LMFDB - Embedded newform 5488.2.a.s.1.9

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 = [9,0,0,0,-13,0,0,0,13,0,-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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 20x^{7} + 131x^{5} - 14x^{4} - 302x^{3} + 91x^{2} + 133x - 7 $ x^9 - 20x^7 + 131x^5 - 14x^4 - 302x^3 + 91x^2 + 133x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2744)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-2.62260$ 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)

$f(q)$	$=$	$q+2.62260 q^{3} -0.341478 q^{5} +3.87801 q^{9} -1.63427 q^{11} -3.79546 q^{13} -0.895559 q^{15} -0.563514 q^{17} +0.326924 q^{19} -6.42051 q^{23} -4.88339 q^{25} +2.30266 q^{27} -2.13340 q^{29} -2.42777 q^{31} -4.28602 q^{33} +9.94934 q^{37} -9.95396 q^{39} -7.60309 q^{41} +3.48964 q^{43} -1.32425 q^{45} -7.79238 q^{47} -1.47787 q^{51} -7.82985 q^{53} +0.558066 q^{55} +0.857388 q^{57} -1.63759 q^{59} +0.163812 q^{61} +1.29607 q^{65} -5.08098 q^{67} -16.8384 q^{69} +2.80377 q^{71} +12.8241 q^{73} -12.8072 q^{75} -15.8939 q^{79} -5.59509 q^{81} +11.6355 q^{83} +0.192428 q^{85} -5.59505 q^{87} -10.6215 q^{89} -6.36706 q^{93} -0.111637 q^{95} +5.19230 q^{97} -6.33770 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 13 q^{5} + 13 q^{9} - 3 q^{11} - 20 q^{13} - 3 q^{15} - 10 q^{17} + 10 q^{19} + 4 q^{23} + 22 q^{25} + 3 q^{29} + 10 q^{31} - 7 q^{33} + 6 q^{37} + 20 q^{39} - 13 q^{41} + 8 q^{43} - 52 q^{45} - 10 q^{47}+ \cdots + 14 q^{99}+O(q^{100}) $ 9 * q - 13 * q^5 + 13 * q^9 - 3 * q^11 - 20 * q^13 - 3 * q^15 - 10 * q^17 + 10 * q^19 + 4 * q^23 + 22 * q^25 + 3 * q^29 + 10 * q^31 - 7 * q^33 + 6 * q^37 + 20 * q^39 - 13 * q^41 + 8 * q^43 - 52 * q^45 - 10 * q^47 + 23 * q^51 - 8 * q^53 + 25 * q^55 - q^57 - 7 * q^59 - 52 * q^61 + 28 * q^65 + 9 * q^67 - 42 * q^69 - 9 * q^71 - 23 * q^73 + 7 * q^75 - 10 * q^79 - 3 * q^81 - 18 * q^83 - 5 * q^85 + 7 * q^87 - 17 * q^89 - 53 * q^93 - 25 * q^95 - 24 * q^97 + 14 * 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
$3$	2.62260	1.51416	0.757078	−	0.653324i	$-0.226626\pi$
$3$	2.62260	1.51416	0.757078	+	0.653324i	$0.226626\pi$
$4$	0	0
$5$	−0.341478	−0.152714	−0.0763568	−	0.997081i	$-0.524329\pi$
$5$	−0.341478	−0.152714	−0.0763568	+	0.997081i	$0.524329\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	3.87801	1.29267
$10$	0	0
$11$	−1.63427	−0.492750	−0.246375	−	0.969175i	$-0.579239\pi$
$11$	−1.63427	−0.492750	−0.246375	+	0.969175i	$0.579239\pi$
$12$	0	0
$13$	−3.79546	−1.05267	−0.526336	−	0.850277i	$-0.676434\pi$
$13$	−3.79546	−1.05267	−0.526336	+	0.850277i	$0.676434\pi$
$14$	0	0
$15$	−0.895559	−0.231232
$16$	0	0
$17$	−0.563514	−0.136672	−0.0683362	−	0.997662i	$-0.521769\pi$
$17$	−0.563514	−0.136672	−0.0683362	+	0.997662i	$0.521769\pi$
$18$	0	0
$19$	0.326924	0.0750014	0.0375007	−	0.999297i	$-0.488060\pi$
$19$	0.326924	0.0750014	0.0375007	+	0.999297i	$0.488060\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.42051	−1.33877	−0.669384	−	0.742917i	$-0.733442\pi$
$23$	−6.42051	−1.33877	−0.669384	+	0.742917i	$0.733442\pi$
$24$	0	0
$25$	−4.88339	−0.976679
$26$	0	0
$27$	2.30266	0.443146
$28$	0	0
$29$	−2.13340	−0.396163	−0.198082	−	0.980186i	$-0.563471\pi$
$29$	−2.13340	−0.396163	−0.198082	+	0.980186i	$0.563471\pi$
$30$	0	0
$31$	−2.42777	−0.436040	−0.218020	−	0.975944i	$-0.569960\pi$
$31$	−2.42777	−0.436040	−0.218020	+	0.975944i	$0.569960\pi$
$32$	0	0
$33$	−4.28602	−0.746101
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.94934	1.63566	0.817831	−	0.575459i	$-0.195176\pi$
$37$	9.94934	1.63566	0.817831	+	0.575459i	$0.195176\pi$
$38$	0	0
$39$	−9.95396	−1.59391
$40$	0	0
$41$	−7.60309	−1.18740	−0.593701	−	0.804685i	$-0.702334\pi$
$41$	−7.60309	−1.18740	−0.593701	+	0.804685i	$0.702334\pi$
$42$	0	0
$43$	3.48964	0.532165	0.266083	−	0.963950i	$-0.414271\pi$
$43$	3.48964	0.532165	0.266083	+	0.963950i	$0.414271\pi$
$44$	0	0
$45$	−1.32425	−0.197408
$46$	0	0
$47$	−7.79238	−1.13664	−0.568318	−	0.822809i	$-0.692406\pi$
$47$	−7.79238	−1.13664	−0.568318	+	0.822809i	$0.692406\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1.47787	−0.206943
$52$	0	0
$53$	−7.82985	−1.07551	−0.537756	−	0.843100i	$-0.680728\pi$
$53$	−7.82985	−1.07551	−0.537756	+	0.843100i	$0.680728\pi$
$54$	0	0
$55$	0.558066	0.0752497
$56$	0	0
$57$	0.857388	0.113564
$58$	0	0
$59$	−1.63759	−0.213196	−0.106598	−	0.994302i	$-0.533996\pi$
$59$	−1.63759	−0.213196	−0.106598	+	0.994302i	$0.533996\pi$
$60$	0	0
$61$	0.163812	0.0209740	0.0104870	−	0.999945i	$-0.496662\pi$
$61$	0.163812	0.0209740	0.0104870	+	0.999945i	$0.496662\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.29607	0.160757
