LMFDB - Embedded newform 5488.2.a.h.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5488,2,Mod(1,5488)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5488.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5488, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5488 = 2^{4} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5488.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-5,0,11,0,0,0,9,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$43.8219006293$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.1279733.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 6x^{4} + 10x^{3} + 10x^{2} - 11x - 1 $ x^6 - 2x^5 - 6x^4 + 10x^3 + 10x^2 - 11*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 343)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.908891$ of defining polynomial
Character	$\chi$	$=$	5488.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.20643 q^{3} -1.84420 q^{5} +1.86834 q^{9} -3.73428 q^{11} +3.52416 q^{13} +4.06910 q^{15} +6.63777 q^{17} -5.54425 q^{19} -0.379398 q^{23} -1.59893 q^{25} +2.49692 q^{27} -3.54898 q^{29} +4.54761 q^{31} +8.23944 q^{33} -11.6650 q^{37} -7.77581 q^{39} +6.30200 q^{41} -6.46316 q^{43} -3.44560 q^{45} -6.14079 q^{47} -14.6458 q^{51} +0.224289 q^{53} +6.88676 q^{55} +12.2330 q^{57} -0.866024 q^{59} +0.566291 q^{61} -6.49924 q^{65} +10.5141 q^{67} +0.837116 q^{69} +0.137550 q^{71} -13.1670 q^{73} +3.52794 q^{75} -3.31595 q^{79} -11.1143 q^{81} -14.9402 q^{83} -12.2414 q^{85} +7.83058 q^{87} +2.14997 q^{89} -10.0340 q^{93} +10.2247 q^{95} -2.84765 q^{97} -6.97693 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 5 q^{3} + 11 q^{5} + 9 q^{9} + q^{11} + 7 q^{13} + 5 q^{15} + 26 q^{17} + 3 q^{19} + 9 q^{23} + 15 q^{25} - 8 q^{27} - 2 q^{29} + 2 q^{31} + 18 q^{33} - 2 q^{37} - 7 q^{39} + 28 q^{41} - 5 q^{43}+ \cdots - 12 q^{99}+O(q^{100}) $ 6 * q - 5 * q^3 + 11 * q^5 + 9 * q^9 + q^11 + 7 * q^13 + 5 * q^15 + 26 * q^17 + 3 * q^19 + 9 * q^23 + 15 * q^25 - 8 * q^27 - 2 * q^29 + 2 * q^31 + 18 * q^33 - 2 * q^37 - 7 * q^39 + 28 * q^41 - 5 * q^43 + 3 * q^45 - 18 * q^47 - 13 * q^51 - q^53 + 22 * q^55 - 26 * q^57 - 5 * q^59 - 17 * q^61 - 14 * q^65 + 22 * q^67 - 20 * q^69 + 2 * q^71 + 12 * q^73 + 27 * q^75 + 2 * q^79 - 30 * q^81 - 7 * q^83 + 37 * q^85 + 25 * q^87 + 39 * q^89 - 22 * q^93 + 11 * q^95 + 35 * q^97 - 12 * q^99

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
$3$	−2.20643	−1.27388	−0.636942	−	0.770912i	$-0.719801\pi$
$3$	−2.20643	−1.27388	−0.636942	+	0.770912i	$0.719801\pi$
$4$	0	0
$5$	−1.84420	−0.824751	−0.412375	−	0.911014i	$-0.635301\pi$
$5$	−1.84420	−0.824751	−0.412375	+	0.911014i	$0.635301\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.86834	0.622781
$10$	0	0
$11$	−3.73428	−1.12593	−0.562964	−	0.826481i	$-0.690339\pi$
$11$	−3.73428	−1.12593	−0.562964	+	0.826481i	$0.690339\pi$
$12$	0	0
$13$	3.52416	0.977425	0.488712	−	0.872445i	$-0.337467\pi$
$13$	3.52416	0.977425	0.488712	+	0.872445i	$0.337467\pi$
$14$	0	0
$15$	4.06910	1.05064
$16$	0	0
$17$	6.63777	1.60989	0.804947	−	0.593346i	$-0.202193\pi$
$17$	6.63777	1.60989	0.804947	+	0.593346i	$0.202193\pi$
$18$	0	0
$19$	−5.54425	−1.27194	−0.635969	−	0.771714i	$-0.719399\pi$
$19$	−5.54425	−1.27194	−0.635969	+	0.771714i	$0.719399\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.379398	−0.0791099	−0.0395550	−	0.999217i	$-0.512594\pi$
$23$	−0.379398	−0.0791099	−0.0395550	+	0.999217i	$0.512594\pi$
$24$	0	0
$25$	−1.59893	−0.319786
$26$	0	0
$27$	2.49692	0.480533
$28$	0	0
$29$	−3.54898	−0.659029	−0.329514	−	0.944151i	$-0.606885\pi$
$29$	−3.54898	−0.659029	−0.329514	+	0.944151i	$0.606885\pi$
$30$	0	0
$31$	4.54761	0.816775	0.408387	−	0.912809i	$-0.366091\pi$
$31$	4.54761	0.816775	0.408387	+	0.912809i	$0.366091\pi$
$32$	0	0
$33$	8.23944	1.43430
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−11.6650	−1.91772	−0.958858	−	0.283885i	$-0.908376\pi$
$37$	−11.6650	−1.91772	−0.958858	+	0.283885i	$0.908376\pi$
$38$	0	0
$39$	−7.77581	−1.24513
$40$	0	0
$41$	6.30200	0.984207	0.492103	−	0.870537i	$-0.336228\pi$
$41$	6.30200	0.984207	0.492103	+	0.870537i	$0.336228\pi$
$42$	0	0
$43$	−6.46316	−0.985622	−0.492811	−	0.870136i	$-0.664031\pi$
$43$	−6.46316	−0.985622	−0.492811	+	0.870136i	$0.664031\pi$
$44$	0	0
$45$	−3.44560	−0.513639
$46$	0	0
$47$	−6.14079	−0.895726	−0.447863	−	0.894102i	$-0.647815\pi$
$47$	−6.14079	−0.895726	−0.447863	+	0.894102i	$0.647815\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−14.6458	−2.05082
$52$	0	0
$53$	0.224289	0.0308085	0.0154043	−	0.999881i	$-0.495096\pi$
$53$	0.224289	0.0308085	0.0154043	+	0.999881i	$0.495096\pi$
$54$	0	0
$55$	6.88676	0.928610
$56$	0	0
$57$	12.2330	1.62030
$58$	0	0
$59$	−0.866024	−0.112747	−0.0563734	−	0.998410i	$-0.517954\pi$
