LMFDB - Embedded newform 5600.2.a.bv.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5600,2,Mod(1,5600)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5600 = 2^{5} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5600.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$44.7162251319$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.504568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 2 $ x^5 - x^4 - 6x^3 + 3x^2 + 7*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.28447$ of defining polynomial
Character	$\chi$	$=$	5600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.63460 q^{3} +1.00000 q^{7} -0.328072 q^{9} +O(q^{10})$	$q+1.63460 q^{3} +1.00000 q^{7} -0.328072 q^{9} +1.24087 q^{11} +4.20355 q^{13} -3.39596 q^{17} +6.46162 q^{19} +1.63460 q^{21} -2.15509 q^{23} -5.44008 q^{27} +3.96490 q^{29} -10.0611 q^{31} +2.02833 q^{33} +6.76815 q^{37} +6.87113 q^{39} -0.131318 q^{41} +7.40498 q^{43} +4.82702 q^{47} +1.00000 q^{49} -5.55105 q^{51} +10.0374 q^{53} +10.5622 q^{57} +10.9864 q^{59} -6.33030 q^{61} -0.328072 q^{63} +2.65614 q^{67} -3.52271 q^{69} +0.754563 q^{71} +6.03736 q^{73} +1.24087 q^{77} +14.6652 q^{79} -7.90815 q^{81} -14.0813 q^{83} +6.48105 q^{87} -12.6742 q^{89} +4.20355 q^{91} -16.4459 q^{93} +0.914215 q^{97} -0.407096 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} + 5 q^{7} + 7 q^{9}+O(q^{10}) $ 5 * q - 2 * q^3 + 5 * q^7 + 7 * q^9	$ 5 q - 2 q^{3} + 5 q^{7} + 7 q^{9} + 4 q^{11} + 4 q^{17} + 12 q^{19} - 2 q^{21} + 8 q^{23} - 14 q^{27} - 12 q^{29} + 12 q^{31} - 8 q^{33} + 12 q^{37} + 32 q^{39} - 2 q^{41} - 8 q^{43} + 14 q^{47} + 5 q^{49} + 12 q^{51} + 8 q^{53} - 8 q^{57} + 16 q^{59} - 10 q^{61} + 7 q^{63} - 4 q^{67} + 4 q^{69} + 4 q^{71} - 12 q^{73} + 4 q^{77} + 32 q^{79} + q^{81} - 24 q^{83} + 2 q^{87} + 2 q^{89} - 20 q^{93} - 12 q^{97} + 40 q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 + 5 * q^7 + 7 * q^9 + 4 * q^11 + 4 * q^17 + 12 * q^19 - 2 * q^21 + 8 * q^23 - 14 * q^27 - 12 * q^29 + 12 * q^31 - 8 * q^33 + 12 * q^37 + 32 * q^39 - 2 * q^41 - 8 * q^43 + 14 * q^47 + 5 * q^49 + 12 * q^51 + 8 * q^53 - 8 * q^57 + 16 * q^59 - 10 * q^61 + 7 * q^63 - 4 * q^67 + 4 * q^69 + 4 * q^71 - 12 * q^73 + 4 * q^77 + 32 * q^79 + q^81 - 24 * q^83 + 2 * q^87 + 2 * q^89 - 20 * q^93 - 12 * q^97 + 40 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.63460	0.943739	0.471869	−	0.881668i	$-0.343579\pi$
$3$	1.63460	0.943739	0.471869	+	0.881668i	$0.343579\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−0.328072	−0.109357
$10$	0	0
$11$	1.24087	0.374137	0.187069	−	0.982347i	$-0.440101\pi$
$11$	1.24087	0.374137	0.187069	+	0.982347i	$0.440101\pi$
$12$	0	0
$13$	4.20355	1.16585	0.582927	−	0.812524i	$-0.301907\pi$
$13$	4.20355	1.16585	0.582927	+	0.812524i	$0.301907\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.39596	−0.823641	−0.411821	−	0.911265i	$-0.635107\pi$
$17$	−3.39596	−0.823641	−0.411821	+	0.911265i	$0.635107\pi$
$18$	0	0
$19$	6.46162	1.48240	0.741198	−	0.671286i	$-0.234258\pi$
$19$	6.46162	1.48240	0.741198	+	0.671286i	$0.234258\pi$
$20$	0	0
$21$	1.63460	0.356700
$22$	0	0
$23$	−2.15509	−0.449367	−0.224683	−	0.974432i	$-0.572135\pi$
$23$	−2.15509	−0.449367	−0.224683	+	0.974432i	$0.572135\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.44008	−1.04694
$28$	0	0
$29$	3.96490	0.736264	0.368132	−	0.929773i	$-0.379997\pi$
$29$	3.96490	0.736264	0.368132	+	0.929773i	$0.379997\pi$
$30$	0	0
$31$	−10.0611	−1.80703	−0.903516	−	0.428555i	$-0.859023\pi$
$31$	−10.0611	−1.80703	−0.903516	+	0.428555i	$0.859023\pi$
$32$	0	0
$33$	2.02833	0.353088
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.76815	1.11268	0.556338	−	0.830956i	$-0.312206\pi$
$37$	6.76815	1.11268	0.556338	+	0.830956i	$0.312206\pi$
$38$	0	0
$39$	6.87113	1.10026
$40$	0	0
$41$	−0.131318	−0.0205084	−0.0102542	−	0.999947i	$-0.503264\pi$
$41$	−0.131318	−0.0205084	−0.0102542	+	0.999947i	$0.503264\pi$
$42$	0	0
$43$	7.40498	1.12925	0.564625	−	0.825348i	$-0.309021\pi$
$43$	7.40498	1.12925	0.564625	+	0.825348i	$0.309021\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.82702	0.704093	0.352046	−	0.935983i	$-0.385486\pi$
$47$	4.82702	0.704093	0.352046	+	0.935983i	$0.385486\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.55105	−0.777302
$52$	0	0
$53$	10.0374	1.37874	0.689368	−	0.724411i	$-0.257888\pi$
$53$	10.0374	1.37874	0.689368	+	0.724411i	$0.257888\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	10.5622	1.39900
$58$	0	0
$59$	10.9864	1.43031	0.715157	−	0.698964i	$-0.246355\pi$
$59$	10.9864	1.43031	0.715157	+	0.698964i	$0.246355\pi$
$60$	0	0
$61$	−6.33030	−0.810512	−0.405256	−	0.914203i	$-0.632818\pi$
$61$	−6.33030	−0.810512	−0.405256	+	0.914203i	$0.632818\pi$
$62$	0	0
$63$	−0.328072	−0.0413332
