LMFDB - Embedded newform 5488.2.a.g.1.6

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,-7,0,7,0,0,0,9,0,-3] 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:	$1$
Dimension:	$6$
Coefficient field:	6.6.2624293.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 9x^{4} + 8x^{3} + 21x^{2} - 14x - 7 $ x^6 - x^5 - 9x^4 + 8x^3 + 21x^2 - 14x - 7
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1372)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.05989$ 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)

$f(q)$	$=$	$q+1.30687 q^{3} +0.966761 q^{5} -1.29208 q^{9} +3.78649 q^{11} -2.23352 q^{13} +1.26343 q^{15} +2.45003 q^{17} -8.23900 q^{19} -4.87463 q^{23} -4.06537 q^{25} -5.60921 q^{27} +2.43817 q^{29} -1.65248 q^{31} +4.94846 q^{33} -3.64773 q^{37} -2.91893 q^{39} -0.608796 q^{41} -3.24779 q^{43} -1.24913 q^{45} -9.46730 q^{47} +3.20188 q^{51} -1.76371 q^{53} +3.66063 q^{55} -10.7673 q^{57} +0.605173 q^{59} +8.41158 q^{61} -2.15928 q^{65} -9.34944 q^{67} -6.37053 q^{69} +5.91185 q^{71} -10.5889 q^{73} -5.31293 q^{75} -11.0462 q^{79} -3.45428 q^{81} -12.9132 q^{83} +2.36859 q^{85} +3.18637 q^{87} +15.9539 q^{89} -2.15959 q^{93} -7.96515 q^{95} -9.28081 q^{97} -4.89245 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 7 q^{3} + 7 q^{5} + 9 q^{9} - 3 q^{11} + 7 q^{13} - 11 q^{15} - 7 q^{19} - 11 q^{23} + 3 q^{25} - 28 q^{27} + 6 q^{29} - 14 q^{31} - 6 q^{37} - 5 q^{39} + 3 q^{43} + 21 q^{45} - 42 q^{47} + 17 q^{51}+ \cdots + 34 q^{99}+O(q^{100}) $ 6 * q - 7 * q^3 + 7 * q^5 + 9 * q^9 - 3 * q^11 + 7 * q^13 - 11 * q^15 - 7 * q^19 - 11 * q^23 + 3 * q^25 - 28 * q^27 + 6 * q^29 - 14 * q^31 - 6 * q^37 - 5 * q^39 + 3 * q^43 + 21 * q^45 - 42 * q^47 + 17 * q^51 - 17 * q^53 - 14 * q^55 - 6 * q^57 - 35 * q^59 + 7 * q^61 - 6 * q^65 + 14 * q^67 + 14 * q^69 + 28 * q^71 - 35 * q^75 - 6 * q^79 + 34 * q^81 - 49 * q^83 - 23 * q^85 - 21 * q^87 - 7 * q^89 - 14 * q^93 + 25 * q^95 - 21 * q^97 + 34 * 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
$3$	1.30687	0.754524	0.377262	−	0.926107i	$-0.376866\pi$
$3$	1.30687	0.754524	0.377262	+	0.926107i	$0.376866\pi$
$4$	0	0
$5$	0.966761	0.432349	0.216174	−	0.976355i	$-0.430642\pi$
$5$	0.966761	0.432349	0.216174	+	0.976355i	$0.430642\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.29208	−0.430694
$10$	0	0
$11$	3.78649	1.14167	0.570834	−	0.821065i	$-0.306620\pi$
$11$	3.78649	1.14167	0.570834	+	0.821065i	$0.306620\pi$
$12$	0	0
$13$	−2.23352	−0.619467	−0.309734	−	0.950823i	$-0.600240\pi$
$13$	−2.23352	−0.619467	−0.309734	+	0.950823i	$0.600240\pi$
$14$	0	0
$15$	1.26343	0.326217
$16$	0	0
$17$	2.45003	0.594219	0.297110	−	0.954843i	$-0.403977\pi$
$17$	2.45003	0.594219	0.297110	+	0.954843i	$0.403977\pi$
$18$	0	0
$19$	−8.23900	−1.89016	−0.945078	−	0.326845i	$-0.894015\pi$
$19$	−8.23900	−1.89016	−0.945078	+	0.326845i	$0.894015\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.87463	−1.01643	−0.508216	−	0.861230i	$-0.669695\pi$
$23$	−4.87463	−1.01643	−0.508216	+	0.861230i	$0.669695\pi$
$24$	0	0
$25$	−4.06537	−0.813075
$26$	0	0
$27$	−5.60921	−1.07949
$28$	0	0
$29$	2.43817	0.452756	0.226378	−	0.974040i	$-0.427312\pi$
$29$	2.43817	0.452756	0.226378	+	0.974040i	$0.427312\pi$
$30$	0	0
$31$	−1.65248	−0.296795	−0.148397	−	0.988928i	$-0.547411\pi$
$31$	−1.65248	−0.296795	−0.148397	+	0.988928i	$0.547411\pi$
$32$	0	0
$33$	4.94846	0.861417
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.64773	−0.599682	−0.299841	−	0.953989i	$-0.596934\pi$
$37$	−3.64773	−0.599682	−0.299841	+	0.953989i	$0.596934\pi$
$38$	0	0
$39$	−2.91893	−0.467403
$40$	0	0
$41$	−0.608796	−0.0950780	−0.0475390	−	0.998869i	$-0.515138\pi$
$41$	−0.608796	−0.0950780	−0.0475390	+	0.998869i	$0.515138\pi$
$42$	0	0
$43$	−3.24779	−0.495283	−0.247641	−	0.968852i	$-0.579655\pi$
$43$	−3.24779	−0.495283	−0.247641	+	0.968852i	$0.579655\pi$
$44$	0	0
$45$	−1.24913	−0.186210
$46$	0	0
$47$	−9.46730	−1.38095	−0.690474	−	0.723357i	$-0.742598\pi$
$47$	−9.46730	−1.38095	−0.690474	+	0.723357i	$0.742598\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	3.20188	0.448353
$52$	0	0
$53$	−1.76371	−0.242264	−0.121132	−	0.992636i	$-0.538653\pi$
$53$	−1.76371	−0.242264	−0.121132	+	0.992636i	$0.538653\pi$
$54$	0	0
$55$	3.66063	0.493599
$56$	0	0
$57$	−10.7673	−1.42617
$58$	0	0
$59$	0.605173	0.0787869	0.0393934	−	0.999224i	$-0.487457\pi$
$59$	0.605173	0.0787869	0.0393934	+	0.999224i	$0.487457\pi$
$60$	0	0
$61$	8.41158	1.07699	0.538497	−	0.842628i	$-0.318993\pi$
$61$	8.41158	1.07699	0.538497	+	0.842628i	$0.318993\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.15928	−0.267826
$66$	0	0
$67$	−9.34944	−1.14222	−0.571108	−	0.820875i	$-0.693486\pi$
