LMFDB - Embedded newform 6048.2.a.bn.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			$-1.55466$ 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)

$f(q)$	$=$	$q-3.95805 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-3.95805 q^{5} +1.00000 q^{7} -0.848723 q^{11} -5.85936 q^{13} -2.75004 q^{17} +7.66613 q^{19} +2.70808 q^{23} +10.6661 q^{25} +5.85936 q^{29} +5.80677 q^{31} -3.95805 q^{35} -3.35929 q^{37} -1.90131 q^{41} +2.35929 q^{43} +3.55681 q^{47} +1.00000 q^{49} -10.2186 q^{53} +3.35929 q^{55} +5.77545 q^{59} -5.50008 q^{61} +23.1916 q^{65} +11.4729 q^{67} -9.45812 q^{71} -7.97282 q^{73} -0.848723 q^{77} +10.0780 q^{79} -9.85936 q^{83} +10.8848 q^{85} +1.81741 q^{89} -5.85936 q^{91} -30.3429 q^{95} +15.8322 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.95805	−1.77009	−0.885046	−	0.465503i	$-0.845873\pi$
$5$	−3.95805	−1.77009	−0.885046	+	0.465503i	$0.845873\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.848723	−0.255900	−0.127950	−	0.991781i	$-0.540840\pi$
$11$	−0.848723	−0.255900	−0.127950	+	0.991781i	$0.540840\pi$
$12$	0	0
$13$	−5.85936	−1.62509	−0.812547	−	0.582895i	$-0.801920\pi$
$13$	−5.85936	−1.62509	−0.812547	+	0.582895i	$0.801920\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.75004	−0.666982	−0.333491	−	0.942753i	$-0.608227\pi$
$17$	−2.75004	−0.666982	−0.333491	+	0.942753i	$0.608227\pi$
$18$	0	0
$19$	7.66613	1.75873	0.879365	−	0.476147i	$-0.157967\pi$
$19$	7.66613	1.75873	0.879365	+	0.476147i	$0.157967\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.70808	0.564675	0.282337	−	0.959315i	$-0.408890\pi$
$23$	2.70808	0.564675	0.282337	+	0.959315i	$0.408890\pi$
$24$	0	0
$25$	10.6661	2.13323
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.85936	1.08806	0.544028	−	0.839067i	$-0.316899\pi$
$29$	5.85936	1.08806	0.544028	+	0.839067i	$0.316899\pi$
$30$	0	0
$31$	5.80677	1.04293	0.521463	−	0.853274i	$-0.325386\pi$
$31$	5.80677	1.04293	0.521463	+	0.853274i	$0.325386\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.95805	−0.669032
$36$	0	0
$37$	−3.35929	−0.552263	−0.276132	−	0.961120i	$-0.589053\pi$
$37$	−3.35929	−0.552263	−0.276132	+	0.961120i	$0.589053\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.90131	−0.296935	−0.148468	−	0.988917i	$-0.547434\pi$
$41$	−1.90131	−0.296935	−0.148468	+	0.988917i	$0.547434\pi$
$42$	0	0
$43$	2.35929	0.359788	0.179894	−	0.983686i	$-0.442425\pi$
$43$	2.35929	0.359788	0.179894	+	0.983686i	$0.442425\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.55681	0.518814	0.259407	−	0.965768i	$-0.416473\pi$
$47$	3.55681	0.518814	0.259407	+	0.965768i	$0.416473\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.2186	−1.40364	−0.701820	−	0.712355i	$-0.747629\pi$
$53$	−10.2186	−1.40364	−0.701820	+	0.712355i	$0.747629\pi$
$54$	0	0
$55$	3.35929	0.452966
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.77545	0.751900	0.375950	−	0.926640i	$-0.377316\pi$
$59$	5.77545	0.751900	0.375950	+	0.926640i	$0.377316\pi$
$60$	0	0
$61$	−5.50008	−0.704212	−0.352106	−	0.935960i	$-0.614534\pi$
$61$	−5.50008	−0.704212	−0.352106	+	0.935960i	$0.614534\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	23.1916	2.87657
$66$	0	0
$67$	11.4729	1.40164	0.700819	−	0.713339i	$-0.252818\pi$
$67$	11.4729	1.40164	0.700819	+	0.713339i	$0.252818\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.45812	−1.12247	−0.561236	−	0.827656i	$-0.689674\pi$
$71$	−9.45812	−1.12247	−0.561236	+	0.827656i	$0.689674\pi$
$72$	0	0
$73$	−7.97282	−0.933148	−0.466574	−	0.884482i	$-0.654512\pi$
$73$	−7.97282	−0.933148	−0.466574	+	0.884482i	$0.654512\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.848723	−0.0967210
$78$	0	0
$79$	10.0780	1.13386	0.566932	−	0.823764i	$-0.308130\pi$
$79$	10.0780	1.13386	0.566932	+	0.823764i	$0.308130\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.85936	−1.08221	−0.541103	−	0.840957i	$-0.681993\pi$
$83$	−9.85936	−1.08221	−0.541103	+	0.840957i	$0.681993\pi$
$84$	0	0
$85$	10.8848	1.18062
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.81741	0.192645	0.0963224	−	0.995350i	$-0.469292\pi$
$89$	1.81741	0.192645	0.0963224	+	0.995350i	$0.469292\pi$
$90$	0	0
$91$	−5.85936	−0.614228
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−30.3429	−3.11312
$96$	0	0
