LMFDB - Embedded newform 6400.2.a.bm.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6400,2,Mod(1,6400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6400 = 2^{8} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6400.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:	$51.1042572936$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 640)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	6400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{3} +1.41421 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.41421 q^{3} +1.41421 q^{7} -1.00000 q^{9} +2.82843 q^{11} -2.00000 q^{13} -2.00000 q^{17} -5.65685 q^{19} +2.00000 q^{21} +1.41421 q^{23} -5.65685 q^{27} +8.00000 q^{29} -8.48528 q^{31} +4.00000 q^{33} -10.0000 q^{37} -2.82843 q^{39} -7.07107 q^{43} -9.89949 q^{47} -5.00000 q^{49} -2.82843 q^{51} -2.00000 q^{53} -8.00000 q^{57} -5.65685 q^{59} +10.0000 q^{61} -1.41421 q^{63} +1.41421 q^{67} +2.00000 q^{69} +14.1421 q^{71} +6.00000 q^{73} +4.00000 q^{77} +11.3137 q^{79} -5.00000 q^{81} -12.7279 q^{83} +11.3137 q^{87} -10.0000 q^{89} -2.82843 q^{91} -12.0000 q^{93} -14.0000 q^{97} -2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} - 4 q^{13} - 4 q^{17} + 4 q^{21} + 16 q^{29} + 8 q^{33} - 20 q^{37} - 10 q^{49} - 4 q^{53} - 16 q^{57} + 20 q^{61} + 4 q^{69} + 12 q^{73} + 8 q^{77} - 10 q^{81} - 20 q^{89} - 24 q^{93} - 28 q^{97}+O(q^{100}) $ 2 * q - 2 * q^9 - 4 * q^13 - 4 * q^17 + 4 * q^21 + 16 * q^29 + 8 * q^33 - 20 * q^37 - 10 * q^49 - 4 * q^53 - 16 * q^57 + 20 * q^61 + 4 * q^69 + 12 * q^73 + 8 * q^77 - 10 * q^81 - 20 * q^89 - 24 * q^93 - 28 * q^97

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
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.41421	0.534522	0.267261	−	0.963624i	$-0.413881\pi$
$7$	1.41421	0.534522	0.267261	+	0.963624i	$0.413881\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.82843	0.852803	0.426401	−	0.904534i	$-0.359781\pi$
$11$	2.82843	0.852803	0.426401	+	0.904534i	$0.359781\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−5.65685	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	1.41421	0.294884	0.147442	−	0.989071i	$-0.452896\pi$
$23$	1.41421	0.294884	0.147442	+	0.989071i	$0.452896\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	−8.48528	−1.52400	−0.762001	−	0.647576i	$-0.775783\pi$
$31$	−8.48528	−1.52400	−0.762001	+	0.647576i	$0.775783\pi$
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	−2.82843	−0.452911
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−7.07107	−1.07833	−0.539164	−	0.842201i	$-0.681260\pi$
$43$	−7.07107	−1.07833	−0.539164	+	0.842201i	$0.681260\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.89949	−1.44399	−0.721995	−	0.691898i	$-0.756775\pi$
$47$	−9.89949	−1.44399	−0.721995	+	0.691898i	$0.756775\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	−2.82843	−0.396059
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−8.00000	−1.05963
$58$	0	0
$59$	−5.65685	−0.736460	−0.368230	−	0.929735i	$-0.620036\pi$
$59$	−5.65685	−0.736460	−0.368230	+	0.929735i	$0.620036\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	−1.41421	−0.178174
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.41421	0.172774	0.0863868	−	0.996262i	$-0.472468\pi$
$67$	1.41421	0.172774	0.0863868	+	0.996262i	$0.472468\pi$
$68$	0	0
$69$	2.00000	0.240772
$70$	0	0
$71$	14.1421	1.67836	0.839181	−	0.543852i	$-0.183035\pi$
$71$	14.1421	1.67836	0.839181	+	0.543852i	$0.183035\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.00000	0.455842
$78$	0	0
$79$	11.3137	1.27289	0.636446	−	0.771321i	$-0.280404\pi$
$79$	11.3137	1.27289	0.636446	+	0.771321i	$0.280404\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−12.7279	−1.39707	−0.698535	−	0.715575i	$-0.746165\pi$
