LMFDB - Embedded newform 640.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [640,2,Mod(1,640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(640, 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("640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 640 = 2^{7} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	640.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:	$5.11042572936$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23607 q^{3} -1.00000 q^{5} -3.23607 q^{7} -1.47214 q^{9} -2.00000 q^{11} -4.47214 q^{13} -1.23607 q^{15} +4.47214 q^{17} -4.47214 q^{19} -4.00000 q^{21} -4.76393 q^{23} +1.00000 q^{25} -5.52786 q^{27} -2.00000 q^{29} +6.47214 q^{31} -2.47214 q^{33} +3.23607 q^{35} +6.94427 q^{37} -5.52786 q^{39} +12.4721 q^{41} -7.70820 q^{43} +1.47214 q^{45} -7.23607 q^{47} +3.47214 q^{49} +5.52786 q^{51} +0.472136 q^{53} +2.00000 q^{55} -5.52786 q^{57} -8.47214 q^{59} -6.00000 q^{61} +4.76393 q^{63} +4.47214 q^{65} +7.70820 q^{67} -5.88854 q^{69} -2.47214 q^{71} +4.47214 q^{73} +1.23607 q^{75} +6.47214 q^{77} +12.9443 q^{79} -2.41641 q^{81} -3.70820 q^{83} -4.47214 q^{85} -2.47214 q^{87} -14.9443 q^{89} +14.4721 q^{91} +8.00000 q^{93} +4.47214 q^{95} -16.4721 q^{97} +2.94427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 6 q^{9} - 4 q^{11} + 2 q^{15} - 8 q^{21} - 14 q^{23} + 2 q^{25} - 20 q^{27} - 4 q^{29} + 4 q^{31} + 4 q^{33} + 2 q^{35} - 4 q^{37} - 20 q^{39} + 16 q^{41} - 2 q^{43}+ \cdots - 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 6 * q^9 - 4 * q^11 + 2 * q^15 - 8 * q^21 - 14 * q^23 + 2 * q^25 - 20 * q^27 - 4 * q^29 + 4 * q^31 + 4 * q^33 + 2 * q^35 - 4 * q^37 - 20 * q^39 + 16 * q^41 - 2 * q^43 - 6 * q^45 - 10 * q^47 - 2 * q^49 + 20 * q^51 - 8 * q^53 + 4 * q^55 - 20 * q^57 - 8 * q^59 - 12 * q^61 + 14 * q^63 + 2 * q^67 + 24 * q^69 + 4 * q^71 - 2 * q^75 + 4 * q^77 + 8 * q^79 + 22 * q^81 + 6 * q^83 + 4 * q^87 - 12 * q^89 + 20 * q^91 + 16 * q^93 - 24 * q^97 - 12 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.23607	0.713644	0.356822	−	0.934172i	$-0.383860\pi$
$3$	1.23607	0.713644	0.356822	+	0.934172i	$0.383860\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.23607	−1.22312	−0.611559	−	0.791199i	$-0.709457\pi$
$7$	−3.23607	−1.22312	−0.611559	+	0.791199i	$0.709457\pi$
$8$	0	0
$9$	−1.47214	−0.490712
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	−1.23607	−0.319151
$16$	0	0
$17$	4.47214	1.08465	0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214	1.08465	0.542326	+	0.840168i	$0.317544\pi$
$18$	0	0
$19$	−4.47214	−1.02598	−0.512989	−	0.858395i	$-0.671462\pi$
$19$	−4.47214	−1.02598	−0.512989	+	0.858395i	$0.671462\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	−4.76393	−0.993348	−0.496674	−	0.867937i	$-0.665446\pi$
$23$	−4.76393	−0.993348	−0.496674	+	0.867937i	$0.665446\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.52786	−1.06384