$97$	15.8322	1.60751	0.803757	−	0.594957i	$-0.202831\pi$
$97$	15.8322	1.60751	0.803757	+	0.594957i	$0.202831\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.5822	−1.45099	−0.725493	−	0.688230i	$-0.758388\pi$
$101$	−14.5822	−1.45099	−0.725493	+	0.688230i	$0.758388\pi$
$102$	0	0
$103$	−5.91195	−0.582522	−0.291261	−	0.956644i	$-0.594075\pi$
$103$	−5.91195	−0.582522	−0.291261	+	0.956644i	$0.594075\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.6661	−1.80452	−0.902261	−	0.431189i	$-0.858094\pi$
$107$	−18.6661	−1.80452	−0.902261	+	0.431189i	$0.858094\pi$
$108$	0	0
$109$	−5.80677	−0.556188	−0.278094	−	0.960554i	$-0.589703\pi$
$109$	−5.80677	−0.556188	−0.278094	+	0.960554i	$0.589703\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	19.9161	1.87355	0.936774	−	0.349934i	$-0.113796\pi$
$113$	19.9161	1.87355	0.936774	+	0.349934i	$0.113796\pi$
$114$	0	0
$115$	−10.7187	−0.999526
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.75004	−0.252096
$120$	0	0
$121$	−10.2797	−0.934515
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−22.4268	−2.00591
$126$	0	0
$127$	−0.359285	−0.0318814	−0.0159407	−	0.999873i	$-0.505074\pi$
$127$	−0.359285	−0.0318814	−0.0159407	+	0.999873i	$0.505074\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.94327	−0.519266	−0.259633	−	0.965707i	$-0.583602\pi$
$131$	−5.94327	−0.519266	−0.259633	+	0.965707i	$0.583602\pi$
$132$	0	0
$133$	7.66613	0.664738
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.9730	−1.10836	−0.554178	−	0.832398i	$-0.686967\pi$
$137$	−12.9730	−1.10836	−0.554178	+	0.832398i	$0.686967\pi$
$138$	0	0
$139$	−5.61354	−0.476134	−0.238067	−	0.971249i	$-0.576514\pi$
$139$	−5.61354	−0.476134	−0.238067	+	0.971249i	$0.576514\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.97298	0.415861
$144$	0	0
$145$	−23.1916	−1.92596
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.80263	0.639216	0.319608	−	0.947550i	$-0.396449\pi$
$149$	7.80263	0.639216	0.319608	+	0.947550i	$0.396449\pi$
$150$	0	0
$151$	−0.499925	−0.0406833	−0.0203416	−	0.999793i	$-0.506475\pi$
$151$	−0.499925	−0.0406833	−0.0203416	+	0.999793i	$0.506475\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−22.9835	−1.84608
$156$	0	0
$157$	−20.1915	−1.61145	−0.805727	−	0.592287i	$-0.798225\pi$
$157$	−20.1915	−1.61145	−0.805727	+	0.592287i	$0.798225\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.70808	0.213427
$162$	0	0
$163$	−9.25425	−0.724849	−0.362425	−	0.932013i	$-0.618051\pi$
$163$	−9.25425	−0.724849	−0.362425	+	0.932013i	$0.618051\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−25.5296	−1.97554	−0.987771	−	0.155911i	$-0.950169\pi$
$167$	−25.5296	−1.97554	−0.987771	+	0.155911i	$0.950169\pi$
$168$	0	0
$169$	21.3321	1.64093
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2.70808	−0.205892	−0.102946	−	0.994687i	$-0.532827\pi$
$173$	−2.70808	−0.205892	−0.102946	+	0.994687i	$0.532827\pi$
$174$	0	0
$175$	10.6661	0.806284
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.16621	0.610371	0.305185	−	0.952293i	$-0.401282\pi$
$179$	8.16621	0.610371	0.305185	+	0.952293i	$0.401282\pi$
$180$	0	0
$181$	−5.39489	−0.400999	−0.200500	−	0.979694i	$-0.564257\pi$
$181$	−5.39489	−0.400999	−0.200500	+	0.979694i	$0.564257\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	13.2962	0.977557
$186$	0	0
$187$	2.33402	0.170680
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.8428	1.21870	0.609352	−	0.792900i	$-0.291430\pi$
$191$	16.8428	1.21870	0.609352	+	0.792900i	$0.291430\pi$
$192$	0	0
$193$	15.6135	1.12389	0.561944	−	0.827176i	$-0.310054\pi$
$193$	15.6135	1.12389	0.561944	+	0.827176i	$0.310054\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.80263	0.270926	0.135463	−	0.990782i	$-0.456748\pi$
$197$	3.80263	0.270926	0.135463	+	0.990782i	$0.456748\pi$
$198$	0	0
$199$	16.3594	1.15969	0.579845	−	0.814727i	$-0.303113\pi$
$199$	16.3594	1.15969	0.579845	+	0.814727i	$0.303113\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.85936	0.411247
$204$	0	0
$205$	7.52549	0.525603
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6.50642	−0.450059
$210$	0	0
$211$	−2.11346	−0.145497	−0.0727484	−	0.997350i	$-0.523177\pi$
$211$	−2.11346	−0.145497	−0.0727484	+	0.997350i	$0.523177\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−9.33816	−0.636857