$83$	−12.7279	−1.39707	−0.698535	+	0.715575i	$0.746165\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	11.3137	1.21296
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−2.82843	−0.296500
$92$	0	0
$93$	−12.0000	−1.24434
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	−2.82843	−0.284268
$100$	0	0
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	0	0
$103$	−4.24264	−0.418040	−0.209020	−	0.977911i	$-0.567027\pi$
$103$	−4.24264	−0.418040	−0.209020	+	0.977911i	$0.567027\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.5563	1.50389	0.751945	−	0.659226i	$-0.229116\pi$
$107$	15.5563	1.50389	0.751945	+	0.659226i	$0.229116\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−14.1421	−1.34231
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−2.82843	−0.259281
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−12.7279	−1.12942	−0.564710	−	0.825289i	$-0.691012\pi$
$127$	−12.7279	−1.12942	−0.564710	+	0.825289i	$0.691012\pi$
$128$	0	0
$129$	−10.0000	−0.880451
$130$	0	0
$131$	19.7990	1.72985	0.864923	−	0.501905i	$-0.167367\pi$
$131$	19.7990	1.72985	0.864923	+	0.501905i	$0.167367\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	−11.3137	−0.959616	−0.479808	−	0.877373i	$-0.659294\pi$
$139$	−11.3137	−0.959616	−0.479808	+	0.877373i	$0.659294\pi$
$140$	0	0
$141$	−14.0000	−1.17901
$142$	0	0
$143$	−5.65685	−0.473050
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−7.07107	−0.583212
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	19.7990	1.61122	0.805609	−	0.592447i	$-0.201838\pi$
$151$	19.7990	1.61122	0.805609	+	0.592447i	$0.201838\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	0	0
$159$	−2.82843	−0.224309
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	−4.24264	−0.332309	−0.166155	−	0.986100i	$-0.553135\pi$
$163$	−4.24264	−0.332309	−0.166155	+	0.986100i	$0.553135\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.5563	1.20379	0.601893	−	0.798577i	$-0.294413\pi$
$167$	15.5563	1.20379	0.601893	+	0.798577i	$0.294413\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	5.65685	0.432590
$172$	0	0
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	11.3137	0.845626	0.422813	−	0.906217i	$-0.361043\pi$
$179$	11.3137	0.845626	0.422813	+	0.906217i	$0.361043\pi$
$180$	0	0
$181$	−4.00000	−0.297318	−0.148659	−	0.988889i	$-0.547496\pi$
$181$	−4.00000	−0.297318	−0.148659	+	0.988889i	$0.547496\pi$
$182$	0	0
$183$	14.1421	1.04542
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5.65685	−0.413670
$188$	0	0
$189$	−8.00000	−0.581914
$190$	0	0
$191$	−8.48528	−0.613973	−0.306987	−	0.951714i	$-0.599321\pi$
$191$	−8.48528	−0.613973	−0.306987	+	0.951714i	$0.599321\pi$
$192$	0	0
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−16.9706	−1.20301	−0.601506	−	0.798869i	$-0.705432\pi$
$199$	−16.9706	−1.20301	−0.601506	+	0.798869i	$0.705432\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	0	0
$203$	11.3137	0.794067
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.41421	−0.0982946
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	−8.48528	−0.584151	−0.292075	−	0.956395i	$-0.594346\pi$
$211$	−8.48528	−0.584151	−0.292075	+	0.956395i	$0.594346\pi$
$212$	0	0
$213$	20.0000	1.37038
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−12.0000	−0.814613
$218$	0	0
$219$	8.48528	0.573382
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	7.07107	0.473514	0.236757	−	0.971569i	$-0.423916\pi$
$223$	7.07107	0.473514	0.236757	+	0.971569i	$0.423916\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.41421	−0.0938647	−0.0469323	−	0.998898i	$-0.514945\pi$
$227$	−1.41421	−0.0938647	−0.0469323	+	0.998898i	$0.514945\pi$
$228$	0	0
$229$	16.0000	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$230$	0	0
$231$	5.65685	0.372194
$232$	0	0
$233$	2.00000	0.131024	0.0655122	−	0.997852i	$-0.479132\pi$