$67$	−9.34944	−1.14222	−0.571108	+	0.820875i	$0.693486\pi$
$68$	0	0
$69$	−6.37053	−0.766922
$70$	0	0
$71$	5.91185	0.701608	0.350804	−	0.936449i	$-0.385908\pi$
$71$	5.91185	0.701608	0.350804	+	0.936449i	$0.385908\pi$
$72$	0	0
$73$	−10.5889	−1.23934	−0.619671	−	0.784861i	$-0.712734\pi$
$73$	−10.5889	−1.23934	−0.619671	+	0.784861i	$0.712734\pi$
$74$	0	0
$75$	−5.31293	−0.613484
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−11.0462	−1.24280	−0.621398	−	0.783495i	$-0.713435\pi$
$79$	−11.0462	−1.24280	−0.621398	+	0.783495i	$0.713435\pi$
$80$	0	0
$81$	−3.45428	−0.383809
$82$	0	0
$83$	−12.9132	−1.41740	−0.708701	−	0.705509i	$-0.750719\pi$
$83$	−12.9132	−1.41740	−0.708701	+	0.705509i	$0.750719\pi$
$84$	0	0
$85$	2.36859	0.256910
$86$	0	0
$87$	3.18637	0.341615
$88$	0	0
$89$	15.9539	1.69111	0.845556	−	0.533886i	$-0.179269\pi$
$89$	15.9539	1.69111	0.845556	+	0.533886i	$0.179269\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.15959	−0.223939
$94$	0	0
$95$	−7.96515	−0.817207
$96$	0	0
$97$	−9.28081	−0.942324	−0.471162	−	0.882047i	$-0.656165\pi$
$97$	−9.28081	−0.942324	−0.471162	+	0.882047i	$0.656165\pi$
$98$	0	0
$99$	−4.89245	−0.491710
$100$	0	0
$101$	18.9030	1.88092	0.940459	−	0.339906i	$-0.110395\pi$
$101$	18.9030	1.88092	0.940459	+	0.339906i	$0.110395\pi$
$102$	0	0
$103$	5.09626	0.502149	0.251074	−	0.967968i	$-0.419216\pi$
$103$	5.09626	0.502149	0.251074	+	0.967968i	$0.419216\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.61977	0.543284	0.271642	−	0.962398i	$-0.412433\pi$
$107$	5.61977	0.543284	0.271642	+	0.962398i	$0.412433\pi$
$108$	0	0
$109$	10.2323	0.980080	0.490040	−	0.871700i	$-0.336982\pi$
$109$	10.2323	0.980080	0.490040	+	0.871700i	$0.336982\pi$
$110$	0	0
$111$	−4.76712	−0.452475
$112$	0	0
$113$	15.1691	1.42699	0.713494	−	0.700662i	$-0.247112\pi$
$113$	15.1691	1.42699	0.713494	+	0.700662i	$0.247112\pi$
$114$	0	0
$115$	−4.71261	−0.439453
$116$	0	0
$117$	2.88589	0.266801
$118$	0	0
$119$	0	0
$120$	0	0
$121$	3.33749	0.303408
$122$	0	0
$123$	−0.795620	−0.0717386
$124$	0	0
$125$	−8.76405	−0.783881
$126$	0	0
$127$	−3.20619	−0.284503	−0.142252	−	0.989831i	$-0.545434\pi$
$127$	−3.20619	−0.284503	−0.142252	+	0.989831i	$0.545434\pi$
$128$	0	0
$129$	−4.24445	−0.373703
$130$	0	0
$131$	5.50744	0.481187	0.240593	−	0.970626i	$-0.422658\pi$
$131$	5.50744	0.481187	0.240593	+	0.970626i	$0.422658\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−5.42277	−0.466717
$136$	0	0
$137$	11.9637	1.02213	0.511064	−	0.859543i	$-0.329252\pi$
$137$	11.9637	1.02213	0.511064	+	0.859543i	$0.329252\pi$
$138$	0	0
$139$	−10.4025	−0.882325	−0.441162	−	0.897427i	$-0.645434\pi$
$139$	−10.4025	−0.882325	−0.441162	+	0.897427i	$0.645434\pi$
$140$	0	0
$141$	−12.3726	−1.04196
$142$	0	0
$143$	−8.45720	−0.707226
$144$	0	0
$145$	2.35712	0.195749
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.7250	−1.37017	−0.685084	−	0.728464i	$-0.740235\pi$
$149$	−16.7250	−1.37017	−0.685084	+	0.728464i	$0.740235\pi$
$150$	0	0
$151$	18.5501	1.50958	0.754791	−	0.655965i	$-0.227738\pi$
$151$	18.5501	1.50958	0.754791	+	0.655965i	$0.227738\pi$
$152$	0	0
$153$	−3.16564	−0.255926
$154$	0	0
$155$	−1.59756	−0.128319
$156$	0	0
$157$	−12.2276	−0.975870	−0.487935	−	0.872880i	$-0.662250\pi$
$157$	−12.2276	−0.975870	−0.487935	+	0.872880i	$0.662250\pi$
$158$	0	0
$159$	−2.30495	−0.182794
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−15.0761	−1.18085	−0.590427	−	0.807091i	$-0.701041\pi$
$163$	−15.0761	−1.18085	−0.590427	+	0.807091i	$0.701041\pi$
$164$	0	0
$165$	4.78398	0.372432
$166$	0	0
$167$	−8.55828	−0.662260	−0.331130	−	0.943585i	$-0.607430\pi$
$167$	−8.55828	−0.662260	−0.331130	+	0.943585i	$0.607430\pi$
$168$	0	0
$169$	−8.01139	−0.616261
$170$	0	0
$171$	10.6455	0.814078
$172$	0	0
$173$	−17.7608	−1.35033	−0.675165	−	0.737666i	$-0.735928\pi$
$173$	−17.7608	−1.35033	−0.675165	+	0.737666i	$0.735928\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.790885	0.0594466
$178$	0	0
$179$	14.0614	1.05100	0.525498	−	0.850795i	$-0.323879\pi$
$179$	14.0614	1.05100	0.525498	+	0.850795i	$0.323879\pi$
$180$	0	0
$181$	3.14416	0.233704	0.116852	−	0.993149i	$-0.462720\pi$
$181$	3.14416	0.233704	0.116852	+	0.993149i	$0.462720\pi$
$182$	0	0
$183$	10.9929	0.812617
$184$	0	0
$185$	−3.52648	−0.259272
$186$	0	0
$187$	9.27700	0.678402
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.9595	1.51658	0.758288	−	0.651920i	$-0.226036\pi$
$191$	20.9595	1.51658	0.758288	+	0.651920i	$0.226036\pi$
$192$	0	0
$193$	−4.84186	−0.348525	−0.174262	−	0.984699i	$-0.555754\pi$
$193$	−4.84186	−0.348525	−0.174262	+	0.984699i	$0.555754\pi$
$194$	0	0
$195$	−2.82191	−0.202081