$66$	0	0
$67$	−5.08098	−0.620740	−0.310370	−	0.950616i	$-0.600453\pi$
$67$	−5.08098	−0.620740	−0.310370	+	0.950616i	$0.600453\pi$
$68$	0	0
$69$	−16.8384	−2.02710
$70$	0	0
$71$	2.80377	0.332746	0.166373	−	0.986063i	$-0.446794\pi$
$71$	2.80377	0.332746	0.166373	+	0.986063i	$0.446794\pi$
$72$	0	0
$73$	12.8241	1.50095	0.750474	−	0.660900i	$-0.229825\pi$
$73$	12.8241	1.50095	0.750474	+	0.660900i	$0.229825\pi$
$74$	0	0
$75$	−12.8072	−1.47884
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−15.8939	−1.78821	−0.894104	−	0.447859i	$-0.852187\pi$
$79$	−15.8939	−1.78821	−0.894104	+	0.447859i	$0.852187\pi$
$80$	0	0
$81$	−5.59509	−0.621676
$82$	0	0
$83$	11.6355	1.27716	0.638578	−	0.769557i	$-0.279523\pi$
$83$	11.6355	1.27716	0.638578	+	0.769557i	$0.279523\pi$
$84$	0	0
$85$	0.192428	0.0208717
$86$	0	0
$87$	−5.59505	−0.599853
$88$	0	0
$89$	−10.6215	−1.12588	−0.562941	−	0.826497i	$-0.690330\pi$
$89$	−10.6215	−1.12588	−0.562941	+	0.826497i	$0.690330\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−6.36706	−0.660233
$94$	0	0
$95$	−0.111637	−0.0114537
$96$	0	0
$97$	5.19230	0.527198	0.263599	−	0.964632i	$-0.415090\pi$
$97$	5.19230	0.527198	0.263599	+	0.964632i	$0.415090\pi$
$98$	0	0
$99$	−6.33770	−0.636963
$100$	0	0
$101$	−13.6974	−1.36294	−0.681469	−	0.731847i	$-0.738659\pi$
$101$	−13.6974	−1.36294	−0.681469	+	0.731847i	$0.738659\pi$
$102$	0	0
$103$	12.9172	1.27277	0.636384	−	0.771372i	$-0.280429\pi$
$103$	12.9172	1.27277	0.636384	+	0.771372i	$0.280429\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0664	1.16650	0.583251	−	0.812292i	$-0.301780\pi$
$107$	12.0664	1.16650	0.583251	+	0.812292i	$0.301780\pi$
$108$	0	0
$109$	−11.6489	−1.11577	−0.557883	−	0.829919i	$-0.688386\pi$
$109$	−11.6489	−1.11577	−0.557883	+	0.829919i	$0.688386\pi$
$110$	0	0
$111$	26.0931	2.47665
$112$	0	0
$113$	14.7037	1.38321	0.691606	−	0.722275i	$-0.256903\pi$
$113$	14.7037	1.38321	0.691606	+	0.722275i	$0.256903\pi$
$114$	0	0
$115$	2.19246	0.204448
$116$	0	0
$117$	−14.7188	−1.36076
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.32917	−0.757197
$122$	0	0
$123$	−19.9398	−1.79791
$124$	0	0
$125$	3.37496	0.301866
$126$	0	0
$127$	−3.73654	−0.331564	−0.165782	−	0.986162i	$-0.553015\pi$
$127$	−3.73654	−0.331564	−0.165782	+	0.986162i	$0.553015\pi$
$128$	0	0
$129$	9.15192	0.805782
$130$	0	0
$131$	−15.3307	−1.33945	−0.669726	−	0.742609i	$-0.733588\pi$
$131$	−15.3307	−1.33945	−0.669726	+	0.742609i	$0.733588\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.786306	−0.0676745
$136$	0	0
$137$	0.632289	0.0540201	0.0270100	−	0.999635i	$-0.491401\pi$
$137$	0.632289	0.0540201	0.0270100	+	0.999635i	$0.491401\pi$
$138$	0	0
$139$	13.8616	1.17573	0.587864	−	0.808960i	$-0.299969\pi$
$139$	13.8616	1.17573	0.587864	+	0.808960i	$0.299969\pi$
$140$	0	0
$141$	−20.4363	−1.72104
$142$	0	0
$143$	6.20280	0.518704
$144$	0	0
$145$	0.728510	0.0604995
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.8516	−1.13477	−0.567383	−	0.823454i	$-0.692044\pi$
$149$	−13.8516	−1.13477	−0.567383	+	0.823454i	$0.692044\pi$
$150$	0	0
$151$	7.90823	0.643563	0.321781	−	0.946814i	$-0.395718\pi$
$151$	7.90823	0.643563	0.321781	+	0.946814i	$0.395718\pi$
$152$	0	0
$153$	−2.18531	−0.176672
$154$	0	0
$155$	0.829030	0.0665893
$156$	0	0
$157$	−20.9292	−1.67033	−0.835167	−	0.549997i	$-0.814629\pi$
$157$	−20.9292	−1.67033	−0.835167	+	0.549997i	$0.814629\pi$
$158$	0	0
$159$	−20.5345	−1.62849
$160$	0	0
$161$	0	0
$162$	0	0
$163$	23.4873	1.83967	0.919833	−	0.392311i	$-0.128324\pi$
$163$	23.4873	1.83967	0.919833	+	0.392311i	$0.128324\pi$
$164$	0	0
$165$	1.46358	0.113940
$166$	0	0
$167$	4.18701	0.324001	0.162000	−	0.986791i	$-0.448205\pi$
$167$	4.18701	0.324001	0.162000	+	0.986791i	$0.448205\pi$
$168$	0	0
$169$	1.40552	0.108117
$170$	0	0
$171$	1.26781	0.0969520
$172$	0	0
$173$	−4.47873	−0.340511	−0.170256	−	0.985400i	$-0.554459\pi$
$173$	−4.47873	−0.340511	−0.170256	+	0.985400i	$0.554459\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.29474	−0.322812
$178$	0	0
$179$	19.3482	1.44616	0.723078	−	0.690767i	$-0.242727\pi$
$179$	19.3482	1.44616	0.723078	+	0.690767i	$0.242727\pi$
$180$	0	0
$181$	−8.72039	−0.648182	−0.324091	−	0.946026i	$-0.605058\pi$
$181$	−8.72039	−0.648182	−0.324091	+	0.946026i	$0.605058\pi$
$182$	0	0
$183$	0.429613	0.0317579
$184$	0	0
$185$	−3.39748	−0.249788
$186$	0	0
$187$	0.920933	0.0673453
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.25043	0.379908	0.189954	−	0.981793i	$-0.439166\pi$
$191$	5.25043	0.379908	0.189954	+	0.981793i	$0.439166\pi$
$192$	0	0
$193$	−5.16764	−0.371975	−0.185988	−	0.982552i	$-0.559548\pi$