$59$	−0.866024	−0.112747	−0.0563734	+	0.998410i	$0.517954\pi$
$60$	0	0
$61$	0.566291	0.0725061	0.0362531	−	0.999343i	$-0.488458\pi$
$61$	0.566291	0.0725061	0.0362531	+	0.999343i	$0.488458\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.49924	−0.806132
$66$	0	0
$67$	10.5141	1.28450	0.642248	−	0.766497i	$-0.278002\pi$
$67$	10.5141	1.28450	0.642248	+	0.766497i	$0.278002\pi$
$68$	0	0
$69$	0.837116	0.100777
$70$	0	0
$71$	0.137550	0.0163241	0.00816207	−	0.999967i	$-0.497402\pi$
$71$	0.137550	0.0163241	0.00816207	+	0.999967i	$0.497402\pi$
$72$	0	0
$73$	−13.1670	−1.54108	−0.770541	−	0.637391i	$-0.780014\pi$
$73$	−13.1670	−1.54108	−0.770541	+	0.637391i	$0.780014\pi$
$74$	0	0
$75$	3.52794	0.407371
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−3.31595	−0.373074	−0.186537	−	0.982448i	$-0.559726\pi$
$79$	−3.31595	−0.373074	−0.186537	+	0.982448i	$0.559726\pi$
$80$	0	0
$81$	−11.1143	−1.23492
$82$	0	0
$83$	−14.9402	−1.63990	−0.819950	−	0.572435i	$-0.805999\pi$
$83$	−14.9402	−1.63990	−0.819950	+	0.572435i	$0.805999\pi$
$84$	0	0
$85$	−12.2414	−1.32776
$86$	0	0
$87$	7.83058	0.839526
$88$	0	0
$89$	2.14997	0.227896	0.113948	−	0.993487i	$-0.463650\pi$
$89$	2.14997	0.227896	0.113948	+	0.993487i	$0.463650\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−10.0340	−1.04048
$94$	0	0
$95$	10.2247	1.04903
$96$	0	0
$97$	−2.84765	−0.289135	−0.144568	−	0.989495i	$-0.546179\pi$
$97$	−2.84765	−0.289135	−0.144568	+	0.989495i	$0.546179\pi$
$98$	0	0
$99$	−6.97693	−0.701207
$100$	0	0
$101$	−3.80643	−0.378753	−0.189377	−	0.981905i	$-0.560647\pi$
$101$	−3.80643	−0.378753	−0.189377	+	0.981905i	$0.560647\pi$
$102$	0	0
$103$	−0.283953	−0.0279787	−0.0139893	−	0.999902i	$-0.504453\pi$
$103$	−0.283953	−0.0279787	−0.0139893	+	0.999902i	$0.504453\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.97848	−0.481288	−0.240644	−	0.970614i	$-0.577359\pi$
$107$	−4.97848	−0.481288	−0.240644	+	0.970614i	$0.577359\pi$
$108$	0	0
$109$	10.7863	1.03314	0.516571	−	0.856244i	$-0.327208\pi$
$109$	10.7863	1.03314	0.516571	+	0.856244i	$0.327208\pi$
$110$	0	0
$111$	25.7381	2.44295
$112$	0	0
$113$	−13.0423	−1.22692	−0.613460	−	0.789726i	$-0.710223\pi$
$113$	−13.0423	−1.22692	−0.613460	+	0.789726i	$0.710223\pi$
$114$	0	0
$115$	0.699685	0.0652459
$116$	0	0
$117$	6.58434	0.608722
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2.94486	0.267715
$122$	0	0
$123$	−13.9049	−1.25377
$124$	0	0
$125$	12.1697	1.08849
$126$	0	0
$127$	16.5277	1.46660	0.733300	−	0.679905i	$-0.237979\pi$
$127$	16.5277	1.46660	0.733300	+	0.679905i	$0.237979\pi$
$128$	0	0
$129$	14.2605	1.25557
$130$	0	0
$131$	10.7096	0.935698	0.467849	−	0.883808i	$-0.345029\pi$
$131$	10.7096	0.935698	0.467849	+	0.883808i	$0.345029\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−4.60482	−0.396320
$136$	0	0
$137$	−10.0277	−0.856722	−0.428361	−	0.903608i	$-0.640909\pi$
$137$	−10.0277	−0.856722	−0.428361	+	0.903608i	$0.640909\pi$
$138$	0	0
$139$	19.2745	1.63484	0.817421	−	0.576040i	$-0.195403\pi$
$139$	19.2745	1.63484	0.817421	+	0.576040i	$0.195403\pi$
$140$	0	0
$141$	13.5492	1.14105
$142$	0	0
$143$	−13.1602	−1.10051
$144$	0	0
$145$	6.54502	0.543534
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.85089	0.725093	0.362546	−	0.931966i	$-0.381907\pi$
$149$	8.85089	0.725093	0.362546	+	0.931966i	$0.381907\pi$
$150$	0	0
$151$	−0.492465	−0.0400762	−0.0200381	−	0.999799i	$-0.506379\pi$
$151$	−0.492465	−0.0400762	−0.0200381	+	0.999799i	$0.506379\pi$
$152$	0	0
$153$	12.4016	1.00261
$154$	0	0
$155$	−8.38669	−0.673635
$156$	0	0
$157$	−7.46685	−0.595920	−0.297960	−	0.954578i	$-0.596306\pi$
$157$	−7.46685	−0.595920	−0.297960	+	0.954578i	$0.596306\pi$
$158$	0	0
$159$	−0.494879	−0.0392465
$160$	0	0
$161$	0	0
$162$	0	0
$163$	3.25350	0.254834	0.127417	−	0.991849i	$-0.459331\pi$
$163$	3.25350	0.254834	0.127417	+	0.991849i	$0.459331\pi$
$164$	0	0
$165$	−15.1952	−1.18294
$166$	0	0
$167$	6.52270	0.504742	0.252371	−	0.967631i	$-0.418790\pi$
$167$	6.52270	0.504742	0.252371	+	0.967631i	$0.418790\pi$
$168$	0	0
$169$	−0.580329	−0.0446407
$170$	0	0
$171$	−10.3586	−0.792140
$172$	0	0
$173$	7.80621	0.593495	0.296748	−	0.954956i	$-0.404098\pi$
$173$	7.80621	0.593495	0.296748	+	0.954956i	$0.404098\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.91082	0.143626
$178$	0	0
$179$	0.743131	0.0555442	0.0277721	−	0.999614i	$-0.491159\pi$
$179$	0.743131	0.0555442	0.0277721	+	0.999614i	$0.491159\pi$
$180$	0	0
$181$	1.36991	0.101824	0.0509122	−	0.998703i	$-0.483787\pi$
$181$	1.36991	0.101824	0.0509122	+	0.998703i	$0.483787\pi$
$182$	0	0
$183$	−1.24948	−0.0923644
$184$	0	0
$185$	21.5126	1.58164
$186$	0	0
$187$	−24.7873	−1.81263
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.58544	0.259434	0.129717	−	0.991551i	$-0.458593\pi$