$196$	0	0
$197$	18.6607	1.32952	0.664759	−	0.747058i	$-0.268534\pi$
$197$	18.6607	1.32952	0.664759	+	0.747058i	$0.268534\pi$
$198$	0	0
$199$	6.39492	0.453324	0.226662	−	0.973973i	$-0.427219\pi$
$199$	6.39492	0.453324	0.226662	+	0.973973i	$0.427219\pi$
$200$	0	0
$201$	−12.2185	−0.861829
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−0.588561	−0.0411069
$206$	0	0
$207$	6.29842	0.437771
$208$	0	0
$209$	−31.1969	−2.15793
$210$	0	0
$211$	−22.2657	−1.53283	−0.766417	−	0.642344i	$-0.777962\pi$
$211$	−22.2657	−1.53283	−0.766417	+	0.642344i	$0.777962\pi$
$212$	0	0
$213$	7.72605	0.529380
$214$	0	0
$215$	−3.13983	−0.214135
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−13.8384	−0.935113
$220$	0	0
$221$	−5.47219	−0.368099
$222$	0	0
$223$	−20.2710	−1.35745	−0.678724	−	0.734393i	$-0.737467\pi$
$223$	−20.2710	−1.35745	−0.678724	+	0.734393i	$0.737467\pi$
$224$	0	0
$225$	5.25279	0.350186
$226$	0	0
$227$	−14.8862	−0.988033	−0.494016	−	0.869453i	$-0.664472\pi$
$227$	−14.8862	−0.988033	−0.494016	+	0.869453i	$0.664472\pi$
$228$	0	0
$229$	−10.6493	−0.703728	−0.351864	−	0.936051i	$-0.614452\pi$
$229$	−10.6493	−0.703728	−0.351864	+	0.936051i	$0.614452\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	25.9915	1.70276	0.851381	−	0.524548i	$-0.175765\pi$
$233$	25.9915	1.70276	0.851381	+	0.524548i	$0.175765\pi$
$234$	0	0
$235$	−9.15262	−0.597051
$236$	0	0
$237$	−14.4360	−0.937720
$238$	0	0
$239$	18.0618	1.16832	0.584162	−	0.811637i	$-0.301423\pi$
$239$	18.0618	1.16832	0.584162	+	0.811637i	$0.301423\pi$
$240$	0	0
$241$	−11.3485	−0.731023	−0.365512	−	0.930807i	$-0.619106\pi$
$241$	−11.3485	−0.731023	−0.365512	+	0.930807i	$0.619106\pi$
$242$	0	0
$243$	12.3133	0.789899
$244$	0	0
$245$	0	0
$246$	0	0
$247$	18.4020	1.17089
$248$	0	0
$249$	−16.8759	−1.06946
$250$	0	0
$251$	−2.21702	−0.139937	−0.0699685	−	0.997549i	$-0.522290\pi$
$251$	−2.21702	−0.139937	−0.0699685	+	0.997549i	$0.522290\pi$
$252$	0	0
$253$	−18.4577	−1.16043
$254$	0	0
$255$	3.09545	0.193845
$256$	0	0
$257$	−31.4261	−1.96031	−0.980153	−	0.198245i	$-0.936476\pi$
$257$	−31.4261	−1.96031	−0.980153	+	0.198245i	$0.936476\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3.15031	−0.194999
$262$	0	0
$263$	−6.52141	−0.402128	−0.201064	−	0.979578i	$-0.564440\pi$
$263$	−6.52141	−0.402128	−0.201064	+	0.979578i	$0.564440\pi$
$264$	0	0
$265$	−1.70509	−0.104743
$266$	0	0
$267$	20.8498	1.27598
$268$	0	0
$269$	30.7308	1.87369	0.936844	−	0.349748i	$-0.113733\pi$
$269$	30.7308	1.87369	0.936844	+	0.349748i	$0.113733\pi$
$270$	0	0
$271$	11.7045	0.710998	0.355499	−	0.934677i	$-0.384311\pi$
$271$	11.7045	0.710998	0.355499	+	0.934677i	$0.384311\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−15.3935	−0.928262
$276$	0	0
$277$	−28.4892	−1.71175	−0.855874	−	0.517184i	$-0.826980\pi$
$277$	−28.4892	−1.71175	−0.855874	+	0.517184i	$0.826980\pi$
$278$	0	0
$279$	2.13514	0.127828
$280$	0	0
$281$	−5.96553	−0.355874	−0.177937	−	0.984042i	$-0.556942\pi$
$281$	−5.96553	−0.355874	−0.177937	+	0.984042i	$0.556942\pi$
$282$	0	0
$283$	−1.07567	−0.0639421	−0.0319711	−	0.999489i	$-0.510178\pi$
$283$	−1.07567	−0.0639421	−0.0319711	+	0.999489i	$0.510178\pi$
$284$	0	0
$285$	−10.4094	−0.616602
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−10.9974	−0.646903
$290$	0	0
$291$	−12.1288	−0.711006
$292$	0	0
$293$	3.88129	0.226747	0.113374	−	0.993552i	$-0.463834\pi$
$293$	3.88129	0.226747	0.113374	+	0.993552i	$0.463834\pi$
$294$	0	0
$295$	0.585058	0.0340634
$296$	0	0
$297$	−21.2392	−1.23242
$298$	0	0
$299$	10.8876	0.629646
$300$	0	0
$301$	0	0
$302$	0	0
$303$	24.7038	1.41920
$304$	0	0
$305$	8.13199	0.465637
$306$	0	0
$307$	−21.6625	−1.23635	−0.618173	−	0.786042i	$-0.712127\pi$
$307$	−21.6625	−1.23635	−0.618173	+	0.786042i	$0.712127\pi$
$308$	0	0
$309$	6.66016	0.378883
$310$	0	0
$311$	−9.72726	−0.551582	−0.275791	−	0.961218i	$-0.588940\pi$
$311$	−9.72726	−0.551582	−0.275791	+	0.961218i	$0.588940\pi$
$312$	0	0
$313$	−10.2315	−0.578317	−0.289158	−	0.957281i	$-0.593375\pi$
$313$	−10.2315	−0.578317	−0.289158	+	0.957281i	$0.593375\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.1837	1.13363	0.566815	−	0.823845i	$-0.308176\pi$
$317$	20.1837	1.13363	0.566815	+	0.823845i	$0.308176\pi$
$318$	0	0
$319$	9.23209	0.516898
$320$	0	0
$321$	7.34433	0.409921
$322$	0	0
$323$	−20.1858	−1.12317
$324$	0	0
$325$	9.08009	0.503673
$326$	0	0
$327$	13.3724	0.739494
$328$	0	0
$329$	0	0
$330$	0	0
$331$	23.5648	1.29524	0.647620	−	0.761964i	$-0.275764\pi$
$331$	23.5648	1.29524	0.647620	+	0.761964i	$0.275764\pi$
$332$	0	0
$333$	4.71316	0.258279
$334$	0	0
$335$	−9.03867	−0.493835
$336$	0	0