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.65614	0.324500	0.162250	−	0.986750i	$-0.448125\pi$
$67$	2.65614	0.324500	0.162250	+	0.986750i	$0.448125\pi$
$68$	0	0
$69$	−3.52271	−0.424085
$70$	0	0
$71$	0.754563	0.0895501	0.0447751	−	0.998997i	$-0.485743\pi$
$71$	0.754563	0.0895501	0.0447751	+	0.998997i	$0.485743\pi$
$72$	0	0
$73$	6.03736	0.706619	0.353310	−	0.935506i	$-0.385056\pi$
$73$	6.03736	0.706619	0.353310	+	0.935506i	$0.385056\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.24087	0.141411
$78$	0	0
$79$	14.6652	1.64996	0.824980	−	0.565162i	$-0.191186\pi$
$79$	14.6652	1.64996	0.824980	+	0.565162i	$0.191186\pi$
$80$	0	0
$81$	−7.90815	−0.878684
$82$	0	0
$83$	−14.0813	−1.54562	−0.772809	−	0.634639i	$-0.781149\pi$
$83$	−14.0813	−1.54562	−0.772809	+	0.634639i	$0.781149\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.48105	0.694841
$88$	0	0
$89$	−12.6742	−1.34346	−0.671731	−	0.740795i	$-0.734449\pi$
$89$	−12.6742	−1.34346	−0.671731	+	0.740795i	$0.734449\pi$
$90$	0	0
$91$	4.20355	0.440652
$92$	0	0
$93$	−16.4459	−1.70537
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.914215	0.0928244	0.0464122	−	0.998922i	$-0.485221\pi$
$97$	0.914215	0.0928244	0.0464122	+	0.998922i	$0.485221\pi$
$98$	0	0
$99$	−0.407096	−0.0409146
$100$	0	0
$101$	13.4818	1.34149	0.670743	−	0.741689i	$-0.265975\pi$
$101$	13.4818	1.34149	0.670743	+	0.741689i	$0.265975\pi$
$102$	0	0
$103$	0.516841	0.0509258	0.0254629	−	0.999676i	$-0.491894\pi$
$103$	0.516841	0.0509258	0.0254629	+	0.999676i	$0.491894\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.8486	−1.82216	−0.911081	−	0.412228i	$-0.864751\pi$
$107$	−18.8486	−1.82216	−0.911081	+	0.412228i	$0.864751\pi$
$108$	0	0
$109$	0.345270	0.0330709	0.0165354	−	0.999863i	$-0.494736\pi$
$109$	0.345270	0.0330709	0.0165354	+	0.999863i	$0.494736\pi$
$110$	0	0
$111$	11.0632	1.05008
$112$	0	0
$113$	3.77026	0.354677	0.177338	−	0.984150i	$-0.443251\pi$
$113$	3.77026	0.354677	0.177338	+	0.984150i	$0.443251\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.37907	−0.127495
$118$	0	0
$119$	−3.39596	−0.311307
$120$	0	0
$121$	−9.46024	−0.860021
$122$	0	0
$123$	−0.214652	−0.0193545
$124$	0	0
$125$	0	0
$126$	0	0
$127$	5.63895	0.500376	0.250188	−	0.968197i	$-0.419508\pi$
$127$	5.63895	0.500376	0.250188	+	0.968197i	$0.419508\pi$
$128$	0	0
$129$	12.1042	1.06572
$130$	0	0
$131$	10.5183	0.918987	0.459493	−	0.888181i	$-0.348031\pi$
$131$	10.5183	0.918987	0.459493	+	0.888181i	$0.348031\pi$
$132$	0	0
$133$	6.46162	0.560293
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.96632	−0.595173	−0.297586	−	0.954695i	$-0.596182\pi$
$137$	−6.96632	−0.595173	−0.297586	+	0.954695i	$0.596182\pi$
$138$	0	0
$139$	10.3258	0.875827	0.437913	−	0.899017i	$-0.355718\pi$
$139$	10.3258	0.875827	0.437913	+	0.899017i	$0.355718\pi$
$140$	0	0
$141$	7.89025	0.664479
$142$	0	0
$143$	5.21607	0.436189
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.63460	0.134820
$148$	0	0
$149$	−0.773875	−0.0633983	−0.0316992	−	0.999497i	$-0.510092\pi$
$149$	−0.773875	−0.0633983	−0.0316992	+	0.999497i	$0.510092\pi$
$150$	0	0
$151$	6.14607	0.500160	0.250080	−	0.968225i	$-0.419543\pi$
$151$	6.14607	0.500160	0.250080	+	0.968225i	$0.419543\pi$
$152$	0	0
$153$	1.11412	0.0900712
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−18.5421	−1.47982	−0.739909	−	0.672707i	$-0.765131\pi$
$157$	−18.5421	−1.47982	−0.739909	+	0.672707i	$0.765131\pi$
$158$	0	0
$159$	16.4071	1.30117
$160$	0	0
$161$	−2.15509	−0.169845
$162$	0	0
$163$	−0.993429	−0.0778113	−0.0389057	−	0.999243i	$-0.512387\pi$
$163$	−0.993429	−0.0778113	−0.0389057	+	0.999243i	$0.512387\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.22052	−0.249212	−0.124606	−	0.992206i	$-0.539767\pi$
$167$	−3.22052	−0.249212	−0.124606	+	0.992206i	$0.539767\pi$
$168$	0	0
$169$	4.66981	0.359216
$170$	0	0
$171$	−2.11988	−0.162111
$172$	0	0
$173$	15.9187	1.21028	0.605138	−	0.796120i	$-0.293118\pi$
$173$	15.9187	1.21028	0.605138	+	0.796120i	$0.293118\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	17.9585	1.34984
$178$	0	0
$179$	−9.84647	−0.735960	−0.367980	−	0.929834i	$-0.619951\pi$
$179$	−9.84647	−0.735960	−0.367980	+	0.929834i	$0.619951\pi$
$180$	0	0
$181$	21.6562	1.60969	0.804845	−	0.593484i	$-0.202248\pi$
$181$	21.6562	1.60969	0.804845	+	0.593484i	$0.202248\pi$
$182$	0	0
$183$	−10.3475	−0.764911
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.21395	−0.308155
$188$	0	0
$189$	−5.44008	−0.395707
$190$	0	0
$191$	17.7197	1.28215	0.641077	−	0.767477i	$-0.278488\pi$
$191$	17.7197	1.28215	0.641077	+	0.767477i	$0.278488\pi$
$192$	0	0
$193$	12.0238	0.865490	0.432745	−	0.901516i	$-0.357545\pi$
$193$	12.0238	0.865490	0.432745	+	0.901516i	$0.357545\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.12069	−0.293587	−0.146794	−	0.989167i	$-0.546895\pi$
