LMFDB - Embedded newform 4400.2.a.bk.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4400,2,Mod(1,4400)] 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("4400.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-1,0,0,0,1,0,-3,0,2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$35.1341768894$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2200)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4400.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{3} -0.618034 q^{7} -0.381966 q^{9} +1.00000 q^{11} -2.23607 q^{13} -4.85410 q^{17} +5.47214 q^{19} +1.00000 q^{21} +6.32624 q^{23} +5.47214 q^{27} -4.38197 q^{29} +4.23607 q^{31} -1.61803 q^{33} -1.76393 q^{37} +3.61803 q^{39} +7.94427 q^{41} -8.47214 q^{43} -4.70820 q^{47} -6.61803 q^{49} +7.85410 q^{51} +3.85410 q^{53} -8.85410 q^{57} +3.76393 q^{59} +7.09017 q^{61} +0.236068 q^{63} -10.2361 q^{69} +0.291796 q^{71} +7.09017 q^{73} -0.618034 q^{77} -2.85410 q^{79} -7.70820 q^{81} -1.14590 q^{83} +7.09017 q^{87} -12.5623 q^{89} +1.38197 q^{91} -6.85410 q^{93} -4.90983 q^{97} -0.381966 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + q^{7} - 3 q^{9} + 2 q^{11} - 3 q^{17} + 2 q^{19} + 2 q^{21} - 3 q^{23} + 2 q^{27} - 11 q^{29} + 4 q^{31} - q^{33} - 8 q^{37} + 5 q^{39} - 2 q^{41} - 8 q^{43} + 4 q^{47} - 11 q^{49} + 9 q^{51}+ \cdots - 3 q^{99}+O(q^{100}) $ 2 * q - q^3 + q^7 - 3 * q^9 + 2 * q^11 - 3 * q^17 + 2 * q^19 + 2 * q^21 - 3 * q^23 + 2 * q^27 - 11 * q^29 + 4 * q^31 - q^33 - 8 * q^37 + 5 * q^39 - 2 * q^41 - 8 * q^43 + 4 * q^47 - 11 * q^49 + 9 * q^51 + q^53 - 11 * q^57 + 12 * q^59 + 3 * q^61 - 4 * q^63 - 16 * q^69 + 14 * q^71 + 3 * q^73 + q^77 + q^79 - 2 * q^81 - 9 * q^83 + 3 * q^87 - 5 * q^89 + 5 * q^91 - 7 * q^93 - 21 * q^97 - 3 * 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
$3$	−1.61803	−0.934172	−0.467086	−	0.884212i	$-0.654696\pi$
$3$	−1.61803	−0.934172	−0.467086	+	0.884212i	$0.654696\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.618034	−0.233595	−0.116797	−	0.993156i	$-0.537263\pi$
$7$	−0.618034	−0.233595	−0.116797	+	0.993156i	$0.537263\pi$
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.23607	−0.620174	−0.310087	−	0.950708i	$-0.600358\pi$
$13$	−2.23607	−0.620174	−0.310087	+	0.950708i	$0.600358\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.85410	−1.17729	−0.588646	−	0.808391i	$-0.700339\pi$
$17$	−4.85410	−1.17729	−0.588646	+	0.808391i	$0.700339\pi$
$18$	0	0
$19$	5.47214	1.25539	0.627697	−	0.778458i	$-0.283998\pi$
$19$	5.47214	1.25539	0.627697	+	0.778458i	$0.283998\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	6.32624	1.31911	0.659556	−	0.751656i	$-0.270744\pi$
$23$	6.32624	1.31911	0.659556	+	0.751656i	$0.270744\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.47214	1.05311
$28$	0	0
$29$	−4.38197	−0.813711	−0.406855	−	0.913493i	$-0.633375\pi$
$29$	−4.38197	−0.813711	−0.406855	+	0.913493i	$0.633375\pi$
$30$	0	0
$31$	4.23607	0.760820	0.380410	−	0.924818i	$-0.375783\pi$
$31$	4.23607	0.760820	0.380410	+	0.924818i	$0.375783\pi$
$32$	0	0
$33$	−1.61803	−0.281664
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.76393	−0.289989	−0.144994	−	0.989432i	$-0.546316\pi$
$37$	−1.76393	−0.289989	−0.144994	+	0.989432i	$0.546316\pi$
$38$	0	0
$39$	3.61803	0.579349
$40$	0	0
$41$	7.94427	1.24069	0.620343	−	0.784330i	$-0.286993\pi$
$41$	7.94427	1.24069	0.620343	+	0.784330i	$0.286993\pi$
$42$	0	0
$43$	−8.47214	−1.29199	−0.645994	−	0.763342i	$-0.723557\pi$
$43$	−8.47214	−1.29199	−0.645994	+	0.763342i	$0.723557\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.70820	−0.686762	−0.343381	−	0.939196i	$-0.611572\pi$
$47$	−4.70820	−0.686762	−0.343381	+	0.939196i	$0.611572\pi$
$48$	0	0
$49$	−6.61803	−0.945433
$50$	0	0
$51$	7.85410	1.09979
$52$	0	0
$53$	3.85410	0.529402	0.264701	−	0.964331i	$-0.414727\pi$
$53$	3.85410	0.529402	0.264701	+	0.964331i	$0.414727\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−8.85410	−1.17275
$58$	0	0
$59$	3.76393	0.490022	0.245011	−	0.969520i	$-0.421208\pi$
$59$	3.76393	0.490022	0.245011	+	0.969520i	$0.421208\pi$
$60$	0	0
$61$	7.09017	0.907803	0.453902	−	0.891052i	$-0.350032\pi$
$61$	7.09017	0.907803	0.453902	+	0.891052i	$0.350032\pi$
$62$	0	0
$63$	0.236068	0.0297418
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	−10.2361	−1.23228
$70$	0	0
$71$	0.291796	0.0346298	0.0173149	−	0.999850i	$-0.494488\pi$
$71$	0.291796	0.0346298	0.0173149	+	0.999850i	$0.494488\pi$
$72$	0	0
$73$	7.09017	0.829842	0.414921	−	0.909858i	$-0.363809\pi$
