LMFDB - Embedded newform 1280.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,2,Mod(1,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, 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("1280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1280.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:	$10.2208514587$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
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			$3.16228$ of defining polynomial
Character	$\chi$	$=$	1280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.16228 q^{3} -1.00000 q^{5} +3.16228 q^{7} +7.00000 q^{9} +O(q^{10})$	$q+3.16228 q^{3} -1.00000 q^{5} +3.16228 q^{7} +7.00000 q^{9} -6.00000 q^{13} -3.16228 q^{15} -2.00000 q^{17} +6.32456 q^{19} +10.0000 q^{21} +3.16228 q^{23} +1.00000 q^{25} +12.6491 q^{27} +4.00000 q^{29} +6.32456 q^{31} -3.16228 q^{35} -2.00000 q^{37} -18.9737 q^{39} -3.16228 q^{43} -7.00000 q^{45} -9.48683 q^{47} +3.00000 q^{49} -6.32456 q^{51} -6.00000 q^{53} +20.0000 q^{57} -6.32456 q^{59} -2.00000 q^{61} +22.1359 q^{63} +6.00000 q^{65} -9.48683 q^{67} +10.0000 q^{69} +6.32456 q^{71} +14.0000 q^{73} +3.16228 q^{75} -12.6491 q^{79} +19.0000 q^{81} -3.16228 q^{83} +2.00000 q^{85} +12.6491 q^{87} -10.0000 q^{89} -18.9737 q^{91} +20.0000 q^{93} -6.32456 q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 14 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 + 14 * q^9	$ 2 q - 2 q^{5} + 14 q^{9} - 12 q^{13} - 4 q^{17} + 20 q^{21} + 2 q^{25} + 8 q^{29} - 4 q^{37} - 14 q^{45} + 6 q^{49} - 12 q^{53} + 40 q^{57} - 4 q^{61} + 12 q^{65} + 20 q^{69} + 28 q^{73} + 38 q^{81} + 4 q^{85} - 20 q^{89} + 40 q^{93} + 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 14 * q^9 - 12 * q^13 - 4 * q^17 + 20 * q^21 + 2 * q^25 + 8 * q^29 - 4 * q^37 - 14 * q^45 + 6 * q^49 - 12 * q^53 + 40 * q^57 - 4 * q^61 + 12 * q^65 + 20 * q^69 + 28 * q^73 + 38 * q^81 + 4 * q^85 - 20 * q^89 + 40 * q^93 + 4 * 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
$3$	3.16228	1.82574	0.912871	−	0.408248i	$-0.133860\pi$
$3$	3.16228	1.82574	0.912871	+	0.408248i	$0.133860\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.16228	1.19523	0.597614	−	0.801784i	$-0.296115\pi$
$7$	3.16228	1.19523	0.597614	+	0.801784i	$0.296115\pi$
$8$	0	0
$9$	7.00000	2.33333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	−3.16228	−0.816497
$16$	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
$19$	6.32456	1.45095	0.725476	−	0.688247i	$-0.241620\pi$
$19$	6.32456	1.45095	0.725476	+	0.688247i	$0.241620\pi$
$20$	0	0
$21$	10.0000	2.18218
$22$	0	0
$23$	3.16228	0.659380	0.329690	−	0.944089i	$-0.393056\pi$
$23$	3.16228	0.659380	0.329690	+	0.944089i	$0.393056\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	12.6491	2.43432
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	6.32456	1.13592	0.567962	−	0.823055i	$-0.307732\pi$
$31$	6.32456	1.13592	0.567962	+	0.823055i	$0.307732\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.16228	−0.534522
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	−18.9737	−3.03822
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−3.16228	−0.482243	−0.241121	−	0.970495i	$-0.577515\pi$
$43$	−3.16228	−0.482243	−0.241121	+	0.970495i	$0.577515\pi$
$44$	0	0
$45$	−7.00000	−1.04350
$46$	0	0
$47$	−9.48683	−1.38380	−0.691898	−	0.721995i	$-0.743225\pi$
$47$	−9.48683	−1.38380	−0.691898	+	0.721995i	$0.743225\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	−6.32456	−0.885615
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	20.0000	2.64906
$58$	0	0
$59$	−6.32456	−0.823387	−0.411693	−	0.911322i	$-0.635063\pi$
$59$	−6.32456	−0.823387	−0.411693	+	0.911322i	$0.635063\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	22.1359	2.78887
$64$	0	0
$65$	6.00000	0.744208
$66$	0	0