$233$	2.00000	0.131024	0.0655122	+	0.997852i	$0.479132\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	16.0000	1.03931
$238$	0	0
$239$	−5.65685	−0.365911	−0.182956	−	0.983121i	$-0.558567\pi$
$239$	−5.65685	−0.365911	−0.182956	+	0.983121i	$0.558567\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	0	0
$243$	9.89949	0.635053
$244$	0	0
$245$	0	0
$246$	0	0
$247$	11.3137	0.719874
$248$	0	0
$249$	−18.0000	−1.14070
$250$	0	0
$251$	−14.1421	−0.892644	−0.446322	−	0.894873i	$-0.647266\pi$
$251$	−14.1421	−0.892644	−0.446322	+	0.894873i	$0.647266\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	0	0
$259$	−14.1421	−0.878750
$260$	0	0
$261$	−8.00000	−0.495188
$262$	0	0
$263$	−1.41421	−0.0872041	−0.0436021	−	0.999049i	$-0.513883\pi$
$263$	−1.41421	−0.0872041	−0.0436021	+	0.999049i	$0.513883\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−14.1421	−0.865485
$268$	0	0
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	−8.48528	−0.515444	−0.257722	−	0.966219i	$-0.582972\pi$
$271$	−8.48528	−0.515444	−0.257722	+	0.966219i	$0.582972\pi$
$272$	0	0
$273$	−4.00000	−0.242091
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	0	0
$279$	8.48528	0.508001
$280$	0	0
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	0	0
$283$	−15.5563	−0.924729	−0.462364	−	0.886690i	$-0.652999\pi$
$283$	−15.5563	−0.924729	−0.462364	+	0.886690i	$0.652999\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−19.7990	−1.16064
$292$	0	0
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−16.0000	−0.928414
$298$	0	0
$299$	−2.82843	−0.163572
$300$	0	0
$301$	−10.0000	−0.576390
$302$	0	0
$303$	5.65685	0.324978
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.7279	0.726421	0.363210	−	0.931707i	$-0.381681\pi$
$307$	12.7279	0.726421	0.363210	+	0.931707i	$0.381681\pi$
$308$	0	0
$309$	−6.00000	−0.341328
$310$	0	0
$311$	−2.82843	−0.160385	−0.0801927	−	0.996779i	$-0.525554\pi$
$311$	−2.82843	−0.160385	−0.0801927	+	0.996779i	$0.525554\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	22.6274	1.26689
$320$	0	0
$321$	22.0000	1.22792
$322$	0	0
$323$	11.3137	0.629512
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.82843	−0.156412
$328$	0	0
$329$	−14.0000	−0.771845
$330$	0	0
$331$	25.4558	1.39918	0.699590	−	0.714545i	$-0.253366\pi$
$331$	25.4558	1.39918	0.699590	+	0.714545i	$0.253366\pi$
$332$	0	0
$333$	10.0000	0.547997
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	−8.48528	−0.460857
$340$	0	0
$341$	−24.0000	−1.29967
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.24264	−0.227757	−0.113878	−	0.993495i	$-0.536327\pi$
$347$	−4.24264	−0.227757	−0.113878	+	0.993495i	$0.536327\pi$
$348$	0	0
$349$	24.0000	1.28469	0.642345	−	0.766415i	$-0.277962\pi$
$349$	24.0000	1.28469	0.642345	+	0.766415i	$0.277962\pi$
$350$	0	0
$351$	11.3137	0.603881
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	5.65685	0.298557	0.149279	−	0.988795i	$-0.452305\pi$
$359$	5.65685	0.298557	0.149279	+	0.988795i	$0.452305\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	−4.24264	−0.222681
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.24264	0.221464	0.110732	−	0.993850i	$-0.464680\pi$
$367$	4.24264	0.221464	0.110732	+	0.993850i	$0.464680\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.82843	−0.146845
$372$	0	0
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−16.0000	−0.824042
$378$	0	0
$379$	33.9411	1.74344	0.871719	−	0.490006i	$-0.163005\pi$
$379$	33.9411	1.74344	0.871719	+	0.490006i	$0.163005\pi$
$380$	0	0
$381$	−18.0000	−0.922168
$382$	0	0
$383$	−26.8701	−1.37300	−0.686498	−	0.727132i	$-0.740853\pi$
$383$	−26.8701	−1.37300	−0.686498	+	0.727132i	$0.740853\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.07107	0.359443
$388$	0	0
$389$	−22.0000	−1.11544	−0.557722	−	0.830028i	$-0.688325\pi$
$389$	−22.0000	−1.11544	−0.557722	+	0.830028i	$0.688325\pi$