$73$	7.09017	0.829842	0.414921	+	0.909858i	$0.363809\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.618034	−0.0704315
$78$	0	0
$79$	−2.85410	−0.321112	−0.160556	−	0.987027i	$-0.551329\pi$
$79$	−2.85410	−0.321112	−0.160556	+	0.987027i	$0.551329\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	−1.14590	−0.125779	−0.0628893	−	0.998021i	$-0.520032\pi$
$83$	−1.14590	−0.125779	−0.0628893	+	0.998021i	$0.520032\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.09017	0.760146
$88$	0	0
$89$	−12.5623	−1.33160	−0.665801	−	0.746129i	$-0.731910\pi$
$89$	−12.5623	−1.33160	−0.665801	+	0.746129i	$0.731910\pi$
$90$	0	0
$91$	1.38197	0.144869
$92$	0	0
$93$	−6.85410	−0.710737
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−4.90983	−0.498518	−0.249259	−	0.968437i	$-0.580187\pi$
$97$	−4.90983	−0.498518	−0.249259	+	0.968437i	$0.580187\pi$
$98$	0	0
$99$	−0.381966	−0.0383890
$100$	0	0
$101$	−7.56231	−0.752478	−0.376239	−	0.926523i	$-0.622783\pi$
$101$	−7.56231	−0.752478	−0.376239	+	0.926523i	$0.622783\pi$
$102$	0	0
$103$	−16.8541	−1.66068	−0.830342	−	0.557254i	$-0.811855\pi$
$103$	−16.8541	−1.66068	−0.830342	+	0.557254i	$0.811855\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.9443	−1.15470	−0.577348	−	0.816498i	$-0.695912\pi$
$107$	−11.9443	−1.15470	−0.577348	+	0.816498i	$0.695912\pi$
$108$	0	0
$109$	5.79837	0.555383	0.277692	−	0.960670i	$-0.410431\pi$
$109$	5.79837	0.555383	0.277692	+	0.960670i	$0.410431\pi$
$110$	0	0
$111$	2.85410	0.270899
$112$	0	0
$113$	−0.236068	−0.0222074	−0.0111037	−	0.999938i	$-0.503534\pi$
$113$	−0.236068	−0.0222074	−0.0111037	+	0.999938i	$0.503534\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.854102	0.0789618
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−12.8541	−1.15902
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0.381966	0.0338940	0.0169470	−	0.999856i	$-0.494605\pi$
$127$	0.381966	0.0338940	0.0169470	+	0.999856i	$0.494605\pi$
$128$	0	0
$129$	13.7082	1.20694
$130$	0	0
$131$	15.0902	1.31843	0.659217	−	0.751953i	$-0.270888\pi$
$131$	15.0902	1.31843	0.659217	+	0.751953i	$0.270888\pi$
$132$	0	0
$133$	−3.38197	−0.293254
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.38197	−0.545248	−0.272624	−	0.962121i	$-0.587892\pi$
$137$	−6.38197	−0.545248	−0.272624	+	0.962121i	$0.587892\pi$
$138$	0	0
$139$	7.00000	0.593732	0.296866	−	0.954919i	$-0.404058\pi$
$139$	7.00000	0.593732	0.296866	+	0.954919i	$0.404058\pi$
$140$	0	0
$141$	7.61803	0.641554
$142$	0	0
$143$	−2.23607	−0.186989
$144$	0	0
$145$	0	0
$146$	0	0
$147$	10.7082	0.883198
$148$	0	0
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	13.6525	1.11102	0.555511	−	0.831509i	$-0.312523\pi$
$151$	13.6525	1.11102	0.555511	+	0.831509i	$0.312523\pi$
$152$	0	0
$153$	1.85410	0.149895
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−17.4164	−1.38998	−0.694990	−	0.719019i	$-0.744591\pi$
$157$	−17.4164	−1.38998	−0.694990	+	0.719019i	$0.744591\pi$
$158$	0	0
$159$	−6.23607	−0.494552
$160$	0	0
$161$	−3.90983	−0.308138
$162$	0	0
$163$	−15.3820	−1.20481	−0.602404	−	0.798191i	$-0.705790\pi$
$163$	−15.3820	−1.20481	−0.602404	+	0.798191i	$0.705790\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.00000	−0.696441	−0.348220	−	0.937413i	$-0.613214\pi$
$167$	−9.00000	−0.696441	−0.348220	+	0.937413i	$0.613214\pi$
$168$	0	0
$169$	−8.00000	−0.615385
$170$	0	0
$171$	−2.09017	−0.159839
$172$	0	0
$173$	14.8885	1.13196	0.565978	−	0.824421i	$-0.308499\pi$
$173$	14.8885	1.13196	0.565978	+	0.824421i	$0.308499\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.09017	−0.457765
$178$	0	0
$179$	13.0902	0.978405	0.489203	−	0.872170i	$-0.337288\pi$
$179$	13.0902	0.978405	0.489203	+	0.872170i	$0.337288\pi$
$180$	0	0
$181$	13.0902	0.972985	0.486492	−	0.873685i	$-0.338276\pi$
$181$	13.0902	0.972985	0.486492	+	0.873685i	$0.338276\pi$
$182$	0	0
$183$	−11.4721	−0.848045
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.85410	−0.354967
$188$	0	0
$189$	−3.38197	−0.246002
$190$	0	0
$191$	10.3820	0.751213	0.375606	−	0.926779i	$-0.377434\pi$
$191$	10.3820	0.751213	0.375606	+	0.926779i	$0.377434\pi$
$192$	0	0
$193$	−12.9443	−0.931749	−0.465875	−	0.884851i	$-0.654260\pi$
$193$	−12.9443	−0.931749	−0.465875	+	0.884851i	$0.654260\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.5623	1.53625	0.768125	−	0.640300i	$-0.221190\pi$
$197$	21.5623	1.53625	0.768125	+	0.640300i	$0.221190\pi$
$198$	0	0
$199$	−20.5623	−1.45762	−0.728812	−	0.684714i	$-0.759927\pi$
$199$	−20.5623	−1.45762	−0.728812	+	0.684714i	$0.759927\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.70820	0.190079