$191$	3.58544	0.259434	0.129717	+	0.991551i	$0.458593\pi$
$192$	0	0
$193$	16.4604	1.18485	0.592423	−	0.805627i	$-0.298171\pi$
$193$	16.4604	1.18485	0.592423	+	0.805627i	$0.298171\pi$
$194$	0	0
$195$	14.3401	1.02692
$196$	0	0
$197$	10.3112	0.734641	0.367320	−	0.930094i	$-0.380275\pi$
$197$	10.3112	0.734641	0.367320	+	0.930094i	$0.380275\pi$
$198$	0	0
$199$	−19.6837	−1.39534	−0.697669	−	0.716421i	$-0.745779\pi$
$199$	−19.6837	−1.39534	−0.697669	+	0.716421i	$0.745779\pi$
$200$	0	0
$201$	−23.1986	−1.63630
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−11.6221	−0.811725
$206$	0	0
$207$	−0.708846	−0.0492682
$208$	0	0
$209$	20.7038	1.43211
$210$	0	0
$211$	−7.19588	−0.495385	−0.247692	−	0.968839i	$-0.579672\pi$
$211$	−7.19588	−0.495385	−0.247692	+	0.968839i	$0.579672\pi$
$212$	0	0
$213$	−0.303494	−0.0207951
$214$	0	0
$215$	11.9193	0.812892
$216$	0	0
$217$	0	0
$218$	0	0
$219$	29.0521	1.96316
$220$	0	0
$221$	23.3925	1.57355
$222$	0	0
$223$	−1.62864	−0.109062	−0.0545309	−	0.998512i	$-0.517366\pi$
$223$	−1.62864	−0.109062	−0.0545309	+	0.998512i	$0.517366\pi$
$224$	0	0
$225$	−2.98736	−0.199157
$226$	0	0
$227$	3.73215	0.247711	0.123856	−	0.992300i	$-0.460474\pi$
$227$	3.73215	0.247711	0.123856	+	0.992300i	$0.460474\pi$
$228$	0	0
$229$	−25.5932	−1.69125	−0.845624	−	0.533779i	$-0.820771\pi$
$229$	−25.5932	−1.69125	−0.845624	+	0.533779i	$0.820771\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.74090	0.638148	0.319074	−	0.947730i	$-0.396628\pi$
$233$	9.74090	0.638148	0.319074	+	0.947730i	$0.396628\pi$
$234$	0	0
$235$	11.3248	0.738751
$236$	0	0
$237$	7.31643	0.475253
$238$	0	0
$239$	29.9105	1.93475	0.967376	−	0.253345i	$-0.0815306\pi$
$239$	29.9105	1.93475	0.967376	+	0.253345i	$0.0815306\pi$
$240$	0	0
$241$	0.780930	0.0503041	0.0251521	−	0.999684i	$-0.491993\pi$
$241$	0.780930	0.0503041	0.0251521	+	0.999684i	$0.491993\pi$
$242$	0	0
$243$	17.0322	1.09262
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−19.5388	−1.24322
$248$	0	0
$249$	32.9646	2.08904
$250$	0	0
$251$	15.2681	0.963713	0.481857	−	0.876250i	$-0.339963\pi$
$251$	15.2681	0.963713	0.481857	+	0.876250i	$0.339963\pi$
$252$	0	0
$253$	1.41678	0.0890721
$254$	0	0
$255$	27.0097	1.69141
$256$	0	0
$257$	0.336019	0.0209603	0.0104801	−	0.999945i	$-0.496664\pi$
$257$	0.336019	0.0209603	0.0104801	+	0.999945i	$0.496664\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6.63071	−0.410431
$262$	0	0
$263$	6.60022	0.406987	0.203494	−	0.979076i	$-0.434770\pi$
$263$	6.60022	0.406987	0.203494	+	0.979076i	$0.434770\pi$
$264$	0	0
$265$	−0.413634	−0.0254093
$266$	0	0
$267$	−4.74377	−0.290314
$268$	0	0
$269$	−17.5132	−1.06780	−0.533901	−	0.845547i	$-0.679274\pi$
$269$	−17.5132	−1.06780	−0.533901	+	0.845547i	$0.679274\pi$
$270$	0	0
$271$	21.2849	1.29296	0.646482	−	0.762929i	$-0.276239\pi$
$271$	21.2849	1.29296	0.646482	+	0.762929i	$0.276239\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.97086	0.360057
$276$	0	0
$277$	−0.845076	−0.0507757	−0.0253878	−	0.999678i	$-0.508082\pi$
$277$	−0.845076	−0.0507757	−0.0253878	+	0.999678i	$0.508082\pi$
$278$	0	0
$279$	8.49650	0.508672
$280$	0	0
$281$	0.184609	0.0110129	0.00550643	−	0.999985i	$-0.498247\pi$
$281$	0.184609	0.0110129	0.00550643	+	0.999985i	$0.498247\pi$
$282$	0	0
$283$	11.9908	0.712780	0.356390	−	0.934337i	$-0.384007\pi$
$283$	11.9908	0.712780	0.356390	+	0.934337i	$0.384007\pi$
$284$	0	0
$285$	−22.5601	−1.33635
$286$	0	0
$287$	0	0
$288$	0	0
$289$	27.0599	1.59176
$290$	0	0
$291$	6.28316	0.368325
$292$	0	0
$293$	21.4080	1.25067	0.625334	−	0.780358i	$-0.284963\pi$
$293$	21.4080	1.25067	0.625334	+	0.780358i	$0.284963\pi$
$294$	0	0
$295$	1.59712	0.0929880
$296$	0	0
$297$	−9.32421	−0.541046
$298$	0	0
$299$	−1.33706	−0.0773240
$300$	0	0
$301$	0	0
$302$	0	0
$303$	8.39862	0.482488
$304$	0	0
$305$	−1.04435	−0.0597995
$306$	0	0
$307$	4.24526	0.242290	0.121145	−	0.992635i	$-0.461343\pi$
$307$	4.24526	0.242290	0.121145	+	0.992635i	$0.461343\pi$
$308$	0	0
$309$	0.626522	0.0356416
$310$	0	0
$311$	5.54344	0.314340	0.157170	−	0.987572i	$-0.449763\pi$
$311$	5.54344	0.314340	0.157170	+	0.987572i	$0.449763\pi$
$312$	0	0
$313$	8.07170	0.456239	0.228120	−	0.973633i	$-0.426742\pi$
$313$	8.07170	0.456239	0.228120	+	0.973633i	$0.426742\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.52010	0.534702	0.267351	−	0.963599i	$-0.413852\pi$
$317$	9.52010	0.534702	0.267351	+	0.963599i	$0.413852\pi$
$318$	0	0
$319$	13.2529	0.742019
$320$	0	0
$321$	10.9847	0.613105
$322$	0	0
$323$	−36.8014	−2.04769
$324$	0	0
$325$	−5.63489	−0.312567
$326$	0	0
$327$	−23.7993	−1.31610
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−0.956186	−0.0525567	−0.0262784	−	0.999655i	$-0.508366\pi$
$331$	−0.956186	−0.0525567	−0.0262784	+	0.999655i	$0.508366\pi$