$390$	0	0
$391$	−2.82843	−0.143040
$392$	0	0
$393$	28.0000	1.41241
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	−11.3137	−0.566394
$400$	0	0
$401$	−38.0000	−1.89763	−0.948815	−	0.315833i	$-0.897716\pi$
$401$	−38.0000	−1.89763	−0.948815	+	0.315833i	$0.897716\pi$
$402$	0	0
$403$	16.9706	0.845364
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−28.2843	−1.40200
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	−19.7990	−0.976612
$412$	0	0
$413$	−8.00000	−0.393654
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−16.0000	−0.783523
$418$	0	0
$419$	16.9706	0.829066	0.414533	−	0.910034i	$-0.363945\pi$
$419$	16.9706	0.829066	0.414533	+	0.910034i	$0.363945\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	0	0
$423$	9.89949	0.481330
$424$	0	0
$425$	0	0
$426$	0	0
$427$	14.1421	0.684386
$428$	0	0
$429$	−8.00000	−0.386244
$430$	0	0
$431$	14.1421	0.681203	0.340601	−	0.940208i	$-0.389369\pi$
$431$	14.1421	0.681203	0.340601	+	0.940208i	$0.389369\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.00000	−0.382692
$438$	0	0
$439$	−5.65685	−0.269987	−0.134993	−	0.990846i	$-0.543101\pi$
$439$	−5.65685	−0.269987	−0.134993	+	0.990846i	$0.543101\pi$
$440$	0	0
$441$	5.00000	0.238095
$442$	0	0
$443$	24.0416	1.14225	0.571126	−	0.820862i	$-0.306507\pi$
$443$	24.0416	1.14225	0.571126	+	0.820862i	$0.306507\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−8.48528	−0.401340
$448$	0	0
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	28.0000	1.31555
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	0	0
$459$	11.3137	0.528079
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	26.8701	1.24876	0.624379	−	0.781122i	$-0.285352\pi$
$463$	26.8701	1.24876	0.624379	+	0.781122i	$0.285352\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.3848	−0.850746	−0.425373	−	0.905018i	$-0.639857\pi$
$467$	−18.3848	−0.850746	−0.425373	+	0.905018i	$0.639857\pi$
$468$	0	0
$469$	2.00000	0.0923514
$470$	0	0
$471$	−25.4558	−1.17294
$472$	0	0
$473$	−20.0000	−0.919601
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	−22.6274	−1.03387	−0.516937	−	0.856024i	$-0.672928\pi$
$479$	−22.6274	−1.03387	−0.516937	+	0.856024i	$0.672928\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	0	0
$483$	2.82843	0.128698
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−26.8701	−1.21760	−0.608799	−	0.793324i	$-0.708349\pi$
$487$	−26.8701	−1.21760	−0.608799	+	0.793324i	$0.708349\pi$
$488$	0	0
$489$	−6.00000	−0.271329
$490$	0	0
$491$	8.48528	0.382935	0.191468	−	0.981499i	$-0.438675\pi$
$491$	8.48528	0.382935	0.191468	+	0.981499i	$0.438675\pi$
$492$	0	0
$493$	−16.0000	−0.720604
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.0000	0.897123
$498$	0	0
$499$	33.9411	1.51941	0.759707	−	0.650266i	$-0.225342\pi$
$499$	33.9411	1.51941	0.759707	+	0.650266i	$0.225342\pi$
$500$	0	0
$501$	22.0000	0.982888
$502$	0	0
$503$	−4.24264	−0.189170	−0.0945850	−	0.995517i	$-0.530152\pi$
$503$	−4.24264	−0.189170	−0.0945850	+	0.995517i	$0.530152\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−12.7279	−0.565267
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	8.48528	0.375367
$512$	0	0
$513$	32.0000	1.41283
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−28.0000	−1.23144
$518$	0	0
$519$	31.1127	1.36570
$520$	0	0
$521$	−2.00000	−0.0876216	−0.0438108	−	0.999040i	$-0.513950\pi$
$521$	−2.00000	−0.0876216	−0.0438108	+	0.999040i	$0.513950\pi$
$522$	0	0
$523$	29.6985	1.29862	0.649312	−	0.760522i	$-0.275057\pi$
$523$	29.6985	1.29862	0.649312	+	0.760522i	$0.275057\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16.9706	0.739249
$528$	0	0
$529$	−21.0000	−0.913043
$530$	0	0
$531$	5.65685	0.245487
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	16.0000	0.690451
$538$	0	0
$539$	−14.1421	−0.609145
$540$	0	0