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.41641	−0.167952
$208$	0	0
$209$	5.47214	0.378516
$210$	0	0
$211$	−6.70820	−0.461812	−0.230906	−	0.972976i	$-0.574169\pi$
$211$	−6.70820	−0.461812	−0.230906	+	0.972976i	$0.574169\pi$
$212$	0	0
$213$	−0.472136	−0.0323502
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.61803	−0.177724
$218$	0	0
$219$	−11.4721	−0.775215
$220$	0	0
$221$	10.8541	0.730126
$222$	0	0
$223$	5.18034	0.346901	0.173451	−	0.984843i	$-0.444508\pi$
$223$	5.18034	0.346901	0.173451	+	0.984843i	$0.444508\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.79837	0.517596	0.258798	−	0.965931i	$-0.416674\pi$
$227$	7.79837	0.517596	0.258798	+	0.965931i	$0.416674\pi$
$228$	0	0
$229$	−27.5066	−1.81769	−0.908843	−	0.417139i	$-0.863033\pi$
$229$	−27.5066	−1.81769	−0.908843	+	0.417139i	$0.863033\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	−18.1459	−1.18878	−0.594389	−	0.804178i	$-0.702606\pi$
$233$	−18.1459	−1.18878	−0.594389	+	0.804178i	$0.702606\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.61803	0.299974
$238$	0	0
$239$	−21.0902	−1.36421	−0.682105	−	0.731254i	$-0.738935\pi$
$239$	−21.0902	−1.36421	−0.682105	+	0.731254i	$0.738935\pi$
$240$	0	0
$241$	−17.0344	−1.09728	−0.548642	−	0.836057i	$-0.684855\pi$
$241$	−17.0344	−1.09728	−0.548642	+	0.836057i	$0.684855\pi$
$242$	0	0
$243$	−3.94427	−0.253025
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−12.2361	−0.778562
$248$	0	0
$249$	1.85410	0.117499
$250$	0	0
$251$	7.14590	0.451045	0.225523	−	0.974238i	$-0.427591\pi$
$251$	7.14590	0.451045	0.225523	+	0.974238i	$0.427591\pi$
$252$	0	0
$253$	6.32624	0.397727
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.05573	−0.0658545	−0.0329273	−	0.999458i	$-0.510483\pi$
$257$	−1.05573	−0.0658545	−0.0329273	+	0.999458i	$0.510483\pi$
$258$	0	0
$259$	1.09017	0.0677399
$260$	0	0
$261$	1.67376	0.103603
$262$	0	0
$263$	−26.1246	−1.61091	−0.805456	−	0.592655i	$-0.798080\pi$
$263$	−26.1246	−1.61091	−0.805456	+	0.592655i	$0.798080\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	20.3262	1.24395
$268$	0	0
$269$	−23.7984	−1.45101	−0.725506	−	0.688216i	$-0.758394\pi$
$269$	−23.7984	−1.45101	−0.725506	+	0.688216i	$0.758394\pi$
$270$	0	0
$271$	−18.4164	−1.11872	−0.559359	−	0.828926i	$-0.688952\pi$
$271$	−18.4164	−1.11872	−0.559359	+	0.828926i	$0.688952\pi$
$272$	0	0
$273$	−2.23607	−0.135333
$274$	0	0
$275$	0	0
$276$	0	0
$277$	15.9443	0.957998	0.478999	−	0.877815i	$-0.341000\pi$
$277$	15.9443	0.957998	0.478999	+	0.877815i	$0.341000\pi$
$278$	0	0
$279$	−1.61803	−0.0968692
$280$	0	0
$281$	−31.3607	−1.87082	−0.935411	−	0.353563i	$-0.884970\pi$
$281$	−31.3607	−1.87082	−0.935411	+	0.353563i	$0.884970\pi$
$282$	0	0
$283$	13.1459	0.781443	0.390721	−	0.920509i	$-0.372226\pi$
$283$	13.1459	0.781443	0.390721	+	0.920509i	$0.372226\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.90983	−0.289818
$288$	0	0
$289$	6.56231	0.386018
$290$	0	0
$291$	7.94427	0.465701
$292$	0	0
$293$	−19.8885	−1.16190	−0.580951	−	0.813939i	$-0.697319\pi$
$293$	−19.8885	−1.16190	−0.580951	+	0.813939i	$0.697319\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.47214	0.317526
$298$	0	0
$299$	−14.1459	−0.818078
$300$	0	0
$301$	5.23607	0.301802
$302$	0	0
$303$	12.2361	0.702944
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−4.85410	−0.277038	−0.138519	−	0.990360i	$-0.544234\pi$
$307$	−4.85410	−0.277038	−0.138519	+	0.990360i	$0.544234\pi$
$308$	0	0
$309$	27.2705	1.55137
$310$	0	0
$311$	−10.8885	−0.617433	−0.308716	−	0.951154i	$-0.599899\pi$
$311$	−10.8885	−0.617433	−0.308716	+	0.951154i	$0.599899\pi$
$312$	0	0
$313$	−27.7082	−1.56616	−0.783080	−	0.621921i	$-0.786353\pi$
$313$	−27.7082	−1.56616	−0.783080	+	0.621921i	$0.786353\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.27051	−0.239856	−0.119928	−	0.992783i	$-0.538266\pi$
$317$	−4.27051	−0.239856	−0.119928	+	0.992783i	$0.538266\pi$
$318$	0	0
$319$	−4.38197	−0.245343
$320$	0	0
$321$	19.3262	1.07869
$322$	0	0
$323$	−26.5623	−1.47797
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−9.38197	−0.518824
$328$	0	0
$329$	2.90983	0.160424
$330$	0	0
$331$	−28.4164	−1.56191	−0.780954	−	0.624589i	$-0.785266\pi$
$331$	−28.4164	−1.56191	−0.780954	+	0.624589i	$0.785266\pi$
$332$	0	0
$333$	0.673762	0.0369219
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−15.5279	−0.845857	−0.422928	−	0.906163i	$-0.638998\pi$
$337$	−15.5279	−0.845857	−0.422928	+	0.906163i	$0.638998\pi$
$338$	0	0
$339$	0.381966	0.0207455
$340$	0	0
$341$	4.23607	0.229396
$342$	0	0
$343$	8.41641	0.454443