$193$	−5.16764	−0.371975	−0.185988	+	0.982552i	$0.559548\pi$
$194$	0	0
$195$	3.39906	0.243412
$196$	0	0
$197$	10.4092	0.741624	0.370812	−	0.928708i	$-0.379079\pi$
$197$	10.4092	0.741624	0.370812	+	0.928708i	$0.379079\pi$
$198$	0	0
$199$	17.6447	1.25080	0.625400	−	0.780304i	$-0.284936\pi$
$199$	17.6447	1.25080	0.625400	+	0.780304i	$0.284936\pi$
$200$	0	0
$201$	−13.3253	−0.939897
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.59629	0.181333
$206$	0	0
$207$	−24.8988	−1.73058
$208$	0	0
$209$	−0.534280	−0.0369569
$210$	0	0
$211$	13.0253	0.896698	0.448349	−	0.893859i	$-0.352012\pi$
$211$	13.0253	0.896698	0.448349	+	0.893859i	$0.352012\pi$
$212$	0	0
$213$	7.35315	0.503830
$214$	0	0
$215$	−1.19164	−0.0812689
$216$	0	0
$217$	0	0
$218$	0	0
$219$	33.6324	2.27267
$220$	0	0
$221$	2.13880	0.143871
$222$	0	0
$223$	−8.95096	−0.599401	−0.299700	−	0.954033i	$-0.596887\pi$
$223$	−8.95096	−0.599401	−0.299700	+	0.954033i	$0.596887\pi$
$224$	0	0
$225$	−18.9378	−1.26252
$226$	0	0
$227$	−15.5647	−1.03307	−0.516533	−	0.856267i	$-0.672778\pi$
$227$	−15.5647	−1.03307	−0.516533	+	0.856267i	$0.672778\pi$
$228$	0	0
$229$	−28.5098	−1.88398	−0.941991	−	0.335639i	$-0.891048\pi$
$229$	−28.5098	−1.88398	−0.941991	+	0.335639i	$0.891048\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.30414	0.0854372	0.0427186	−	0.999087i	$-0.486398\pi$
$233$	1.30414	0.0854372	0.0427186	+	0.999087i	$0.486398\pi$
$234$	0	0
$235$	2.66093	0.173580
$236$	0	0
$237$	−41.6834	−2.70763
$238$	0	0
$239$	8.22806	0.532229	0.266114	−	0.963941i	$-0.414260\pi$
$239$	8.22806	0.532229	0.266114	+	0.963941i	$0.414260\pi$
$240$	0	0
$241$	3.43859	0.221499	0.110750	−	0.993848i	$-0.464675\pi$
$241$	3.43859	0.221499	0.110750	+	0.993848i	$0.464675\pi$
$242$	0	0
$243$	−21.5816	−1.38446
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.24083	−0.0789518
$248$	0	0
$249$	30.5151	1.93381
$250$	0	0
$251$	−17.8262	−1.12518	−0.562588	−	0.826737i	$-0.690194\pi$
$251$	−17.8262	−1.12518	−0.562588	+	0.826737i	$0.690194\pi$
$252$	0	0
$253$	10.4928	0.659678
$254$	0	0
$255$	0.504660	0.0316030
$256$	0	0
$257$	23.9218	1.49220	0.746099	−	0.665835i	$-0.231924\pi$
$257$	23.9218	1.49220	0.746099	+	0.665835i	$0.231924\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−8.27335	−0.512108
$262$	0	0
$263$	26.1350	1.61156	0.805778	−	0.592218i	$-0.201748\pi$
$263$	26.1350	1.61156	0.805778	+	0.592218i	$0.201748\pi$
$264$	0	0
$265$	2.67372	0.164245
$266$	0	0
$267$	−27.8560	−1.70476
$268$	0	0
$269$	29.2146	1.78124	0.890622	−	0.454744i	$-0.150269\pi$
$269$	29.2146	1.78124	0.890622	+	0.454744i	$0.150269\pi$
$270$	0	0
$271$	−17.2758	−1.04943	−0.524716	−	0.851277i	$-0.675829\pi$
$271$	−17.2758	−1.04943	−0.524716	+	0.851277i	$0.675829\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.98077	0.481258
$276$	0	0
$277$	−1.75070	−0.105189	−0.0525946	−	0.998616i	$-0.516749\pi$
$277$	−1.75070	−0.105189	−0.0525946	+	0.998616i	$0.516749\pi$
$278$	0	0
$279$	−9.41491	−0.563656
$280$	0	0
$281$	−21.6688	−1.29265	−0.646325	−	0.763062i	$-0.723695\pi$
$281$	−21.6688	−1.29265	−0.646325	+	0.763062i	$0.723695\pi$
$282$	0	0
$283$	−25.8465	−1.53641	−0.768207	−	0.640201i	$-0.778851\pi$
$283$	−25.8465	−1.53641	−0.768207	+	0.640201i	$0.778851\pi$
$284$	0	0
$285$	−0.292779	−0.0173427
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.6825	−0.981321
$290$	0	0
$291$	13.6173	0.798260
$292$	0	0
$293$	5.00644	0.292479	0.146240	−	0.989249i	$-0.453283\pi$
$293$	5.00644	0.292479	0.146240	+	0.989249i	$0.453283\pi$
$294$	0	0
$295$	0.559201	0.0325580
$296$	0	0
$297$	−3.76315	−0.218360
$298$	0	0
$299$	24.3688	1.40928
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−35.9226	−2.06370
$304$	0	0
$305$	−0.0559382	−0.00320301
$306$	0	0
$307$	15.4740	0.883148	0.441574	−	0.897225i	$-0.354420\pi$
$307$	15.4740	0.883148	0.441574	+	0.897225i	$0.354420\pi$
$308$	0	0
$309$	33.8765	1.92717
$310$	0	0
$311$	−13.4393	−0.762075	−0.381037	−	0.924560i	$-0.624433\pi$
$311$	−13.4393	−0.762075	−0.381037	+	0.924560i	$0.624433\pi$
$312$	0	0
$313$	−29.6339	−1.67501	−0.837504	−	0.546431i	$-0.815986\pi$
$313$	−29.6339	−1.67501	−0.837504	+	0.546431i	$0.815986\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.92997	0.164564	0.0822818	−	0.996609i	$-0.473779\pi$
$317$	2.92997	0.164564	0.0822818	+	0.996609i	$0.473779\pi$
$318$	0	0
$319$	3.48655	0.195209
$320$	0	0
$321$	31.6453	1.76627
$322$	0	0
$323$	−0.184226	−0.0102506
$324$	0	0
$325$	18.5347	1.02812
$326$	0	0
$327$	−30.5505	−1.68945
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.94913	0.491888	0.245944	−	0.969284i	$-0.420902\pi$
$331$	8.94913	0.491888	0.245944	+	0.969284i	$0.420902\pi$