$197$	−4.12069	−0.293587	−0.146794	+	0.989167i	$0.546895\pi$
$198$	0	0
$199$	9.86868	0.699572	0.349786	−	0.936830i	$-0.386254\pi$
$199$	9.86868	0.699572	0.349786	+	0.936830i	$0.386254\pi$
$200$	0	0
$201$	4.34174	0.306243
$202$	0	0
$203$	3.96490	0.278282
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.707024	0.0491416
$208$	0	0
$209$	8.01804	0.554620
$210$	0	0
$211$	−6.93076	−0.477133	−0.238567	−	0.971126i	$-0.576678\pi$
$211$	−6.93076	−0.477133	−0.238567	+	0.971126i	$0.576678\pi$
$212$	0	0
$213$	1.23341	0.0845119
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−10.0611	−0.682994
$218$	0	0
$219$	9.86868	0.666864
$220$	0	0
$221$	−14.2751	−0.960246
$222$	0	0
$223$	23.0625	1.54438	0.772191	−	0.635391i	$-0.219161\pi$
$223$	23.0625	1.54438	0.772191	+	0.635391i	$0.219161\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.86073	0.189873	0.0949366	−	0.995483i	$-0.469735\pi$
$227$	2.86073	0.189873	0.0949366	+	0.995483i	$0.469735\pi$
$228$	0	0
$229$	18.5227	1.22402	0.612009	−	0.790851i	$-0.290362\pi$
$229$	18.5227	1.22402	0.612009	+	0.790851i	$0.290362\pi$
$230$	0	0
$231$	2.02833	0.133455
$232$	0	0
$233$	−23.1112	−1.51407	−0.757033	−	0.653376i	$-0.773352\pi$
$233$	−23.1112	−1.51407	−0.757033	+	0.653376i	$0.773352\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	23.9717	1.55713
$238$	0	0
$239$	−9.01270	−0.582983	−0.291491	−	0.956573i	$-0.594151\pi$
$239$	−9.01270	−0.582983	−0.291491	+	0.956573i	$0.594151\pi$
$240$	0	0
$241$	−12.4415	−0.801427	−0.400713	−	0.916203i	$-0.631238\pi$
$241$	−12.4415	−0.801427	−0.400713	+	0.916203i	$0.631238\pi$
$242$	0	0
$243$	3.39354	0.217696
$244$	0	0
$245$	0	0
$246$	0	0
$247$	27.1617	1.72826
$248$	0	0
$249$	−23.0173	−1.45866
$250$	0	0
$251$	21.3984	1.35066	0.675329	−	0.737517i	$-0.264002\pi$
$251$	21.3984	1.35066	0.675329	+	0.737517i	$0.264002\pi$
$252$	0	0
$253$	−2.67419	−0.168125
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−27.5737	−1.72000	−0.859999	−	0.510296i	$-0.829536\pi$
$257$	−27.5737	−1.72000	−0.859999	+	0.510296i	$0.829536\pi$
$258$	0	0
$259$	6.76815	0.420552
$260$	0	0
$261$	−1.30077	−0.0805159
$262$	0	0
$263$	6.77388	0.417695	0.208848	−	0.977948i	$-0.433029\pi$
$263$	6.77388	0.417695	0.208848	+	0.977948i	$0.433029\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−20.7173	−1.26788
$268$	0	0
$269$	−14.5794	−0.888922	−0.444461	−	0.895798i	$-0.646605\pi$
$269$	−14.5794	−0.888922	−0.444461	+	0.895798i	$0.646605\pi$
$270$	0	0
$271$	23.3842	1.42049	0.710244	−	0.703956i	$-0.248585\pi$
$271$	23.3842	1.42049	0.710244	+	0.703956i	$0.248585\pi$
$272$	0	0
$273$	6.87113	0.415860
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−8.57577	−0.515268	−0.257634	−	0.966243i	$-0.582943\pi$
$277$	−8.57577	−0.515268	−0.257634	+	0.966243i	$0.582943\pi$
$278$	0	0
$279$	3.30077	0.197612
$280$	0	0
$281$	−3.49675	−0.208598	−0.104299	−	0.994546i	$-0.533260\pi$
$281$	−3.49675	−0.208598	−0.104299	+	0.994546i	$0.533260\pi$
$282$	0	0
$283$	−18.2952	−1.08754	−0.543769	−	0.839235i	$-0.683003\pi$
$283$	−18.2952	−1.08754	−0.543769	+	0.839235i	$0.683003\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.131318	−0.00775143
$288$	0	0
$289$	−5.46746	−0.321615
$290$	0	0
$291$	1.49438	0.0876020
$292$	0	0
$293$	−30.4752	−1.78038	−0.890189	−	0.455592i	$-0.849428\pi$
$293$	−30.4752	−1.78038	−0.890189	+	0.455592i	$0.849428\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−6.75044	−0.391700
$298$	0	0
$299$	−9.05901	−0.523896
$300$	0	0
$301$	7.40498	0.426816
$302$	0	0
$303$	22.0374	1.26601
$304$	0	0
$305$	0	0
$306$	0	0
$307$	24.8725	1.41955	0.709773	−	0.704430i	$-0.248797\pi$
$307$	24.8725	1.41955	0.709773	+	0.704430i	$0.248797\pi$
$308$	0	0
$309$	0.844830	0.0480607
$310$	0	0
$311$	7.42579	0.421078	0.210539	−	0.977585i	$-0.432478\pi$
$311$	7.42579	0.421078	0.210539	+	0.977585i	$0.432478\pi$
$312$	0	0
$313$	−8.10594	−0.458175	−0.229088	−	0.973406i	$-0.573574\pi$
$313$	−8.10594	−0.458175	−0.229088	+	0.973406i	$0.573574\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.6010	0.820076	0.410038	−	0.912068i	$-0.365515\pi$
$317$	14.6010	0.820076	0.410038	+	0.912068i	$0.365515\pi$
$318$	0	0
$319$	4.91994	0.275464
$320$	0	0
$321$	−30.8100	−1.71964
$322$	0	0
$323$	−21.9434	−1.22096
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.564380	0.0312103
$328$	0	0
$329$	4.82702	0.266122
$330$	0	0
$331$	29.7374	1.63452	0.817258	−	0.576271i	$-0.195493\pi$
$331$	29.7374	1.63452	0.817258	+	0.576271i	$0.195493\pi$
$332$	0	0
$333$	−2.22044	−0.121679
$334$	0	0
$335$	0	0
$336$	0	0
$337$	7.88799	0.429686	0.214843	−	0.976649i	$-0.431076\pi$
$337$	7.88799	0.429686	0.214843	+	0.976649i	$0.431076\pi$
$338$	0	0
$339$	6.16289	0.334722
$340$	0	0