$337$	17.6174	0.959680	0.479840	−	0.877356i	$-0.340695\pi$
$337$	17.6174	0.959680	0.479840	+	0.877356i	$0.340695\pi$
$338$	0	0
$339$	19.8241	1.07670
$340$	0	0
$341$	−6.25711	−0.338842
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−6.15878	−0.331578
$346$	0	0
$347$	26.8286	1.44023	0.720116	−	0.693853i	$-0.244088\pi$
$347$	26.8286	1.44023	0.720116	+	0.693853i	$0.244088\pi$
$348$	0	0
$349$	−28.0841	−1.50331	−0.751654	−	0.659557i	$-0.770744\pi$
$349$	−28.0841	−1.50331	−0.751654	+	0.659557i	$0.770744\pi$
$350$	0	0
$351$	12.5283	0.668710
$352$	0	0
$353$	11.8381	0.630077	0.315039	−	0.949079i	$-0.397982\pi$
$353$	11.8381	0.630077	0.315039	+	0.949079i	$0.397982\pi$
$354$	0	0
$355$	5.71535	0.303339
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10.5460	−0.556596	−0.278298	−	0.960495i	$-0.589770\pi$
$359$	−10.5460	−0.556596	−0.278298	+	0.960495i	$0.589770\pi$
$360$	0	0
$361$	48.8811	2.57269
$362$	0	0
$363$	4.36168	0.228929
$364$	0	0
$365$	−10.2370	−0.535828
$366$	0	0
$367$	23.0457	1.20297	0.601487	−	0.798882i	$-0.294575\pi$
$367$	23.0457	1.20297	0.601487	+	0.798882i	$0.294575\pi$
$368$	0	0
$369$	0.786614	0.0409495
$370$	0	0
$371$	0	0
$372$	0	0
$373$	31.4969	1.63085	0.815424	−	0.578865i	$-0.196504\pi$
$373$	31.4969	1.63085	0.815424	+	0.578865i	$0.196504\pi$
$374$	0	0
$375$	−11.4535	−0.591457
$376$	0	0
$377$	−5.44569	−0.280467
$378$	0	0
$379$	−24.2707	−1.24670	−0.623351	−	0.781942i	$-0.714229\pi$
$379$	−24.2707	−1.24670	−0.623351	+	0.781942i	$0.714229\pi$
$380$	0	0
$381$	−4.19008	−0.214665
$382$	0	0
$383$	4.97658	0.254291	0.127146	−	0.991884i	$-0.459418\pi$
$383$	4.97658	0.254291	0.127146	+	0.991884i	$0.459418\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.19640	0.213315
$388$	0	0
$389$	13.9757	0.708596	0.354298	−	0.935133i	$-0.384720\pi$
$389$	13.9757	0.708596	0.354298	+	0.935133i	$0.384720\pi$
$390$	0	0
$391$	−11.9430	−0.603983
$392$	0	0
$393$	7.19752	0.363067
$394$	0	0
$395$	−10.6791	−0.537322
$396$	0	0
$397$	5.35212	0.268615	0.134308	−	0.990940i	$-0.457119\pi$
$397$	5.35212	0.268615	0.134308	+	0.990940i	$0.457119\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.81339	0.0905561	0.0452781	−	0.998974i	$-0.485583\pi$
$401$	1.81339	0.0905561	0.0452781	+	0.998974i	$0.485583\pi$
$402$	0	0
$403$	3.69086	0.183855
$404$	0	0
$405$	−3.33947	−0.165939
$406$	0	0
$407$	−13.8121	−0.684639
$408$	0	0
$409$	−33.9007	−1.67628	−0.838140	−	0.545455i	$-0.816357\pi$
$409$	−33.9007	−1.67628	−0.838140	+	0.545455i	$0.816357\pi$
$410$	0	0
$411$	15.6350	0.771220
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−12.4839	−0.612812
$416$	0	0
$417$	−13.5947	−0.665735
$418$	0	0
$419$	15.6842	0.766226	0.383113	−	0.923702i	$-0.374852\pi$
$419$	15.6842	0.766226	0.383113	+	0.923702i	$0.374852\pi$
$420$	0	0
$421$	−1.36544	−0.0665477	−0.0332739	−	0.999446i	$-0.510593\pi$
$421$	−1.36544	−0.0665477	−0.0332739	+	0.999446i	$0.510593\pi$
$422$	0	0
$423$	12.2325	0.594766
$424$	0	0
$425$	−9.96028	−0.483145
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−11.0525	−0.533619
$430$	0	0
$431$	−27.8744	−1.34266	−0.671331	−	0.741158i	$-0.734277\pi$
$431$	−27.8744	−1.34266	−0.671331	+	0.741158i	$0.734277\pi$
$432$	0	0
$433$	10.5777	0.508334	0.254167	−	0.967160i	$-0.418199\pi$
$433$	10.5777	0.508334	0.254167	+	0.967160i	$0.418199\pi$
$434$	0	0
$435$	3.08046	0.147697
$436$	0	0
$437$	40.1621	1.92121
$438$	0	0
$439$	29.5293	1.40936	0.704679	−	0.709526i	$-0.251091\pi$
$439$	29.5293	1.40936	0.704679	+	0.709526i	$0.251091\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	15.4800	0.735475	0.367737	−	0.929930i	$-0.380132\pi$
$443$	15.4800	0.735475	0.367737	+	0.929930i	$0.380132\pi$
$444$	0	0
$445$	15.4236	0.731150
$446$	0	0
$447$	−21.8575	−1.03383
$448$	0	0
$449$	−36.5746	−1.72606	−0.863032	−	0.505149i	$-0.831438\pi$
$449$	−36.5746	−1.72606	−0.863032	+	0.505149i	$0.831438\pi$
$450$	0	0
$451$	−2.30520	−0.108548
$452$	0	0
$453$	24.2426	1.13902
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.69743	−0.360070	−0.180035	−	0.983660i	$-0.557621\pi$
$457$	−7.69743	−0.360070	−0.180035	+	0.983660i	$0.557621\pi$
$458$	0	0
$459$	−13.7427	−0.641455
$460$	0	0
$461$	13.4571	0.626761	0.313381	−	0.949628i	$-0.398538\pi$
$461$	13.4571	0.626761	0.313381	+	0.949628i	$0.398538\pi$
$462$	0	0
$463$	27.7916	1.29158	0.645792	−	0.763514i	$-0.276527\pi$
$463$	27.7916	1.29158	0.645792	+	0.763514i	$0.276527\pi$
$464$	0	0
$465$	−2.08781	−0.0968197
$466$	0	0
$467$	−31.2557	−1.44634	−0.723171	−	0.690669i	$-0.757316\pi$
$467$	−31.2557	−1.44634	−0.723171	+	0.690669i	$0.757316\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−15.9799	−0.736317
$472$	0	0
$473$	−12.2977	−0.565449
$474$	0	0