$332$	0	0
$333$	−21.7943	−1.19432
$334$	0	0
$335$	−19.3900	−1.05939
$336$	0	0
$337$	−1.86777	−0.101744	−0.0508720	−	0.998705i	$-0.516200\pi$
$337$	−1.86777	−0.101744	−0.0508720	+	0.998705i	$0.516200\pi$
$338$	0	0
$339$	28.7770	1.56295
$340$	0	0
$341$	−16.9821	−0.919630
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1.54381	−0.0831158
$346$	0	0
$347$	−18.5068	−0.993499	−0.496750	−	0.867894i	$-0.665473\pi$
$347$	−18.5068	−0.993499	−0.496750	+	0.867894i	$0.665473\pi$
$348$	0	0
$349$	−13.8862	−0.743311	−0.371656	−	0.928371i	$-0.621210\pi$
$349$	−13.8862	−0.743311	−0.371656	+	0.928371i	$0.621210\pi$
$350$	0	0
$351$	8.79954	0.469685
$352$	0	0
$353$	31.1881	1.65997	0.829987	−	0.557783i	$-0.188348\pi$
$353$	31.1881	1.65997	0.829987	+	0.557783i	$0.188348\pi$
$354$	0	0
$355$	−0.253669	−0.0134633
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.64177	0.139427	0.0697136	−	0.997567i	$-0.477791\pi$
$359$	2.64177	0.139427	0.0697136	+	0.997567i	$0.477791\pi$
$360$	0	0
$361$	11.7387	0.617827
$362$	0	0
$363$	−6.49764	−0.341038
$364$	0	0
$365$	24.2826	1.27101
$366$	0	0
$367$	−11.6596	−0.608626	−0.304313	−	0.952572i	$-0.598427\pi$
$367$	−11.6596	−0.608626	−0.304313	+	0.952572i	$0.598427\pi$
$368$	0	0
$369$	11.7743	0.612946
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−22.0116	−1.13972	−0.569859	−	0.821743i	$-0.693002\pi$
$373$	−22.0116	−1.13972	−0.569859	+	0.821743i	$0.693002\pi$
$374$	0	0
$375$	−26.8517	−1.38662
$376$	0	0
$377$	−12.5071	−0.644151
$378$	0	0
$379$	−17.4248	−0.895054	−0.447527	−	0.894270i	$-0.647695\pi$
$379$	−17.4248	−0.895054	−0.447527	+	0.894270i	$0.647695\pi$
$380$	0	0
$381$	−36.4673	−1.86828
$382$	0	0
$383$	8.16376	0.417149	0.208574	−	0.978007i	$-0.433118\pi$
$383$	8.16376	0.417149	0.208574	+	0.978007i	$0.433118\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−12.0754	−0.613827
$388$	0	0
$389$	−21.6439	−1.09739	−0.548696	−	0.836022i	$-0.684876\pi$
$389$	−21.6439	−1.09739	−0.548696	+	0.836022i	$0.684876\pi$
$390$	0	0
$391$	−2.51835	−0.127359
$392$	0	0
$393$	−23.6299	−1.19197
$394$	0	0
$395$	6.11528	0.307693
$396$	0	0
$397$	35.6323	1.78834	0.894168	−	0.447731i	$-0.147768\pi$
$397$	35.6323	1.78834	0.894168	+	0.447731i	$0.147768\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.6190	1.17948	0.589739	−	0.807594i	$-0.299231\pi$
$401$	23.6190	1.17948	0.589739	+	0.807594i	$0.299231\pi$
$402$	0	0
$403$	16.0265	0.798336
$404$	0	0
$405$	20.4970	1.01850
$406$	0	0
$407$	43.5605	2.15921
$408$	0	0
$409$	16.3760	0.809740	0.404870	−	0.914374i	$-0.367317\pi$
$409$	16.3760	0.809740	0.404870	+	0.914374i	$0.367317\pi$
$410$	0	0
$411$	22.1254	1.09136
$412$	0	0
$413$	0	0
$414$	0	0
$415$	27.5527	1.35251
$416$	0	0
$417$	−42.5279	−2.08260
$418$	0	0
$419$	36.1778	1.76740	0.883700	−	0.468053i	$-0.155045\pi$
$419$	36.1778	1.76740	0.883700	+	0.468053i	$0.155045\pi$
$420$	0	0
$421$	−28.8191	−1.40456	−0.702279	−	0.711901i	$-0.747834\pi$
$421$	−28.8191	−1.40456	−0.702279	+	0.711901i	$0.747834\pi$
$422$	0	0
$423$	−11.4731	−0.557842
$424$	0	0
$425$	−10.6133	−0.514823
$426$	0	0
$427$	0	0
$428$	0	0
$429$	29.0371	1.40192
$430$	0	0
$431$	13.3922	0.645081	0.322541	−	0.946556i	$-0.395463\pi$
$431$	13.3922	0.645081	0.322541	+	0.946556i	$0.395463\pi$
$432$	0	0
$433$	27.0966	1.30218	0.651090	−	0.759001i	$-0.274312\pi$
$433$	27.0966	1.30218	0.651090	+	0.759001i	$0.274312\pi$
$434$	0	0
$435$	−14.4411	−0.692400
$436$	0	0
$437$	2.10348	0.100623
$438$	0	0
$439$	−20.7508	−0.990384	−0.495192	−	0.868784i	$-0.664902\pi$
$439$	−20.7508	−0.990384	−0.495192	+	0.868784i	$0.664902\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	31.2488	1.48467	0.742337	−	0.670027i	$-0.233717\pi$
$443$	31.2488	1.48467	0.742337	+	0.670027i	$0.233717\pi$
$444$	0	0
$445$	−3.96497	−0.187958
$446$	0	0
$447$	−19.5289	−0.923684
$448$	0	0
$449$	−22.4427	−1.05914	−0.529569	−	0.848267i	$-0.677646\pi$
$449$	−22.4427	−1.05914	−0.529569	+	0.848267i	$0.677646\pi$
$450$	0	0
$451$	−23.5334	−1.10815
$452$	0	0
$453$	1.08659	0.0510525
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.3509	1.32620	0.663100	−	0.748531i	$-0.269241\pi$
$457$	28.3509	1.32620	0.663100	+	0.748531i	$0.269241\pi$
$458$	0	0
$459$	16.5740	0.773607
$460$	0	0
$461$	5.42872	0.252841	0.126420	−	0.991977i	$-0.459651\pi$
$461$	5.42872	0.252841	0.126420	+	0.991977i	$0.459651\pi$
$462$	0	0
$463$	−5.13489	−0.238638	−0.119319	−	0.992856i	$-0.538071\pi$
$463$	−5.13489	−0.238638	−0.119319	+	0.992856i	$0.538071\pi$
$464$	0	0
$465$	18.5047	0.858134
$466$	0	0
$467$	30.2927	1.40178	0.700889	−	0.713270i	$-0.252787\pi$
$467$	30.2927	1.40178	0.700889	+	0.713270i	$0.252787\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	16.4751	0.759133
$472$	0	0
$473$	24.1352	1.10974
$474$	0	0