$216$	0	0
$217$	5.80677	0.394189
$218$	0	0
$219$	0	0
$220$	0	0
$221$	16.1135	1.08391
$222$	0	0
$223$	−5.35929	−0.358884	−0.179442	−	0.983769i	$-0.557429\pi$
$223$	−5.35929	−0.358884	−0.179442	+	0.983769i	$0.557429\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.80263	−0.517879	−0.258939	−	0.965894i	$-0.583373\pi$
$227$	−7.80263	−0.517879	−0.258939	+	0.965894i	$0.583373\pi$
$228$	0	0
$229$	−10.4999	−0.693855	−0.346927	−	0.937892i	$-0.612775\pi$
$229$	−10.4999	−0.693855	−0.346927	+	0.937892i	$0.612775\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−27.0297	−1.77077	−0.885387	−	0.464854i	$-0.846107\pi$
$233$	−27.0297	−1.77077	−0.885387	+	0.464854i	$0.846107\pi$
$234$	0	0
$235$	−14.0780	−0.918348
$236$	0	0
$237$	0	0
$238$	0	0
$239$	10.3849	0.671740	0.335870	−	0.941908i	$-0.390970\pi$
$239$	10.3849	0.671740	0.335870	+	0.941908i	$0.390970\pi$
$240$	0	0
$241$	−18.1915	−1.17182	−0.585908	−	0.810378i	$-0.699262\pi$
$241$	−18.1915	−1.17182	−0.585908	+	0.810378i	$0.699262\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.95805	−0.252870
$246$	0	0
$247$	−44.9186	−2.85810
$248$	0	0
$249$	0	0
$250$	0	0
$251$	29.1644	1.84084	0.920422	−	0.390927i	$-0.127845\pi$
$251$	29.1644	1.84084	0.920422	+	0.390927i	$0.127845\pi$
$252$	0	0
$253$	−2.29841	−0.144500
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.8700	0.678052	0.339026	−	0.940777i	$-0.389903\pi$
$257$	10.8700	0.678052	0.339026	+	0.940777i	$0.389903\pi$
$258$	0	0
$259$	−3.35929	−0.208736
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.6455	0.779752	0.389876	−	0.920867i	$-0.372518\pi$
$263$	12.6455	0.779752	0.389876	+	0.920867i	$0.372518\pi$
$264$	0	0
$265$	40.4459	2.48457
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.2590	1.35715	0.678577	−	0.734529i	$-0.262597\pi$
$269$	22.2590	1.35715	0.678577	+	0.734529i	$0.262597\pi$
$270$	0	0
$271$	−25.0510	−1.52174	−0.760869	−	0.648905i	$-0.775227\pi$
$271$	−25.0510	−1.52174	−0.760869	+	0.648905i	$0.775227\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−9.05259	−0.545892
$276$	0	0
$277$	20.3051	1.22001	0.610007	−	0.792396i	$-0.291167\pi$
$277$	20.3051	1.22001	0.610007	+	0.792396i	$0.291167\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	9.94327	0.593166	0.296583	−	0.955007i	$-0.404153\pi$
$281$	9.94327	0.593166	0.296583	+	0.955007i	$0.404153\pi$
$282$	0	0
$283$	−15.7187	−0.934381	−0.467191	−	0.884157i	$-0.654734\pi$
$283$	−15.7187	−0.934381	−0.467191	+	0.884157i	$0.654734\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.90131	−0.112231
$288$	0	0
$289$	−9.43729	−0.555135
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−24.1958	−1.41353	−0.706766	−	0.707448i	$-0.749847\pi$
$293$	−24.1958	−1.41353	−0.706766	+	0.707448i	$0.749847\pi$
$294$	0	0
$295$	−22.8595	−1.33093
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−15.8676	−0.917649
$300$	0	0
$301$	2.35929	0.135987
$302$	0	0
$303$	0	0
$304$	0	0
$305$	21.7696	1.24652
$306$	0	0
$307$	−17.6661	−1.00826	−0.504130	−	0.863628i	$-0.668187\pi$
$307$	−17.6661	−1.00826	−0.504130	+	0.863628i	$0.668187\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−25.8050	−1.46327	−0.731634	−	0.681698i	$-0.761242\pi$
$311$	−25.8050	−1.46327	−0.731634	+	0.681698i	$0.761242\pi$
$312$	0	0
$313$	25.8594	1.46166	0.730829	−	0.682561i	$-0.239134\pi$
$313$	25.8594	1.46166	0.730829	+	0.682561i	$0.239134\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.329730	−0.0185195	−0.00925973	−	0.999957i	$-0.502948\pi$
$317$	−0.329730	−0.0185195	−0.00925973	+	0.999957i	$0.502948\pi$
$318$	0	0
$319$	−4.97298	−0.278433
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−21.0821	−1.17304
$324$	0	0
$325$	−62.4967	−3.46669
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.55681	0.196093
$330$	0	0
$331$	25.6135	1.40785	0.703924	−	0.710276i	$-0.251430\pi$
$331$	25.6135	1.40785	0.703924	+	0.710276i	$0.251430\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−45.4103	−2.48103
$336$	0	0
$337$	−8.71872	−0.474939	−0.237470	−	0.971395i	$-0.576318\pi$
$337$	−8.71872	−0.474939	−0.237470	+	0.971395i	$0.576318\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4.92834	−0.266885
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−17.5066	−0.939802	−0.469901	−	0.882719i	$-0.655710\pi$