$475$	33.4946	1.53684
$476$	0	0
$477$	2.27886	0.104342
$478$	0	0
$479$	−41.1302	−1.87929	−0.939643	−	0.342156i	$-0.888843\pi$
$479$	−41.1302	−1.87929	−0.939643	+	0.342156i	$0.888843\pi$
$480$	0	0
$481$	8.14727	0.371484
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−8.97233	−0.407412
$486$	0	0
$487$	−2.11949	−0.0960434	−0.0480217	−	0.998846i	$-0.515292\pi$
$487$	−2.11949	−0.0960434	−0.0480217	+	0.998846i	$0.515292\pi$
$488$	0	0
$489$	−19.7026	−0.890982
$490$	0	0
$491$	−27.0278	−1.21975	−0.609873	−	0.792499i	$-0.708780\pi$
$491$	−27.0278	−1.21975	−0.609873	+	0.792499i	$0.708780\pi$
$492$	0	0
$493$	5.97358	0.269036
$494$	0	0
$495$	−4.72983	−0.212590
$496$	0	0
$497$	0	0
$498$	0	0
$499$	29.8469	1.33613	0.668065	−	0.744103i	$-0.267123\pi$
$499$	29.8469	1.33613	0.668065	+	0.744103i	$0.267123\pi$
$500$	0	0
$501$	−11.1846	−0.499691
$502$	0	0
$503$	−30.5276	−1.36116	−0.680580	−	0.732674i	$-0.738272\pi$
$503$	−30.5276	−1.36116	−0.680580	+	0.732674i	$0.738272\pi$
$504$	0	0
$505$	18.2747	0.813213
$506$	0	0
$507$	−10.4699	−0.464983
$508$	0	0
$509$	−12.7661	−0.565848	−0.282924	−	0.959142i	$-0.591304\pi$
$509$	−12.7661	−0.565848	−0.282924	+	0.959142i	$0.591304\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	46.2143	2.04041
$514$	0	0
$515$	4.92686	0.217104
$516$	0	0
$517$	−35.8478	−1.57659
$518$	0	0
$519$	−23.2112	−1.01886
$520$	0	0
$521$	−16.5210	−0.723799	−0.361900	−	0.932217i	$-0.617872\pi$
$521$	−16.5210	−0.723799	−0.361900	+	0.932217i	$0.617872\pi$
$522$	0	0
$523$	−10.4297	−0.456058	−0.228029	−	0.973654i	$-0.573228\pi$
$523$	−10.4297	−0.456058	−0.228029	+	0.973654i	$0.573228\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.04863	−0.176361
$528$	0	0
$529$	0.762053	0.0331328
$530$	0	0
$531$	−0.781933	−0.0339330
$532$	0	0
$533$	1.35976	0.0588977
$534$	0	0
$535$	5.43298	0.234888
$536$	0	0
$537$	18.3764	0.793002
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−26.4047	−1.13523	−0.567614	−	0.823295i	$-0.692133\pi$
$541$	−26.4047	−1.13523	−0.567614	+	0.823295i	$0.692133\pi$
$542$	0	0
$543$	4.10902	0.176335
$544$	0	0
$545$	9.89223	0.423736
$546$	0	0
$547$	23.1444	0.989584	0.494792	−	0.869012i	$-0.335244\pi$
$547$	23.1444	0.989584	0.494792	+	0.869012i	$0.335244\pi$
$548$	0	0
$549$	−10.8684	−0.463854
$550$	0	0
$551$	−20.0880	−0.855780
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−4.60866	−0.195627
$556$	0	0
$557$	4.95075	0.209770	0.104885	−	0.994484i	$-0.466553\pi$
$557$	4.95075	0.209770	0.104885	+	0.994484i	$0.466553\pi$
$558$	0	0
$559$	7.25400	0.306811
$560$	0	0
$561$	12.1239	0.511870
$562$	0	0
$563$	−12.5784	−0.530116	−0.265058	−	0.964232i	$-0.585391\pi$
$563$	−12.5784	−0.530116	−0.265058	+	0.964232i	$0.585391\pi$
$564$	0	0
$565$	14.6649	0.616956
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.5059	−0.440428	−0.220214	−	0.975452i	$-0.570676\pi$
$569$	−10.5059	−0.440428	−0.220214	+	0.975452i	$0.570676\pi$
$570$	0	0
$571$	3.21931	0.134724	0.0673620	−	0.997729i	$-0.478542\pi$
$571$	3.21931	0.134724	0.0673620	+	0.997729i	$0.478542\pi$
$572$	0	0
$573$	27.3914	1.14429
$574$	0	0
$575$	19.8172	0.826434
$576$	0	0
$577$	−25.6436	−1.06756	−0.533779	−	0.845624i	$-0.679229\pi$
$577$	−25.6436	−1.06756	−0.533779	+	0.845624i	$0.679229\pi$
$578$	0	0
$579$	−6.32770	−0.262970
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−6.67827	−0.276586
$584$	0	0
$585$	2.78997	0.115351
$586$	0	0
$587$	−13.6017	−0.561402	−0.280701	−	0.959795i	$-0.590567\pi$
$587$	−13.6017	−0.561402	−0.280701	+	0.959795i	$0.590567\pi$
$588$	0	0
$589$	13.6148	0.560989
$590$	0	0
$591$	24.3871	1.00315
$592$	0	0
$593$	31.9715	1.31291	0.656455	−	0.754365i	$-0.272055\pi$
$593$	31.9715	1.31291	0.656455	+	0.754365i	$0.272055\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.35736	0.342044
$598$	0	0
$599$	7.93619	0.324264	0.162132	−	0.986769i	$-0.448163\pi$
$599$	7.93619	0.324264	0.162132	+	0.986769i	$0.448163\pi$
$600$	0	0
$601$	2.41626	0.0985612	0.0492806	−	0.998785i	$-0.484307\pi$
$601$	2.41626	0.0985612	0.0492806	+	0.998785i	$0.484307\pi$
$602$	0	0
$603$	12.0802	0.491945
$604$	0	0
$605$	3.22655	0.131178
$606$	0	0
$607$	−13.3356	−0.541274	−0.270637	−	0.962681i	$-0.587234\pi$
$607$	−13.3356	−0.541274	−0.270637	+	0.962681i	$0.587234\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	21.1454	0.855452
$612$	0	0
$613$	−24.7392	−0.999205	−0.499602	−	0.866255i	$-0.666521\pi$
$613$	−24.7392	−0.999205	−0.499602	+	0.866255i	$0.666521\pi$
$614$	0	0
$615$	−0.769174	−0.0310161
$616$	0	0
$617$	−13.6111	−0.547963	−0.273982	−	0.961735i	$-0.588341\pi$
$617$	−13.6111	−0.547963	−0.273982	+	0.961735i	$0.588341\pi$
$618$	0	0