$67$	−9.48683	−1.15900	−0.579501	−	0.814972i	$-0.696752\pi$
$67$	−9.48683	−1.15900	−0.579501	+	0.814972i	$0.696752\pi$
$68$	0	0
$69$	10.0000	1.20386
$70$	0	0
$71$	6.32456	0.750587	0.375293	−	0.926906i	$-0.377542\pi$
$71$	6.32456	0.750587	0.375293	+	0.926906i	$0.377542\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	0	0
$75$	3.16228	0.365148
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−12.6491	−1.42314	−0.711568	−	0.702617i	$-0.752015\pi$
$79$	−12.6491	−1.42314	−0.711568	+	0.702617i	$0.752015\pi$
$80$	0	0
$81$	19.0000	2.11111
$82$	0	0
$83$	−3.16228	−0.347105	−0.173553	−	0.984825i	$-0.555525\pi$
$83$	−3.16228	−0.347105	−0.173553	+	0.984825i	$0.555525\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	12.6491	1.35613
$88$	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
$91$	−18.9737	−1.98898
$92$	0	0
$93$	20.0000	2.07390
$94$	0	0
$95$	−6.32456	−0.648886
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	15.8114	1.55794	0.778971	−	0.627060i	$-0.215742\pi$
$103$	15.8114	1.55794	0.778971	+	0.627060i	$0.215742\pi$
$104$	0	0
$105$	−10.0000	−0.975900
$106$	0	0
$107$	−15.8114	−1.52854	−0.764272	−	0.644894i	$-0.776902\pi$
$107$	−15.8114	−1.52854	−0.764272	+	0.644894i	$0.776902\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−6.32456	−0.600300
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−3.16228	−0.294884
$116$	0	0
$117$	−42.0000	−3.88290
$118$	0	0
$119$	−6.32456	−0.579771
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.48683	0.841820	0.420910	−	0.907102i	$-0.361711\pi$
$127$	9.48683	0.841820	0.420910	+	0.907102i	$0.361711\pi$
$128$	0	0
$129$	−10.0000	−0.880451
$130$	0	0
$131$	12.6491	1.10516	0.552579	−	0.833461i	$-0.313644\pi$
$131$	12.6491	1.10516	0.552579	+	0.833461i	$0.313644\pi$
$132$	0	0
$133$	20.0000	1.73422
$134$	0	0
$135$	−12.6491	−1.08866
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	−6.32456	−0.536442	−0.268221	−	0.963357i	$-0.586436\pi$
$139$	−6.32456	−0.536442	−0.268221	+	0.963357i	$0.586436\pi$
$140$	0	0
$141$	−30.0000	−2.52646
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	9.48683	0.782461
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	6.32456	0.514685	0.257343	−	0.966320i	$-0.417153\pi$
$151$	6.32456	0.514685	0.257343	+	0.966320i	$0.417153\pi$
$152$	0	0
$153$	−14.0000	−1.13183
$154$	0	0
$155$	−6.32456	−0.508001
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	−18.9737	−1.50471
$160$	0	0
$161$	10.0000	0.788110
$162$	0	0
$163$	−9.48683	−0.743066	−0.371533	−	0.928420i	$-0.621168\pi$
$163$	−9.48683	−0.743066	−0.371533	+	0.928420i	$0.621168\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.8114	−1.22352	−0.611761	−	0.791043i	$-0.709539\pi$
$167$	−15.8114	−1.22352	−0.611761	+	0.791043i	$0.709539\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	44.2719	3.38556
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	3.16228	0.239046
$176$	0	0
$177$	−20.0000	−1.50329
$178$	0	0
$179$	−18.9737	−1.41816	−0.709079	−	0.705129i	$-0.750889\pi$
$179$	−18.9737	−1.41816	−0.709079	+	0.705129i	$0.750889\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	−6.32456	−0.467525
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	0	0
$188$	0	0
$189$	40.0000	2.90957
$190$	0	0
$191$	18.9737	1.37289	0.686443	−	0.727183i	$-0.259171\pi$
$191$	18.9737	1.37289	0.686443	+	0.727183i	$0.259171\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	18.9737	1.35873
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−30.0000	−2.11604
$202$	0	0
$203$	12.6491	0.887794
$204$	0	0
$205$	0	0
$206$	0	0
$207$	22.1359	1.53855
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	20.0000	1.37038
$214$	0	0
$215$	3.16228	0.215666
$216$	0	0
$217$	20.0000	1.35769
$218$	0	0
$219$	44.2719	2.99162