$347$	−17.5066	−0.939802	−0.469901	+	0.882719i	$0.655710\pi$
$348$	0	0
$349$	−9.71872	−0.520231	−0.260116	−	0.965577i	$-0.583761\pi$
$349$	−9.71872	−0.520231	−0.260116	+	0.965577i	$0.583761\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0.682517	0.0363267	0.0181634	−	0.999835i	$-0.494218\pi$
$353$	0.682517	0.0363267	0.0181634	+	0.999835i	$0.494218\pi$
$354$	0	0
$355$	37.4357	1.98688
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.55252	−0.398607	−0.199303	−	0.979938i	$-0.563868\pi$
$359$	−7.55252	−0.398607	−0.199303	+	0.979938i	$0.563868\pi$
$360$	0	0
$361$	39.7696	2.09313
$362$	0	0
$363$	0	0
$364$	0	0
$365$	31.5568	1.65176
$366$	0	0
$367$	−24.6915	−1.28889	−0.644444	−	0.764651i	$-0.722911\pi$
$367$	−24.6915	−1.28889	−0.644444	+	0.764651i	$0.722911\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−10.2186	−0.530526
$372$	0	0
$373$	−35.1390	−1.81943	−0.909715	−	0.415233i	$-0.863700\pi$
$373$	−35.1390	−1.81943	−0.909715	+	0.415233i	$0.863700\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−34.3321	−1.76819
$378$	0	0
$379$	5.35944	0.275296	0.137648	−	0.990481i	$-0.456046\pi$
$379$	5.35944	0.275296	0.137648	+	0.990481i	$0.456046\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−23.6348	−1.20768	−0.603841	−	0.797105i	$-0.706364\pi$
$383$	−23.6348	−1.20768	−0.603841	+	0.797105i	$0.706364\pi$
$384$	0	0
$385$	3.35929	0.171205
$386$	0	0
$387$	0	0
$388$	0	0
$389$	12.8950	0.653801	0.326900	−	0.945059i	$-0.393996\pi$
$389$	12.8950	0.653801	0.326900	+	0.945059i	$0.393996\pi$
$390$	0	0
$391$	−7.44733	−0.376628
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−39.8892	−2.00704
$396$	0	0
$397$	26.3324	1.32159	0.660793	−	0.750568i	$-0.270220\pi$
$397$	26.3324	1.32159	0.660793	+	0.750568i	$0.270220\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−28.6917	−1.43279	−0.716397	−	0.697692i	$-0.754210\pi$
$401$	−28.6917	−1.43279	−0.716397	+	0.697692i	$0.754210\pi$
$402$	0	0
$403$	−34.0240	−1.69485
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.85110	0.141324
$408$	0	0
$409$	−22.5239	−1.11373	−0.556867	−	0.830602i	$-0.687997\pi$
$409$	−22.5239	−1.11373	−0.556867	+	0.830602i	$0.687997\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.77545	0.284191
$414$	0	0
$415$	39.0238	1.91560
$416$	0	0
$417$	0	0
$418$	0	0
$419$	37.6856	1.84106	0.920532	−	0.390667i	$-0.127756\pi$
$419$	37.6856	1.84106	0.920532	+	0.390667i	$0.127756\pi$
$420$	0	0
$421$	2.69155	0.131178	0.0655890	−	0.997847i	$-0.479107\pi$
$421$	2.69155	0.131178	0.0655890	+	0.997847i	$0.479107\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−29.3323	−1.42282
$426$	0	0
$427$	−5.50008	−0.266167
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.70808	−0.130444	−0.0652219	−	0.997871i	$-0.520776\pi$
$431$	−2.70808	−0.130444	−0.0652219	+	0.997871i	$0.520776\pi$
$432$	0	0
$433$	0.964393	0.0463458	0.0231729	−	0.999731i	$-0.492623\pi$
$433$	0.964393	0.0463458	0.0231729	+	0.999731i	$0.492623\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	20.7605	0.993111
$438$	0	0
$439$	−20.4373	−0.975419	−0.487709	−	0.873006i	$-0.662167\pi$
$439$	−20.4373	−0.975419	−0.487709	+	0.873006i	$0.662167\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−30.8156	−1.46410	−0.732048	−	0.681253i	$-0.761435\pi$
$443$	−30.8156	−1.46410	−0.732048	+	0.681253i	$0.761435\pi$
$444$	0	0
$445$	−7.19338	−0.340999
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−23.3535	−1.10212	−0.551061	−	0.834465i	$-0.685777\pi$
$449$	−23.3535	−1.10212	−0.551061	+	0.834465i	$0.685777\pi$
$450$	0	0
$451$	1.61369	0.0759857
$452$	0	0
$453$	0	0
$454$	0	0
$455$	23.1916	1.08724
$456$	0	0
$457$	−18.6644	−0.873082	−0.436541	−	0.899684i	$-0.643797\pi$
$457$	−18.6644	−0.873082	−0.436541	+	0.899684i	$0.643797\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	26.7802	1.24728	0.623639	−	0.781713i	$-0.285654\pi$
$461$	26.7802	1.24728	0.623639	+	0.781713i	$0.285654\pi$
$462$	0	0
$463$	5.71872	0.265772	0.132886	−	0.991131i	$-0.457576\pi$
$463$	5.71872	0.265772	0.132886	+	0.991131i	$0.457576\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−39.7128	−1.83769	−0.918845	−	0.394618i	$-0.870877\pi$
$467$	−39.7128	−1.83769	−0.918845	+	0.394618i	$0.870877\pi$
$468$	0	0
$469$	11.4729	0.529769
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.00238	−0.0920695