$619$	13.4489	0.540557	0.270278	−	0.962782i	$-0.412884\pi$
$619$	13.4489	0.540557	0.270278	+	0.962782i	$0.412884\pi$
$620$	0	0
$621$	27.3428	1.09723
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.8541	0.474165
$626$	0	0
$627$	−40.7704	−1.62821
$628$	0	0
$629$	−8.93703	−0.356343
$630$	0	0
$631$	34.2018	1.36155	0.680777	−	0.732491i	$-0.261642\pi$
$631$	34.2018	1.36155	0.680777	+	0.732491i	$0.261642\pi$
$632$	0	0
$633$	−29.0984	−1.15656
$634$	0	0
$635$	−3.09962	−0.123005
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−7.63859	−0.302178
$640$	0	0
$641$	−42.2560	−1.66901	−0.834506	−	0.550999i	$-0.814247\pi$
$641$	−42.2560	−1.66901	−0.834506	+	0.550999i	$0.814247\pi$
$642$	0	0
$643$	−21.6806	−0.855000	−0.427500	−	0.904015i	$-0.640606\pi$
$643$	−21.6806	−0.855000	−0.427500	+	0.904015i	$0.640606\pi$
$644$	0	0
$645$	−4.10337	−0.161570
$646$	0	0
$647$	−33.5817	−1.32023	−0.660117	−	0.751162i	$-0.729493\pi$
$647$	−33.5817	−1.32023	−0.660117	+	0.751162i	$0.729493\pi$
$648$	0	0
$649$	2.29148	0.0899485
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.74289	0.185603	0.0928017	−	0.995685i	$-0.470418\pi$
$653$	4.74289	0.185603	0.0928017	+	0.995685i	$0.470418\pi$
$654$	0	0
$655$	5.32438	0.208041
$656$	0	0
$657$	13.6818	0.533777
$658$	0	0
$659$	11.1917	0.435968	0.217984	−	0.975952i	$-0.430052\pi$
$659$	11.1917	0.435968	0.217984	+	0.975952i	$0.430052\pi$
$660$	0	0
$661$	44.4382	1.72844	0.864222	−	0.503110i	$-0.167811\pi$
$661$	44.4382	1.72844	0.864222	+	0.503110i	$0.167811\pi$
$662$	0	0
$663$	−7.15146	−0.277740
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−11.8852	−0.460195
$668$	0	0
$669$	−26.4917	−1.02423
$670$	0	0
$671$	31.8504	1.22957
$672$	0	0
$673$	−29.0230	−1.11875	−0.559377	−	0.828913i	$-0.688960\pi$
$673$	−29.0230	−1.11875	−0.559377	+	0.828913i	$0.688960\pi$
$674$	0	0
$675$	22.8035	0.877708
$676$	0	0
$677$	2.78883	0.107183	0.0535916	−	0.998563i	$-0.482933\pi$
$677$	2.78883	0.107183	0.0535916	+	0.998563i	$0.482933\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−19.4544	−0.745494
$682$	0	0
$683$	5.71003	0.218488	0.109244	−	0.994015i	$-0.465157\pi$
$683$	5.71003	0.218488	0.109244	+	0.994015i	$0.465157\pi$
$684$	0	0
$685$	11.5660	0.441916
$686$	0	0
$687$	−13.9173	−0.530980
$688$	0	0
$689$	3.93929	0.150075
$690$	0	0
$691$	−51.3054	−1.95175	−0.975875	−	0.218329i	$-0.929939\pi$
$691$	−51.3054	−1.95175	−0.975875	+	0.218329i	$0.929939\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10.0567	−0.381472
$696$	0	0
$697$	−1.49157	−0.0564972
$698$	0	0
$699$	33.9677	1.28477
$700$	0	0
$701$	30.6669	1.15827	0.579137	−	0.815230i	$-0.303390\pi$
$701$	30.6669	1.15827	0.579137	+	0.815230i	$0.303390\pi$
$702$	0	0
$703$	30.0536	1.13349
$704$	0	0
$705$	−11.9613	−0.450489
$706$	0	0
$707$	0	0
$708$	0	0
$709$	27.2154	1.02210	0.511049	−	0.859552i	$-0.329257\pi$
$709$	27.2154	1.02210	0.511049	+	0.859552i	$0.329257\pi$
$710$	0	0
$711$	14.2726	0.535265
$712$	0	0
$713$	8.05526	0.301672
$714$	0	0
$715$	−8.17609	−0.305768
$716$	0	0
$717$	23.6045	0.881528
$718$	0	0
$719$	−49.4926	−1.84576	−0.922881	−	0.385084i	$-0.874172\pi$
$719$	−49.4926	−1.84576	−0.922881	+	0.385084i	$0.874172\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−14.8311	−0.551574
$724$	0	0
$725$	−9.91205	−0.368124
$726$	0	0
$727$	34.1732	1.26742	0.633708	−	0.773573i	$-0.281532\pi$
$727$	34.1732	1.26742	0.633708	+	0.773573i	$0.281532\pi$
$728$	0	0
$729$	26.4548	0.979807
$730$	0	0
$731$	−7.95717	−0.294306
$732$	0	0
$733$	38.9796	1.43975	0.719873	−	0.694106i	$-0.244200\pi$
$733$	38.9796	1.43975	0.719873	+	0.694106i	$0.244200\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−35.4015	−1.30403
$738$	0	0
$739$	28.2772	1.04019	0.520096	−	0.854108i	$-0.325896\pi$
$739$	28.2772	1.04019	0.520096	+	0.854108i	$0.325896\pi$
$740$	0	0
$741$	24.0491	0.883464
$742$	0	0
$743$	−27.4681	−1.00771	−0.503854	−	0.863789i	$-0.668085\pi$
$743$	−27.4681	−1.00771	−0.503854	+	0.863789i	$0.668085\pi$
$744$	0	0
$745$	−16.1691	−0.592391
$746$	0	0
$747$	16.6848	0.610466
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0.200008	0.00729840	0.00364920	−	0.999993i	$-0.498838\pi$
$751$	0.200008	0.00729840	0.00364920	+	0.999993i	$0.498838\pi$
$752$	0	0
$753$	−2.89736	−0.105586
$754$	0	0
$755$	17.9335	0.652666
$756$	0	0
$757$	−9.94418	−0.361427	−0.180714	−	0.983536i	$-0.557841\pi$
$757$	−9.94418	−0.361427	−0.180714	+	0.983536i	$0.557841\pi$
$758$	0	0
$759$	−24.1219	−0.875571
$760$	0	0
$761$	−5.86755	−0.212699	−0.106349	−	0.994329i	$-0.533916\pi$
$761$	−5.86755	−0.212699	−0.106349	+	0.994329i	$0.533916\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−3.06041	−0.110650
$766$	0	0
$767$	−1.35167	−0.0488059