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	−22.1359	−1.48233	−0.741166	−	0.671322i	$-0.765727\pi$
$223$	−22.1359	−1.48233	−0.741166	+	0.671322i	$0.765727\pi$
$224$	0	0
$225$	7.00000	0.466667
$226$	0	0
$227$	9.48683	0.629663	0.314832	−	0.949148i	$-0.398052\pi$
$227$	9.48683	0.629663	0.314832	+	0.949148i	$0.398052\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	9.48683	0.618853
$236$	0	0
$237$	−40.0000	−2.59828
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	22.1359	1.42002
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−37.9473	−2.41453
$248$	0	0
$249$	−10.0000	−0.633724
$250$	0	0
$251$	−12.6491	−0.798405	−0.399202	−	0.916863i	$-0.630713\pi$
$251$	−12.6491	−0.798405	−0.399202	+	0.916863i	$0.630713\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	6.32456	0.396059
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−6.32456	−0.392989
$260$	0	0
$261$	28.0000	1.73316
$262$	0	0
$263$	22.1359	1.36496	0.682480	−	0.730904i	$-0.260901\pi$
$263$	22.1359	1.36496	0.682480	+	0.730904i	$0.260901\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	−31.6228	−1.93528
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	6.32456	0.384189	0.192095	−	0.981376i	$-0.438472\pi$
$271$	6.32456	0.384189	0.192095	+	0.981376i	$0.438472\pi$
$272$	0	0
$273$	−60.0000	−3.63137
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	44.2719	2.65049
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	3.16228	0.187978	0.0939889	−	0.995573i	$-0.470038\pi$
$283$	3.16228	0.187978	0.0939889	+	0.995573i	$0.470038\pi$
$284$	0	0
$285$	−20.0000	−1.18470
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	6.32456	0.370752
$292$	0	0
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	0	0
$295$	6.32456	0.368230
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−18.9737	−1.09728
$300$	0	0
$301$	−10.0000	−0.576390
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	15.8114	0.902404	0.451202	−	0.892422i	$-0.350995\pi$
$307$	15.8114	0.902404	0.451202	+	0.892422i	$0.350995\pi$
$308$	0	0
$309$	50.0000	2.84440
$310$	0	0
$311$	−18.9737	−1.07590	−0.537949	−	0.842977i	$-0.680801\pi$
$311$	−18.9737	−1.07590	−0.537949	+	0.842977i	$0.680801\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	−22.1359	−1.24722
$316$	0	0
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−50.0000	−2.79073
$322$	0	0
$323$	−12.6491	−0.703815
$324$	0	0
$325$	−6.00000	−0.332820
$326$	0	0
$327$	31.6228	1.74874
$328$	0	0
$329$	−30.0000	−1.65395
$330$	0	0
$331$	12.6491	0.695258	0.347629	−	0.937632i	$-0.386987\pi$
$331$	12.6491	0.695258	0.347629	+	0.937632i	$0.386987\pi$
$332$	0	0
$333$	−14.0000	−0.767195
$334$	0	0
$335$	9.48683	0.518321
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	18.9737	1.03051
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−12.6491	−0.682988
$344$	0	0
$345$	−10.0000	−0.538382
$346$	0	0
$347$	15.8114	0.848800	0.424400	−	0.905475i	$-0.360485\pi$
$347$	15.8114	0.848800	0.424400	+	0.905475i	$0.360485\pi$
$348$	0	0
$349$	−4.00000	−0.214115	−0.107058	−	0.994253i	$-0.534143\pi$
$349$	−4.00000	−0.214115	−0.107058	+	0.994253i	$0.534143\pi$
$350$	0	0
$351$	−75.8947	−4.05096
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	−6.32456	−0.335673
$356$	0	0
$357$	−20.0000	−1.05851
$358$	0	0
$359$	12.6491	0.667595	0.333797	−	0.942645i	$-0.391670\pi$
$359$	12.6491	0.667595	0.333797	+	0.942645i	$0.391670\pi$
$360$	0	0
$361$	21.0000	1.10526
$362$	0	0
$363$	−34.7851	−1.82574
$364$	0	0
$365$	−14.0000	−0.732793
$366$	0	0
$367$	−3.16228	−0.165070	−0.0825348	−	0.996588i	$-0.526302\pi$
$367$	−3.16228	−0.165070	−0.0825348	+	0.996588i	$0.526302\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−18.9737	−0.985064
$372$	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
$375$	−3.16228	−0.163299
$376$	0	0
$377$	−24.0000	−1.23606
$378$	0	0