$341$	−12.4846	−0.676078
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.28725	0.176469	0.0882344	−	0.996100i	$-0.471878\pi$
$347$	3.28725	0.176469	0.0882344	+	0.996100i	$0.471878\pi$
$348$	0	0
$349$	6.68073	0.357611	0.178806	−	0.983884i	$-0.442777\pi$
$349$	6.68073	0.357611	0.178806	+	0.983884i	$0.442777\pi$
$350$	0	0
$351$	−22.8676	−1.22058
$352$	0	0
$353$	−26.3234	−1.40105	−0.700527	−	0.713626i	$-0.747052\pi$
$353$	−26.3234	−1.40105	−0.700527	+	0.713626i	$0.747052\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−5.55105	−0.293793
$358$	0	0
$359$	−8.27155	−0.436556	−0.218278	−	0.975887i	$-0.570044\pi$
$359$	−8.27155	−0.436556	−0.218278	+	0.975887i	$0.570044\pi$
$360$	0	0
$361$	22.7525	1.19750
$362$	0	0
$363$	−15.4637	−0.811635
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−33.7548	−1.76198	−0.880992	−	0.473130i	$-0.843124\pi$
$367$	−33.7548	−1.76198	−0.880992	+	0.473130i	$0.843124\pi$
$368$	0	0
$369$	0.0430816	0.00224274
$370$	0	0
$371$	10.0374	0.521114
$372$	0	0
$373$	−28.0582	−1.45280	−0.726400	−	0.687272i	$-0.758808\pi$
$373$	−28.0582	−1.45280	−0.726400	+	0.687272i	$0.758808\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	16.6667	0.858377
$378$	0	0
$379$	15.6585	0.804322	0.402161	−	0.915569i	$-0.368259\pi$
$379$	15.6585	0.804322	0.402161	+	0.915569i	$0.368259\pi$
$380$	0	0
$381$	9.21744	0.472224
$382$	0	0
$383$	18.6950	0.955270	0.477635	−	0.878558i	$-0.341494\pi$
$383$	18.6950	0.955270	0.477635	+	0.878558i	$0.341494\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.42937	−0.123492
$388$	0	0
$389$	33.7798	1.71270	0.856352	−	0.516392i	$-0.172725\pi$
$389$	33.7798	1.71270	0.856352	+	0.516392i	$0.172725\pi$
$390$	0	0
$391$	7.31859	0.370117
$392$	0	0
$393$	17.1932	0.867284
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−0.466183	−0.0233971	−0.0116985	−	0.999932i	$-0.503724\pi$
$397$	−0.466183	−0.0233971	−0.0116985	+	0.999932i	$0.503724\pi$
$398$	0	0
$399$	10.5622	0.528771
$400$	0	0
$401$	33.2221	1.65903	0.829516	−	0.558483i	$-0.188616\pi$
$401$	33.2221	1.65903	0.829516	+	0.558483i	$0.188616\pi$
$402$	0	0
$403$	−42.2924	−2.10674
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.39841	0.416294
$408$	0	0
$409$	−15.4588	−0.764390	−0.382195	−	0.924082i	$-0.624832\pi$
$409$	−15.4588	−0.764390	−0.382195	+	0.924082i	$0.624832\pi$
$410$	0	0
$411$	−11.3872	−0.561688
$412$	0	0
$413$	10.9864	0.540608
$414$	0	0
$415$	0	0
$416$	0	0
$417$	16.8787	0.826551
$418$	0	0
$419$	29.0268	1.41805	0.709025	−	0.705183i	$-0.249135\pi$
$419$	29.0268	1.41805	0.709025	+	0.705183i	$0.249135\pi$
$420$	0	0
$421$	−5.31745	−0.259156	−0.129578	−	0.991569i	$-0.541362\pi$
$421$	−5.31745	−0.259156	−0.129578	+	0.991569i	$0.541362\pi$
$422$	0	0
$423$	−1.58361	−0.0769977
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.33030	−0.306345
$428$	0	0
$429$	8.52620	0.411649
$430$	0	0
$431$	−17.6480	−0.850073	−0.425036	−	0.905176i	$-0.639739\pi$
$431$	−17.6480	−0.850073	−0.425036	+	0.905176i	$0.639739\pi$
$432$	0	0
$433$	−18.5011	−0.889107	−0.444554	−	0.895752i	$-0.646638\pi$
$433$	−18.5011	−0.889107	−0.444554	+	0.895752i	$0.646638\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−13.9254	−0.666140
$438$	0	0
$439$	14.7444	0.703711	0.351855	−	0.936054i	$-0.385551\pi$
$439$	14.7444	0.703711	0.351855	+	0.936054i	$0.385551\pi$
$440$	0	0
$441$	−0.328072	−0.0156225
$442$	0	0
$443$	−18.0223	−0.856264	−0.428132	−	0.903716i	$-0.640828\pi$
$443$	−18.0223	−0.856264	−0.428132	+	0.903716i	$0.640828\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1.26498	−0.0598314
$448$	0	0
$449$	−2.46746	−0.116447	−0.0582233	−	0.998304i	$-0.518544\pi$
$449$	−2.46746	−0.116447	−0.0582233	+	0.998304i	$0.518544\pi$
$450$	0	0
$451$	−0.162948	−0.00767294
$452$	0	0
$453$	10.0464	0.472020
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.73587	−0.0812004	−0.0406002	−	0.999175i	$-0.512927\pi$
$457$	−1.73587	−0.0812004	−0.0406002	+	0.999175i	$0.512927\pi$
$458$	0	0
$459$	18.4743	0.862306
$460$	0	0
$461$	9.00725	0.419510	0.209755	−	0.977754i	$-0.432733\pi$
$461$	9.00725	0.419510	0.209755	+	0.977754i	$0.432733\pi$
$462$	0	0
$463$	39.1293	1.81849	0.909246	−	0.416260i	$-0.136659\pi$
$463$	39.1293	1.81849	0.909246	+	0.416260i	$0.136659\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.47873	0.299800	0.149900	−	0.988701i	$-0.452105\pi$
$467$	6.47873	0.299800	0.149900	+	0.988701i	$0.452105\pi$
$468$	0	0
$469$	2.65614	0.122649
$470$	0	0
$471$	−30.3089	−1.39656
$472$	0	0
$473$	9.18864	0.422494
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−3.29298	−0.150775
$478$	0	0
$479$	−22.5651	−1.03103	−0.515514	−	0.856881i	$-0.672399\pi$
$479$	−22.5651	−1.03103	−0.515514	+	0.856881i	$0.672399\pi$
$480$	0	0
$481$	28.4502	1.29722
$482$	0	0
$483$	−3.52271	−0.160289