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	6.47214	1.16243	0.581215	−	0.813750i	$-0.302578\pi$
$31$	6.47214	1.16243	0.581215	+	0.813750i	$0.302578\pi$
$32$	0	0
$33$	−2.47214	−0.430344
$34$	0	0
$35$	3.23607	0.546995
$36$	0	0
$37$	6.94427	1.14163	0.570816	−	0.821078i	$-0.306627\pi$
$37$	6.94427	1.14163	0.570816	+	0.821078i	$0.306627\pi$
$38$	0	0
$39$	−5.52786	−0.885167
$40$	0	0
$41$	12.4721	1.94782	0.973910	−	0.226934i	$-0.0728701\pi$
$41$	12.4721	1.94782	0.973910	+	0.226934i	$0.0728701\pi$
$42$	0	0
$43$	−7.70820	−1.17549	−0.587745	−	0.809046i	$-0.699984\pi$
$43$	−7.70820	−1.17549	−0.587745	+	0.809046i	$0.699984\pi$
$44$	0	0
$45$	1.47214	0.219453
$46$	0	0
$47$	−7.23607	−1.05549	−0.527744	−	0.849403i	$-0.676962\pi$
$47$	−7.23607	−1.05549	−0.527744	+	0.849403i	$0.676962\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	0	0
$51$	5.52786	0.774056
$52$	0	0
$53$	0.472136	0.0648529	0.0324264	−	0.999474i	$-0.489677\pi$
$53$	0.472136	0.0648529	0.0324264	+	0.999474i	$0.489677\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	−5.52786	−0.732183
$58$	0	0
$59$	−8.47214	−1.10298	−0.551489	−	0.834182i	$-0.685940\pi$
$59$	−8.47214	−1.10298	−0.551489	+	0.834182i	$0.685940\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	4.76393	0.600199
$64$	0	0
$65$	4.47214	0.554700
$66$	0	0
$67$	7.70820	0.941707	0.470853	−	0.882211i	$-0.343946\pi$
$67$	7.70820	0.941707	0.470853	+	0.882211i	$0.343946\pi$
$68$	0	0
$69$	−5.88854	−0.708897
$70$	0	0
$71$	−2.47214	−0.293389	−0.146694	−	0.989182i	$-0.546863\pi$
$71$	−2.47214	−0.293389	−0.146694	+	0.989182i	$0.546863\pi$
$72$	0	0
$73$	4.47214	0.523424	0.261712	−	0.965146i	$-0.415713\pi$
$73$	4.47214	0.523424	0.261712	+	0.965146i	$0.415713\pi$
$74$	0	0
$75$	1.23607	0.142729
$76$	0	0
$77$	6.47214	0.737568
$78$	0	0
$79$	12.9443	1.45634	0.728172	−	0.685394i	$-0.240370\pi$
$79$	12.9443	1.45634	0.728172	+	0.685394i	$0.240370\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	−3.70820	−0.407028	−0.203514	−	0.979072i	$-0.565236\pi$
$83$	−3.70820	−0.407028	−0.203514	+	0.979072i	$0.565236\pi$
$84$	0	0
$85$	−4.47214	−0.485071
$86$	0	0
$87$	−2.47214	−0.265041
$88$	0	0
$89$	−14.9443	−1.58409	−0.792045	−	0.610463i	$-0.790983\pi$
$89$	−14.9443	−1.58409	−0.792045	+	0.610463i	$0.790983\pi$
$90$	0	0
$91$	14.4721	1.51709
$92$	0	0
$93$	8.00000	0.829561
$94$	0	0
$95$	4.47214	0.458831
$96$	0	0
$97$	−16.4721	−1.67249	−0.836246	−	0.548354i	$-0.815254\pi$
$97$	−16.4721	−1.67249	−0.836246	+	0.548354i	$0.815254\pi$
$98$	0	0
$99$	2.94427	0.295910
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	3.23607	0.318859	0.159430	−	0.987209i	$-0.449034\pi$
$103$	3.23607	0.318859	0.159430	+	0.987209i	$0.449034\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	−17.2361	−1.66627	−0.833137	−	0.553067i	$-0.813457\pi$
$107$	−17.2361	−1.66627	−0.833137	+	0.553067i	$0.813457\pi$
$108$	0	0
$109$	14.9443	1.43140	0.715701	−	0.698407i	$-0.246107\pi$
$109$	14.9443	1.43140	0.715701	+	0.698407i	$0.246107\pi$