$379$	−6.32456	−0.324871	−0.162435	−	0.986719i	$-0.551935\pi$
$379$	−6.32456	−0.324871	−0.162435	+	0.986719i	$0.551935\pi$
$380$	0	0
$381$	30.0000	1.53695
$382$	0	0
$383$	−22.1359	−1.13109	−0.565547	−	0.824716i	$-0.691335\pi$
$383$	−22.1359	−1.13109	−0.565547	+	0.824716i	$0.691335\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−22.1359	−1.12523
$388$	0	0
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	−6.32456	−0.319847
$392$	0	0
$393$	40.0000	2.01773
$394$	0	0
$395$	12.6491	0.636446
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	63.2456	3.16624
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	−37.9473	−1.89029
$404$	0	0
$405$	−19.0000	−0.944118
$406$	0	0
$407$	0	0
$408$	0	0
$409$	36.0000	1.78009	0.890043	−	0.455877i	$-0.150674\pi$
$409$	36.0000	1.78009	0.890043	+	0.455877i	$0.150674\pi$
$410$	0	0
$411$	−56.9210	−2.80771
$412$	0	0
$413$	−20.0000	−0.984136
$414$	0	0
$415$	3.16228	0.155230
$416$	0	0
$417$	−20.0000	−0.979404
$418$	0	0
$419$	−31.6228	−1.54487	−0.772437	−	0.635092i	$-0.780962\pi$
$419$	−31.6228	−1.54487	−0.772437	+	0.635092i	$0.780962\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	−66.4078	−3.22886
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	−6.32456	−0.306067
$428$	0	0
$429$	0	0
$430$	0	0
$431$	31.6228	1.52322	0.761608	−	0.648038i	$-0.224410\pi$
$431$	31.6228	1.52322	0.761608	+	0.648038i	$0.224410\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	0	0
$435$	−12.6491	−0.606478
$436$	0	0
$437$	20.0000	0.956730
$438$	0	0
$439$	−12.6491	−0.603709	−0.301855	−	0.953354i	$-0.597606\pi$
$439$	−12.6491	−0.603709	−0.301855	+	0.953354i	$0.597606\pi$
$440$	0	0
$441$	21.0000	1.00000
$442$	0	0
$443$	41.1096	1.95318	0.976588	−	0.215117i	$-0.0690133\pi$
$443$	41.1096	1.95318	0.976588	+	0.215117i	$0.0690133\pi$
$444$	0	0
$445$	10.0000	0.474045
$446$	0	0
$447$	−31.6228	−1.49571
$448$	0	0
$449$	4.00000	0.188772	0.0943858	−	0.995536i	$-0.469911\pi$
$449$	4.00000	0.188772	0.0943858	+	0.995536i	$0.469911\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	20.0000	0.939682
$454$	0	0
$455$	18.9737	0.889499
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	0	0
$459$	−25.2982	−1.18082
$460$	0	0
$461$	40.0000	1.86299	0.931493	−	0.363760i	$-0.118507\pi$
$461$	40.0000	1.86299	0.931493	+	0.363760i	$0.118507\pi$
$462$	0	0
$463$	−3.16228	−0.146964	−0.0734818	−	0.997297i	$-0.523411\pi$
$463$	−3.16228	−0.146964	−0.0734818	+	0.997297i	$0.523411\pi$
$464$	0	0
$465$	−20.0000	−0.927478
$466$	0	0
$467$	−3.16228	−0.146333	−0.0731664	−	0.997320i	$-0.523310\pi$
$467$	−3.16228	−0.146333	−0.0731664	+	0.997320i	$0.523310\pi$
$468$	0	0
$469$	−30.0000	−1.38527
$470$	0	0
$471$	−6.32456	−0.291420
$472$	0	0
$473$	0	0
$474$	0	0
$475$	6.32456	0.290191
$476$	0	0
$477$	−42.0000	−1.92305
$478$	0	0
$479$	25.2982	1.15591	0.577953	−	0.816070i	$-0.303852\pi$
$479$	25.2982	1.15591	0.577953	+	0.816070i	$0.303852\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	0	0
$483$	31.6228	1.43889
$484$	0	0
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	−9.48683	−0.429889	−0.214945	−	0.976626i	$-0.568957\pi$
$487$	−9.48683	−0.429889	−0.214945	+	0.976626i	$0.568957\pi$
$488$	0	0
$489$	−30.0000	−1.35665
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	−8.00000	−0.360302
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.0000	0.897123
$498$	0	0
$499$	31.6228	1.41563	0.707815	−	0.706398i	$-0.249681\pi$
$499$	31.6228	1.41563	0.707815	+	0.706398i	$0.249681\pi$
$500$	0	0
$501$	−50.0000	−2.23384
$502$	0	0
$503$	−34.7851	−1.55099	−0.775494	−	0.631354i	$-0.782499\pi$
$503$	−34.7851	−1.55099	−0.775494	+	0.631354i	$0.782499\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	72.7324	3.23016
$508$	0	0
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	0	0
$511$	44.2719	1.95847
$512$	0	0