$332$	0	0
$333$	38.5836	2.11437
$334$	0	0
$335$	1.73504	0.0947955
$336$	0	0
$337$	16.5302	0.900457	0.450228	−	0.892913i	$-0.351343\pi$
$337$	16.5302	0.900457	0.450228	+	0.892913i	$0.351343\pi$
$338$	0	0
$339$	38.5620	2.09440
$340$	0	0
$341$	3.96762	0.214859
$342$	0	0
$343$	0	0
$344$	0	0
$345$	5.74994	0.309566
$346$	0	0
$347$	4.43312	0.237982	0.118991	−	0.992895i	$-0.462034\pi$
$347$	4.43312	0.237982	0.118991	+	0.992895i	$0.462034\pi$
$348$	0	0
$349$	−8.12723	−0.435041	−0.217520	−	0.976056i	$-0.569797\pi$
$349$	−8.12723	−0.435041	−0.217520	+	0.976056i	$0.569797\pi$
$350$	0	0
$351$	−8.73964	−0.466487
$352$	0	0
$353$	−27.6271	−1.47044	−0.735220	−	0.677828i	$-0.762921\pi$
$353$	−27.6271	−1.47044	−0.735220	+	0.677828i	$0.762921\pi$
$354$	0	0
$355$	−0.957426	−0.0508149
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−37.3723	−1.97243	−0.986216	−	0.165463i	$-0.947088\pi$
$359$	−37.3723	−1.97243	−0.986216	+	0.165463i	$0.947088\pi$
$360$	0	0
$361$	−18.8931	−0.994375
$362$	0	0
$363$	−21.8440	−1.14651
$364$	0	0
$365$	−4.37915	−0.229215
$366$	0	0
$367$	13.4219	0.700619	0.350310	−	0.936634i	$-0.386076\pi$
$367$	13.4219	0.700619	0.350310	+	0.936634i	$0.386076\pi$
$368$	0	0
$369$	−29.4848	−1.53492
$370$	0	0
$371$	0	0
$372$	0	0
$373$	4.18853	0.216874	0.108437	−	0.994103i	$-0.465415\pi$
$373$	4.18853	0.216874	0.108437	+	0.994103i	$0.465415\pi$
$374$	0	0
$375$	8.85116	0.457072
$376$	0	0
$377$	8.09725	0.417029
$378$	0	0
$379$	23.8997	1.22765	0.613824	−	0.789443i	$-0.289631\pi$
$379$	23.8997	1.22765	0.613824	+	0.789443i	$0.289631\pi$
$380$	0	0
$381$	−9.79944	−0.502040
$382$	0	0
$383$	−13.5860	−0.694211	−0.347106	−	0.937826i	$-0.612835\pi$
$383$	−13.5860	−0.694211	−0.347106	+	0.937826i	$0.612835\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	13.5329	0.687914
$388$	0	0
$389$	7.35797	0.373064	0.186532	−	0.982449i	$-0.440275\pi$
$389$	7.35797	0.373064	0.186532	+	0.982449i	$0.440275\pi$
$390$	0	0
$391$	3.61805	0.182973
$392$	0	0
$393$	−40.2063	−2.02814
$394$	0	0
$395$	5.42743	0.273084
$396$	0	0
$397$	−6.00128	−0.301196	−0.150598	−	0.988595i	$-0.548120\pi$
$397$	−6.00128	−0.301196	−0.150598	+	0.988595i	$0.548120\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	32.3935	1.61766	0.808828	−	0.588045i	$-0.200102\pi$
$401$	32.3935	1.61766	0.808828	+	0.588045i	$0.200102\pi$
$402$	0	0
$403$	9.21450	0.459007
$404$	0	0
$405$	1.91060	0.0949384
$406$	0	0
$407$	−16.2599	−0.805972
$408$	0	0
$409$	−2.80630	−0.138763	−0.0693813	−	0.997590i	$-0.522103\pi$
$409$	−2.80630	−0.138763	−0.0693813	+	0.997590i	$0.522103\pi$
$410$	0	0
$411$	1.65824	0.0817948
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−3.97325	−0.195039
$416$	0	0
$417$	36.3534	1.78023
$418$	0	0
$419$	−27.9948	−1.36764	−0.683818	−	0.729653i	$-0.739682\pi$
$419$	−27.9948	−1.36764	−0.683818	+	0.729653i	$0.739682\pi$
$420$	0	0
$421$	−11.0697	−0.539504	−0.269752	−	0.962930i	$-0.586942\pi$
$421$	−11.0697	−0.539504	−0.269752	+	0.962930i	$0.586942\pi$
$422$	0	0
$423$	−30.2189	−1.46929
$424$	0	0
$425$	2.75186	0.133485
$426$	0	0
$427$	0	0
$428$	0	0
$429$	16.2674	0.785399
$430$	0	0
$431$	17.0623	0.821861	0.410931	−	0.911667i	$-0.365204\pi$
$431$	17.0623	0.821861	0.410931	+	0.911667i	$0.365204\pi$
$432$	0	0
$433$	20.0310	0.962629	0.481315	−	0.876548i	$-0.340159\pi$
$433$	20.0310	0.962629	0.481315	+	0.876548i	$0.340159\pi$
$434$	0	0
$435$	1.91059	0.0916057
$436$	0	0
$437$	−2.09901	−0.100409
$438$	0	0
$439$	27.7079	1.32242	0.661212	−	0.750199i	$-0.270042\pi$
$439$	27.7079	1.32242	0.661212	+	0.750199i	$0.270042\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10.1876	−0.484027	−0.242013	−	0.970273i	$-0.577808\pi$
$443$	−10.1876	−0.484027	−0.242013	+	0.970273i	$0.577808\pi$
$444$	0	0
$445$	3.62703	0.171938
$446$	0	0
$447$	−36.3271	−1.71821
$448$	0	0
$449$	37.2385	1.75740	0.878698	−	0.477379i	$-0.158413\pi$
$449$	37.2385	1.75740	0.878698	+	0.477379i	$0.158413\pi$
$450$	0	0
$451$	12.4255	0.585093
$452$	0	0
$453$	20.7401	0.974455
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.82527	0.225716	0.112858	−	0.993611i	$-0.463999\pi$
$457$	4.82527	0.225716	0.112858	+	0.993611i	$0.463999\pi$
$458$	0	0
$459$	−1.29758	−0.0605658
$460$	0	0
$461$	−25.9002	−1.20629	−0.603146	−	0.797631i	$-0.706086\pi$
$461$	−25.9002	−1.20629	−0.603146	+	0.797631i	$0.706086\pi$
$462$	0	0
$463$	9.30833	0.432595	0.216297	−	0.976328i	$-0.430602\pi$
$463$	9.30833	0.432595	0.216297	+	0.976328i	$0.430602\pi$
$464$	0	0
$465$	2.17421	0.100827
$466$	0	0
$467$	−39.2149	−1.81465	−0.907324	−	0.420432i	$-0.861879\pi$
$467$	−39.2149	−1.81465	−0.907324	+	0.420432i	$0.861879\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−54.8889	−2.52915