$475$	8.86488	0.406749
$476$	0	0
$477$	0.419050	0.0191870
$478$	0	0
$479$	−25.1959	−1.15123	−0.575616	−	0.817720i	$-0.695238\pi$
$479$	−25.1959	−1.15123	−0.575616	+	0.817720i	$0.695238\pi$
$480$	0	0
$481$	−41.1093	−1.87442
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.25164	0.238465
$486$	0	0
$487$	−24.3018	−1.10122	−0.550609	−	0.834763i	$-0.685604\pi$
$487$	−24.3018	−1.10122	−0.550609	+	0.834763i	$0.685604\pi$
$488$	0	0
$489$	−7.17864	−0.324629
$490$	0	0
$491$	27.5779	1.24457	0.622286	−	0.782790i	$-0.286204\pi$
$491$	27.5779	1.24457	0.622286	+	0.782790i	$0.286204\pi$
$492$	0	0
$493$	−23.5573	−1.06097
$494$	0	0
$495$	12.8668	0.578321
$496$	0	0
$497$	0	0
$498$	0	0
$499$	26.6883	1.19473	0.597366	−	0.801969i	$-0.296214\pi$
$499$	26.6883	1.19473	0.597366	+	0.801969i	$0.296214\pi$
$500$	0	0
$501$	−14.3919	−0.642983
$502$	0	0
$503$	32.4635	1.44747	0.723737	−	0.690076i	$-0.242423\pi$
$503$	32.4635	1.44747	0.723737	+	0.690076i	$0.242423\pi$
$504$	0	0
$505$	7.01980	0.312377
$506$	0	0
$507$	1.28046	0.0568670
$508$	0	0
$509$	40.4630	1.79349	0.896745	−	0.442547i	$-0.145925\pi$
$509$	40.4630	1.79349	0.896745	+	0.442547i	$0.145925\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−13.8436	−0.611208
$514$	0	0
$515$	0.523665	0.0230754
$516$	0	0
$517$	22.9314	1.00852
$518$	0	0
$519$	−17.2239	−0.756044
$520$	0	0
$521$	21.1552	0.926827	0.463413	−	0.886142i	$-0.346624\pi$
$521$	21.1552	0.926827	0.463413	+	0.886142i	$0.346624\pi$
$522$	0	0
$523$	−7.87515	−0.344356	−0.172178	−	0.985066i	$-0.555080\pi$
$523$	−7.87515	−0.344356	−0.172178	+	0.985066i	$0.555080\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	30.1860	1.31492
$528$	0	0
$529$	−22.8561	−0.993742
$530$	0	0
$531$	−1.61803	−0.0702166
$532$	0	0
$533$	22.2092	0.961988
$534$	0	0
$535$	9.18130	0.396942
$536$	0	0
$537$	−1.63967	−0.0707569
$538$	0	0
$539$	0	0
$540$	0	0
$541$	26.8628	1.15492	0.577462	−	0.816418i	$-0.304043\pi$
$541$	26.8628	1.15492	0.577462	+	0.816418i	$0.304043\pi$
$542$	0	0
$543$	−3.02261	−0.129712
$544$	0	0
$545$	−19.8921	−0.852084
$546$	0	0
$547$	−41.5859	−1.77808	−0.889042	−	0.457826i	$-0.848628\pi$
$547$	−41.5859	−1.77808	−0.889042	+	0.457826i	$0.848628\pi$
$548$	0	0
$549$	1.05803	0.0451555
$550$	0	0
$551$	19.6764	0.838244
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−47.4661	−2.01482
$556$	0	0
$557$	40.8401	1.73045	0.865226	−	0.501382i	$-0.167175\pi$
$557$	40.8401	1.73045	0.865226	+	0.501382i	$0.167175\pi$
$558$	0	0
$559$	−22.7772	−0.963372
$560$	0	0
$561$	54.6915	2.30908
$562$	0	0
$563$	−31.7933	−1.33993	−0.669964	−	0.742394i	$-0.733690\pi$
$563$	−31.7933	−1.33993	−0.669964	+	0.742394i	$0.733690\pi$
$564$	0	0
$565$	24.0526	1.01190
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−39.6010	−1.66016	−0.830080	−	0.557645i	$-0.811705\pi$
$569$	−39.6010	−1.66016	−0.830080	+	0.557645i	$0.811705\pi$
$570$	0	0
$571$	−26.8217	−1.12245	−0.561226	−	0.827662i	$-0.689670\pi$
$571$	−26.8217	−1.12245	−0.561226	+	0.827662i	$0.689670\pi$
$572$	0	0
$573$	−7.91104	−0.330488
$574$	0	0
$575$	0.606631	0.0252983
$576$	0	0
$577$	23.7753	0.989778	0.494889	−	0.868956i	$-0.335209\pi$
$577$	23.7753	0.989778	0.494889	+	0.868956i	$0.335209\pi$
$578$	0	0
$579$	−36.3188	−1.50936
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−0.837560	−0.0346882
$584$	0	0
$585$	−12.1428	−0.502044
$586$	0	0
$587$	5.32973	0.219982	0.109991	−	0.993933i	$-0.464918\pi$
$587$	5.32973	0.219982	0.109991	+	0.993933i	$0.464918\pi$
$588$	0	0
$589$	−25.2131	−1.03889
$590$	0	0
$591$	−22.7509	−0.935847
$592$	0	0
$593$	15.9452	0.654792	0.327396	−	0.944887i	$-0.393829\pi$
$593$	15.9452	0.654792	0.327396	+	0.944887i	$0.393829\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	43.4306	1.77750
$598$	0	0
$599$	30.0532	1.22794	0.613971	−	0.789329i	$-0.289571\pi$
$599$	30.0532	1.22794	0.613971	+	0.789329i	$0.289571\pi$
$600$	0	0
$601$	26.2894	1.07237	0.536183	−	0.844102i	$-0.319866\pi$
$601$	26.2894	1.07237	0.536183	+	0.844102i	$0.319866\pi$
$602$	0	0
$603$	19.6439	0.799960
$604$	0	0
$605$	−5.43091	−0.220798
$606$	0	0
$607$	17.5340	0.711683	0.355841	−	0.934546i	$-0.384194\pi$
$607$	17.5340	0.711683	0.355841	+	0.934546i	$0.384194\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−21.6411	−0.875505
$612$	0	0
$613$	−13.6953	−0.553148	−0.276574	−	0.960993i	$-0.589199\pi$
$613$	−13.6953	−0.553148	−0.276574	+	0.960993i	$0.589199\pi$
$614$	0	0
$615$	25.6435	1.03404
$616$	0	0
$617$	38.9699	1.56887	0.784435	−	0.620211i	$-0.212953\pi$
$617$	38.9699	1.56887	0.784435	+	0.620211i	$0.212953\pi$
$618$	0	0
$619$	19.2030	0.771835	0.385918	−	0.922533i	$-0.373885\pi$
$619$	19.2030	0.771835	0.385918	+	0.922533i	$0.373885\pi$
$620$	0	0
$621$	−0.947327	−0.0380149
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−14.4488	−0.577950