$513$	80.0000	3.53209
$514$	0	0
$515$	−15.8114	−0.696733
$516$	0	0
$517$	0	0
$518$	0	0
$519$	44.2719	1.94332
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	3.16228	0.138277	0.0691384	−	0.997607i	$-0.477975\pi$
$523$	3.16228	0.138277	0.0691384	+	0.997607i	$0.477975\pi$
$524$	0	0
$525$	10.0000	0.436436
$526$	0	0
$527$	−12.6491	−0.551004
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	−44.2719	−1.92124
$532$	0	0
$533$	0	0
$534$	0	0
$535$	15.8114	0.683586
$536$	0	0
$537$	−60.0000	−2.58919
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−40.0000	−1.71973	−0.859867	−	0.510518i	$-0.829454\pi$
$541$	−40.0000	−1.71973	−0.859867	+	0.510518i	$0.829454\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−10.0000	−0.428353
$546$	0	0
$547$	−15.8114	−0.676046	−0.338023	−	0.941138i	$-0.609758\pi$
$547$	−15.8114	−0.676046	−0.338023	+	0.941138i	$0.609758\pi$
$548$	0	0
$549$	−14.0000	−0.597505
$550$	0	0
$551$	25.2982	1.07774
$552$	0	0
$553$	−40.0000	−1.70097
$554$	0	0
$555$	6.32456	0.268462
$556$	0	0
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	0	0
$559$	18.9737	0.802501
$560$	0	0
$561$	0	0
$562$	0	0
$563$	28.4605	1.19947	0.599734	−	0.800200i	$-0.295273\pi$
$563$	28.4605	1.19947	0.599734	+	0.800200i	$0.295273\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	0	0
$567$	60.0833	2.52326
$568$	0	0
$569$	4.00000	0.167689	0.0838444	−	0.996479i	$-0.473280\pi$
$569$	4.00000	0.167689	0.0838444	+	0.996479i	$0.473280\pi$
$570$	0	0
$571$	−12.6491	−0.529349	−0.264674	−	0.964338i	$-0.585264\pi$
$571$	−12.6491	−0.529349	−0.264674	+	0.964338i	$0.585264\pi$
$572$	0	0
$573$	60.0000	2.50654
$574$	0	0
$575$	3.16228	0.131876
$576$	0	0
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	0	0
$579$	44.2719	1.83988
$580$	0	0
$581$	−10.0000	−0.414870
$582$	0	0
$583$	0	0
$584$	0	0
$585$	42.0000	1.73649
$586$	0	0
$587$	3.16228	0.130521	0.0652606	−	0.997868i	$-0.479212\pi$
$587$	3.16228	0.130521	0.0652606	+	0.997868i	$0.479212\pi$
$588$	0	0
$589$	40.0000	1.64817
$590$	0	0
$591$	−56.9210	−2.34142
$592$	0	0
$593$	−46.0000	−1.88899	−0.944497	−	0.328521i	$-0.893450\pi$
$593$	−46.0000	−1.88899	−0.944497	+	0.328521i	$0.893450\pi$
$594$	0	0
$595$	6.32456	0.259281
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−37.9473	−1.55049	−0.775243	−	0.631663i	$-0.782373\pi$
$599$	−37.9473	−1.55049	−0.775243	+	0.631663i	$0.782373\pi$
$600$	0	0
$601$	40.0000	1.63163	0.815817	−	0.578310i	$-0.196288\pi$
$601$	40.0000	1.63163	0.815817	+	0.578310i	$0.196288\pi$
$602$	0	0
$603$	−66.4078	−2.70434
$604$	0	0
$605$	11.0000	0.447214
$606$	0	0
$607$	−34.7851	−1.41188	−0.705941	−	0.708271i	$-0.749476\pi$
$607$	−34.7851	−1.41188	−0.705941	+	0.708271i	$0.749476\pi$
$608$	0	0
$609$	40.0000	1.62088
$610$	0	0
$611$	56.9210	2.30278
$612$	0	0
$613$	46.0000	1.85792	0.928961	−	0.370177i	$-0.120703\pi$
$613$	46.0000	1.85792	0.928961	+	0.370177i	$0.120703\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	0	0
$619$	31.6228	1.27103	0.635513	−	0.772090i	$-0.280789\pi$
$619$	31.6228	1.27103	0.635513	+	0.772090i	$0.280789\pi$
$620$	0	0
$621$	40.0000	1.60514
$622$	0	0
$623$	−31.6228	−1.26694
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.00000	0.159490
$630$	0	0
$631$	−6.32456	−0.251777	−0.125888	−	0.992044i	$-0.540178\pi$
$631$	−6.32456	−0.251777	−0.125888	+	0.992044i	$0.540178\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−9.48683	−0.376473
$636$	0	0
$637$	−18.0000	−0.713186
$638$	0	0
$639$	44.2719	1.75137
$640$	0	0
$641$	−40.0000	−1.57991	−0.789953	−	0.613168i	$-0.789895\pi$
$641$	−40.0000	−1.57991	−0.789953	+	0.613168i	$0.789895\pi$
$642$	0	0
$643$	9.48683	0.374124	0.187062	−	0.982348i	$-0.440103\pi$
$643$	9.48683	0.374124	0.187062	+	0.982348i	$0.440103\pi$
$644$	0	0
$645$	10.0000	0.393750
$646$	0	0