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.2705	0.927130	0.463565	−	0.886063i	$-0.346570\pi$
$347$	17.2705	0.927130	0.463565	+	0.886063i	$0.346570\pi$
$348$	0	0
$349$	26.5279	1.42000	0.710002	−	0.704200i	$-0.248694\pi$
$349$	26.5279	1.42000	0.710002	+	0.704200i	$0.248694\pi$
$350$	0	0
$351$	−12.2361	−0.653113
$352$	0	0
$353$	−36.4164	−1.93825	−0.969125	−	0.246570i	$-0.920696\pi$
$353$	−36.4164	−1.93825	−0.969125	+	0.246570i	$0.920696\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.85410	−0.256906
$358$	0	0
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	10.9443	0.576014
$362$	0	0
$363$	−1.61803	−0.0849248
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−16.7984	−0.876868	−0.438434	−	0.898763i	$-0.644467\pi$
$367$	−16.7984	−0.876868	−0.438434	+	0.898763i	$0.644467\pi$
$368$	0	0
$369$	−3.03444	−0.157967
$370$	0	0
$371$	−2.38197	−0.123666
$372$	0	0
$373$	−4.70820	−0.243782	−0.121891	−	0.992544i	$-0.538896\pi$
$373$	−4.70820	−0.243782	−0.121891	+	0.992544i	$0.538896\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.79837	0.504642
$378$	0	0
$379$	26.3050	1.35119	0.675597	−	0.737271i	$-0.263886\pi$
$379$	26.3050	1.35119	0.675597	+	0.737271i	$0.263886\pi$
$380$	0	0
$381$	−0.618034	−0.0316628
$382$	0	0
$383$	29.4164	1.50311	0.751554	−	0.659671i	$-0.229305\pi$
$383$	29.4164	1.50311	0.751554	+	0.659671i	$0.229305\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.23607	0.164499
$388$	0	0
$389$	−17.5279	−0.888698	−0.444349	−	0.895854i	$-0.646565\pi$
$389$	−17.5279	−0.888698	−0.444349	+	0.895854i	$0.646565\pi$
$390$	0	0
$391$	−30.7082	−1.55298
$392$	0	0
$393$	−24.4164	−1.23164
$394$	0	0
$395$	0	0
$396$	0	0
$397$	26.8541	1.34777	0.673884	−	0.738837i	$-0.264625\pi$
$397$	26.8541	1.34777	0.673884	+	0.738837i	$0.264625\pi$
$398$	0	0
$399$	5.47214	0.273949
$400$	0	0
$401$	26.3607	1.31639	0.658195	−	0.752848i	$-0.271320\pi$
$401$	26.3607	1.31639	0.658195	+	0.752848i	$0.271320\pi$
$402$	0	0
$403$	−9.47214	−0.471841
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.76393	−0.0874349
$408$	0	0
$409$	−6.12461	−0.302843	−0.151421	−	0.988469i	$-0.548385\pi$
$409$	−6.12461	−0.302843	−0.151421	+	0.988469i	$0.548385\pi$
$410$	0	0
$411$	10.3262	0.509356
$412$	0	0
$413$	−2.32624	−0.114467
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−11.3262	−0.554648
$418$	0	0
$419$	0.888544	0.0434082	0.0217041	−	0.999764i	$-0.493091\pi$
$419$	0.888544	0.0434082	0.0217041	+	0.999764i	$0.493091\pi$
$420$	0	0
$421$	−3.09017	−0.150606	−0.0753028	−	0.997161i	$-0.523992\pi$
$421$	−3.09017	−0.150606	−0.0753028	+	0.997161i	$0.523992\pi$
$422$	0	0
$423$	1.79837	0.0874399
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−4.38197	−0.212058
$428$	0	0
$429$	3.61803	0.174680
$430$	0	0
$431$	1.00000	0.0481683	0.0240842	−	0.999710i	$-0.492333\pi$
$431$	1.00000	0.0481683	0.0240842	+	0.999710i	$0.492333\pi$
$432$	0	0
$433$	32.7771	1.57517	0.787583	−	0.616208i	$-0.211332\pi$
$433$	32.7771	1.57517	0.787583	+	0.616208i	$0.211332\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	34.6180	1.65601
$438$	0	0
$439$	−23.3262	−1.11330	−0.556650	−	0.830747i	$-0.687914\pi$
$439$	−23.3262	−1.11330	−0.556650	+	0.830747i	$0.687914\pi$
$440$	0	0
$441$	2.52786	0.120374
$442$	0	0
$443$	14.2918	0.679024	0.339512	−	0.940602i	$-0.389738\pi$
$443$	14.2918	0.679024	0.339512	+	0.940602i	$0.389738\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	19.4164	0.918365
$448$	0	0
$449$	31.1591	1.47049	0.735243	−	0.677803i	$-0.237068\pi$
$449$	31.1591	1.47049	0.735243	+	0.677803i	$0.237068\pi$
$450$	0	0
$451$	7.94427	0.374081
$452$	0	0
$453$	−22.0902	−1.03789
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.43769	−0.207587	−0.103793	−	0.994599i	$-0.533098\pi$
$457$	−4.43769	−0.207587	−0.103793	+	0.994599i	$0.533098\pi$
$458$	0	0
$459$	−26.5623	−1.23982
$460$	0	0
$461$	−18.1246	−0.844147	−0.422074	−	0.906562i	$-0.638698\pi$
$461$	−18.1246	−0.844147	−0.422074	+	0.906562i	$0.638698\pi$
$462$	0	0
$463$	18.7082	0.869444	0.434722	−	0.900565i	$-0.356847\pi$
$463$	18.7082	0.869444	0.434722	+	0.900565i	$0.356847\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−33.6525	−1.55725	−0.778625	−	0.627489i	$-0.784083\pi$
$467$	−33.6525	−1.55725	−0.778625	+	0.627489i	$0.784083\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	28.1803	1.29848
$472$	0	0
$473$	−8.47214	−0.389549
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.47214	−0.0674045
$478$	0	0
$479$	−12.5967	−0.575560	−0.287780	−	0.957697i	$-0.592917\pi$