$626$	0	0
$627$	−45.6815	−1.82434
$628$	0	0
$629$	−77.4296	−3.08732
$630$	0	0
$631$	6.72503	0.267719	0.133860	−	0.991000i	$-0.457263\pi$
$631$	6.72503	0.267719	0.133860	+	0.991000i	$0.457263\pi$
$632$	0	0
$633$	15.8772	0.631063
$634$	0	0
$635$	−30.4804	−1.20958
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0.256990	0.0101664
$640$	0	0
$641$	1.87232	0.0739522	0.0369761	−	0.999316i	$-0.488227\pi$
$641$	1.87232	0.0739522	0.0369761	+	0.999316i	$0.488227\pi$
$642$	0	0
$643$	−12.7541	−0.502974	−0.251487	−	0.967861i	$-0.580920\pi$
$643$	−12.7541	−0.502974	−0.251487	+	0.967861i	$0.580920\pi$
$644$	0	0
$645$	−26.2992	−1.03553
$646$	0	0
$647$	27.5115	1.08159	0.540795	−	0.841154i	$-0.318123\pi$
$647$	27.5115	1.08159	0.540795	+	0.841154i	$0.318123\pi$
$648$	0	0
$649$	3.23398	0.126945
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−42.6671	−1.66969	−0.834847	−	0.550482i	$-0.814444\pi$
$653$	−42.6671	−1.66969	−0.834847	+	0.550482i	$0.814444\pi$
$654$	0	0
$655$	−19.7505	−0.771717
$656$	0	0
$657$	−24.6005	−0.959757
$658$	0	0
$659$	1.61927	0.0630777	0.0315388	−	0.999503i	$-0.489959\pi$
$659$	1.61927	0.0630777	0.0315388	+	0.999503i	$0.489959\pi$
$660$	0	0
$661$	−31.7881	−1.23641	−0.618207	−	0.786016i	$-0.712141\pi$
$661$	−31.7881	−1.23641	−0.618207	+	0.786016i	$0.712141\pi$
$662$	0	0
$663$	−51.6140	−2.00452
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.34647	0.0521357
$668$	0	0
$669$	3.59348	0.138932
$670$	0	0
$671$	−2.11469	−0.0816367
$672$	0	0
$673$	−4.13772	−0.159497	−0.0797487	−	0.996815i	$-0.525412\pi$
$673$	−4.13772	−0.159497	−0.0797487	+	0.996815i	$0.525412\pi$
$674$	0	0
$675$	−3.99241	−0.153668
$676$	0	0
$677$	28.5136	1.09587	0.547933	−	0.836522i	$-0.315415\pi$
$677$	28.5136	1.09587	0.547933	+	0.836522i	$0.315415\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−8.23473	−0.315555
$682$	0	0
$683$	16.3795	0.626743	0.313372	−	0.949631i	$-0.398541\pi$
$683$	16.3795	0.626743	0.313372	+	0.949631i	$0.398541\pi$
$684$	0	0
$685$	18.4930	0.706582
$686$	0	0
$687$	56.4697	2.15445
$688$	0	0
$689$	0.790430	0.0301130
$690$	0	0
$691$	−28.8843	−1.09881	−0.549406	−	0.835555i	$-0.685146\pi$
$691$	−28.8843	−1.09881	−0.549406	+	0.835555i	$0.685146\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−35.5460	−1.34834
$696$	0	0
$697$	41.8312	1.58447
$698$	0	0
$699$	−21.4926	−0.812926
$700$	0	0
$701$	−14.4916	−0.547338	−0.273669	−	0.961824i	$-0.588237\pi$
$701$	−14.4916	−0.547338	−0.273669	+	0.961824i	$0.588237\pi$
$702$	0	0
$703$	64.6738	2.43922
$704$	0	0
$705$	−24.9875	−0.941083
$706$	0	0
$707$	0	0
$708$	0	0
$709$	3.96069	0.148747	0.0743735	−	0.997230i	$-0.476304\pi$
$709$	3.96069	0.148747	0.0743735	+	0.997230i	$0.476304\pi$
$710$	0	0
$711$	−6.19534	−0.232344
$712$	0	0
$713$	−1.72535	−0.0646150
$714$	0	0
$715$	24.2700	0.907647
$716$	0	0
$717$	−65.9956	−2.46465
$718$	0	0
$719$	−13.3939	−0.499508	−0.249754	−	0.968309i	$-0.580350\pi$
$719$	−13.3939	−0.499508	−0.249754	+	0.968309i	$0.580350\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.72307	−0.0640816
$724$	0	0
$725$	5.67457	0.210748
$726$	0	0
$727$	−5.08669	−0.188655	−0.0943275	−	0.995541i	$-0.530070\pi$
$727$	−5.08669	−0.188655	−0.0943275	+	0.995541i	$0.530070\pi$
$728$	0	0
$729$	−4.23751	−0.156945
$730$	0	0
$731$	−42.9009	−1.58675
$732$	0	0
$733$	−46.0447	−1.70070	−0.850351	−	0.526217i	$-0.823610\pi$
$733$	−46.0447	−1.70070	−0.850351	+	0.526217i	$0.823610\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−39.2624	−1.44625
$738$	0	0
$739$	8.62542	0.317291	0.158646	−	0.987336i	$-0.449287\pi$
$739$	8.62542	0.317291	0.158646	+	0.987336i	$0.449287\pi$
$740$	0	0
$741$	43.1110	1.58372
$742$	0	0
$743$	−47.1158	−1.72851	−0.864255	−	0.503054i	$-0.832210\pi$
$743$	−47.1158	−1.72851	−0.864255	+	0.503054i	$0.832210\pi$
$744$	0	0
$745$	−16.3228	−0.598021
$746$	0	0
$747$	−27.9134	−1.02130
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−44.0986	−1.60918	−0.804591	−	0.593830i	$-0.797615\pi$
$751$	−44.0986	−1.60918	−0.804591	+	0.593830i	$0.797615\pi$
$752$	0	0
$753$	−33.6880	−1.22766
$754$	0	0
$755$	0.908203	0.0330529
$756$	0	0
$757$	16.0489	0.583309	0.291654	−	0.956524i	$-0.405794\pi$
$757$	16.0489	0.583309	0.291654	+	0.956524i	$0.405794\pi$
$758$	0	0
$759$	−3.12603	−0.113468
$760$	0	0
$761$	33.3388	1.20853	0.604266	−	0.796783i	$-0.293466\pi$
$761$	33.3388	1.20853	0.604266	+	0.796783i	$0.293466\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−22.8711	−0.826905
$766$	0	0
$767$	−3.05200	−0.110201
$768$	0	0
$769$	−24.5500	−0.885296	−0.442648	−	0.896695i	$-0.645961\pi$
$769$	−24.5500	−0.885296	−0.442648	+	0.896695i	$0.645961\pi$
$770$	0	0
$771$	−0.741404	−0.0267010
$772$	0	0
$773$	−2.90051	−0.104324	−0.0521621	−	0.998639i	$-0.516611\pi$