$647$	−9.48683	−0.372966	−0.186483	−	0.982458i	$-0.559709\pi$
$647$	−9.48683	−0.372966	−0.186483	+	0.982458i	$0.559709\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	63.2456	2.47879
$652$	0	0
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	−12.6491	−0.494242
$656$	0	0
$657$	98.0000	3.82334
$658$	0	0
$659$	44.2719	1.72459	0.862294	−	0.506408i	$-0.169027\pi$
$659$	44.2719	1.72459	0.862294	+	0.506408i	$0.169027\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	0	0
$663$	37.9473	1.47375
$664$	0	0
$665$	−20.0000	−0.775567
$666$	0	0
$667$	12.6491	0.489776
$668$	0	0
$669$	−70.0000	−2.70636
$670$	0	0
$671$	0	0
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	0	0
$675$	12.6491	0.486864
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	6.32456	0.242714
$680$	0	0
$681$	30.0000	1.14960
$682$	0	0
$683$	22.1359	0.847008	0.423504	−	0.905894i	$-0.360800\pi$
$683$	22.1359	0.847008	0.423504	+	0.905894i	$0.360800\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	0	0
$687$	12.6491	0.482594
$688$	0	0
$689$	36.0000	1.37149
$690$	0	0
$691$	37.9473	1.44358	0.721792	−	0.692110i	$-0.243319\pi$
$691$	37.9473	1.44358	0.721792	+	0.692110i	$0.243319\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.32456	0.239904
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−44.2719	−1.67452
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	−12.6491	−0.477070
$704$	0	0
$705$	30.0000	1.12987
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	0	0
$711$	−88.5438	−3.32065
$712$	0	0
$713$	20.0000	0.749006
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−50.5964	−1.88693	−0.943464	−	0.331474i	$-0.892454\pi$
$719$	−50.5964	−1.88693	−0.943464	+	0.331474i	$0.892454\pi$
$720$	0	0
$721$	50.0000	1.86210
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.00000	0.148556
$726$	0	0
$727$	3.16228	0.117282	0.0586412	−	0.998279i	$-0.481323\pi$
$727$	3.16228	0.117282	0.0586412	+	0.998279i	$0.481323\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	6.32456	0.233922
$732$	0	0
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	0	0
$735$	−9.48683	−0.349927
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−18.9737	−0.697958	−0.348979	−	0.937131i	$-0.613471\pi$
$739$	−18.9737	−0.697958	−0.348979	+	0.937131i	$0.613471\pi$
$740$	0	0
$741$	−120.000	−4.40831
$742$	0	0
$743$	9.48683	0.348038	0.174019	−	0.984742i	$-0.444325\pi$
$743$	9.48683	0.348038	0.174019	+	0.984742i	$0.444325\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	0	0
$747$	−22.1359	−0.809912
$748$	0	0
$749$	−50.0000	−1.82696
$750$	0	0
$751$	6.32456	0.230786	0.115393	−	0.993320i	$-0.463187\pi$
$751$	6.32456	0.230786	0.115393	+	0.993320i	$0.463187\pi$
$752$	0	0
$753$	−40.0000	−1.45768
$754$	0	0
$755$	−6.32456	−0.230174
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	31.6228	1.14482
$764$	0	0
$765$	14.0000	0.506171
$766$	0	0
$767$	37.9473	1.37020
$768$	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
$771$	−56.9210	−2.04996
$772$	0	0
$773$	−34.0000	−1.22290	−0.611448	−	0.791285i	$-0.709412\pi$
$773$	−34.0000	−1.22290	−0.611448	+	0.791285i	$0.709412\pi$
$774$	0	0
$775$	6.32456	0.227185
$776$	0	0
$777$	−20.0000	−0.717496
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	50.5964	1.80817
$784$	0	0
$785$	2.00000	0.0713831
$786$	0	0
$787$	3.16228	0.112723	0.0563615	−	0.998410i	$-0.482050\pi$
$787$	3.16228	0.112723	0.0563615	+	0.998410i	$0.482050\pi$
$788$	0	0
$789$	70.0000	2.49207
$790$	0	0
$791$	18.9737	0.674626
$792$	0	0
$793$	12.0000	0.426132
$794$	0	0
$795$	18.9737	0.672927
$796$	0	0
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	18.9737	0.671240
$800$	0	0
$801$	−70.0000	−2.47333
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−10.0000	−0.352454
$806$	0	0
$807$	−31.6228	−1.11317