$472$	0	0
$473$	−5.70301	−0.262225
$474$	0	0
$475$	−1.59650	−0.0732523
$476$	0	0
$477$	−30.3642	−1.39028
$478$	0	0
$479$	−5.97179	−0.272858	−0.136429	−	0.990650i	$-0.543563\pi$
$479$	−5.97179	−0.272858	−0.136429	+	0.990650i	$0.543563\pi$
$480$	0	0
$481$	−37.7623	−1.72181
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.77306	−0.0805103
$486$	0	0
$487$	42.3336	1.91832	0.959160	−	0.282864i	$-0.0912846\pi$
$487$	42.3336	1.91832	0.959160	+	0.282864i	$0.0912846\pi$
$488$	0	0
$489$	61.5976	2.78554
$490$	0	0
$491$	29.7412	1.34220	0.671102	−	0.741365i	$-0.265821\pi$
$491$	29.7412	1.34220	0.671102	+	0.741365i	$0.265821\pi$
$492$	0	0
$493$	1.20220	0.0541445
$494$	0	0
$495$	2.16418	0.0972729
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−13.2096	−0.591341	−0.295670	−	0.955290i	$-0.595543\pi$
$499$	−13.2096	−0.591341	−0.295670	+	0.955290i	$0.595543\pi$
$500$	0	0
$501$	10.9808	0.490587
$502$	0	0
$503$	8.41289	0.375112	0.187556	−	0.982254i	$-0.439943\pi$
$503$	8.41289	0.375112	0.187556	+	0.982254i	$0.439943\pi$
$504$	0	0
$505$	4.67735	0.208139
$506$	0	0
$507$	3.68611	0.163706
$508$	0	0
$509$	4.22347	0.187202	0.0936009	−	0.995610i	$-0.470162\pi$
$509$	4.22347	0.187202	0.0936009	+	0.995610i	$0.470162\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0.752792	0.0332366
$514$	0	0
$515$	−4.41093	−0.194369
$516$	0	0
$517$	12.7348	0.560077
$518$	0	0
$519$	−11.7459	−0.515587
$520$	0	0
$521$	−38.7713	−1.69860	−0.849300	−	0.527910i	$-0.822976\pi$
$521$	−38.7713	−1.69860	−0.849300	+	0.527910i	$0.822976\pi$
$522$	0	0
$523$	17.4755	0.764151	0.382075	−	0.924131i	$-0.375210\pi$
$523$	17.4755	0.764151	0.382075	+	0.924131i	$0.375210\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.36808	0.0595946
$528$	0	0
$529$	18.2229	0.792300
$530$	0	0
$531$	−6.35059	−0.275592
$532$	0	0
$533$	28.8572	1.24995
$534$	0	0
$535$	−4.12041	−0.178141
$536$	0	0
$537$	50.7426	2.18971
$538$	0	0
$539$	0	0
$540$	0	0
$541$	13.6449	0.586640	0.293320	−	0.956014i	$-0.405240\pi$
$541$	13.6449	0.586640	0.293320	+	0.956014i	$0.405240\pi$
$542$	0	0
$543$	−22.8701	−0.981448
$544$	0	0
$545$	3.97786	0.170393
$546$	0	0
$547$	−34.6289	−1.48062	−0.740312	−	0.672264i	$-0.765322\pi$
$547$	−34.6289	−1.48062	−0.740312	+	0.672264i	$0.765322\pi$
$548$	0	0
$549$	0.635264	0.0271124
$550$	0	0
$551$	−0.697460	−0.0297128
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−8.91022	−0.378218
$556$	0	0
$557$	−30.4501	−1.29021	−0.645107	−	0.764092i	$-0.723187\pi$
$557$	−30.4501	−1.29021	−0.645107	+	0.764092i	$0.723187\pi$
$558$	0	0
$559$	−13.2448	−0.560195
$560$	0	0
$561$	2.41523	0.101971
$562$	0	0
$563$	−16.1062	−0.678794	−0.339397	−	0.940643i	$-0.610223\pi$
$563$	−16.1062	−0.678794	−0.339397	+	0.940643i	$0.610223\pi$
$564$	0	0
$565$	−5.02101	−0.211235
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.85811	0.119818	0.0599092	−	0.998204i	$-0.480919\pi$
$569$	2.85811	0.119818	0.0599092	+	0.998204i	$0.480919\pi$
$570$	0	0
$571$	22.3920	0.937076	0.468538	−	0.883443i	$-0.344781\pi$
$571$	22.3920	0.937076	0.468538	+	0.883443i	$0.344781\pi$
$572$	0	0
$573$	13.7698	0.575240
$574$	0	0
$575$	31.3539	1.30755
$576$	0	0
$577$	−7.90066	−0.328909	−0.164454	−	0.986385i	$-0.552586\pi$
$577$	−7.90066	−0.328909	−0.164454	+	0.986385i	$0.552586\pi$
$578$	0	0
$579$	−13.5526	−0.563228
$580$	0	0
$581$	0	0
$582$	0	0
$583$	12.7961	0.529959
$584$	0	0
$585$	5.02615	0.207806
$586$	0	0
$587$	13.2704	0.547730	0.273865	−	0.961768i	$-0.411698\pi$
$587$	13.2704	0.547730	0.273865	+	0.961768i	$0.411698\pi$
$588$	0	0
$589$	−0.793695	−0.0327036
$590$	0	0
$591$	27.2991	1.12293
$592$	0	0
$593$	10.8623	0.446060	0.223030	−	0.974812i	$-0.428405\pi$
$593$	10.8623	0.446060	0.223030	+	0.974812i	$0.428405\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	46.2750	1.89391
$598$	0	0
$599$	−16.3445	−0.667819	−0.333909	−	0.942605i	$-0.608368\pi$
$599$	−16.3445	−0.667819	−0.333909	+	0.942605i	$0.608368\pi$
$600$	0	0
$601$	45.3426	1.84956	0.924781	−	0.380500i	$-0.124248\pi$
$601$	45.3426	1.84956	0.924781	+	0.380500i	$0.124248\pi$
$602$	0	0
$603$	−19.7041	−0.802411
$604$	0	0
$605$	2.84423	0.115634
$606$	0	0
$607$	7.92339	0.321601	0.160800	−	0.986987i	$-0.448592\pi$
$607$	7.92339	0.321601	0.160800	+	0.986987i	$0.448592\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	29.5757	1.19650
$612$	0	0
$613$	13.3520	0.539284	0.269642	−	0.962961i	$-0.413095\pi$
$613$	13.3520	0.539284	0.269642	+	0.962961i	$0.413095\pi$
$614$	0	0
$615$	6.80901	0.274566
$616$	0	0
$617$	−6.50284	−0.261794	−0.130897	−	0.991396i	$-0.541786\pi$