$474$	0	0
$475$	81.7679	3.75177
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10.8323	−0.494942	−0.247471	−	0.968895i	$-0.579600\pi$
$479$	−10.8323	−0.494942	−0.247471	+	0.968895i	$0.579600\pi$
$480$	0	0
$481$	19.6833	0.897480
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−62.6645	−2.84545
$486$	0	0
$487$	31.5781	1.43094	0.715470	−	0.698644i	$-0.246213\pi$
$487$	31.5781	1.43094	0.715470	+	0.698644i	$0.246213\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	21.5361	0.971912	0.485956	−	0.873983i	$-0.338472\pi$
$491$	21.5361	0.971912	0.485956	+	0.873983i	$0.338472\pi$
$492$	0	0
$493$	−16.1135	−0.725714
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.45812	−0.424255
$498$	0	0
$499$	−35.2425	−1.57767	−0.788834	−	0.614606i	$-0.789315\pi$
$499$	−35.2425	−1.57767	−0.788834	+	0.614606i	$0.789315\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−16.1915	−0.721942	−0.360971	−	0.932577i	$-0.617555\pi$
$503$	−16.1915	−0.721942	−0.360971	+	0.932577i	$0.617555\pi$
$504$	0	0
$505$	57.7171	2.56838
$506$	0	0
$507$	0	0
$508$	0	0
$509$	18.6957	0.828672	0.414336	−	0.910124i	$-0.364014\pi$
$509$	18.6957	0.828672	0.414336	+	0.910124i	$0.364014\pi$
$510$	0	0
$511$	−7.97282	−0.352697
$512$	0	0
$513$	0	0
$514$	0	0
$515$	23.3998	1.03112
$516$	0	0
$517$	−3.01874	−0.132764
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10.3698	−0.454308	−0.227154	−	0.973859i	$-0.572942\pi$
$521$	−10.3698	−0.454308	−0.227154	+	0.973859i	$0.572942\pi$
$522$	0	0
$523$	−23.6137	−1.03255	−0.516277	−	0.856421i	$-0.672683\pi$
$523$	−23.6137	−1.03255	−0.516277	+	0.856421i	$0.672683\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.9688	−0.695613
$528$	0	0
$529$	−15.6663	−0.681143
$530$	0	0
$531$	0	0
$532$	0	0
$533$	11.1405	0.482548
$534$	0	0
$535$	73.8814	3.19417
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.848723	−0.0365571
$540$	0	0
$541$	−23.9730	−1.03068	−0.515339	−	0.856986i	$-0.672334\pi$
$541$	−23.9730	−1.03068	−0.515339	+	0.856986i	$0.672334\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	22.9835	0.984503
$546$	0	0
$547$	38.9458	1.66520	0.832601	−	0.553873i	$-0.186851\pi$
$547$	38.9458	1.66520	0.832601	+	0.553873i	$0.186851\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	44.9186	1.91360
$552$	0	0
$553$	10.0780	0.428560
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−15.9161	−0.674386	−0.337193	−	0.941435i	$-0.609478\pi$
$557$	−15.9161	−0.674386	−0.337193	+	0.941435i	$0.609478\pi$
$558$	0	0
$559$	−13.8239	−0.584689
$560$	0	0
$561$	0	0
$562$	0	0
$563$	29.4103	1.23949	0.619747	−	0.784801i	$-0.287235\pi$
$563$	29.4103	1.23949	0.619747	+	0.784801i	$0.287235\pi$
$564$	0	0
$565$	−78.8288	−3.31635
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−14.5295	−0.609108	−0.304554	−	0.952495i	$-0.598507\pi$
$569$	−14.5295	−0.609108	−0.304554	+	0.952495i	$0.598507\pi$
$570$	0	0
$571$	9.55091	0.399693	0.199847	−	0.979827i	$-0.435956\pi$
$571$	9.55091	0.399693	0.199847	+	0.979827i	$0.435956\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	28.8848	1.20458
$576$	0	0
$577$	31.0510	1.29267	0.646335	−	0.763054i	$-0.276301\pi$
$577$	31.0510	1.29267	0.646335	+	0.763054i	$0.276301\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−9.85936	−0.409035
$582$	0	0
$583$	8.67280	0.359191
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11.7483	0.484903	0.242452	−	0.970163i	$-0.422048\pi$
$587$	11.7483	0.484903	0.242452	+	0.970163i	$0.422048\pi$
$588$	0	0
$589$	44.5155	1.83423
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−36.6514	−1.50509	−0.752545	−	0.658540i	$-0.771174\pi$
$593$	−36.6514	−1.50509	−0.752545	+	0.658540i	$0.771174\pi$
$594$	0	0
$595$	10.8848	0.446232
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−17.6453	−0.720967	−0.360484	−	0.932766i	$-0.617388\pi$
$599$	−17.6453	−0.720967	−0.360484	+	0.932766i	$0.617388\pi$
$600$	0	0
$601$	11.6407	0.474835	0.237417	−	0.971408i	$-0.423699\pi$
$601$	11.6407	0.474835	0.237417	+	0.971408i	$0.423699\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	40.6874	1.65418
$606$	0	0
$607$	42.0508	1.70679	0.853395	−	0.521264i	$-0.174539\pi$
$607$	42.0508	1.70679	0.853395	+	0.521264i	$0.174539\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−20.8406	−0.843121