$479$	−12.5967	−0.575560	−0.287780	+	0.957697i	$0.592917\pi$
$480$	0	0
$481$	3.94427	0.179843
$482$	0	0
$483$	6.32624	0.287854
$484$	0	0
$485$	0	0
$486$	0	0
$487$	15.1803	0.687887	0.343943	−	0.938990i	$-0.388237\pi$
$487$	15.1803	0.687887	0.343943	+	0.938990i	$0.388237\pi$
$488$	0	0
$489$	24.8885	1.12550
$490$	0	0
$491$	23.8328	1.07556	0.537780	−	0.843085i	$-0.319263\pi$
$491$	23.8328	1.07556	0.537780	+	0.843085i	$0.319263\pi$
$492$	0	0
$493$	21.2705	0.957976
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.180340	−0.00808935
$498$	0	0
$499$	−22.6869	−1.01561	−0.507803	−	0.861473i	$-0.669542\pi$
$499$	−22.6869	−1.01561	−0.507803	+	0.861473i	$0.669542\pi$
$500$	0	0
$501$	14.5623	0.650596
$502$	0	0
$503$	37.2361	1.66027	0.830137	−	0.557559i	$-0.188262\pi$
$503$	37.2361	1.66027	0.830137	+	0.557559i	$0.188262\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.9443	0.574875
$508$	0	0
$509$	−36.2148	−1.60519	−0.802596	−	0.596523i	$-0.796548\pi$
$509$	−36.2148	−1.60519	−0.802596	+	0.596523i	$0.796548\pi$
$510$	0	0
$511$	−4.38197	−0.193847
$512$	0	0
$513$	29.9443	1.32207
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4.70820	−0.207067
$518$	0	0
$519$	−24.0902	−1.05744
$520$	0	0
$521$	−38.6525	−1.69340	−0.846698	−	0.532074i	$-0.821413\pi$
$521$	−38.6525	−1.69340	−0.846698	+	0.532074i	$0.821413\pi$
$522$	0	0
$523$	−0.472136	−0.0206451	−0.0103225	−	0.999947i	$-0.503286\pi$
$523$	−0.472136	−0.0206451	−0.0103225	+	0.999947i	$0.503286\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−20.5623	−0.895708
$528$	0	0
$529$	17.0213	0.740056
$530$	0	0
$531$	−1.43769	−0.0623906
$532$	0	0
$533$	−17.7639	−0.769441
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−21.1803	−0.913999
$538$	0	0
$539$	−6.61803	−0.285059
$540$	0	0
$541$	28.5066	1.22559	0.612797	−	0.790241i	$-0.290044\pi$
$541$	28.5066	1.22559	0.612797	+	0.790241i	$0.290044\pi$
$542$	0	0
$543$	−21.1803	−0.908935
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−22.4377	−0.959367	−0.479683	−	0.877442i	$-0.659248\pi$
$547$	−22.4377	−0.959367	−0.479683	+	0.877442i	$0.659248\pi$
$548$	0	0
$549$	−2.70820	−0.115583
$550$	0	0
$551$	−23.9787	−1.02153
$552$	0	0
$553$	1.76393	0.0750100
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.7082	0.623207	0.311603	−	0.950212i	$-0.399134\pi$
$557$	14.7082	0.623207	0.311603	+	0.950212i	$0.399134\pi$
$558$	0	0
$559$	18.9443	0.801257
$560$	0	0
$561$	7.85410	0.331600
$562$	0	0
$563$	6.61803	0.278917	0.139458	−	0.990228i	$-0.455464\pi$
$563$	6.61803	0.278917	0.139458	+	0.990228i	$0.455464\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.76393	0.200066
$568$	0	0
$569$	32.9098	1.37965	0.689826	−	0.723975i	$-0.257687\pi$
$569$	32.9098	1.37965	0.689826	+	0.723975i	$0.257687\pi$
$570$	0	0
$571$	−41.4508	−1.73466	−0.867332	−	0.497730i	$-0.834167\pi$
$571$	−41.4508	−1.73466	−0.867332	+	0.497730i	$0.834167\pi$
$572$	0	0
$573$	−16.7984	−0.701762
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−31.7984	−1.32378	−0.661892	−	0.749599i	$-0.730246\pi$
$577$	−31.7984	−1.32378	−0.661892	+	0.749599i	$0.730246\pi$
$578$	0	0
$579$	20.9443	0.870414
$580$	0	0
$581$	0.708204	0.0293812
$582$	0	0
$583$	3.85410	0.159621
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.27051	−0.0937140	−0.0468570	−	0.998902i	$-0.514921\pi$
$587$	−2.27051	−0.0937140	−0.0468570	+	0.998902i	$0.514921\pi$
$588$	0	0
$589$	23.1803	0.955129
$590$	0	0
$591$	−34.8885	−1.43512
$592$	0	0
$593$	6.88854	0.282879	0.141439	−	0.989947i	$-0.454827\pi$
$593$	6.88854	0.282879	0.141439	+	0.989947i	$0.454827\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	33.2705	1.36167
$598$	0	0
$599$	−26.0902	−1.06602	−0.533008	−	0.846110i	$-0.678938\pi$
$599$	−26.0902	−1.06602	−0.533008	+	0.846110i	$0.678938\pi$
$600$	0	0
$601$	−25.7426	−1.05006	−0.525032	−	0.851082i	$-0.675947\pi$
$601$	−25.7426	−1.05006	−0.525032	+	0.851082i	$0.675947\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−9.05573	−0.367561	−0.183780	−	0.982967i	$-0.558834\pi$
$607$	−9.05573	−0.367561	−0.183780	+	0.982967i	$0.558834\pi$
$608$	0	0
$609$	−4.38197	−0.177566
$610$	0	0
$611$	10.5279	0.425912
$612$	0	0
$613$	24.0902	0.972993	0.486496	−	0.873683i	$-0.338275\pi$
$613$	24.0902	0.972993	0.486496	+	0.873683i	$0.338275\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19.2361	0.774415	0.387207	−	0.921993i	$-0.373440\pi$
$617$	19.2361	0.774415	0.387207	+	0.921993i	$0.373440\pi$
$618$	0	0
$619$	21.2918	0.855790	0.427895	−	0.903829i	$-0.359255\pi$
$619$	21.2918	0.855790	0.427895	+	0.903829i	$0.359255\pi$
$620$	0	0