$773$	−2.90051	−0.104324	−0.0521621	+	0.998639i	$0.516611\pi$
$774$	0	0
$775$	−7.27132	−0.261193
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−34.9399	−1.25185
$780$	0	0
$781$	−0.513649	−0.0183798
$782$	0	0
$783$	−8.86152	−0.316685
$784$	0	0
$785$	13.7704	0.491485
$786$	0	0
$787$	33.7397	1.20269	0.601345	−	0.798989i	$-0.294632\pi$
$787$	33.7397	1.20269	0.601345	+	0.798989i	$0.294632\pi$
$788$	0	0
$789$	−14.5629	−0.518455
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1.99570	0.0708693
$794$	0	0
$795$	0.912655	0.0323686
$796$	0	0
$797$	−30.9331	−1.09571	−0.547853	−	0.836575i	$-0.684555\pi$
$797$	−30.9331	−1.09571	−0.547853	+	0.836575i	$0.684555\pi$
$798$	0	0
$799$	−40.7611	−1.44202
$800$	0	0
$801$	4.01689	0.141930
$802$	0	0
$803$	49.1693	1.73515
$804$	0	0
$805$	0	0
$806$	0	0
$807$	38.6418	1.36026
$808$	0	0
$809$	13.1237	0.461404	0.230702	−	0.973024i	$-0.425898\pi$
$809$	13.1237	0.461404	0.230702	+	0.973024i	$0.425898\pi$
$810$	0	0
$811$	−4.05213	−0.142289	−0.0711447	−	0.997466i	$-0.522665\pi$
$811$	−4.05213	−0.142289	−0.0711447	+	0.997466i	$0.522665\pi$
$812$	0	0
$813$	−46.9637	−1.64709
$814$	0	0
$815$	−6.00011	−0.210175
$816$	0	0
$817$	35.8334	1.25365
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.1792	1.08816	0.544081	−	0.839032i	$-0.316878\pi$
$821$	31.1792	1.08816	0.544081	+	0.839032i	$0.316878\pi$
$822$	0	0
$823$	13.9653	0.486800	0.243400	−	0.969926i	$-0.421737\pi$
$823$	13.9653	0.486800	0.243400	+	0.969926i	$0.421737\pi$
$824$	0	0
$825$	−13.1743	−0.458671
$826$	0	0
$827$	10.2868	0.357707	0.178853	−	0.983876i	$-0.442761\pi$
$827$	10.2868	0.357707	0.178853	+	0.983876i	$0.442761\pi$
$828$	0	0
$829$	−5.41383	−0.188030	−0.0940150	−	0.995571i	$-0.529970\pi$
$829$	−5.41383	−0.188030	−0.0940150	+	0.995571i	$0.529970\pi$
$830$	0	0
$831$	1.86460	0.0646823
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−12.0292	−0.416286
$836$	0	0
$837$	11.3550	0.392487
$838$	0	0
$839$	−36.3155	−1.25375	−0.626875	−	0.779120i	$-0.715666\pi$
$839$	−36.3155	−1.25375	−0.626875	+	0.779120i	$0.715666\pi$
$840$	0	0
$841$	−16.4048	−0.565681
$842$	0	0
$843$	−0.407328	−0.0140291
$844$	0	0
$845$	1.07024	0.0368174
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−26.4569	−0.907999
$850$	0	0
$851$	4.42568	0.151710
$852$	0	0
$853$	−47.6293	−1.63080	−0.815398	−	0.578901i	$-0.803482\pi$
$853$	−47.6293	−1.63080	−0.815398	+	0.578901i	$0.803482\pi$
$854$	0	0
$855$	19.1033	0.653318
$856$	0	0
$857$	6.32479	0.216051	0.108025	−	0.994148i	$-0.465547\pi$
$857$	6.32479	0.216051	0.108025	+	0.994148i	$0.465547\pi$
$858$	0	0
$859$	17.1359	0.584668	0.292334	−	0.956316i	$-0.405568\pi$
$859$	17.1359	0.584668	0.292334	+	0.956316i	$0.405568\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−35.5864	−1.21138	−0.605688	−	0.795702i	$-0.707102\pi$
$863$	−35.5864	−1.21138	−0.605688	+	0.795702i	$0.707102\pi$
$864$	0	0
$865$	−14.3962	−0.489486
$866$	0	0
$867$	−59.7059	−2.02772
$868$	0	0
$869$	12.3827	0.420055
$870$	0	0
$871$	37.0532	1.25550
$872$	0	0
$873$	−5.32040	−0.180068
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−9.35708	−0.315966	−0.157983	−	0.987442i	$-0.550499\pi$
$877$	−9.35708	−0.315966	−0.157983	+	0.987442i	$0.550499\pi$
$878$	0	0
$879$	−47.2353	−1.59321
$880$	0	0
$881$	−8.26401	−0.278422	−0.139211	−	0.990263i	$-0.544457\pi$
$881$	−8.26401	−0.278422	−0.139211	+	0.990263i	$0.544457\pi$
$882$	0	0
$883$	−27.8225	−0.936301	−0.468151	−	0.883649i	$-0.655080\pi$
$883$	−27.8225	−0.936301	−0.468151	+	0.883649i	$0.655080\pi$
$884$	0	0
$885$	−3.52394	−0.118456
$886$	0	0
$887$	4.18643	0.140567	0.0702833	−	0.997527i	$-0.477610\pi$
$887$	4.18643	0.140567	0.0702833	+	0.997527i	$0.477610\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	41.5040	1.39044
$892$	0	0
$893$	34.0461	1.13931
$894$	0	0
$895$	−1.37048	−0.0458101
$896$	0	0
$897$	2.95013	0.0985018
$898$	0	0
$899$	−16.1394	−0.538278
$900$	0	0
$901$	1.48878	0.0495985
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.52638	−0.0839797
$906$	0	0
$907$	6.94057	0.230458	0.115229	−	0.993339i	$-0.463240\pi$
$907$	6.94057	0.230458	0.115229	+	0.993339i	$0.463240\pi$
$908$	0	0
$909$	−7.11171	−0.235881
$910$	0	0
$911$	−21.7159	−0.719480	−0.359740	−	0.933053i	$-0.617135\pi$
$911$	−21.7159	−0.719480	−0.359740	+	0.933053i	$0.617135\pi$
$912$	0	0
$913$	55.7909	1.84641
$914$	0	0
$915$	2.30429	0.0761776
$916$	0	0
$917$	0	0
$918$	0	0
$919$	8.02713	0.264790	0.132395	−	0.991197i	$-0.457733\pi$
$919$	8.02713	0.264790	0.132395	+	0.991197i	$0.457733\pi$
$920$	0	0
$921$	−9.36688	−0.308649
$922$	0	0
$923$	0.484747	0.0159556
$924$	0	0
$925$	18.6516	0.613260
$926$	0	0
$927$	−0.530521	−0.0174246
$928$	0	0
$929$	3.37852	0.110846	0.0554229	−	0.998463i	$-0.482349\pi$