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−39.2500	−1.77859	−0.889293	−	0.457339i	$-0.848803\pi$
$487$	−39.2500	−1.77859	−0.889293	+	0.457339i	$0.848803\pi$
$488$	0	0
$489$	−1.62386	−0.0734336
$490$	0	0
$491$	−33.7793	−1.52444	−0.762220	−	0.647318i	$-0.775891\pi$
$491$	−33.7793	−1.52444	−0.762220	+	0.647318i	$0.775891\pi$
$492$	0	0
$493$	−13.4647	−0.606418
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.754563	0.0338468
$498$	0	0
$499$	11.6529	0.521654	0.260827	−	0.965386i	$-0.416005\pi$
$499$	11.6529	0.521654	0.260827	+	0.965386i	$0.416005\pi$
$500$	0	0
$501$	−5.26428	−0.235191
$502$	0	0
$503$	15.0639	0.671668	0.335834	−	0.941921i	$-0.390982\pi$
$503$	15.0639	0.671668	0.335834	+	0.941921i	$0.390982\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7.63329	0.339006
$508$	0	0
$509$	−19.8458	−0.879649	−0.439825	−	0.898084i	$-0.644959\pi$
$509$	−19.8458	−0.879649	−0.439825	+	0.898084i	$0.644959\pi$
$510$	0	0
$511$	6.03736	0.267077
$512$	0	0
$513$	−35.1517	−1.55199
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.98971	0.263427
$518$	0	0
$519$	26.0208	1.14218
$520$	0	0
$521$	−18.7964	−0.823484	−0.411742	−	0.911300i	$-0.635080\pi$
$521$	−18.7964	−0.823484	−0.411742	+	0.911300i	$0.635080\pi$
$522$	0	0
$523$	−31.0907	−1.35950	−0.679750	−	0.733444i	$-0.737912\pi$
$523$	−31.0907	−1.35950	−0.679750	+	0.733444i	$0.737912\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	34.1672	1.48835
$528$	0	0
$529$	−18.3556	−0.798070
$530$	0	0
$531$	−3.60435	−0.156415
$532$	0	0
$533$	−0.552000	−0.0239098
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−16.0951	−0.694554
$538$	0	0
$539$	1.24087	0.0534482
$540$	0	0
$541$	−35.8816	−1.54267	−0.771336	−	0.636428i	$-0.780411\pi$
$541$	−35.8816	−1.54267	−0.771336	+	0.636428i	$0.780411\pi$
$542$	0	0
$543$	35.3993	1.51913
$544$	0	0
$545$	0	0
$546$	0	0
$547$	25.4869	1.08974	0.544871	−	0.838520i	$-0.316579\pi$
$547$	25.4869	1.08974	0.544871	+	0.838520i	$0.316579\pi$
$548$	0	0
$549$	2.07679	0.0886354
$550$	0	0
$551$	25.6197	1.09144
$552$	0	0
$553$	14.6652	0.623626
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−23.5917	−0.999612	−0.499806	−	0.866137i	$-0.666595\pi$
$557$	−23.5917	−0.999612	−0.499806	+	0.866137i	$0.666595\pi$
$558$	0	0
$559$	31.1272	1.31654
$560$	0	0
$561$	−6.88814	−0.290818
$562$	0	0
$563$	−1.25815	−0.0530245	−0.0265123	−	0.999648i	$-0.508440\pi$
$563$	−1.25815	−0.0530245	−0.0265123	+	0.999648i	$0.508440\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−7.90815	−0.332111
$568$	0	0
$569$	23.4899	0.984747	0.492373	−	0.870384i	$-0.336129\pi$
$569$	23.4899	0.984747	0.492373	+	0.870384i	$0.336129\pi$
$570$	0	0
$571$	−18.0476	−0.755269	−0.377634	−	0.925955i	$-0.623262\pi$
$571$	−18.0476	−0.755269	−0.377634	+	0.925955i	$0.623262\pi$
$572$	0	0
$573$	28.9647	1.21002
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−45.4141	−1.89061	−0.945306	−	0.326184i	$-0.894237\pi$
$577$	−45.4141	−1.89061	−0.945306	+	0.326184i	$0.894237\pi$
$578$	0	0
$579$	19.6541	0.816796
$580$	0	0
$581$	−14.0813	−0.584189
$582$	0	0
$583$	12.4551	0.515837
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−38.1946	−1.57646	−0.788230	−	0.615381i	$-0.789002\pi$
$587$	−38.1946	−1.57646	−0.788230	+	0.615381i	$0.789002\pi$
$588$	0	0
$589$	−65.0112	−2.67874
$590$	0	0
$591$	−6.73569	−0.277069
$592$	0	0
$593$	11.0544	0.453952	0.226976	−	0.973900i	$-0.427116\pi$
$593$	11.0544	0.453952	0.226976	+	0.973900i	$0.427116\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.1314	0.660213
$598$	0	0
$599$	10.4206	0.425773	0.212887	−	0.977077i	$-0.431714\pi$
$599$	10.4206	0.425773	0.212887	+	0.977077i	$0.431714\pi$
$600$	0	0
$601$	−31.5047	−1.28510	−0.642552	−	0.766242i	$-0.722125\pi$
$601$	−31.5047	−1.28510	−0.642552	+	0.766242i	$0.722125\pi$
$602$	0	0
$603$	−0.871407	−0.0354864
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−34.9493	−1.41855	−0.709275	−	0.704932i	$-0.750977\pi$
$607$	−34.9493	−1.41855	−0.709275	+	0.704932i	$0.750977\pi$
$608$	0	0
$609$	6.48105	0.262625
$610$	0	0
$611$	20.2906	0.820869
$612$	0	0
$613$	37.9464	1.53264	0.766319	−	0.642460i	$-0.222086\pi$
$613$	37.9464	1.53264	0.766319	+	0.642460i	$0.222086\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27.0081	1.08731	0.543653	−	0.839310i	$-0.317041\pi$
$617$	27.0081	1.08731	0.543653	+	0.839310i	$0.317041\pi$
$618$	0	0
$619$	−29.0656	−1.16825	−0.584123	−	0.811665i	$-0.698561\pi$
$619$	−29.0656	−1.16825	−0.584123	+	0.811665i	$0.698561\pi$
$620$	0	0
$621$	11.7238	0.470462
$622$	0	0
$623$	−12.6742	−0.507781
$624$	0	0
$625$	0	0
$626$	0	0
$627$	13.1063	0.523416
$628$	0	0
$629$	−22.9844	−0.916447
$630$	0	0
$631$	−38.8146	−1.54518	−0.772592	−	0.634903i	$-0.781040\pi$
$631$	−38.8146	−1.54518	−0.772592	+	0.634903i	$0.781040\pi$