$621$	34.6180	1.38917
$622$	0	0
$623$	7.76393	0.311055
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−8.85410	−0.353599
$628$	0	0
$629$	8.56231	0.341401
$630$	0	0
$631$	32.5066	1.29407	0.647033	−	0.762462i	$-0.276009\pi$
$631$	32.5066	1.29407	0.647033	+	0.762462i	$0.276009\pi$
$632$	0	0
$633$	10.8541	0.431412
$634$	0	0
$635$	0	0
$636$	0	0
$637$	14.7984	0.586333
$638$	0	0
$639$	−0.111456	−0.00440914
$640$	0	0
$641$	−1.11146	−0.0438999	−0.0219499	−	0.999759i	$-0.506987\pi$
$641$	−1.11146	−0.0438999	−0.0219499	+	0.999759i	$0.506987\pi$
$642$	0	0
$643$	−9.05573	−0.357123	−0.178562	−	0.983929i	$-0.557144\pi$
$643$	−9.05573	−0.357123	−0.178562	+	0.983929i	$0.557144\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13.9443	−0.548206	−0.274103	−	0.961700i	$-0.588381\pi$
$647$	−13.9443	−0.548206	−0.274103	+	0.961700i	$0.588381\pi$
$648$	0	0
$649$	3.76393	0.147747
$650$	0	0
$651$	4.23607	0.166025
$652$	0	0
$653$	19.3262	0.756294	0.378147	−	0.925746i	$-0.376561\pi$
$653$	19.3262	0.756294	0.378147	+	0.925746i	$0.376561\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2.70820	−0.105657
$658$	0	0
$659$	−31.7984	−1.23869	−0.619344	−	0.785119i	$-0.712602\pi$
$659$	−31.7984	−1.23869	−0.619344	+	0.785119i	$0.712602\pi$
$660$	0	0
$661$	−26.5967	−1.03449	−0.517247	−	0.855836i	$-0.673043\pi$
$661$	−26.5967	−1.03449	−0.517247	+	0.855836i	$0.673043\pi$
$662$	0	0
$663$	−17.5623	−0.682063
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−27.7214	−1.07338
$668$	0	0
$669$	−8.38197	−0.324066
$670$	0	0
$671$	7.09017	0.273713
$672$	0	0
$673$	−15.4721	−0.596407	−0.298204	−	0.954502i	$-0.596387\pi$
$673$	−15.4721	−0.596407	−0.298204	+	0.954502i	$0.596387\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	32.7639	1.25922	0.629610	−	0.776911i	$-0.283215\pi$
$677$	32.7639	1.25922	0.629610	+	0.776911i	$0.283215\pi$
$678$	0	0
$679$	3.03444	0.116451
$680$	0	0
$681$	−12.6180	−0.483524
$682$	0	0
$683$	35.7639	1.36847	0.684234	−	0.729262i	$-0.260137\pi$
$683$	35.7639	1.36847	0.684234	+	0.729262i	$0.260137\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	44.5066	1.69803
$688$	0	0
$689$	−8.61803	−0.328321
$690$	0	0
$691$	−1.20163	−0.0457120	−0.0228560	−	0.999739i	$-0.507276\pi$
$691$	−1.20163	−0.0457120	−0.0228560	+	0.999739i	$0.507276\pi$
$692$	0	0
$693$	0.236068	0.00896748
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−38.5623	−1.46065
$698$	0	0
$699$	29.3607	1.11052
$700$	0	0
$701$	5.52786	0.208785	0.104392	−	0.994536i	$-0.466710\pi$
$701$	5.52786	0.208785	0.104392	+	0.994536i	$0.466710\pi$
$702$	0	0
$703$	−9.65248	−0.364050
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.67376	0.175775
$708$	0	0
$709$	16.3607	0.614438	0.307219	−	0.951639i	$-0.400602\pi$
$709$	16.3607	0.614438	0.307219	+	0.951639i	$0.400602\pi$
$710$	0	0
$711$	1.09017	0.0408846
$712$	0	0
$713$	26.7984	1.00361
$714$	0	0
$715$	0	0
$716$	0	0
$717$	34.1246	1.27441
$718$	0	0
$719$	41.9443	1.56426	0.782129	−	0.623117i	$-0.214134\pi$
$719$	41.9443	1.56426	0.782129	+	0.623117i	$0.214134\pi$
$720$	0	0
$721$	10.4164	0.387927
$722$	0	0
$723$	27.5623	1.02505
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−41.5066	−1.53939	−0.769697	−	0.638410i	$-0.779593\pi$
$727$	−41.5066	−1.53939	−0.769697	+	0.638410i	$0.779593\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	41.1246	1.52105
$732$	0	0
$733$	45.4721	1.67955	0.839776	−	0.542933i	$-0.182686\pi$
$733$	45.4721	1.67955	0.839776	+	0.542933i	$0.182686\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−36.0344	−1.32555	−0.662774	−	0.748819i	$-0.730621\pi$
$739$	−36.0344	−1.32555	−0.662774	+	0.748819i	$0.730621\pi$
$740$	0	0
$741$	19.7984	0.727311
$742$	0	0
$743$	11.9656	0.438974	0.219487	−	0.975615i	$-0.429562\pi$
$743$	11.9656	0.438974	0.219487	+	0.975615i	$0.429562\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.437694	0.0160144
$748$	0	0
$749$	7.38197	0.269731
$750$	0	0
$751$	34.1591	1.24648	0.623241	−	0.782030i	$-0.285816\pi$
$751$	34.1591	1.24648	0.623241	+	0.782030i	$0.285816\pi$
$752$	0	0
$753$	−11.5623	−0.421354
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−51.8328	−1.88390	−0.941948	−	0.335759i	$-0.891007\pi$
$757$	−51.8328	−1.88390	−0.941948	+	0.335759i	$0.891007\pi$
$758$	0	0
$759$	−10.2361	−0.371546
$760$	0	0
$761$	−37.8885	−1.37346	−0.686729	−	0.726913i	$-0.740954\pi$
$761$	−37.8885	−1.37346	−0.686729	+	0.726913i	$0.740954\pi$
$762$	0	0
$763$	−3.58359	−0.129735
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.41641	−0.303899
$768$	0	0
$769$	29.9443	1.07982	0.539909	−	0.841723i	$-0.318459\pi$