$541$	44.0000	1.89171	0.945854	−	0.324593i	$-0.105227\pi$
$541$	44.0000	1.89171	0.945854	+	0.324593i	$0.105227\pi$
$542$	0	0
$543$	−5.65685	−0.242759
$544$	0	0
$545$	0	0
$546$	0	0
$547$	21.2132	0.907011	0.453506	−	0.891253i	$-0.350173\pi$
$547$	21.2132	0.907011	0.453506	+	0.891253i	$0.350173\pi$
$548$	0	0
$549$	−10.0000	−0.426790
$550$	0	0
$551$	−45.2548	−1.92792
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.0000	0.423714	0.211857	−	0.977301i	$-0.432049\pi$
$557$	10.0000	0.423714	0.211857	+	0.977301i	$0.432049\pi$
$558$	0	0
$559$	14.1421	0.598149
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	1.41421	0.0596020	0.0298010	−	0.999556i	$-0.490513\pi$
$563$	1.41421	0.0596020	0.0298010	+	0.999556i	$0.490513\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−7.07107	−0.296957
$568$	0	0
$569$	−12.0000	−0.503066	−0.251533	−	0.967849i	$-0.580935\pi$
$569$	−12.0000	−0.503066	−0.251533	+	0.967849i	$0.580935\pi$
$570$	0	0
$571$	−25.4558	−1.06529	−0.532647	−	0.846338i	$-0.678803\pi$
$571$	−25.4558	−1.06529	−0.532647	+	0.846338i	$0.678803\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	0	0
$579$	31.1127	1.29300
$580$	0	0
$581$	−18.0000	−0.746766
$582$	0	0
$583$	−5.65685	−0.234283
$584$	0	0
$585$	0	0
$586$	0	0
$587$	35.3553	1.45927	0.729636	−	0.683836i	$-0.239690\pi$
$587$	35.3553	1.45927	0.729636	+	0.683836i	$0.239690\pi$
$588$	0	0
$589$	48.0000	1.97781
$590$	0	0
$591$	25.4558	1.04711
$592$	0	0
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−24.0000	−0.982255
$598$	0	0
$599$	11.3137	0.462266	0.231133	−	0.972922i	$-0.425757\pi$
$599$	11.3137	0.462266	0.231133	+	0.972922i	$0.425757\pi$
$600$	0	0
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	0	0
$603$	−1.41421	−0.0575912
$604$	0	0
$605$	0	0
$606$	0	0
$607$	46.6690	1.89424	0.947119	−	0.320882i	$-0.103979\pi$
$607$	46.6690	1.89424	0.947119	+	0.320882i	$0.103979\pi$
$608$	0	0
$609$	16.0000	0.648353
$610$	0	0
$611$	19.7990	0.800981
$612$	0	0
$613$	−46.0000	−1.85792	−0.928961	−	0.370177i	$-0.879297\pi$
$613$	−46.0000	−1.85792	−0.928961	+	0.370177i	$0.879297\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	0	0
$623$	−14.1421	−0.566593
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−22.6274	−0.903652
$628$	0	0
$629$	20.0000	0.797452
$630$	0	0
$631$	−19.7990	−0.788185	−0.394093	−	0.919071i	$-0.628941\pi$
$631$	−19.7990	−0.788185	−0.394093	+	0.919071i	$0.628941\pi$
$632$	0	0
$633$	−12.0000	−0.476957
$634$	0	0
$635$	0	0
$636$	0	0
$637$	10.0000	0.396214
$638$	0	0
$639$	−14.1421	−0.559454
$640$	0	0
$641$	16.0000	0.631962	0.315981	−	0.948766i	$-0.397666\pi$
$641$	16.0000	0.631962	0.315981	+	0.948766i	$0.397666\pi$
$642$	0	0
$643$	−29.6985	−1.17119	−0.585597	−	0.810602i	$-0.699140\pi$
$643$	−29.6985	−1.17119	−0.585597	+	0.810602i	$0.699140\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.07107	0.277992	0.138996	−	0.990293i	$-0.455612\pi$
$647$	7.07107	0.277992	0.138996	+	0.990293i	$0.455612\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	−16.9706	−0.665129
$652$	0	0
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	−16.9706	−0.661079	−0.330540	−	0.943792i	$-0.607231\pi$
$659$	−16.9706	−0.661079	−0.330540	+	0.943792i	$0.607231\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	5.65685	0.219694
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11.3137	0.438069
$668$	0	0
$669$	10.0000	0.386622
$670$	0	0
$671$	28.2843	1.09190
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	−19.7990	−0.759815
$680$	0	0
$681$	−2.00000	−0.0766402
$682$	0	0
$683$	−7.07107	−0.270567	−0.135283	−	0.990807i	$-0.543195\pi$
$683$	−7.07107	−0.270567	−0.135283	+	0.990807i	$0.543195\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	22.6274	0.863290
$688$	0	0
$689$	4.00000	0.152388
$690$	0	0
$691$	8.48528	0.322795	0.161398	−	0.986889i	$-0.448400\pi$