$632$	0	0
$633$	−11.3290	−0.450289
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.20355	0.166551
$638$	0	0
$639$	−0.247551	−0.00979297
$640$	0	0
$641$	40.2837	1.59111	0.795555	−	0.605881i	$-0.207179\pi$
$641$	40.2837	1.59111	0.795555	+	0.605881i	$0.207179\pi$
$642$	0	0
$643$	−2.02226	−0.0797500	−0.0398750	−	0.999205i	$-0.512696\pi$
$643$	−2.02226	−0.0797500	−0.0398750	+	0.999205i	$0.512696\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.08779	0.317964	0.158982	−	0.987281i	$-0.449179\pi$
$647$	8.08779	0.317964	0.158982	+	0.987281i	$0.449179\pi$
$648$	0	0
$649$	13.6328	0.535133
$650$	0	0
$651$	−16.4459	−0.644568
$652$	0	0
$653$	−25.0259	−0.979338	−0.489669	−	0.871908i	$-0.662882\pi$
$653$	−25.0259	−0.979338	−0.489669	+	0.871908i	$0.662882\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.98069	−0.0772740
$658$	0	0
$659$	−35.1054	−1.36751	−0.683756	−	0.729711i	$-0.739655\pi$
$659$	−35.1054	−1.36751	−0.683756	+	0.729711i	$0.739655\pi$
$660$	0	0
$661$	−7.69197	−0.299183	−0.149592	−	0.988748i	$-0.547796\pi$
$661$	−7.69197	−0.299183	−0.149592	+	0.988748i	$0.547796\pi$
$662$	0	0
$663$	−23.3341	−0.906221
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.54471	−0.330853
$668$	0	0
$669$	37.6981	1.45749
$670$	0	0
$671$	−7.85510	−0.303243
$672$	0	0
$673$	4.07019	0.156894	0.0784472	−	0.996918i	$-0.475004\pi$
$673$	4.07019	0.156894	0.0784472	+	0.996918i	$0.475004\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−32.6451	−1.25465	−0.627327	−	0.778756i	$-0.715851\pi$
$677$	−32.6451	−1.25465	−0.627327	+	0.778756i	$0.715851\pi$
$678$	0	0
$679$	0.914215	0.0350843
$680$	0	0
$681$	4.67616	0.179191
$682$	0	0
$683$	35.7284	1.36711	0.683554	−	0.729900i	$-0.260433\pi$
$683$	35.7284	1.36711	0.683554	+	0.729900i	$0.260433\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	30.2773	1.15515
$688$	0	0
$689$	42.1925	1.60741
$690$	0	0
$691$	27.9663	1.06389	0.531945	−	0.846779i	$-0.321461\pi$
$691$	27.9663	1.06389	0.531945	+	0.846779i	$0.321461\pi$
$692$	0	0
$693$	−0.407096	−0.0154643
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0.445949	0.0168915
$698$	0	0
$699$	−37.7777	−1.42888
$700$	0	0
$701$	−19.7996	−0.747822	−0.373911	−	0.927465i	$-0.621983\pi$
$701$	−19.7996	−0.747822	−0.373911	+	0.927465i	$0.621983\pi$
$702$	0	0
$703$	43.7332	1.64943
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.4818	0.507034
$708$	0	0
$709$	8.46892	0.318057	0.159029	−	0.987274i	$-0.449164\pi$
$709$	8.46892	0.318057	0.159029	+	0.987274i	$0.449164\pi$
$710$	0	0
$711$	−4.81123	−0.180435
$712$	0	0
$713$	21.6826	0.812020
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−14.7322	−0.550183
$718$	0	0
$719$	−11.2512	−0.419598	−0.209799	−	0.977745i	$-0.567281\pi$
$719$	−11.2512	−0.419598	−0.209799	+	0.977745i	$0.567281\pi$
$720$	0	0
$721$	0.516841	0.0192482
$722$	0	0
$723$	−20.3369	−0.756338
$724$	0	0
$725$	0	0
$726$	0	0
$727$	26.2716	0.974360	0.487180	−	0.873302i	$-0.338026\pi$
$727$	26.2716	0.974360	0.487180	+	0.873302i	$0.338026\pi$
$728$	0	0
$729$	29.2716	1.08413
$730$	0	0
$731$	−25.1470	−0.930096
$732$	0	0
$733$	−48.7804	−1.80175	−0.900873	−	0.434082i	$-0.857073\pi$
$733$	−48.7804	−1.80175	−0.900873	+	0.434082i	$0.857073\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.29594	0.121407
$738$	0	0
$739$	37.5871	1.38267	0.691333	−	0.722537i	$-0.257024\pi$
$739$	37.5871	1.38267	0.691333	+	0.722537i	$0.257024\pi$
$740$	0	0
$741$	44.3986	1.63102
$742$	0	0
$743$	29.2572	1.07334	0.536671	−	0.843792i	$-0.319682\pi$
$743$	29.2572	1.07334	0.536671	+	0.843792i	$0.319682\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.61967	0.169025
$748$	0	0
$749$	−18.8486	−0.688712
$750$	0	0
$751$	41.4125	1.51116	0.755582	−	0.655054i	$-0.227354\pi$
$751$	41.4125	1.51116	0.755582	+	0.655054i	$0.227354\pi$
$752$	0	0
$753$	34.9780	1.27467
$754$	0	0
$755$	0	0
$756$	0	0
$757$	33.1411	1.20453	0.602266	−	0.798295i	$-0.294264\pi$
$757$	33.1411	1.20453	0.602266	+	0.798295i	$0.294264\pi$
$758$	0	0
$759$	−4.37124	−0.158666
$760$	0	0
$761$	7.08166	0.256710	0.128355	−	0.991728i	$-0.459030\pi$
$761$	7.08166	0.256710	0.128355	+	0.991728i	$0.459030\pi$
$762$	0	0
$763$	0.345270	0.0124996
$764$	0	0
$765$	0	0
$766$	0	0
$767$	46.1820	1.66754
$768$	0	0
$769$	20.8844	0.753112	0.376556	−	0.926394i	$-0.377108\pi$
$769$	20.8844	0.753112	0.376556	+	0.926394i	$0.377108\pi$
$770$	0	0
$771$	−45.0720	−1.62323
$772$	0	0
$773$	−50.0409	−1.79985	−0.899923	−	0.436049i	$-0.856377\pi$
$773$	−50.0409	−1.79985	−0.899923	+	0.436049i	$0.856377\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	11.0632	0.396892
$778$	0	0
$779$	−0.848524	−0.0304015
$780$	0	0
$781$	0.936316	0.0335040
$782$	0	0
$783$	−21.5694	−0.770827
$784$	0	0
$785$	0	0
$786$	0	0