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	−12.6491	−0.444170	−0.222085	−	0.975027i	$-0.571286\pi$
$811$	−12.6491	−0.444170	−0.222085	+	0.975027i	$0.571286\pi$
$812$	0	0
$813$	20.0000	0.701431
$814$	0	0
$815$	9.48683	0.332309
$816$	0	0
$817$	−20.0000	−0.699711
$818$	0	0
$819$	−132.816	−4.64095
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	0	0
$823$	28.4605	0.992071	0.496035	−	0.868302i	$-0.334789\pi$
$823$	28.4605	0.992071	0.496035	+	0.868302i	$0.334789\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.48683	−0.329890	−0.164945	−	0.986303i	$-0.552745\pi$
$827$	−9.48683	−0.329890	−0.164945	+	0.986303i	$0.552745\pi$
$828$	0	0
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	6.32456	0.219396
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	15.8114	0.547176
$836$	0	0
$837$	80.0000	2.76520
$838$	0	0
$839$	−12.6491	−0.436696	−0.218348	−	0.975871i	$-0.570067\pi$
$839$	−12.6491	−0.436696	−0.218348	+	0.975871i	$0.570067\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−23.0000	−0.791224
$846$	0	0
$847$	−34.7851	−1.19523
$848$	0	0
$849$	10.0000	0.343199
$850$	0	0
$851$	−6.32456	−0.216803
$852$	0	0
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	0	0
$855$	−44.2719	−1.51407
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	44.2719	1.51054	0.755269	−	0.655415i	$-0.227506\pi$
$859$	44.2719	1.51054	0.755269	+	0.655415i	$0.227506\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	34.7851	1.18410	0.592049	−	0.805902i	$-0.298319\pi$
$863$	34.7851	1.18410	0.592049	+	0.805902i	$0.298319\pi$
$864$	0	0
$865$	−14.0000	−0.476014
$866$	0	0
$867$	−41.1096	−1.39616
$868$	0	0
$869$	0	0
$870$	0	0
$871$	56.9210	1.92869
$872$	0	0
$873$	14.0000	0.473828
$874$	0	0
$875$	−3.16228	−0.106904
$876$	0	0
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	0	0
$879$	−82.2192	−2.77319
$880$	0	0
$881$	−40.0000	−1.34763	−0.673817	−	0.738898i	$-0.735346\pi$
$881$	−40.0000	−1.34763	−0.673817	+	0.738898i	$0.735346\pi$
$882$	0	0
$883$	3.16228	0.106419	0.0532096	−	0.998583i	$-0.483055\pi$
$883$	3.16228	0.106419	0.0532096	+	0.998583i	$0.483055\pi$
$884$	0	0
$885$	20.0000	0.672293
$886$	0	0
$887$	−47.4342	−1.59268	−0.796342	−	0.604847i	$-0.793234\pi$
$887$	−47.4342	−1.59268	−0.796342	+	0.604847i	$0.793234\pi$
$888$	0	0
$889$	30.0000	1.00617
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−60.0000	−2.00782
$894$	0	0
$895$	18.9737	0.634220
$896$	0	0
$897$	−60.0000	−2.00334
$898$	0	0
$899$	25.2982	0.843743
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	−31.6228	−1.05234
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−53.7587	−1.78503	−0.892515	−	0.451019i	$-0.851061\pi$
$907$	−53.7587	−1.78503	−0.892515	+	0.451019i	$0.851061\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	31.6228	1.04771	0.523855	−	0.851808i	$-0.324493\pi$
$911$	31.6228	1.04771	0.523855	+	0.851808i	$0.324493\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	6.32456	0.209083
$916$	0	0
$917$	40.0000	1.32092
$918$	0	0
$919$	−12.6491	−0.417256	−0.208628	−	0.977995i	$-0.566900\pi$
$919$	−12.6491	−0.417256	−0.208628	+	0.977995i	$0.566900\pi$
$920$	0	0
$921$	50.0000	1.64756
$922$	0	0
$923$	−37.9473	−1.24905
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	110.680	3.63520
$928$	0	0
$929$	−4.00000	−0.131236	−0.0656179	−	0.997845i	$-0.520902\pi$
$929$	−4.00000	−0.131236	−0.0656179	+	0.997845i	$0.520902\pi$
$930$	0	0
$931$	18.9737	0.621837
$932$	0	0
$933$	−60.0000	−1.96431
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	0	0
$939$	−18.9737	−0.619182
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−40.0000	−1.30120
$946$	0	0
$947$	15.8114	0.513801	0.256901	−	0.966438i	$-0.417299\pi$
$947$	15.8114	0.513801	0.256901	+	0.966438i	$0.417299\pi$
$948$	0	0
$949$	−84.0000	−2.72676
$950$	0	0
$951$	69.5701	2.25597