$769$	29.9443	1.07982	0.539909	+	0.841723i	$0.318459\pi$
$770$	0	0
$771$	1.70820	0.0615195
$772$	0	0
$773$	32.9098	1.18368	0.591842	−	0.806054i	$-0.298401\pi$
$773$	32.9098	1.18368	0.591842	+	0.806054i	$0.298401\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.76393	−0.0632807
$778$	0	0
$779$	43.4721	1.55755
$780$	0	0
$781$	0.291796	0.0104413
$782$	0	0
$783$	−23.9787	−0.856929
$784$	0	0
$785$	0	0
$786$	0	0
$787$	14.5279	0.517862	0.258931	−	0.965896i	$-0.416630\pi$
$787$	14.5279	0.517862	0.258931	+	0.965896i	$0.416630\pi$
$788$	0	0
$789$	42.2705	1.50487
$790$	0	0
$791$	0.145898	0.00518754
$792$	0	0
$793$	−15.8541	−0.562996
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.20163	0.148829	0.0744146	−	0.997227i	$-0.476291\pi$
$797$	4.20163	0.148829	0.0744146	+	0.997227i	$0.476291\pi$
$798$	0	0
$799$	22.8541	0.808520
$800$	0	0
$801$	4.79837	0.169542
$802$	0	0
$803$	7.09017	0.250207
$804$	0	0
$805$	0	0
$806$	0	0
$807$	38.5066	1.35550
$808$	0	0
$809$	−11.4164	−0.401380	−0.200690	−	0.979655i	$-0.564318\pi$
$809$	−11.4164	−0.401380	−0.200690	+	0.979655i	$0.564318\pi$
$810$	0	0
$811$	−32.8197	−1.15245	−0.576227	−	0.817290i	$-0.695476\pi$
$811$	−32.8197	−1.15245	−0.576227	+	0.817290i	$0.695476\pi$
$812$	0	0
$813$	29.7984	1.04507
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−46.3607	−1.62195
$818$	0	0
$819$	−0.527864	−0.0184451
$820$	0	0
$821$	8.36068	0.291790	0.145895	−	0.989300i	$-0.453394\pi$
$821$	8.36068	0.291790	0.145895	+	0.989300i	$0.453394\pi$
$822$	0	0
$823$	−21.5279	−0.750414	−0.375207	−	0.926941i	$-0.622428\pi$
$823$	−21.5279	−0.750414	−0.375207	+	0.926941i	$0.622428\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.5967	0.820539	0.410270	−	0.911964i	$-0.365435\pi$
$827$	23.5967	0.820539	0.410270	+	0.911964i	$0.365435\pi$
$828$	0	0
$829$	−33.6180	−1.16760	−0.583801	−	0.811897i	$-0.698435\pi$
$829$	−33.6180	−1.16760	−0.583801	+	0.811897i	$0.698435\pi$
$830$	0	0
$831$	−25.7984	−0.894936
$832$	0	0
$833$	32.1246	1.11305
$834$	0	0
$835$	0	0
$836$	0	0
$837$	23.1803	0.801230
$838$	0	0
$839$	23.6738	0.817309	0.408655	−	0.912689i	$-0.365998\pi$
$839$	23.6738	0.817309	0.408655	+	0.912689i	$0.365998\pi$
$840$	0	0
$841$	−9.79837	−0.337875
$842$	0	0
$843$	50.7426	1.74767
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−0.618034	−0.0212359
$848$	0	0
$849$	−21.2705	−0.730002
$850$	0	0
$851$	−11.1591	−0.382527
$852$	0	0
$853$	29.6180	1.01410	0.507051	−	0.861916i	$-0.330736\pi$
$853$	29.6180	1.01410	0.507051	+	0.861916i	$0.330736\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.6525	0.671316	0.335658	−	0.941984i	$-0.391041\pi$
$857$	19.6525	0.671316	0.335658	+	0.941984i	$0.391041\pi$
$858$	0	0
$859$	−43.4853	−1.48370	−0.741850	−	0.670566i	$-0.766051\pi$
$859$	−43.4853	−1.48370	−0.741850	+	0.670566i	$0.766051\pi$
$860$	0	0
$861$	7.94427	0.270740
$862$	0	0
$863$	−9.94427	−0.338507	−0.169253	−	0.985573i	$-0.554136\pi$
$863$	−9.94427	−0.338507	−0.169253	+	0.985573i	$0.554136\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−10.6180	−0.360607
$868$	0	0
$869$	−2.85410	−0.0968188
$870$	0	0
$871$	0	0
$872$	0	0
$873$	1.87539	0.0634723
$874$	0	0
$875$	0	0
$876$	0	0
$877$	29.5279	0.997085	0.498543	−	0.866865i	$-0.333869\pi$
$877$	29.5279	0.997085	0.498543	+	0.866865i	$0.333869\pi$
$878$	0	0
$879$	32.1803	1.08542
$880$	0	0
$881$	−13.6738	−0.460681	−0.230340	−	0.973110i	$-0.573984\pi$
$881$	−13.6738	−0.460681	−0.230340	+	0.973110i	$0.573984\pi$
$882$	0	0
$883$	19.3607	0.651539	0.325769	−	0.945449i	$-0.394377\pi$
$883$	19.3607	0.651539	0.325769	+	0.945449i	$0.394377\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28.2361	0.948074	0.474037	−	0.880505i	$-0.342796\pi$
$887$	28.2361	0.948074	0.474037	+	0.880505i	$0.342796\pi$
$888$	0	0
$889$	−0.236068	−0.00791747
$890$	0	0
$891$	−7.70820	−0.258235
$892$	0	0
$893$	−25.7639	−0.862157
$894$	0	0
$895$	0	0
$896$	0	0
$897$	22.8885	0.764226
$898$	0	0
$899$	−18.5623	−0.619088
$900$	0	0
$901$	−18.7082	−0.623261
$902$	0	0
$903$	−8.47214	−0.281935
$904$	0	0
$905$	0	0
$906$	0	0
$907$	14.8328	0.492516	0.246258	−	0.969204i	$-0.420799\pi$
$907$	14.8328	0.492516	0.246258	+	0.969204i	$0.420799\pi$
$908$	0	0
$909$	2.88854	0.0958070
$910$	0	0
$911$	55.3050	1.83233	0.916167	−	0.400796i	$-0.131266\pi$
$911$	55.3050	1.83233	0.916167	+	0.400796i	$0.131266\pi$
$912$	0	0
$913$	−1.14590	−0.0379237
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9.32624	−0.307980
$918$	0	0
$919$	52.3607	1.72722	0.863610	−	0.504161i	$-0.168198\pi$