$787$	23.0831	0.822824	0.411412	−	0.911449i	$-0.365036\pi$
$787$	23.0831	0.822824	0.411412	+	0.911449i	$0.365036\pi$
$788$	0	0
$789$	11.0726	0.394195
$790$	0	0
$791$	3.77026	0.134055
$792$	0	0
$793$	−26.6097	−0.944939
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−0.553975	−0.0196228	−0.00981140	−	0.999952i	$-0.503123\pi$
$797$	−0.553975	−0.0196228	−0.00981140	+	0.999952i	$0.503123\pi$
$798$	0	0
$799$	−16.3923	−0.579920
$800$	0	0
$801$	4.15805	0.146917
$802$	0	0
$803$	7.49159	0.264372
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−23.8316	−0.838910
$808$	0	0
$809$	−42.8647	−1.50704	−0.753521	−	0.657424i	$-0.771646\pi$
$809$	−42.8647	−1.50704	−0.753521	+	0.657424i	$0.771646\pi$
$810$	0	0
$811$	3.28042	0.115191	0.0575955	−	0.998340i	$-0.481657\pi$
$811$	3.28042	0.115191	0.0575955	+	0.998340i	$0.481657\pi$
$812$	0	0
$813$	38.2238	1.34057
$814$	0	0
$815$	0	0
$816$	0	0
$817$	47.8482	1.67400
$818$	0	0
$819$	−1.37907	−0.0481885
$820$	0	0
$821$	−7.73361	−0.269905	−0.134952	−	0.990852i	$-0.543088\pi$
$821$	−7.73361	−0.269905	−0.134952	+	0.990852i	$0.543088\pi$
$822$	0	0
$823$	−22.9368	−0.799527	−0.399764	−	0.916618i	$-0.630908\pi$
$823$	−22.9368	−0.799527	−0.399764	+	0.916618i	$0.630908\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−48.4640	−1.68526	−0.842629	−	0.538494i	$-0.818993\pi$
$827$	−48.4640	−1.68526	−0.842629	+	0.538494i	$0.818993\pi$
$828$	0	0
$829$	−34.4482	−1.19643	−0.598217	−	0.801334i	$-0.704124\pi$
$829$	−34.4482	−1.19643	−0.598217	+	0.801334i	$0.704124\pi$
$830$	0	0
$831$	−14.0180	−0.486278
$832$	0	0
$833$	−3.39596	−0.117663
$834$	0	0
$835$	0	0
$836$	0	0
$837$	54.7333	1.89186
$838$	0	0
$839$	−9.81974	−0.339015	−0.169508	−	0.985529i	$-0.554218\pi$
$839$	−9.81974	−0.339015	−0.169508	+	0.985529i	$0.554218\pi$
$840$	0	0
$841$	−13.2795	−0.457915
$842$	0	0
$843$	−5.71579	−0.196862
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−9.46024	−0.325058
$848$	0	0
$849$	−29.9054	−1.02635
$850$	0	0
$851$	−14.5860	−0.500000
$852$	0	0
$853$	42.3139	1.44880	0.724400	−	0.689380i	$-0.242117\pi$
$853$	42.3139	1.44880	0.724400	+	0.689380i	$0.242117\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.54342	−0.0527224	−0.0263612	−	0.999652i	$-0.508392\pi$
$857$	−1.54342	−0.0527224	−0.0263612	+	0.999652i	$0.508392\pi$
$858$	0	0
$859$	4.41854	0.150759	0.0753793	−	0.997155i	$-0.475983\pi$
$859$	4.41854	0.150759	0.0753793	+	0.997155i	$0.475983\pi$
$860$	0	0
$861$	−0.214652	−0.00731533
$862$	0	0
$863$	4.91388	0.167270	0.0836352	−	0.996496i	$-0.473347\pi$
$863$	4.91388	0.167270	0.0836352	+	0.996496i	$0.473347\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−8.93713	−0.303521
$868$	0	0
$869$	18.1976	0.617311
$870$	0	0
$871$	11.1652	0.378319
$872$	0	0
$873$	−0.299928	−0.0101510
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−11.7021	−0.395153	−0.197577	−	0.980287i	$-0.563307\pi$
$877$	−11.7021	−0.395153	−0.197577	+	0.980287i	$0.563307\pi$
$878$	0	0
$879$	−49.8148	−1.68021
$880$	0	0
$881$	−29.8823	−1.00676	−0.503380	−	0.864065i	$-0.667910\pi$
$881$	−29.8823	−1.00676	−0.503380	+	0.864065i	$0.667910\pi$
$882$	0	0
$883$	−44.6829	−1.50370	−0.751850	−	0.659334i	$-0.770838\pi$
$883$	−44.6829	−1.50370	−0.751850	+	0.659334i	$0.770838\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−15.8523	−0.532269	−0.266135	−	0.963936i	$-0.585747\pi$
$887$	−15.8523	−0.532269	−0.266135	+	0.963936i	$0.585747\pi$
$888$	0	0
$889$	5.63895	0.189124
$890$	0	0
$891$	−9.81301	−0.328748
$892$	0	0
$893$	31.1903	1.04374
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−14.8079	−0.494421
$898$	0	0
$899$	−39.8914	−1.33045
$900$	0	0
$901$	−34.0865	−1.13558
$902$	0	0
$903$	12.1042	0.402803
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−26.5453	−0.881421	−0.440710	−	0.897649i	$-0.645273\pi$
$907$	−26.5453	−0.881421	−0.440710	+	0.897649i	$0.645273\pi$
$908$	0	0
$909$	−4.42299	−0.146701
$910$	0	0
$911$	0.855810	0.0283543	0.0141771	−	0.999899i	$-0.495487\pi$
$911$	0.855810	0.0283543	0.0141771	+	0.999899i	$0.495487\pi$
$912$	0	0
$913$	−17.4730	−0.578273
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.5183	0.347344
$918$	0	0
$919$	−13.6925	−0.451675	−0.225838	−	0.974165i	$-0.572512\pi$
$919$	−13.6925	−0.451675	−0.225838	+	0.974165i	$0.572512\pi$
$920$	0	0
$921$	40.6566	1.33968
$922$	0	0
$923$	3.17184	0.104402
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−0.169561	−0.00556911
$928$	0	0
$929$	25.2648	0.828912	0.414456	−	0.910069i	$-0.363972\pi$
$929$	25.2648	0.828912	0.414456	+	0.910069i	$0.363972\pi$
$930$	0	0
$931$	6.46162	0.211771
$932$	0	0
$933$	12.1382	0.397388
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−28.8534	−0.942599	−0.471299	−	0.881973i	$-0.656215\pi$
$937$	−28.8534	−0.942599	−0.471299	+	0.881973i	$0.656215\pi$