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	−18.9737	−0.613973
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−56.9210	−1.83807
$960$	0	0
$961$	9.00000	0.290323
$962$	0	0
$963$	−110.680	−3.56660
$964$	0	0
$965$	−14.0000	−0.450676
$966$	0	0
$967$	22.1359	0.711844	0.355922	−	0.934516i	$-0.384167\pi$
$967$	22.1359	0.711844	0.355922	+	0.934516i	$0.384167\pi$
$968$	0	0
$969$	−40.0000	−1.28499
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−20.0000	−0.641171
$974$	0	0
$975$	−18.9737	−0.607644
$976$	0	0
$977$	2.00000	0.0639857	0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000	0.0639857	0.0319928	+	0.999488i	$0.489815\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	70.0000	2.23493
$982$	0	0
$983$	47.4342	1.51291	0.756457	−	0.654043i	$-0.226928\pi$
$983$	47.4342	1.51291	0.756457	+	0.654043i	$0.226928\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	−94.8683	−3.01969
$988$	0	0
$989$	−10.0000	−0.317982
$990$	0	0
$991$	44.2719	1.40634	0.703171	−	0.711020i	$-0.251767\pi$
$991$	44.2719	1.40634	0.703171	+	0.711020i	$0.251767\pi$
$992$	0	0
$993$	40.0000	1.26936
$994$	0	0
$995$	0	0
$996$	0	0
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	0	0
$999$	−25.2982	−0.800400

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.2.d.a.321.1	✓	4	16.5	even	4
640.2.d.a.321.2	yes	4	16.3	odd	4
640.2.d.a.321.3	yes	4	16.11	odd	4
640.2.d.a.321.4	yes	4	16.13	even	4
1280.2.a.h.1.1		2	4.3	odd	2	inner
1280.2.a.h.1.2		2	1.1	even	1	trivial
1280.2.a.l.1.1		2	8.5	even	2
1280.2.a.l.1.2		2	8.3	odd	2
3200.2.d.j.1601.1		4	80.59	odd	4
3200.2.d.j.1601.2		4	80.29	even	4
3200.2.d.j.1601.3		4	80.19	odd	4
3200.2.d.j.1601.4		4	80.69	even	4
3200.2.f.p.449.1		4	80.43	even	4
3200.2.f.p.449.2		4	80.67	even	4
3200.2.f.p.449.3		4	80.77	odd	4
3200.2.f.p.449.4		4	80.53	odd	4
3200.2.f.q.449.1		4	80.37	odd	4
3200.2.f.q.449.2		4	80.13	odd	4
3200.2.f.q.449.3		4	80.3	even	4
3200.2.f.q.449.4		4	80.27	even	4
5760.2.k.t.2881.1		4	48.29	odd	4
5760.2.k.t.2881.2		4	48.35	even	4
5760.2.k.t.2881.3		4	48.5	odd	4
5760.2.k.t.2881.4		4	48.11	even	4
6400.2.a.bz.1.1		2	40.19	odd	2
6400.2.a.bz.1.2		2	40.29	even	2
6400.2.a.ca.1.1		2	5.4	even	2
6400.2.a.ca.1.2		2	20.19	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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1280.a (trivial)

\(f(q)\)	\(=\)	\(q+3.16228 q^{3} -1.00000 q^{5} +3.16228 q^{7} +7.00000 q^{9} +O(q^{10})\)	\(q+3.16228 q^{3} -1.00000 q^{5} +3.16228 q^{7} +7.00000 q^{9} -6.00000 q^{13} -3.16228 q^{15} -2.00000 q^{17} +6.32456 q^{19} +10.0000 q^{21} +3.16228 q^{23} +1.00000 q^{25} +12.6491 q^{27} +4.00000 q^{29} +6.32456 q^{31} -3.16228 q^{35} -2.00000 q^{37} -18.9737 q^{39} -3.16228 q^{43} -7.00000 q^{45} -9.48683 q^{47} +3.00000 q^{49} -6.32456 q^{51} -6.00000 q^{53} +20.0000 q^{57} -6.32456 q^{59} -2.00000 q^{61} +22.1359 q^{63} +6.00000 q^{65} -9.48683 q^{67} +10.0000 q^{69} +6.32456 q^{71} +14.0000 q^{73} +3.16228 q^{75} -12.6491 q^{79} +19.0000 q^{81} -3.16228 q^{83} +2.00000 q^{85} +12.6491 q^{87} -10.0000 q^{89} -18.9737 q^{91} +20.0000 q^{93} -6.32456 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 14 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 + 14 * q^9	\( 2 q - 2 q^{5} + 14 q^{9} - 12 q^{13} - 4 q^{17} + 20 q^{21} + 2 q^{25} + 8 q^{29} - 4 q^{37} - 14 q^{45} + 6 q^{49} - 12 q^{53} + 40 q^{57} - 4 q^{61} + 12 q^{65} + 20 q^{69} + 28 q^{73} + 38 q^{81} + 4 q^{85} - 20 q^{89} + 40 q^{93} + 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 14 * q^9 - 12 * q^13 - 4 * q^17 + 20 * q^21 + 2 * q^25 + 8 * q^29 - 4 * q^37 - 14 * q^45 + 6 * q^49 - 12 * q^53 + 40 * q^57 - 4 * q^61 + 12 * q^65 + 20 * q^69 + 28 * q^73 + 38 * q^81 + 4 * q^85 - 20 * q^89 + 40 * q^93 + 4 * q^97

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