$919$	52.3607	1.72722	0.863610	+	0.504161i	$0.168198\pi$
$920$	0	0
$921$	7.85410	0.258801
$922$	0	0
$923$	−0.652476	−0.0214765
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.43769	0.211442
$928$	0	0
$929$	−23.1803	−0.760522	−0.380261	−	0.924879i	$-0.624166\pi$
$929$	−23.1803	−0.760522	−0.380261	+	0.924879i	$0.624166\pi$
$930$	0	0
$931$	−36.2148	−1.18689
$932$	0	0
$933$	17.6180	0.576789
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−35.0557	−1.14522	−0.572610	−	0.819828i	$-0.694069\pi$
$937$	−35.0557	−1.14522	−0.572610	+	0.819828i	$0.694069\pi$
$938$	0	0
$939$	44.8328	1.46306
$940$	0	0
$941$	−21.5836	−0.703605	−0.351802	−	0.936074i	$-0.614431\pi$
$941$	−21.5836	−0.703605	−0.351802	+	0.936074i	$0.614431\pi$
$942$	0	0
$943$	50.2574	1.63660
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−11.2918	−0.366934	−0.183467	−	0.983026i	$-0.558732\pi$
$947$	−11.2918	−0.366934	−0.183467	+	0.983026i	$0.558732\pi$
$948$	0	0
$949$	−15.8541	−0.514646
$950$	0	0
$951$	6.90983	0.224067
$952$	0	0
$953$	19.5279	0.632570	0.316285	−	0.948664i	$-0.397564\pi$
$953$	19.5279	0.632570	0.316285	+	0.948664i	$0.397564\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	7.09017	0.229193
$958$	0	0
$959$	3.94427	0.127367
$960$	0	0
$961$	−13.0557	−0.421153
$962$	0	0
$963$	4.56231	0.147018
$964$	0	0
$965$	0	0
$966$	0	0
$967$	24.1591	0.776903	0.388451	−	0.921469i	$-0.373010\pi$
$967$	24.1591	0.776903	0.388451	+	0.921469i	$0.373010\pi$
$968$	0	0
$969$	42.9787	1.38068
$970$	0	0
$971$	−3.85410	−0.123684	−0.0618420	−	0.998086i	$-0.519697\pi$
$971$	−3.85410	−0.123684	−0.0618420	+	0.998086i	$0.519697\pi$
$972$	0	0
$973$	−4.32624	−0.138693
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10.5967	0.339020	0.169510	−	0.985528i	$-0.445781\pi$
$977$	10.5967	0.339020	0.169510	+	0.985528i	$0.445781\pi$
$978$	0	0
$979$	−12.5623	−0.401493
$980$	0	0
$981$	−2.21478	−0.0707125
$982$	0	0
$983$	30.2492	0.964800	0.482400	−	0.875951i	$-0.339765\pi$
$983$	30.2492	0.964800	0.482400	+	0.875951i	$0.339765\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−4.70820	−0.149864
$988$	0	0
$989$	−53.5967	−1.70428
$990$	0	0
$991$	−48.5623	−1.54263	−0.771316	−	0.636452i	$-0.780401\pi$
$991$	−48.5623	−1.54263	−0.771316	+	0.636452i	$0.780401\pi$
$992$	0	0
$993$	45.9787	1.45909
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−7.97871	−0.252688	−0.126344	−	0.991986i	$-0.540324\pi$
$997$	−7.97871	−0.252688	−0.126344	+	0.991986i	$0.540324\pi$
$998$	0	0
$999$	−9.65248	−0.305391

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.bk.1.1		2
4.3	odd	2		2200.2.a.r.1.2	yes	2
5.2	odd	4		4400.2.b.ba.4049.4		4
5.3	odd	4		4400.2.b.ba.4049.1		4
5.4	even	2		4400.2.a.bq.1.2		2
20.3	even	4		2200.2.b.j.1849.4		4
20.7	even	4		2200.2.b.j.1849.1		4
20.19	odd	2		2200.2.a.n.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2200.2.a.n.1.1	✓	2	20.19	odd	2
2200.2.a.r.1.2	yes	2	4.3	odd	2
2200.2.b.j.1849.1		4	20.7	even	4
2200.2.b.j.1849.4		4	20.3	even	4
4400.2.a.bk.1.1		2	1.1	even	1	trivial
4400.2.a.bq.1.2		2	5.4	even	2
4400.2.b.ba.4049.1		4	5.3	odd	4
4400.2.b.ba.4049.4		4	5.2	odd	4

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:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{3} -0.618034 q^{7} -0.381966 q^{9} +1.00000 q^{11} -2.23607 q^{13} -4.85410 q^{17} +5.47214 q^{19} +1.00000 q^{21} +6.32624 q^{23} +5.47214 q^{27} -4.38197 q^{29} +4.23607 q^{31} -1.61803 q^{33} -1.76393 q^{37} +3.61803 q^{39} +7.94427 q^{41} -8.47214 q^{43} -4.70820 q^{47} -6.61803 q^{49} +7.85410 q^{51} +3.85410 q^{53} -8.85410 q^{57} +3.76393 q^{59} +7.09017 q^{61} +0.236068 q^{63} -10.2361 q^{69} +0.291796 q^{71} +7.09017 q^{73} -0.618034 q^{77} -2.85410 q^{79} -7.70820 q^{81} -1.14590 q^{83} +7.09017 q^{87} -12.5623 q^{89} +1.38197 q^{91} -6.85410 q^{93} -4.90983 q^{97} -0.381966 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + q^{7} - 3 q^{9} + 2 q^{11} - 3 q^{17} + 2 q^{19} + 2 q^{21} - 3 q^{23} + 2 q^{27} - 11 q^{29} + 4 q^{31} - q^{33} - 8 q^{37} + 5 q^{39} - 2 q^{41} - 8 q^{43} + 4 q^{47} - 11 q^{49} + 9 q^{51}+ \cdots - 3 q^{99}+O(q^{100}) \) 2 * q - q^3 + q^7 - 3 * q^9 + 2 * q^11 - 3 * q^17 + 2 * q^19 + 2 * q^21 - 3 * q^23 + 2 * q^27 - 11 * q^29 + 4 * q^31 - q^33 - 8 * q^37 + 5 * q^39 - 2 * q^41 - 8 * q^43 + 4 * q^47 - 11 * q^49 + 9 * q^51 + q^53 - 11 * q^57 + 12 * q^59 + 3 * q^61 - 4 * q^63 - 16 * q^69 + 14 * q^71 + 3 * q^73 + q^77 + q^79 - 2 * q^81 - 9 * q^83 + 3 * q^87 - 5 * q^89 + 5 * q^91 - 7 * q^93 - 21 * q^97 - 3 * q^99

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