$691$	8.48528	0.322795	0.161398	+	0.986889i	$0.448400\pi$
$692$	0	0
$693$	−4.00000	−0.151947
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	2.82843	0.106981
$700$	0	0
$701$	−26.0000	−0.982006	−0.491003	−	0.871158i	$-0.663370\pi$
$701$	−26.0000	−0.982006	−0.491003	+	0.871158i	$0.663370\pi$
$702$	0	0
$703$	56.5685	2.13352
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.65685	0.212748
$708$	0	0
$709$	16.0000	0.600893	0.300446	−	0.953799i	$-0.402864\pi$
$709$	16.0000	0.600893	0.300446	+	0.953799i	$0.402864\pi$
$710$	0	0
$711$	−11.3137	−0.424297
$712$	0	0
$713$	−12.0000	−0.449404
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.00000	−0.298765
$718$	0	0
$719$	−39.5980	−1.47676	−0.738378	−	0.674387i	$-0.764408\pi$
$719$	−39.5980	−1.47676	−0.738378	+	0.674387i	$0.764408\pi$
$720$	0	0
$721$	−6.00000	−0.223452
$722$	0	0
$723$	−11.3137	−0.420761
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−21.2132	−0.786754	−0.393377	−	0.919377i	$-0.628693\pi$
$727$	−21.2132	−0.786754	−0.393377	+	0.919377i	$0.628693\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	14.1421	0.523066
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.00000	0.147342
$738$	0	0
$739$	−11.3137	−0.416181	−0.208091	−	0.978110i	$-0.566725\pi$
$739$	−11.3137	−0.416181	−0.208091	+	0.978110i	$0.566725\pi$
$740$	0	0
$741$	16.0000	0.587775
$742$	0	0
$743$	15.5563	0.570707	0.285354	−	0.958422i	$-0.407889\pi$
$743$	15.5563	0.570707	0.285354	+	0.958422i	$0.407889\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.7279	0.465690
$748$	0	0
$749$	22.0000	0.803863
$750$	0	0
$751$	8.48528	0.309632	0.154816	−	0.987943i	$-0.450521\pi$
$751$	8.48528	0.309632	0.154816	+	0.987943i	$0.450521\pi$
$752$	0	0
$753$	−20.0000	−0.728841
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	0	0
$759$	5.65685	0.205331
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	−2.82843	−0.102396
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11.3137	0.408514
$768$	0	0
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	−31.1127	−1.12050
$772$	0	0
$773$	26.0000	0.935155	0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000	0.935155	0.467578	+	0.883952i	$0.345127\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−20.0000	−0.717496
$778$	0	0
$779$	0	0
$780$	0	0
$781$	40.0000	1.43131
$782$	0	0
$783$	−45.2548	−1.61728
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−4.24264	−0.151234	−0.0756169	−	0.997137i	$-0.524093\pi$
$787$	−4.24264	−0.151234	−0.0756169	+	0.997137i	$0.524093\pi$
$788$	0	0
$789$	−2.00000	−0.0712019
$790$	0	0
$791$	−8.48528	−0.301702
$792$	0	0
$793$	−20.0000	−0.710221
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−22.0000	−0.779280	−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000	−0.779280	−0.389640	+	0.920967i	$0.627401\pi$
$798$	0	0
$799$	19.7990	0.700438
$800$	0	0
$801$	10.0000	0.353333
$802$	0	0
$803$	16.9706	0.598878
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−42.4264	−1.49348
$808$	0	0
$809$	38.0000	1.33601	0.668004	−	0.744157i	$-0.267149\pi$
$809$	38.0000	1.33601	0.668004	+	0.744157i	$0.267149\pi$
$810$	0	0
$811$	19.7990	0.695237	0.347618	−	0.937636i	$-0.386991\pi$
$811$	19.7990	0.695237	0.347618	+	0.937636i	$0.386991\pi$
$812$	0	0
$813$	−12.0000	−0.420858
$814$	0	0
$815$	0	0
$816$	0	0
$817$	40.0000	1.39942
$818$	0	0
$819$	2.82843	0.0988332
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	−32.5269	−1.13382	−0.566908	−	0.823781i	$-0.691861\pi$
$823$	−32.5269	−1.13382	−0.566908	+	0.823781i	$0.691861\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−49.4975	−1.72120	−0.860598	−	0.509285i	$-0.829910\pi$
$827$	−49.4975	−1.72120	−0.860598	+	0.509285i	$0.829910\pi$
$828$	0	0
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	0	0
$831$	25.4558	0.883053
$832$	0	0
$833$	10.0000	0.346479