$768$	0	0
$769$	−12.9114	−0.465596	−0.232798	−	0.972525i	$-0.574788\pi$
$769$	−12.9114	−0.465596	−0.232798	+	0.972525i	$0.574788\pi$
$770$	0	0
$771$	−41.0699	−1.47910
$772$	0	0
$773$	39.9116	1.43552	0.717760	−	0.696290i	$-0.245167\pi$
$773$	39.9116	1.43552	0.717760	+	0.696290i	$0.245167\pi$
$774$	0	0
$775$	6.71796	0.241316
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.01587	0.179712
$780$	0	0
$781$	22.3852	0.801004
$782$	0	0
$783$	−13.6762	−0.488747
$784$	0	0
$785$	−11.8212	−0.421916
$786$	0	0
$787$	11.1899	0.398878	0.199439	−	0.979910i	$-0.436088\pi$
$787$	11.1899	0.398878	0.199439	+	0.979910i	$0.436088\pi$
$788$	0	0
$789$	−8.52267	−0.303415
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−18.7874	−0.667162
$794$	0	0
$795$	−2.22834	−0.0790309
$796$	0	0
$797$	27.4646	0.972846	0.486423	−	0.873723i	$-0.338301\pi$
$797$	27.4646	0.972846	0.486423	+	0.873723i	$0.338301\pi$
$798$	0	0
$799$	−23.1952	−0.820586
$800$	0	0
$801$	−20.6138	−0.728351
$802$	0	0
$803$	−40.0949	−1.41492
$804$	0	0
$805$	0	0
$806$	0	0
$807$	40.1612	1.41374
$808$	0	0
$809$	38.9064	1.36788	0.683938	−	0.729540i	$-0.260266\pi$
$809$	38.9064	1.36788	0.683938	+	0.729540i	$0.260266\pi$
$810$	0	0
$811$	−24.3485	−0.854992	−0.427496	−	0.904017i	$-0.640604\pi$
$811$	−24.3485	−0.854992	−0.427496	+	0.904017i	$0.640604\pi$
$812$	0	0
$813$	15.2963	0.536465
$814$	0	0
$815$	−14.5750	−0.510541
$816$	0	0
$817$	26.7585	0.936162
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.37632	0.0829341	0.0414670	−	0.999140i	$-0.486797\pi$
$821$	2.37632	0.0829341	0.0414670	+	0.999140i	$0.486797\pi$
$822$	0	0
$823$	8.71831	0.303901	0.151951	−	0.988388i	$-0.451445\pi$
$823$	8.71831	0.303901	0.151951	+	0.988388i	$0.451445\pi$
$824$	0	0
$825$	−20.1173	−0.700396
$826$	0	0
$827$	37.7669	1.31328	0.656642	−	0.754202i	$-0.271976\pi$
$827$	37.7669	1.31328	0.656642	+	0.754202i	$0.271976\pi$
$828$	0	0
$829$	15.3392	0.532752	0.266376	−	0.963869i	$-0.414174\pi$
$829$	15.3392	0.532752	0.266376	+	0.963869i	$0.414174\pi$
$830$	0	0
$831$	−37.2318	−1.29156
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−8.27382	−0.286327
$836$	0	0
$837$	9.26913	0.320388
$838$	0	0
$839$	44.2930	1.52916	0.764582	−	0.644526i	$-0.222945\pi$
$839$	44.2930	1.52916	0.764582	+	0.644526i	$0.222945\pi$
$840$	0	0
$841$	−23.0553	−0.795012
$842$	0	0
$843$	−7.79619	−0.268515
$844$	0	0
$845$	−7.74510	−0.266439
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.40577	−0.0482458
$850$	0	0
$851$	17.7813	0.609536
$852$	0	0
$853$	28.4674	0.974706	0.487353	−	0.873205i	$-0.337962\pi$
$853$	28.4674	0.974706	0.487353	+	0.873205i	$0.337962\pi$
$854$	0	0
$855$	10.2916	0.351966
$856$	0	0
$857$	−24.4097	−0.833819	−0.416909	−	0.908948i	$-0.636887\pi$
$857$	−24.4097	−0.833819	−0.416909	+	0.908948i	$0.636887\pi$
$858$	0	0
$859$	4.18081	0.142647	0.0713237	−	0.997453i	$-0.477278\pi$
$859$	4.18081	0.142647	0.0713237	+	0.997453i	$0.477278\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−49.6632	−1.69055	−0.845277	−	0.534328i	$-0.820565\pi$
$863$	−49.6632	−1.69055	−0.845277	+	0.534328i	$0.820565\pi$
$864$	0	0
$865$	−17.1705	−0.583814
$866$	0	0
$867$	−14.3722	−0.488104
$868$	0	0
$869$	−41.8264	−1.41886
$870$	0	0
$871$	20.8822	0.707565
$872$	0	0
$873$	11.9916	0.405853
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−7.62435	−0.257456	−0.128728	−	0.991680i	$-0.541089\pi$
$877$	−7.62435	−0.257456	−0.128728	+	0.991680i	$0.541089\pi$
$878$	0	0
$879$	5.07235	0.171086
$880$	0	0
$881$	23.6086	0.795393	0.397697	−	0.917517i	$-0.369810\pi$
$881$	23.6086	0.795393	0.397697	+	0.917517i	$0.369810\pi$
$882$	0	0
$883$	1.58826	0.0534491	0.0267245	−	0.999643i	$-0.491492\pi$
$883$	1.58826	0.0534491	0.0267245	+	0.999643i	$0.491492\pi$
$884$	0	0
$885$	0.764597	0.0257016
$886$	0	0
$887$	−7.33270	−0.246208	−0.123104	−	0.992394i	$-0.539285\pi$
$887$	−7.33270	−0.246208	−0.123104	+	0.992394i	$0.539285\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−13.0796	−0.438183
$892$	0	0
$893$	78.0011	2.61021
$894$	0	0
$895$	13.5940	0.454397
$896$	0	0
$897$	14.2287	0.475083
$898$	0	0
$899$	−4.02903	−0.134376
$900$	0	0
$901$	−4.32115	−0.143958
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.03965	0.101041
$906$	0	0
$907$	−27.2999	−0.906478	−0.453239	−	0.891389i	$-0.649732\pi$
$907$	−27.2999	−0.906478	−0.453239	+	0.891389i	$0.649732\pi$
$908$	0	0
$909$	−24.4242	−0.810100
$910$	0	0
$911$	13.1608	0.436036	0.218018	−	0.975945i	$-0.430041\pi$
$911$	13.1608	0.436036	0.218018	+	0.975945i	$0.430041\pi$
$912$	0	0
$913$	−48.8955	−1.61820
$914$	0	0
$915$	10.6275	0.351334
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−6.23584	−0.205702	−0.102851	−	0.994697i	$-0.532796\pi$