$612$	0	0
$613$	−21.1934	−0.855993	−0.427996	−	0.903780i	$-0.640780\pi$
$613$	−21.1934	−0.855993	−0.427996	+	0.903780i	$0.640780\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.2245	0.572659	0.286329	−	0.958131i	$-0.407565\pi$
$617$	14.2245	0.572659	0.286329	+	0.958131i	$0.407565\pi$
$618$	0	0
$619$	8.82390	0.354663	0.177331	−	0.984151i	$-0.443254\pi$
$619$	8.82390	0.354663	0.177331	+	0.984151i	$0.443254\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.81741	0.0728129
$624$	0	0
$625$	35.4357	1.41743
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9.23816	0.368350
$630$	0	0
$631$	12.1406	0.483311	0.241656	−	0.970362i	$-0.422310\pi$
$631$	12.1406	0.483311	0.241656	+	0.970362i	$0.422310\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.42207	0.0564331
$636$	0	0
$637$	−5.85936	−0.232156
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.30270	0.209444	0.104722	−	0.994502i	$-0.466605\pi$
$641$	5.30270	0.209444	0.104722	+	0.994502i	$0.466605\pi$
$642$	0	0
$643$	−37.5409	−1.48047	−0.740234	−	0.672350i	$-0.765285\pi$
$643$	−37.5409	−1.48047	−0.740234	+	0.672350i	$0.765285\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−45.6076	−1.79302	−0.896511	−	0.443022i	$-0.853906\pi$
$647$	−45.6076	−1.79302	−0.896511	+	0.443022i	$0.853906\pi$
$648$	0	0
$649$	−4.90176	−0.192411
$650$	0	0
$651$	0	0
$652$	0	0
$653$	28.5652	1.11784	0.558922	−	0.829220i	$-0.311215\pi$
$653$	28.5652	1.11784	0.558922	+	0.829220i	$0.311215\pi$
$654$	0	0
$655$	23.5237	0.919148
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−12.2436	−0.476944	−0.238472	−	0.971149i	$-0.576646\pi$
$659$	−12.2436	−0.476944	−0.238472	+	0.971149i	$0.576646\pi$
$660$	0	0
$661$	−29.9728	−1.16581	−0.582904	−	0.812541i	$-0.698084\pi$
$661$	−29.9728	−1.16581	−0.582904	+	0.812541i	$0.698084\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−30.3429	−1.17665
$666$	0	0
$667$	15.8676	0.614398
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.66804	0.180208
$672$	0	0
$673$	3.38661	0.130544	0.0652722	−	0.997867i	$-0.479208\pi$
$673$	3.38661	0.130544	0.0652722	+	0.997867i	$0.479208\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18.9792	−0.729429	−0.364714	−	0.931119i	$-0.618833\pi$
$677$	−18.9792	−0.729429	−0.364714	+	0.931119i	$0.618833\pi$
$678$	0	0
$679$	15.8322	0.607583
$680$	0	0
$681$	0	0
$682$	0	0
$683$	49.5555	1.89619	0.948094	−	0.317990i	$-0.103008\pi$
$683$	49.5555	1.89619	0.948094	+	0.317990i	$0.103008\pi$
$684$	0	0
$685$	51.3476	1.96189
$686$	0	0
$687$	0	0
$688$	0	0
$689$	59.8747	2.28105
$690$	0	0
$691$	−28.5593	−1.08645	−0.543224	−	0.839588i	$-0.682797\pi$
$691$	−28.5593	−1.08645	−0.543224	+	0.839588i	$0.682797\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	22.2186	0.842801
$696$	0	0
$697$	5.22869	0.198051
$698$	0	0
$699$	0	0
$700$	0	0
$701$	17.1913	0.649307	0.324654	−	0.945833i	$-0.394752\pi$
$701$	17.1913	0.649307	0.324654	+	0.945833i	$0.394752\pi$
$702$	0	0
$703$	−25.7527	−0.971282
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−14.5822	−0.548421
$708$	0	0
$709$	−46.8578	−1.75978	−0.879890	−	0.475178i	$-0.842384\pi$
$709$	−46.8578	−1.75978	−0.879890	+	0.475178i	$0.842384\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	15.7252	0.588914
$714$	0	0
$715$	−19.6833	−0.736112
$716$	0	0
$717$	0	0
$718$	0	0
$719$	48.3265	1.80227	0.901137	−	0.433534i	$-0.142733\pi$
$719$	48.3265	1.80227	0.901137	+	0.433534i	$0.142733\pi$
$720$	0	0
$721$	−5.91195	−0.220173
$722$	0	0
$723$	0	0
$724$	0	0
$725$	62.4967	2.32107
$726$	0	0
$727$	−15.2814	−0.566757	−0.283378	−	0.959008i	$-0.591455\pi$
$727$	−15.2814	−0.566757	−0.283378	+	0.959008i	$0.591455\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−6.48812	−0.239972
$732$	0	0
$733$	−9.07816	−0.335309	−0.167655	−	0.985846i	$-0.553619\pi$
$733$	−9.07816	−0.335309	−0.167655	+	0.985846i	$0.553619\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.73732	−0.358679
$738$	0	0
$739$	−19.7814	−0.727669	−0.363834	−	0.931464i	$-0.618533\pi$
$739$	−19.7814	−0.727669	−0.363834	+	0.931464i	$0.618533\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.1554	0.592685	0.296342	−	0.955082i	$-0.404233\pi$
$743$	16.1554	0.592685	0.296342	+	0.955082i	$0.404233\pi$
$744$	0	0
$745$	−30.8832	−1.13147