$834$	0	0
$835$	0	0
$836$	0	0
$837$	48.0000	1.65912
$838$	0	0
$839$	28.2843	0.976481	0.488241	−	0.872709i	$-0.337639\pi$
$839$	28.2843	0.976481	0.488241	+	0.872709i	$0.337639\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	−22.6274	−0.779330
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−4.24264	−0.145779
$848$	0	0
$849$	−22.0000	−0.755038
$850$	0	0
$851$	−14.1421	−0.484786
$852$	0	0
$853$	2.00000	0.0684787	0.0342393	−	0.999414i	$-0.489099\pi$
$853$	2.00000	0.0684787	0.0342393	+	0.999414i	$0.489099\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	−39.5980	−1.35107	−0.675533	−	0.737330i	$-0.736086\pi$
$859$	−39.5980	−1.35107	−0.675533	+	0.737330i	$0.736086\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	21.2132	0.722106	0.361053	−	0.932545i	$-0.382417\pi$
$863$	21.2132	0.722106	0.361053	+	0.932545i	$0.382417\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−18.3848	−0.624380
$868$	0	0
$869$	32.0000	1.08553
$870$	0	0
$871$	−2.82843	−0.0958376
$872$	0	0
$873$	14.0000	0.473828
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	0	0
$879$	−2.82843	−0.0954005
$880$	0	0
$881$	−48.0000	−1.61716	−0.808581	−	0.588386i	$-0.799764\pi$
$881$	−48.0000	−1.61716	−0.808581	+	0.588386i	$0.799764\pi$
$882$	0	0
$883$	24.0416	0.809065	0.404533	−	0.914524i	$-0.367434\pi$
$883$	24.0416	0.809065	0.404533	+	0.914524i	$0.367434\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.89949	−0.332393	−0.166196	−	0.986093i	$-0.553149\pi$
$887$	−9.89949	−0.332393	−0.166196	+	0.986093i	$0.553149\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	0	0
$891$	−14.1421	−0.473779
$892$	0	0
$893$	56.0000	1.87397
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.00000	−0.133556
$898$	0	0
$899$	−67.8823	−2.26400
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	−14.1421	−0.470621
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−24.0416	−0.798289	−0.399145	−	0.916888i	$-0.630693\pi$
$907$	−24.0416	−0.798289	−0.399145	+	0.916888i	$0.630693\pi$
$908$	0	0
$909$	−4.00000	−0.132672
$910$	0	0
$911$	48.0833	1.59307	0.796535	−	0.604593i	$-0.206664\pi$
$911$	48.0833	1.59307	0.796535	+	0.604593i	$0.206664\pi$
$912$	0	0
$913$	−36.0000	−1.19143
$914$	0	0
$915$	0	0
$916$	0	0
$917$	28.0000	0.924641
$918$	0	0
$919$	−45.2548	−1.49282	−0.746410	−	0.665487i	$-0.768224\pi$
$919$	−45.2548	−1.49282	−0.746410	+	0.665487i	$0.768224\pi$
$920$	0	0
$921$	18.0000	0.593120
$922$	0	0
$923$	−28.2843	−0.930988
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.24264	0.139347
$928$	0	0
$929$	12.0000	0.393707	0.196854	−	0.980433i	$-0.436928\pi$
$929$	12.0000	0.393707	0.196854	+	0.980433i	$0.436928\pi$
$930$	0	0
$931$	28.2843	0.926980
$932$	0	0
$933$	−4.00000	−0.130954
$934$	0	0
$935$	0	0
$936$	0	0
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	0	0
$939$	8.48528	0.276907
$940$	0	0
$941$	−4.00000	−0.130396	−0.0651981	−	0.997872i	$-0.520768\pi$
$941$	−4.00000	−0.130396	−0.0651981	+	0.997872i	$0.520768\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46.6690	1.51654	0.758270	−	0.651940i	$-0.226045\pi$
$947$	46.6690	1.51654	0.758270	+	0.651940i	$0.226045\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	0	0
$951$	2.82843	0.0917180
$952$	0	0
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	32.0000	1.03441
$958$	0	0
$959$	−19.7990	−0.639343
$960$	0	0
$961$	41.0000	1.32258
$962$	0	0
$963$	−15.5563	−0.501296
$964$	0	0
$965$	0	0
$966$	0	0
$967$	21.2132	0.682171	0.341085	−	0.940032i	$-0.389205\pi$
$967$	21.2132	0.682171	0.341085	+	0.940032i	$0.389205\pi$
$968$	0	0
$969$	16.0000	0.513994
$970$	0	0
$971$	19.7990	0.635380	0.317690	−	0.948195i	$-0.397093\pi$
$971$	19.7990	0.635380	0.317690	+	0.948195i	$0.397093\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	−28.2843	−0.903969