$617$	−6.50284	−0.261794	−0.130897	+	0.991396i	$0.541786\pi$
$618$	0	0
$619$	5.91993	0.237942	0.118971	−	0.992898i	$-0.462040\pi$
$619$	5.91993	0.237942	0.118971	+	0.992898i	$0.462040\pi$
$620$	0	0
$621$	−14.7842	−0.593270
$622$	0	0
$623$	0	0
$624$	0	0
$625$	23.2645	0.930580
$626$	0	0
$627$	−1.40120	−0.0559586
$628$	0	0
$629$	−5.60660	−0.223550
$630$	0	0
$631$	44.2970	1.76344	0.881718	−	0.471777i	$-0.156387\pi$
$631$	44.2970	1.76344	0.881718	+	0.471777i	$0.156387\pi$
$632$	0	0
$633$	34.1601	1.35774
$634$	0	0
$635$	1.27595	0.0506344
$636$	0	0
$637$	0	0
$638$	0	0
$639$	10.8730	0.430131
$640$	0	0
$641$	−8.62706	−0.340748	−0.170374	−	0.985379i	$-0.554498\pi$
$641$	−8.62706	−0.340748	−0.170374	+	0.985379i	$0.554498\pi$
$642$	0	0
$643$	−28.5734	−1.12683	−0.563413	−	0.826175i	$-0.690512\pi$
$643$	−28.5734	−1.12683	−0.563413	+	0.826175i	$0.690512\pi$
$644$	0	0
$645$	−3.12518	−0.123054
$646$	0	0
$647$	−31.5065	−1.23865	−0.619324	−	0.785136i	$-0.712593\pi$
$647$	−31.5065	−1.23865	−0.619324	+	0.785136i	$0.712593\pi$
$648$	0	0
$649$	2.67626	0.105052
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−34.4150	−1.34676	−0.673381	−	0.739296i	$-0.735159\pi$
$653$	−34.4150	−1.34676	−0.673381	+	0.739296i	$0.735159\pi$
$654$	0	0
$655$	5.23510	0.204552
$656$	0	0
$657$	49.7320	1.94023
$658$	0	0
$659$	−13.7941	−0.537343	−0.268672	−	0.963232i	$-0.586585\pi$
$659$	−13.7941	−0.537343	−0.268672	+	0.963232i	$0.586585\pi$
$660$	0	0
$661$	34.0803	1.32557	0.662785	−	0.748810i	$-0.269374\pi$
$661$	34.0803	1.32557	0.662785	+	0.748810i	$0.269374\pi$
$662$	0	0
$663$	5.60920	0.217843
$664$	0	0
$665$	0	0
$666$	0	0
$667$	13.6975	0.530370
$668$	0	0
$669$	−23.4748	−0.907586
$670$	0	0
$671$	−0.267713	−0.0103349
$672$	0	0
$673$	30.1088	1.16061	0.580305	−	0.814399i	$-0.302933\pi$
$673$	30.1088	1.16061	0.580305	+	0.814399i	$0.302933\pi$
$674$	0	0
$675$	−11.2448	−0.432811
$676$	0	0
$677$	−0.297193	−0.0114221	−0.00571103	−	0.999984i	$-0.501818\pi$
$677$	−0.297193	−0.0114221	−0.00571103	+	0.999984i	$0.501818\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−40.8200	−1.56422
$682$	0	0
$683$	9.07142	0.347108	0.173554	−	0.984824i	$-0.444475\pi$
$683$	9.07142	0.347108	0.173554	+	0.984824i	$0.444475\pi$
$684$	0	0
$685$	−0.215913	−0.00824960
$686$	0	0
$687$	−74.7697	−2.85264
$688$	0	0
$689$	29.7179	1.13216
$690$	0	0
$691$	27.8498	1.05946	0.529729	−	0.848167i	$-0.322294\pi$
$691$	27.8498	1.05946	0.529729	+	0.848167i	$0.322294\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.73344	−0.179550
$696$	0	0
$697$	4.28445	0.162285
$698$	0	0
$699$	3.42024	0.129365
$700$	0	0
$701$	−12.9467	−0.488989	−0.244494	−	0.969651i	$-0.578622\pi$
$701$	−12.9467	−0.488989	−0.244494	+	0.969651i	$0.578622\pi$
$702$	0	0
$703$	3.25267	0.122677
$704$	0	0
$705$	6.97853	0.262827
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−18.2567	−0.685644	−0.342822	−	0.939400i	$-0.611383\pi$
$709$	−18.2567	−0.685644	−0.342822	+	0.939400i	$0.611383\pi$
$710$	0	0
$711$	−61.6368	−2.31156
$712$	0	0
$713$	15.5875	0.583757
$714$	0	0
$715$	−2.11812	−0.0792132
$716$	0	0
$717$	21.5789	0.805877
$718$	0	0
$719$	−11.1446	−0.415623	−0.207811	−	0.978169i	$-0.566634\pi$
$719$	−11.1446	−0.415623	−0.207811	+	0.978169i	$0.566634\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	9.01804	0.335385
$724$	0	0
$725$	10.4182	0.386924
$726$	0	0
$727$	−47.3726	−1.75695	−0.878476	−	0.477786i	$-0.841439\pi$
$727$	−47.3726	−1.75695	−0.878476	+	0.477786i	$0.841439\pi$
$728$	0	0
$729$	−39.8146	−1.47461
$730$	0	0
$731$	−1.96646	−0.0727323
$732$	0	0
$733$	−30.0558	−1.11014	−0.555069	−	0.831805i	$-0.687308\pi$
$733$	−30.0558	−1.11014	−0.555069	+	0.831805i	$0.687308\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.30367	0.305870
$738$	0	0
$739$	−19.2090	−0.706614	−0.353307	−	0.935507i	$-0.614943\pi$
$739$	−19.2090	−0.706614	−0.353307	+	0.935507i	$0.614943\pi$
$740$	0	0
$741$	−3.25418	−0.119545
$742$	0	0
$743$	5.71688	0.209732	0.104866	−	0.994486i	$-0.466559\pi$
$743$	5.71688	0.209732	0.104866	+	0.994486i	$0.466559\pi$
$744$	0	0
$745$	4.73001	0.173294
$746$	0	0
$747$	45.1224	1.65094
$748$	0	0
$749$	0	0
$750$	0	0
$751$	7.12152	0.259868	0.129934	−	0.991523i	$-0.458523\pi$
$751$	7.12152	0.259868	0.129934	+	0.991523i	$0.458523\pi$
$752$	0	0
$753$	−46.7508	−1.70369
$754$	0	0
$755$	−2.70049	−0.0982808
$756$	0	0
$757$	40.3533	1.46667	0.733333	−	0.679870i	$-0.237964\pi$
$757$	40.3533	1.46667	0.733333	+	0.679870i	$0.237964\pi$
$758$	0	0
$759$	27.5184	0.998856
$760$	0	0
$761$	0.595020	0.0215695	0.0107847	−	0.999942i	$-0.496567\pi$
$761$	0.595020	0.0215695	0.0107847	+	0.999942i	$0.496567\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0.746236	0.0269802