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−18.6661	−0.682046
$750$	0	0
$751$	−21.4100	−0.781261	−0.390630	−	0.920548i	$-0.627743\pi$
$751$	−21.4100	−0.781261	−0.390630	+	0.920548i	$0.627743\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.97872	0.0720132
$756$	0	0
$757$	−6.49164	−0.235943	−0.117971	−	0.993017i	$-0.537639\pi$
$757$	−6.49164	−0.235943	−0.117971	+	0.993017i	$0.537639\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−50.5704	−1.83318	−0.916588	−	0.399833i	$-0.869068\pi$
$761$	−50.5704	−1.83318	−0.916588	+	0.399833i	$0.869068\pi$
$762$	0	0
$763$	−5.80677	−0.210219
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−33.8405	−1.22191
$768$	0	0
$769$	−21.1290	−0.761931	−0.380965	−	0.924589i	$-0.624408\pi$
$769$	−21.1290	−0.761931	−0.380965	+	0.924589i	$0.624408\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−27.3116	−0.982329	−0.491165	−	0.871067i	$-0.663429\pi$
$773$	−27.3116	−0.982329	−0.491165	+	0.871067i	$0.663429\pi$
$774$	0	0
$775$	61.9358	2.22480
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−14.5757	−0.522230
$780$	0	0
$781$	8.02733	0.287240
$782$	0	0
$783$	0	0
$784$	0	0
$785$	79.9188	2.85242
$786$	0	0
$787$	22.7694	0.811642	0.405821	−	0.913953i	$-0.366986\pi$
$787$	22.7694	0.811642	0.405821	+	0.913953i	$0.366986\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	19.9161	0.708135
$792$	0	0
$793$	32.2269	1.14441
$794$	0	0
$795$	0	0
$796$	0	0
$797$	48.9041	1.73227	0.866137	−	0.499807i	$-0.166596\pi$
$797$	48.9041	1.73227	0.866137	+	0.499807i	$0.166596\pi$
$798$	0	0
$799$	−9.78135	−0.346039
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.76672	0.238792
$804$	0	0
$805$	−10.7187	−0.377785
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−7.24597	−0.254755	−0.127377	−	0.991854i	$-0.540656\pi$
$809$	−7.24597	−0.254755	−0.127377	+	0.991854i	$0.540656\pi$
$810$	0	0
$811$	53.3832	1.87454	0.937270	−	0.348605i	$-0.113345\pi$
$811$	53.3832	1.87454	0.937270	+	0.348605i	$0.113345\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	36.6288	1.28305
$816$	0	0
$817$	18.0866	0.632770
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8.99395	0.313891	0.156945	−	0.987607i	$-0.449835\pi$
$821$	8.99395	0.313891	0.156945	+	0.987607i	$0.449835\pi$
$822$	0	0
$823$	48.6645	1.69634	0.848169	−	0.529725i	$-0.177705\pi$
$823$	48.6645	1.69634	0.848169	+	0.529725i	$0.177705\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.3046	0.914702	0.457351	−	0.889286i	$-0.348798\pi$
$827$	26.3046	0.914702	0.457351	+	0.889286i	$0.348798\pi$
$828$	0	0
$829$	26.8052	0.930982	0.465491	−	0.885053i	$-0.345878\pi$
$829$	26.8052	0.930982	0.465491	+	0.885053i	$0.345878\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.75004	−0.0952832
$834$	0	0
$835$	101.047	3.49689
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.85936	−0.0641923	−0.0320961	−	0.999485i	$-0.510218\pi$
$839$	−1.85936	−0.0641923	−0.0320961	+	0.999485i	$0.510218\pi$
$840$	0	0
$841$	5.33211	0.183866
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−84.4335	−2.90460
$846$	0	0
$847$	−10.2797	−0.353214
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−9.09723	−0.311849
$852$	0	0
$853$	10.8323	0.370892	0.185446	−	0.982654i	$-0.440627\pi$
$853$	10.8323	0.370892	0.185446	+	0.982654i	$0.440627\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0.598610	0.0204481	0.0102241	−	0.999948i	$-0.496746\pi$
$857$	0.598610	0.0204481	0.0102241	+	0.999948i	$0.496746\pi$
$858$	0	0
$859$	−18.4391	−0.629132	−0.314566	−	0.949236i	$-0.601859\pi$
$859$	−18.4391	−0.629132	−0.314566	+	0.949236i	$0.601859\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	50.2761	1.71142	0.855710	−	0.517456i	$-0.173121\pi$
$863$	50.2761	1.71142	0.855710	+	0.517456i	$0.173121\pi$
$864$	0	0
$865$	10.7187	0.364447
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8.55344	−0.290156
$870$	0	0
$871$	−67.2239	−2.27779
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−22.4268	−0.758164
$876$	0	0
$877$	3.66789	0.123856	0.0619279	−	0.998081i	$-0.480275\pi$
$877$	3.66789	0.123856	0.0619279	+	0.998081i	$0.480275\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−23.7122	−0.798885	−0.399443	−	0.916758i	$-0.630796\pi$
$881$	−23.7122	−0.798885	−0.399443	+	0.916758i	$0.630796\pi$
$882$	0	0