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	32.5269	1.03745	0.518724	−	0.854942i	$-0.326407\pi$
$983$	32.5269	1.03745	0.518724	+	0.854942i	$0.326407\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−19.7990	−0.630209
$988$	0	0
$989$	−10.0000	−0.317982
$990$	0	0
$991$	−8.48528	−0.269544	−0.134772	−	0.990877i	$-0.543030\pi$
$991$	−8.48528	−0.269544	−0.134772	+	0.990877i	$0.543030\pi$
$992$	0	0
$993$	36.0000	1.14243
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	0	0
$999$	56.5685	1.78975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6400.2.a.bm.1.2		2
4.3	odd	2	inner	6400.2.a.bm.1.1		2
5.4	even	2		1280.2.a.k.1.1		2
8.3	odd	2		6400.2.a.bo.1.2		2
8.5	even	2		6400.2.a.bo.1.1		2
16.3	odd	4		3200.2.d.t.1601.2		4
16.5	even	4		3200.2.d.t.1601.1		4
16.11	odd	4		3200.2.d.t.1601.4		4
16.13	even	4		3200.2.d.t.1601.3		4
20.19	odd	2		1280.2.a.k.1.2		2
40.19	odd	2		1280.2.a.e.1.1		2
40.29	even	2		1280.2.a.e.1.2		2
80.3	even	4		3200.2.f.k.449.3		4
80.13	odd	4		3200.2.f.k.449.2		4
80.19	odd	4		640.2.d.c.321.3	yes	4
80.27	even	4		3200.2.f.k.449.4		4
80.29	even	4		640.2.d.c.321.1	✓	4
80.37	odd	4		3200.2.f.k.449.1		4
80.43	even	4		3200.2.f.h.449.1		4
80.53	odd	4		3200.2.f.h.449.4		4
80.59	odd	4		640.2.d.c.321.2	yes	4
80.67	even	4		3200.2.f.h.449.2		4
80.69	even	4		640.2.d.c.321.4	yes	4
80.77	odd	4		3200.2.f.h.449.3		4
240.29	odd	4		5760.2.k.q.2881.4		4
240.59	even	4		5760.2.k.q.2881.1		4
240.149	odd	4		5760.2.k.q.2881.2		4
240.179	even	4		5760.2.k.q.2881.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.2.d.c.321.1	✓	4	80.29	even	4
640.2.d.c.321.2	yes	4	80.59	odd	4
640.2.d.c.321.3	yes	4	80.19	odd	4
640.2.d.c.321.4	yes	4	80.69	even	4
1280.2.a.e.1.1		2	40.19	odd	2
1280.2.a.e.1.2		2	40.29	even	2
1280.2.a.k.1.1		2	5.4	even	2
1280.2.a.k.1.2		2	20.19	odd	2
3200.2.d.t.1601.1		4	16.5	even	4
3200.2.d.t.1601.2		4	16.3	odd	4
3200.2.d.t.1601.3		4	16.13	even	4
3200.2.d.t.1601.4		4	16.11	odd	4
3200.2.f.h.449.1		4	80.43	even	4
3200.2.f.h.449.2		4	80.67	even	4
3200.2.f.h.449.3		4	80.77	odd	4
3200.2.f.h.449.4		4	80.53	odd	4
3200.2.f.k.449.1		4	80.37	odd	4
3200.2.f.k.449.2		4	80.13	odd	4
3200.2.f.k.449.3		4	80.3	even	4
3200.2.f.k.449.4		4	80.27	even	4
5760.2.k.q.2881.1		4	240.59	even	4
5760.2.k.q.2881.2		4	240.149	odd	4
5760.2.k.q.2881.3		4	240.179	even	4
5760.2.k.q.2881.4		4	240.29	odd	4
6400.2.a.bm.1.1		2	4.3	odd	2	inner
6400.2.a.bm.1.2		2	1.1	even	1	trivial
6400.2.a.bo.1.1		2	8.5	even	2
6400.2.a.bo.1.2		2	8.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6400 = 2^{8} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6400.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{3} +1.41421 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{3} +1.41421 q^{7} -1.00000 q^{9} +2.82843 q^{11} -2.00000 q^{13} -2.00000 q^{17} -5.65685 q^{19} +2.00000 q^{21} +1.41421 q^{23} -5.65685 q^{27} +8.00000 q^{29} -8.48528 q^{31} +4.00000 q^{33} -10.0000 q^{37} -2.82843 q^{39} -7.07107 q^{43} -9.89949 q^{47} -5.00000 q^{49} -2.82843 q^{51} -2.00000 q^{53} -8.00000 q^{57} -5.65685 q^{59} +10.0000 q^{61} -1.41421 q^{63} +1.41421 q^{67} +2.00000 q^{69} +14.1421 q^{71} +6.00000 q^{73} +4.00000 q^{77} +11.3137 q^{79} -5.00000 q^{81} -12.7279 q^{83} +11.3137 q^{87} -10.0000 q^{89} -2.82843 q^{91} -12.0000 q^{93} -14.0000 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} - 4 q^{13} - 4 q^{17} + 4 q^{21} + 16 q^{29} + 8 q^{33} - 20 q^{37} - 10 q^{49} - 4 q^{53} - 16 q^{57} + 20 q^{61} + 4 q^{69} + 12 q^{73} + 8 q^{77} - 10 q^{81} - 20 q^{89} - 24 q^{93} - 28 q^{97}+O(q^{100}) \) 2 * q - 2 * q^9 - 4 * q^13 - 4 * q^17 + 4 * q^21 + 16 * q^29 + 8 * q^33 - 20 * q^37 - 10 * q^49 - 4 * q^53 - 16 * q^57 + 20 * q^61 + 4 * q^69 + 12 * q^73 + 8 * q^77 - 10 * q^81 - 20 * q^89 - 24 * q^93 - 28 * q^97

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