$919$	−6.23584	−0.205702	−0.102851	+	0.994697i	$0.532796\pi$
$920$	0	0
$921$	−28.3102	−0.932853
$922$	0	0
$923$	−13.2042	−0.434623
$924$	0	0
$925$	14.8294	0.487587
$926$	0	0
$927$	−6.58478	−0.216272
$928$	0	0
$929$	−43.3372	−1.42185	−0.710923	−	0.703270i	$-0.751723\pi$
$929$	−43.3372	−1.42185	−0.710923	+	0.703270i	$0.751723\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−12.7123	−0.416182
$934$	0	0
$935$	8.96865	0.293306
$936$	0	0
$937$	47.4200	1.54914	0.774572	−	0.632486i	$-0.217965\pi$
$937$	47.4200	1.54914	0.774572	+	0.632486i	$0.217965\pi$
$938$	0	0
$939$	−13.3712	−0.436354
$940$	0	0
$941$	37.9074	1.23575	0.617874	−	0.786278i	$-0.287994\pi$
$941$	37.9074	1.23575	0.617874	+	0.786278i	$0.287994\pi$
$942$	0	0
$943$	2.96766	0.0966402
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−48.5076	−1.57628	−0.788142	−	0.615493i	$-0.788957\pi$
$947$	−48.5076	−1.57628	−0.788142	+	0.615493i	$0.788957\pi$
$948$	0	0
$949$	23.6506	0.767732
$950$	0	0
$951$	26.3775	0.855351
$952$	0	0
$953$	33.8440	1.09631	0.548157	−	0.836376i	$-0.315330\pi$
$953$	33.8440	1.09631	0.548157	+	0.836376i	$0.315330\pi$
$954$	0	0
$955$	20.2628	0.655690
$956$	0	0
$957$	12.0652	0.390012
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28.2693	−0.911913
$962$	0	0
$963$	−7.26120	−0.233989
$964$	0	0
$965$	−4.68093	−0.150684
$966$	0	0
$967$	−28.6546	−0.921471	−0.460735	−	0.887538i	$-0.652414\pi$
$967$	−28.6546	−0.921471	−0.460735	+	0.887538i	$0.652414\pi$
$968$	0	0
$969$	−26.3803	−0.847456
$970$	0	0
$971$	38.4769	1.23478	0.617391	−	0.786656i	$-0.288190\pi$
$971$	38.4769	1.23478	0.617391	+	0.786656i	$0.288190\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	11.8665	0.380033
$976$	0	0
$977$	6.87253	0.219872	0.109936	−	0.993939i	$-0.464935\pi$
$977$	6.87253	0.219872	0.109936	+	0.993939i	$0.464935\pi$
$978$	0	0
$979$	60.4093	1.93069
$980$	0	0
$981$	−13.2210	−0.422114
$982$	0	0
$983$	11.9531	0.381244	0.190622	−	0.981664i	$-0.438950\pi$
$983$	11.9531	0.381244	0.190622	+	0.981664i	$0.438950\pi$
$984$	0	0
$985$	18.0404	0.574816
$986$	0	0
$987$	0	0
$988$	0	0
$989$	15.8318	0.503421
$990$	0	0
$991$	29.1548	0.926132	0.463066	−	0.886324i	$-0.346749\pi$
$991$	29.1548	0.926132	0.463066	+	0.886324i	$0.346749\pi$
$992$	0	0
$993$	30.7962	0.977289
$994$	0	0
$995$	6.18236	0.195994
$996$	0	0
$997$	2.99371	0.0948118	0.0474059	−	0.998876i	$-0.484905\pi$
$997$	2.99371	0.0948118	0.0474059	+	0.998876i	$0.484905\pi$
$998$	0	0
$999$	20.4609	0.647353

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5488.2.a.g.1.6		6
4.3	odd	2		1372.2.a.d.1.1	yes	6
7.6	odd	2		5488.2.a.q.1.1		6
28.3	even	6		1372.2.e.d.1353.1		12
28.11	odd	6		1372.2.e.a.1353.6		12
28.19	even	6		1372.2.e.d.361.1		12
28.23	odd	6		1372.2.e.a.361.6		12
28.27	even	2		1372.2.a.a.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1372.2.a.a.1.6	✓	6	28.27	even	2
1372.2.a.d.1.1	yes	6	4.3	odd	2
1372.2.e.a.361.6		12	28.23	odd	6
1372.2.e.a.1353.6		12	28.11	odd	6
1372.2.e.d.361.1		12	28.19	even	6
1372.2.e.d.1353.1		12	28.3	even	6
5488.2.a.g.1.6		6	1.1	even	1	trivial
5488.2.a.q.1.1		6	7.6	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.30687 q^{3} +0.966761 q^{5} -1.29208 q^{9} +3.78649 q^{11} -2.23352 q^{13} +1.26343 q^{15} +2.45003 q^{17} -8.23900 q^{19} -4.87463 q^{23} -4.06537 q^{25} -5.60921 q^{27} +2.43817 q^{29} -1.65248 q^{31} +4.94846 q^{33} -3.64773 q^{37} -2.91893 q^{39} -0.608796 q^{41} -3.24779 q^{43} -1.24913 q^{45} -9.46730 q^{47} +3.20188 q^{51} -1.76371 q^{53} +3.66063 q^{55} -10.7673 q^{57} +0.605173 q^{59} +8.41158 q^{61} -2.15928 q^{65} -9.34944 q^{67} -6.37053 q^{69} +5.91185 q^{71} -10.5889 q^{73} -5.31293 q^{75} -11.0462 q^{79} -3.45428 q^{81} -12.9132 q^{83} +2.36859 q^{85} +3.18637 q^{87} +15.9539 q^{89} -2.15959 q^{93} -7.96515 q^{95} -9.28081 q^{97} -4.89245 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 7 q^{3} + 7 q^{5} + 9 q^{9} - 3 q^{11} + 7 q^{13} - 11 q^{15} - 7 q^{19} - 11 q^{23} + 3 q^{25} - 28 q^{27} + 6 q^{29} - 14 q^{31} - 6 q^{37} - 5 q^{39} + 3 q^{43} + 21 q^{45} - 42 q^{47} + 17 q^{51}+ \cdots + 34 q^{99}+O(q^{100}) \) 6 * q - 7 * q^3 + 7 * q^5 + 9 * q^9 - 3 * q^11 + 7 * q^13 - 11 * q^15 - 7 * q^19 - 11 * q^23 + 3 * q^25 - 28 * q^27 + 6 * q^29 - 14 * q^31 - 6 * q^37 - 5 * q^39 + 3 * q^43 + 21 * q^45 - 42 * q^47 + 17 * q^51 - 17 * q^53 - 14 * q^55 - 6 * q^57 - 35 * q^59 + 7 * q^61 - 6 * q^65 + 14 * q^67 + 14 * q^69 + 28 * q^71 - 35 * q^75 - 6 * q^79 + 34 * q^81 - 49 * q^83 - 23 * q^85 - 21 * q^87 - 7 * q^89 - 14 * q^93 + 25 * q^95 - 21 * q^97 + 34 * q^99

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