$883$	13.9374	0.469030	0.234515	−	0.972113i	$-0.424650\pi$
$883$	13.9374	0.469030	0.234515	+	0.972113i	$0.424650\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6.64087	−0.222978	−0.111489	−	0.993766i	$-0.535562\pi$
$887$	−6.64087	−0.222978	−0.111489	+	0.993766i	$0.535562\pi$
$888$	0	0
$889$	−0.359285	−0.0120500
$890$	0	0
$891$	0	0
$892$	0	0
$893$	27.2669	0.912454
$894$	0	0
$895$	−32.3222	−1.08041
$896$	0	0
$897$	0	0
$898$	0	0
$899$	34.0240	1.13476
$900$	0	0
$901$	28.1017	0.936202
$902$	0	0
$903$	0	0
$904$	0	0
$905$	21.3532	0.709806
$906$	0	0
$907$	−32.2778	−1.07177	−0.535883	−	0.844292i	$-0.680021\pi$
$907$	−32.2778	−1.07177	−0.535883	+	0.844292i	$0.680021\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−17.7168	−0.586984	−0.293492	−	0.955962i	$-0.594817\pi$
$911$	−17.7168	−0.586984	−0.293492	+	0.955962i	$0.594817\pi$
$912$	0	0
$913$	8.36787	0.276936
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.94327	−0.196264
$918$	0	0
$919$	−54.7071	−1.80462	−0.902310	−	0.431088i	$-0.858130\pi$
$919$	−54.7071	−1.80462	−0.902310	+	0.431088i	$0.858130\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	55.4185	1.82412
$924$	0	0
$925$	−35.8306	−1.17810
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−13.4264	−0.440505	−0.220252	−	0.975443i	$-0.570688\pi$
$929$	−13.4264	−0.440505	−0.220252	+	0.975443i	$0.570688\pi$
$930$	0	0
$931$	7.66613	0.251247
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9.23816	−0.302120
$936$	0	0
$937$	−43.1647	−1.41013	−0.705065	−	0.709142i	$-0.749082\pi$
$937$	−43.1647	−1.41013	−0.705065	+	0.709142i	$0.749082\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−50.1735	−1.63561	−0.817804	−	0.575496i	$-0.804809\pi$
$941$	−50.1735	−1.63561	−0.817804	+	0.575496i	$0.804809\pi$
$942$	0	0
$943$	−5.14892	−0.167672
$944$	0	0
$945$	0	0
$946$	0	0
$947$	49.7421	1.61640	0.808200	−	0.588908i	$-0.200442\pi$
$947$	49.7421	1.61640	0.808200	+	0.588908i	$0.200442\pi$
$948$	0	0
$949$	46.7157	1.51645
$950$	0	0
$951$	0	0
$952$	0	0
$953$	20.8595	0.675706	0.337853	−	0.941199i	$-0.390299\pi$
$953$	20.8595	0.675706	0.337853	+	0.941199i	$0.390299\pi$
$954$	0	0
$955$	−66.6647	−2.15722
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12.9730	−0.418919
$960$	0	0
$961$	2.71857	0.0876958
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−61.7991	−1.98938
$966$	0	0
$967$	−44.2068	−1.42160	−0.710798	−	0.703396i	$-0.751666\pi$
$967$	−44.2068	−1.42160	−0.710798	+	0.703396i	$0.751666\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	35.3680	1.13501	0.567507	−	0.823369i	$-0.307908\pi$
$971$	35.3680	1.13501	0.567507	+	0.823369i	$0.307908\pi$
$972$	0	0
$973$	−5.61354	−0.179962
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−6.94580	−0.222216	−0.111108	−	0.993808i	$-0.535440\pi$
$977$	−6.94580	−0.222216	−0.111108	+	0.993808i	$0.535440\pi$
$978$	0	0
$979$	−1.54248	−0.0492977
$980$	0	0
$981$	0	0
$982$	0	0
$983$	27.4433	0.875307	0.437653	−	0.899144i	$-0.355810\pi$
$983$	27.4433	0.875307	0.437653	+	0.899144i	$0.355810\pi$
$984$	0	0
$985$	−15.0510	−0.479564
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6.38914	0.203163
$990$	0	0
$991$	−19.9102	−0.632468	−0.316234	−	0.948681i	$-0.602418\pi$
$991$	−19.9102	−0.632468	−0.316234	+	0.948681i	$0.602418\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−64.7514	−2.05276
$996$	0	0
$997$	−28.6645	−0.907814	−0.453907	−	0.891049i	$-0.649970\pi$
$997$	−28.6645	−0.907814	−0.453907	+	0.891049i	$0.649970\pi$
$998$	0	0
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-3.95805 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-3.95805 q^{5} +1.00000 q^{7} -0.848723 q^{11} -5.85936 q^{13} -2.75004 q^{17} +7.66613 q^{19} +2.70808 q^{23} +10.6661 q^{25} +5.85936 q^{29} +5.80677 q^{31} -3.95805 q^{35} -3.35929 q^{37} -1.90131 q^{41} +2.35929 q^{43} +3.55681 q^{47} +1.00000 q^{49} -10.2186 q^{53} +3.35929 q^{55} +5.77545 q^{59} -5.50008 q^{61} +23.1916 q^{65} +11.4729 q^{67} -9.45812 q^{71} -7.97282 q^{73} -0.848723 q^{77} +10.0780 q^{79} -9.85936 q^{83} +10.8848 q^{85} +1.81741 q^{89} -5.85936 q^{91} -30.3429 q^{95} +15.8322 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} + 4 q^{7}+O(q^{10}) \) 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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