$929$	3.37852	0.110846	0.0554229	+	0.998463i	$0.482349\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−12.2312	−0.400432
$934$	0	0
$935$	45.7127	1.49496
$936$	0	0
$937$	−14.0973	−0.460538	−0.230269	−	0.973127i	$-0.573961\pi$
$937$	−14.0973	−0.460538	−0.230269	+	0.973127i	$0.573961\pi$
$938$	0	0
$939$	−17.8097	−0.581196
$940$	0	0
$941$	51.7670	1.68756	0.843779	−	0.536691i	$-0.180326\pi$
$941$	51.7670	1.68756	0.843779	+	0.536691i	$0.180326\pi$
$942$	0	0
$943$	−2.39096	−0.0778605
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22.4993	0.731128	0.365564	−	0.930786i	$-0.380876\pi$
$947$	22.4993	0.731128	0.365564	+	0.930786i	$0.380876\pi$
$948$	0	0
$949$	−46.4026	−1.50629
$950$	0	0
$951$	−21.0055	−0.681149
$952$	0	0
$953$	−49.0717	−1.58959	−0.794794	−	0.606880i	$-0.792421\pi$
$953$	−49.0717	−1.58959	−0.794794	+	0.606880i	$0.792421\pi$
$954$	0	0
$955$	−6.61227	−0.213968
$956$	0	0
$957$	−29.2416	−0.945246
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−10.3193	−0.332879
$962$	0	0
$963$	−9.30151	−0.299737
$964$	0	0
$965$	−30.3563	−0.977203
$966$	0	0
$967$	−2.06335	−0.0663530	−0.0331765	−	0.999450i	$-0.510562\pi$
$967$	−2.06335	−0.0663530	−0.0331765	+	0.999450i	$0.510562\pi$
$968$	0	0
$969$	81.1999	2.60852
$970$	0	0
$971$	21.2584	0.682213	0.341107	−	0.940025i	$-0.389198\pi$
$971$	21.2584	0.682213	0.341107	+	0.940025i	$0.389198\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	12.4330	0.398175
$976$	0	0
$977$	−56.6371	−1.81198	−0.905990	−	0.423298i	$-0.860872\pi$
$977$	−56.6371	−1.81198	−0.905990	+	0.423298i	$0.860872\pi$
$978$	0	0
$979$	−8.02860	−0.256595
$980$	0	0
$981$	20.1526	0.643421
$982$	0	0
$983$	−23.9416	−0.763618	−0.381809	−	0.924241i	$-0.624699\pi$
$983$	−23.9416	−0.763618	−0.381809	+	0.924241i	$0.624699\pi$
$984$	0	0
$985$	−19.0159	−0.605895
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.45211	0.0779725
$990$	0	0
$991$	53.0929	1.68655	0.843276	−	0.537480i	$-0.180624\pi$
$991$	53.0929	1.68655	0.843276	+	0.537480i	$0.180624\pi$
$992$	0	0
$993$	2.10976	0.0669512
$994$	0	0
$995$	36.3006	1.15081
$996$	0	0
$997$	−11.2226	−0.355422	−0.177711	−	0.984083i	$-0.556869\pi$
$997$	−11.2226	−0.355422	−0.177711	+	0.984083i	$0.556869\pi$
$998$	0	0
$999$	−29.1266	−0.921526

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5488.2.a.h.1.3		6
4.3	odd	2		343.2.a.d.1.3	yes	6
7.6	odd	2		5488.2.a.p.1.4		6
12.11	even	2		3087.2.a.j.1.4		6
20.19	odd	2		8575.2.a.n.1.4		6
28.3	even	6		343.2.c.e.324.4		12
28.11	odd	6		343.2.c.d.324.4		12
28.19	even	6		343.2.c.e.18.4		12
28.23	odd	6		343.2.c.d.18.4		12
28.27	even	2		343.2.a.c.1.3	✓	6
84.83	odd	2		3087.2.a.k.1.4		6
140.139	even	2		8575.2.a.o.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
343.2.a.c.1.3	✓	6	28.27	even	2
343.2.a.d.1.3	yes	6	4.3	odd	2
343.2.c.d.18.4		12	28.23	odd	6
343.2.c.d.324.4		12	28.11	odd	6
343.2.c.e.18.4		12	28.19	even	6
343.2.c.e.324.4		12	28.3	even	6
3087.2.a.j.1.4		6	12.11	even	2
3087.2.a.k.1.4		6	84.83	odd	2
5488.2.a.h.1.3		6	1.1	even	1	trivial
5488.2.a.p.1.4		6	7.6	odd	2
8575.2.a.n.1.4		6	20.19	odd	2
8575.2.a.o.1.4		6	140.139	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5488 = 2^{4} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5488.a (trivial)

\(f(q)\)	\(=\)	\(q-2.20643 q^{3} -1.84420 q^{5} +1.86834 q^{9} -3.73428 q^{11} +3.52416 q^{13} +4.06910 q^{15} +6.63777 q^{17} -5.54425 q^{19} -0.379398 q^{23} -1.59893 q^{25} +2.49692 q^{27} -3.54898 q^{29} +4.54761 q^{31} +8.23944 q^{33} -11.6650 q^{37} -7.77581 q^{39} +6.30200 q^{41} -6.46316 q^{43} -3.44560 q^{45} -6.14079 q^{47} -14.6458 q^{51} +0.224289 q^{53} +6.88676 q^{55} +12.2330 q^{57} -0.866024 q^{59} +0.566291 q^{61} -6.49924 q^{65} +10.5141 q^{67} +0.837116 q^{69} +0.137550 q^{71} -13.1670 q^{73} +3.52794 q^{75} -3.31595 q^{79} -11.1143 q^{81} -14.9402 q^{83} -12.2414 q^{85} +7.83058 q^{87} +2.14997 q^{89} -10.0340 q^{93} +10.2247 q^{95} -2.84765 q^{97} -6.97693 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 5 q^{3} + 11 q^{5} + 9 q^{9} + q^{11} + 7 q^{13} + 5 q^{15} + 26 q^{17} + 3 q^{19} + 9 q^{23} + 15 q^{25} - 8 q^{27} - 2 q^{29} + 2 q^{31} + 18 q^{33} - 2 q^{37} - 7 q^{39} + 28 q^{41} - 5 q^{43}+ \cdots - 12 q^{99}+O(q^{100}) \) 6 * q - 5 * q^3 + 11 * q^5 + 9 * q^9 + q^11 + 7 * q^13 + 5 * q^15 + 26 * q^17 + 3 * q^19 + 9 * q^23 + 15 * q^25 - 8 * q^27 - 2 * q^29 + 2 * q^31 + 18 * q^33 - 2 * q^37 - 7 * q^39 + 28 * q^41 - 5 * q^43 + 3 * q^45 - 18 * q^47 - 13 * q^51 - q^53 + 22 * q^55 - 26 * q^57 - 5 * q^59 - 17 * q^61 - 14 * q^65 + 22 * q^67 - 20 * q^69 + 2 * q^71 + 12 * q^73 + 27 * q^75 + 2 * q^79 - 30 * q^81 - 7 * q^83 + 37 * q^85 + 25 * q^87 + 39 * q^89 - 22 * q^93 + 11 * q^95 + 35 * q^97 - 12 * q^99

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