$938$	0	0
$939$	−13.2500	−0.432397
$940$	0	0
$941$	4.28998	0.139850	0.0699248	−	0.997552i	$-0.477724\pi$
$941$	4.28998	0.139850	0.0699248	+	0.997552i	$0.477724\pi$
$942$	0	0
$943$	0.283001	0.00921578
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−48.2836	−1.56901	−0.784503	−	0.620126i	$-0.787082\pi$
$947$	−48.2836	−1.56901	−0.784503	+	0.620126i	$0.787082\pi$
$948$	0	0
$949$	25.3783	0.823815
$950$	0	0
$951$	23.8669	0.773937
$952$	0	0
$953$	34.2147	1.10832	0.554162	−	0.832409i	$-0.313039\pi$
$953$	34.2147	1.10832	0.554162	+	0.832409i	$0.313039\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	8.04215	0.259966
$958$	0	0
$959$	−6.96632	−0.224954
$960$	0	0
$961$	70.2263	2.26536
$962$	0	0
$963$	6.18369	0.199267
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−49.1870	−1.58175	−0.790873	−	0.611980i	$-0.790373\pi$
$967$	−49.1870	−1.58175	−0.790873	+	0.611980i	$0.790373\pi$
$968$	0	0
$969$	−35.8687	−1.15227
$970$	0	0
$971$	25.2107	0.809051	0.404526	−	0.914527i	$-0.367437\pi$
$971$	25.2107	0.809051	0.404526	+	0.914527i	$0.367437\pi$
$972$	0	0
$973$	10.3258	0.331031
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−18.0389	−0.577116	−0.288558	−	0.957462i	$-0.593176\pi$
$977$	−18.0389	−0.577116	−0.288558	+	0.957462i	$0.593176\pi$
$978$	0	0
$979$	−15.7271	−0.502639
$980$	0	0
$981$	−0.113274	−0.00361654
$982$	0	0
$983$	−51.6099	−1.64610	−0.823051	−	0.567968i	$-0.807730\pi$
$983$	−51.6099	−1.64610	−0.823051	+	0.567968i	$0.807730\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	7.89025	0.251150
$988$	0	0
$989$	−15.9584	−0.507447
$990$	0	0
$991$	−15.4063	−0.489398	−0.244699	−	0.969599i	$-0.578689\pi$
$991$	−15.4063	−0.489398	−0.244699	+	0.969599i	$0.578689\pi$
$992$	0	0
$993$	48.6089	1.54256
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−13.6176	−0.431274	−0.215637	−	0.976474i	$-0.569183\pi$
$997$	−13.6176	−0.431274	−0.215637	+	0.976474i	$0.569183\pi$
$998$	0	0
$999$	−36.8193	−1.16491

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5600.2.a.bv.1.4		5
4.3	odd	2		5600.2.a.bw.1.2		5
5.2	odd	4		1120.2.g.c.449.4	yes	10
5.3	odd	4		1120.2.g.c.449.7	yes	10
5.4	even	2		5600.2.a.bx.1.2		5
20.3	even	4		1120.2.g.b.449.4	✓	10
20.7	even	4		1120.2.g.b.449.7	yes	10
20.19	odd	2		5600.2.a.bu.1.4		5
40.3	even	4		2240.2.g.o.449.7		10
40.13	odd	4		2240.2.g.n.449.4		10
40.27	even	4		2240.2.g.o.449.4		10
40.37	odd	4		2240.2.g.n.449.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.g.b.449.4	✓	10	20.3	even	4
1120.2.g.b.449.7	yes	10	20.7	even	4
1120.2.g.c.449.4	yes	10	5.2	odd	4
1120.2.g.c.449.7	yes	10	5.3	odd	4
2240.2.g.n.449.4		10	40.13	odd	4
2240.2.g.n.449.7		10	40.37	odd	4
2240.2.g.o.449.4		10	40.27	even	4
2240.2.g.o.449.7		10	40.3	even	4
5600.2.a.bu.1.4		5	20.19	odd	2
5600.2.a.bv.1.4		5	1.1	even	1	trivial
5600.2.a.bw.1.2		5	4.3	odd	2
5600.2.a.bx.1.2		5	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5600 = 2^{5} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5600.a (trivial)

\(f(q)\)	\(=\)	\(q+1.63460 q^{3} +1.00000 q^{7} -0.328072 q^{9} +O(q^{10})\)	\(q+1.63460 q^{3} +1.00000 q^{7} -0.328072 q^{9} +1.24087 q^{11} +4.20355 q^{13} -3.39596 q^{17} +6.46162 q^{19} +1.63460 q^{21} -2.15509 q^{23} -5.44008 q^{27} +3.96490 q^{29} -10.0611 q^{31} +2.02833 q^{33} +6.76815 q^{37} +6.87113 q^{39} -0.131318 q^{41} +7.40498 q^{43} +4.82702 q^{47} +1.00000 q^{49} -5.55105 q^{51} +10.0374 q^{53} +10.5622 q^{57} +10.9864 q^{59} -6.33030 q^{61} -0.328072 q^{63} +2.65614 q^{67} -3.52271 q^{69} +0.754563 q^{71} +6.03736 q^{73} +1.24087 q^{77} +14.6652 q^{79} -7.90815 q^{81} -14.0813 q^{83} +6.48105 q^{87} -12.6742 q^{89} +4.20355 q^{91} -16.4459 q^{93} +0.914215 q^{97} -0.407096 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} + 5 q^{7} + 7 q^{9}+O(q^{10}) \) 5 * q - 2 * q^3 + 5 * q^7 + 7 * q^9	\( 5 q - 2 q^{3} + 5 q^{7} + 7 q^{9} + 4 q^{11} + 4 q^{17} + 12 q^{19} - 2 q^{21} + 8 q^{23} - 14 q^{27} - 12 q^{29} + 12 q^{31} - 8 q^{33} + 12 q^{37} + 32 q^{39} - 2 q^{41} - 8 q^{43} + 14 q^{47} + 5 q^{49} + 12 q^{51} + 8 q^{53} - 8 q^{57} + 16 q^{59} - 10 q^{61} + 7 q^{63} - 4 q^{67} + 4 q^{69} + 4 q^{71} - 12 q^{73} + 4 q^{77} + 32 q^{79} + q^{81} - 24 q^{83} + 2 q^{87} + 2 q^{89} - 20 q^{93} - 12 q^{97} + 40 q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 + 5 * q^7 + 7 * q^9 + 4 * q^11 + 4 * q^17 + 12 * q^19 - 2 * q^21 + 8 * q^23 - 14 * q^27 - 12 * q^29 + 12 * q^31 - 8 * q^33 + 12 * q^37 + 32 * q^39 - 2 * q^41 - 8 * q^43 + 14 * q^47 + 5 * q^49 + 12 * q^51 + 8 * q^53 - 8 * q^57 + 16 * q^59 - 10 * q^61 + 7 * q^63 - 4 * q^67 + 4 * q^69 + 4 * q^71 - 12 * q^73 + 4 * q^77 + 32 * q^79 + q^81 - 24 * q^83 + 2 * q^87 + 2 * q^89 - 20 * q^93 - 12 * q^97 + 40 * q^99

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