$766$	0	0
$767$	6.21541	0.224425
$768$	0	0
$769$	−16.0044	−0.577133	−0.288566	−	0.957460i	$-0.593179\pi$
$769$	−16.0044	−0.577133	−0.288566	+	0.957460i	$0.593179\pi$
$770$	0	0
$771$	62.7371	2.25942
$772$	0	0
$773$	−21.1336	−0.760124	−0.380062	−	0.924961i	$-0.624097\pi$
$773$	−21.1336	−0.760124	−0.380062	+	0.924961i	$0.624097\pi$
$774$	0	0
$775$	11.8558	0.425871
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.48563	−0.0890569
$780$	0	0
$781$	−4.58211	−0.163961
$782$	0	0
$783$	−4.91249	−0.175558
$784$	0	0
$785$	7.14687	0.255083
$786$	0	0
$787$	31.3184	1.11638	0.558190	−	0.829713i	$-0.311496\pi$
$787$	31.3184	1.11638	0.558190	+	0.829713i	$0.311496\pi$
$788$	0	0
$789$	68.5416	2.44015
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−0.621742	−0.0220787
$794$	0	0
$795$	7.01209	0.248693
$796$	0	0
$797$	−3.83229	−0.135747	−0.0678734	−	0.997694i	$-0.521621\pi$
$797$	−3.83229	−0.135747	−0.0678734	+	0.997694i	$0.521621\pi$
$798$	0	0
$799$	4.39112	0.155347
$800$	0	0
$801$	−41.1904	−1.45539
$802$	0	0
$803$	−20.9580	−0.739592
$804$	0	0
$805$	0	0
$806$	0	0
$807$	76.6180	2.69708
$808$	0	0
$809$	−35.3464	−1.24271	−0.621356	−	0.783528i	$-0.713418\pi$
$809$	−35.3464	−1.24271	−0.621356	+	0.783528i	$0.713418\pi$
$810$	0	0
$811$	21.3577	0.749970	0.374985	−	0.927031i	$-0.377648\pi$
$811$	21.3577	0.749970	0.374985	+	0.927031i	$0.377648\pi$
$812$	0	0
$813$	−45.3075	−1.58900
$814$	0	0
$815$	−8.02039	−0.280942
$816$	0	0
$817$	1.14085	0.0399132
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−18.2335	−0.636354	−0.318177	−	0.948031i	$-0.603071\pi$
$821$	−18.2335	−0.636354	−0.318177	+	0.948031i	$0.603071\pi$
$822$	0	0
$823$	−5.77735	−0.201386	−0.100693	−	0.994918i	$-0.532106\pi$
$823$	−5.77735	−0.201386	−0.100693	+	0.994918i	$0.532106\pi$
$824$	0	0
$825$	20.9303	0.728700
$826$	0	0
$827$	19.6592	0.683618	0.341809	−	0.939769i	$-0.388960\pi$
$827$	19.6592	0.683618	0.341809	+	0.939769i	$0.388960\pi$
$828$	0	0
$829$	7.81464	0.271414	0.135707	−	0.990749i	$-0.456669\pi$
$829$	7.81464	0.271414	0.135707	+	0.990749i	$0.456669\pi$
$830$	0	0
$831$	−4.59137	−0.159273
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.42977	−0.0494793
$836$	0	0
$837$	−5.59032	−0.193230
$838$	0	0
$839$	−12.8507	−0.443657	−0.221828	−	0.975086i	$-0.571202\pi$
$839$	−12.8507	−0.443657	−0.221828	+	0.975086i	$0.571202\pi$
$840$	0	0
$841$	−24.4486	−0.843055
$842$	0	0
$843$	−56.8284	−1.95727
$844$	0	0
$845$	−0.479954	−0.0165109
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−67.7849	−2.32637
$850$	0	0
$851$	−63.8798	−2.18977
$852$	0	0
$853$	−57.1224	−1.95583	−0.977917	−	0.208992i	$-0.932982\pi$
$853$	−57.1224	−1.95583	−0.977917	+	0.208992i	$0.932982\pi$
$854$	0	0
$855$	−0.432930	−0.0148059
$856$	0	0
$857$	−44.3906	−1.51635	−0.758176	−	0.652049i	$-0.773909\pi$
$857$	−44.3906	−1.51635	−0.758176	+	0.652049i	$0.773909\pi$
$858$	0	0
$859$	14.5395	0.496082	0.248041	−	0.968750i	$-0.420213\pi$
$859$	14.5395	0.496082	0.248041	+	0.968750i	$0.420213\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.6137	−0.395334	−0.197667	−	0.980269i	$-0.563336\pi$
$863$	−11.6137	−0.395334	−0.197667	+	0.980269i	$0.563336\pi$
$864$	0	0
$865$	1.52939	0.0520007
$866$	0	0
$867$	−43.7513	−1.48587
$868$	0	0
$869$	25.9749	0.881140
$870$	0	0
$871$	19.2846	0.653435
$872$	0	0
$873$	20.1358	0.681492
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−21.4186	−0.723253	−0.361627	−	0.932323i	$-0.617779\pi$
$877$	−21.4186	−0.723253	−0.361627	+	0.932323i	$0.617779\pi$
$878$	0	0
$879$	13.1299	0.442859
$880$	0	0
$881$	41.6340	1.40269	0.701343	−	0.712824i	$-0.252584\pi$
$881$	41.6340	1.40269	0.701343	+	0.712824i	$0.252584\pi$
$882$	0	0
$883$	−11.7539	−0.395551	−0.197776	−	0.980247i	$-0.563372\pi$
$883$	−11.7539	−0.395551	−0.197776	+	0.980247i	$0.563372\pi$
$884$	0	0
$885$	1.46656	0.0492978
$886$	0	0
$887$	5.80095	0.194777	0.0973884	−	0.995246i	$-0.468951\pi$
$887$	5.80095	0.194777	0.0973884	+	0.995246i	$0.468951\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	9.14387	0.306331
$892$	0	0
$893$	−2.54751	−0.0852493
$894$	0	0
$895$	−6.60700	−0.220848
$896$	0	0
$897$	63.9094	2.13387
$898$	0	0
$899$	5.17941	0.172743
$900$	0	0
$901$	4.41223	0.146993
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.97782	0.0989862
$906$	0	0
$907$	−41.0318	−1.36244	−0.681220	−	0.732079i	$-0.738550\pi$
$907$	−41.0318	−1.36244	−0.681220	+	0.732079i	$0.738550\pi$
$908$	0	0
$909$	−53.1185	−1.76183
$910$	0	0
$911$	35.2351	1.16739	0.583695	−	0.811973i	$-0.301606\pi$
$911$	35.2351	1.16739	0.583695	+	0.811973i	$0.301606\pi$
$912$	0	0
$913$	−19.0154	−0.629319
