	0.989625i	$0.454107\pi$
$858$	0	0
$859$	22.7822	0.777320	0.388660	−	0.921381i	$-0.372938\pi$
$859$	22.7822	0.777320	0.388660	+	0.921381i	$0.372938\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−52.0105	−1.77046	−0.885229	−	0.465155i	$-0.845998\pi$
$863$	−52.0105	−1.77046	−0.885229	+	0.465155i	$0.845998\pi$
$864$	0	0
$865$	−0.619624	−0.0210679
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−61.0358	−2.07050
$870$	0	0
$871$	−0.165687	−0.00561409
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−8.62048	−0.291426
$876$	0	0
$877$	37.9638	1.28195	0.640974	−	0.767563i	$-0.278531\pi$
$877$	37.9638	1.28195	0.640974	+	0.767563i	$0.278531\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16.1678	0.544709	0.272354	−	0.962197i	$-0.412198\pi$
$881$	16.1678	0.544709	0.272354	+	0.962197i	$0.412198\pi$
$882$	0	0
$883$	16.8807	0.568080	0.284040	−	0.958812i	$-0.408325\pi$
$883$	16.8807	0.568080	0.284040	+	0.958812i	$0.408325\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−36.7341	−1.23341	−0.616705	−	0.787195i	$-0.711533\pi$
$887$	−36.7341	−1.23341	−0.616705	+	0.787195i	$0.711533\pi$
$888$	0	0
$889$	13.4512	0.451140
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.75076	0.192442
$894$	0	0
$895$	−18.8408	−0.629778
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.469634	0.0156632
$900$	0	0
$901$	−63.1669	−2.10439
$902$	0	0
$903$	0	0
$904$	0	0
$905$	46.9661	1.56121
$906$	0	0
$907$	−42.9890	−1.42743	−0.713713	−	0.700438i	$-0.752988\pi$
$907$	−42.9890	−1.42743	−0.713713	+	0.700438i	$0.752988\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	53.2068	1.76282	0.881410	−	0.472352i	$-0.156595\pi$
$911$	53.2068	1.76282	0.881410	+	0.472352i	$0.156595\pi$
$912$	0	0
$913$	16.9699	0.561623
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13.3514	0.440902
$918$	0	0
$919$	−16.8009	−0.554211	−0.277105	−	0.960840i	$-0.589375\pi$
$919$	−16.8009	−0.554211	−0.277105	+	0.960840i	$0.589375\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.10363	0.0692419
$924$	0	0
$925$	17.3439	0.570264
$926$	0	0
$927$	0	0
$928$	0	0
$929$	57.9229	1.90039	0.950194	−	0.311660i	$-0.100885\pi$
$929$	57.9229	1.90039	0.950194	+	0.311660i	$0.100885\pi$
$930$	0	0
$931$	1.34049	0.0439329
$932$	0	0
$933$	0	0
$934$	0	0
$935$	71.2848	2.33126
$936$	0	0
$937$	−31.5623	−1.03110	−0.515548	−	0.856861i	$-0.672411\pi$
$937$	−31.5623	−1.03110	−0.515548	+	0.856861i	$0.672411\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−29.8846	−0.974210	−0.487105	−	0.873343i	$-0.661947\pi$
$941$	−29.8846	−0.974210	−0.487105	+	0.873343i	$0.661947\pi$
$942$	0	0
$943$	0.665290	0.0216648
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46.8695	1.52306	0.761528	−	0.648132i	$-0.224450\pi$
$947$	46.8695	1.52306	0.761528	+	0.648132i	$0.224450\pi$
$948$	0	0
$949$	−2.37974	−0.0772498
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−31.4730	−1.01951	−0.509756	−	0.860319i	$-0.670264\pi$
$953$	−31.4730	−1.01951	−0.509756	+	0.860319i	$0.670264\pi$
$954$	0	0
$955$	46.9209	1.51832
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.77025	−0.283206
$960$	0	0
$961$	−24.9025	−0.803305
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−23.0671	−0.742555
$966$	0	0
$967$	−51.7252	−1.66337	−0.831685	−	0.555248i	$-0.812623\pi$
$967$	−51.7252	−1.66337	−0.831685	+	0.555248i	$0.812623\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	60.2528	1.93360	0.966802	−	0.255528i	$-0.0822493\pi$
$971$	60.2528	1.93360	0.966802	+	0.255528i	$0.0822493\pi$
$972$	0	0
$973$	1.06136	0.0340257
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−17.7423	−0.567628	−0.283814	−	0.958879i	$-0.591600\pi$
$977$	−17.7423	−0.567628	−0.283814	+	0.958879i	$0.591600\pi$
$978$	0	0
$979$	−42.0804	−1.34489
$980$	0	0
$981$	0	0
$982$	0	0
$983$	42.9928	1.37126	0.685629	−	0.727951i	$-0.259527\pi$
$983$	42.9928	1.37126	0.685629	+	0.727951i	$0.259527\pi$
$984$	0	0
$985$	14.3006	0.455655
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.74954	0.0874302
$990$	0	0
$991$	−31.3933	−0.997240	−0.498620	−	0.866821i	$-0.666160\pi$
$991$	−31.3933	−0.997240	−0.498620	+	0.866821i	$0.666160\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	48.5969	1.54062
$996$	0	0
$997$	7.36197	0.233156	0.116578	−	0.993182i	$-0.462807\pi$
$997$	7.36197	0.233156	0.116578	+	0.993182i	$0.462807\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.a.bq.1.1	yes	4
3.2	odd	2		6048.2.a.bl.1.4	✓	4
4.3	odd	2		6048.2.a.bu.1.1	yes	4
12.11	even	2		6048.2.a.bn.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.a.bl.1.4	✓	4	3.2	odd	2
6048.2.a.bn.1.4	yes	4	12.11	even	2
6048.2.a.bq.1.1	yes	4	1.1	even	1	trivial
6048.2.a.bu.1.1	yes	4	4.3	odd	2

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-2.58060 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-2.58060 q^{5} -1.00000 q^{7} -4.04992 q^{11} -0.190188 q^{13} +6.82071 q^{17} +1.34049 q^{19} +0.240108 q^{23} +1.65951 q^{25} -0.190188 q^{29} -2.46932 q^{31} +2.58060 q^{35} +10.4512 q^{37} +2.77079 q^{41} +11.4512 q^{43} +4.29003 q^{47} +1.00000 q^{49} -9.26104 q^{53} +10.4512 q^{55} -12.9710 q^{59} -13.6414 q^{61} +0.490799 q^{65} +0.871175 q^{67} -11.0608 q^{71} +12.5126 q^{73} +4.04992 q^{77} +15.0709 q^{79} -4.19019 q^{83} -17.6015 q^{85} +10.3904 q^{89} +0.190188 q^{91} -3.45928 q^{95} -10.3224 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5} - 4 q^{7}+O(q^{10}) \) 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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