$914$	0	0
$915$	−0.146703	−0.00484986
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−37.9117	−1.25059	−0.625296	−	0.780387i	$-0.715022\pi$
$919$	−37.9117	−1.25059	−0.625296	+	0.780387i	$0.715022\pi$
$920$	0	0
$921$	40.5820	1.33722
$922$	0	0
$923$	−10.6416	−0.350272
$924$	0	0
$925$	−48.5865	−1.59752
$926$	0	0
$927$	50.0929	1.64527
$928$	0	0
$929$	37.0891	1.21685	0.608426	−	0.793610i	$-0.291801\pi$
$929$	37.0891	1.21685	0.608426	+	0.793610i	$0.291801\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−35.2459	−1.15390
$934$	0	0
$935$	−0.314478	−0.0102845
$936$	0	0
$937$	−43.5333	−1.42217	−0.711085	−	0.703106i	$-0.751796\pi$
$937$	−43.5333	−1.42217	−0.711085	+	0.703106i	$0.751796\pi$
$938$	0	0
$939$	−77.7178	−2.53622
$940$	0	0
$941$	−36.3914	−1.18633	−0.593163	−	0.805082i	$-0.702121\pi$
$941$	−36.3914	−1.18633	−0.593163	+	0.805082i	$0.702121\pi$
$942$	0	0
$943$	48.8157	1.58966
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.08304	0.132681	0.0663405	−	0.997797i	$-0.478868\pi$
$947$	4.08304	0.132681	0.0663405	+	0.997797i	$0.478868\pi$
$948$	0	0
$949$	−48.6734	−1.58000
$950$	0	0
$951$	7.68413	0.249175
$952$	0	0
$953$	−1.25957	−0.0408013	−0.0204007	−	0.999792i	$-0.506494\pi$
$953$	−1.25957	−0.0408013	−0.0204007	+	0.999792i	$0.506494\pi$
$954$	0	0
$955$	−1.79291	−0.0580171
$956$	0	0
$957$	9.14381	0.295577
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−25.1059	−0.809869
$962$	0	0
$963$	46.7935	1.50790
$964$	0	0
$965$	1.76464	0.0568057
$966$	0	0
$967$	−3.63836	−0.117002	−0.0585008	−	0.998287i	$-0.518632\pi$
$967$	−3.63836	−0.117002	−0.0585008	+	0.998287i	$0.518632\pi$
$968$	0	0
$969$	−0.483150	−0.0155210
$970$	0	0
$971$	−0.952377	−0.0305632	−0.0152816	−	0.999883i	$-0.504864\pi$
$971$	−0.952377	−0.0305632	−0.0152816	+	0.999883i	$0.504864\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	48.6091	1.55674
$976$	0	0
$977$	−26.4969	−0.847711	−0.423855	−	0.905730i	$-0.639323\pi$
$977$	−26.4969	−0.847711	−0.423855	+	0.905730i	$0.639323\pi$
$978$	0	0
$979$	17.3584	0.554778
$980$	0	0
$981$	−45.1747	−1.44232
$982$	0	0
$983$	−56.3652	−1.79777	−0.898885	−	0.438184i	$-0.855622\pi$
$983$	−56.3652	−1.79777	−0.898885	+	0.438184i	$0.855622\pi$
$984$	0	0
$985$	−3.55451	−0.113256
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−22.4053	−0.712446
$990$	0	0
$991$	−16.1823	−0.514049	−0.257025	−	0.966405i	$-0.582742\pi$
$991$	−16.1823	−0.514049	−0.257025	+	0.966405i	$0.582742\pi$
$992$	0	0
$993$	23.4699	0.744796
$994$	0	0
$995$	−6.02528	−0.191014
$996$	0	0
$997$	6.63773	0.210219	0.105109	−	0.994461i	$-0.466481\pi$
$997$	6.63773	0.210219	0.105109	+	0.994461i	$0.466481\pi$
$998$	0	0
$999$	22.9099	0.724837

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5488.2.a.s.1.9		9
4.3	odd	2		2744.2.a.d.1.1	✓	9
7.6	odd	2		5488.2.a.t.1.1		9
28.27	even	2		2744.2.a.e.1.9	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2744.2.a.d.1.1	✓	9	4.3	odd	2
2744.2.a.e.1.9	yes	9	28.27	even	2
5488.2.a.s.1.9		9	1.1	even	1	trivial
5488.2.a.t.1.1		9	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)

\(f(q)\)	\(=\)	\(q+2.62260 q^{3} -0.341478 q^{5} +3.87801 q^{9} -1.63427 q^{11} -3.79546 q^{13} -0.895559 q^{15} -0.563514 q^{17} +0.326924 q^{19} -6.42051 q^{23} -4.88339 q^{25} +2.30266 q^{27} -2.13340 q^{29} -2.42777 q^{31} -4.28602 q^{33} +9.94934 q^{37} -9.95396 q^{39} -7.60309 q^{41} +3.48964 q^{43} -1.32425 q^{45} -7.79238 q^{47} -1.47787 q^{51} -7.82985 q^{53} +0.558066 q^{55} +0.857388 q^{57} -1.63759 q^{59} +0.163812 q^{61} +1.29607 q^{65} -5.08098 q^{67} -16.8384 q^{69} +2.80377 q^{71} +12.8241 q^{73} -12.8072 q^{75} -15.8939 q^{79} -5.59509 q^{81} +11.6355 q^{83} +0.192428 q^{85} -5.59505 q^{87} -10.6215 q^{89} -6.36706 q^{93} -0.111637 q^{95} +5.19230 q^{97} -6.33770 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 13 q^{5} + 13 q^{9} - 3 q^{11} - 20 q^{13} - 3 q^{15} - 10 q^{17} + 10 q^{19} + 4 q^{23} + 22 q^{25} + 3 q^{29} + 10 q^{31} - 7 q^{33} + 6 q^{37} + 20 q^{39} - 13 q^{41} + 8 q^{43} - 52 q^{45} - 10 q^{47}+ \cdots + 14 q^{99}+O(q^{100}) \) 9 * q - 13 * q^5 + 13 * q^9 - 3 * q^11 - 20 * q^13 - 3 * q^15 - 10 * q^17 + 10 * q^19 + 4 * q^23 + 22 * q^25 + 3 * q^29 + 10 * q^31 - 7 * q^33 + 6 * q^37 + 20 * q^39 - 13 * q^41 + 8 * q^43 - 52 * q^45 - 10 * q^47 + 23 * q^51 - 8 * q^53 + 25 * q^55 - q^57 - 7 * q^59 - 52 * q^61 + 28 * q^65 + 9 * q^67 - 42 * q^69 - 9 * q^71 - 23 * q^73 + 7 * q^75 - 10 * q^79 - 3 * q^81 - 18 * q^83 - 5 * q^85 + 7 * q^87 - 17 * q^89 - 53 * q^93 - 25 * q^95 - 24 * q^97 + 14 * q^99

Introduction

L-functions

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