$110$	0	0
$111$	8.58359	0.814719
$112$	0	0
$113$	−2.94427	−0.276974	−0.138487	−	0.990364i	$-0.544224\pi$
$113$	−2.94427	−0.276974	−0.138487	+	0.990364i	$0.544224\pi$
$114$	0	0
$115$	4.76393	0.444239
$116$	0	0
$117$	6.58359	0.608653
$118$	0	0
$119$	−14.4721	−1.32666
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	15.4164	1.39005
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−20.1803	−1.79072	−0.895358	−	0.445348i	$-0.853080\pi$
$127$	−20.1803	−1.79072	−0.895358	+	0.445348i	$0.853080\pi$
$128$	0	0
$129$	−9.52786	−0.838882
$130$	0	0
$131$	14.9443	1.30569	0.652844	−	0.757493i	$-0.273576\pi$
$131$	14.9443	1.30569	0.652844	+	0.757493i	$0.273576\pi$
$132$	0	0
$133$	14.4721	1.25489
$134$	0	0
$135$	5.52786	0.475763
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	20.4721	1.73642	0.868212	−	0.496194i	$-0.165269\pi$
$139$	20.4721	1.73642	0.868212	+	0.496194i	$0.165269\pi$
$140$	0	0
$141$	−8.94427	−0.753244
$142$	0	0
$143$	8.94427	0.747958
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	4.29180	0.353981
$148$	0	0
$149$	6.94427	0.568897	0.284448	−	0.958691i	$-0.408190\pi$
$149$	6.94427	0.568897	0.284448	+	0.958691i	$0.408190\pi$
$150$	0	0
$151$	23.4164	1.90560	0.952800	−	0.303598i	$-0.0981881\pi$
$151$	23.4164	1.90560	0.952800	+	0.303598i	$0.0981881\pi$
$152$	0	0
$153$	−6.58359	−0.532252
$154$	0	0
$155$	−6.47214	−0.519854
$156$	0	0
$157$	5.05573	0.403491	0.201746	−	0.979438i	$-0.435339\pi$
$157$	5.05573	0.403491	0.201746	+	0.979438i	$0.435339\pi$
$158$	0	0
$159$	0.583592	0.0462819
$160$	0	0
$161$	15.4164	1.21498
$162$	0	0
$163$	−11.7082	−0.917057	−0.458529	−	0.888680i	$-0.651623\pi$
$163$	−11.7082	−0.917057	−0.458529	+	0.888680i	$0.651623\pi$
$164$	0	0
$165$	2.47214	0.192456
$166$	0	0
$167$	14.6525	1.13384	0.566921	−	0.823772i	$-0.308134\pi$
$167$	14.6525	1.13384	0.566921	+	0.823772i	$0.308134\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	6.58359	0.503460
$172$	0	0
$173$	−10.9443	−0.832078	−0.416039	−	0.909347i	$-0.636582\pi$
$173$	−10.9443	−0.832078	−0.416039	+	0.909347i	$0.636582\pi$
$174$	0	0
$175$	−3.23607	−0.244624
$176$	0	0
$177$	−10.4721	−0.787134
$178$	0	0
$179$	−20.4721	−1.53016	−0.765080	−	0.643936i	$-0.777300\pi$
$179$	−20.4721	−1.53016	−0.765080	+	0.643936i	$0.777300\pi$
$180$	0	0
$181$	10.9443	0.813481	0.406741	−	0.913544i	$-0.366665\pi$
$181$	10.9443	0.813481	0.406741	+	0.913544i	$0.366665\pi$
$182$	0	0
$183$	−7.41641	−0.548237
$184$	0	0
$185$	−6.94427	−0.510553
$186$	0	0
$187$	−8.94427	−0.654070
$188$	0	0
$189$	17.8885	1.30120
$190$	0	0
$191$	11.4164	0.826062	0.413031	−	0.910717i	$-0.364470\pi$
$191$	11.4164	0.826062	0.413031	+	0.910717i	$0.364470\pi$
$192$	0	0
$193$	−11.5279	−0.829794	−0.414897	−	0.909868i	$-0.636182\pi$
$193$	−11.5279	−0.829794	−0.414897	+	0.909868i	$0.636182\pi$
$194$	0	0
$195$	5.52786	0.395859
$196$	0	0
$197$	0.472136	0.0336383	0.0168191	−	0.999859i	$-0.494646\pi$
$197$	0.472136	0.0336383	0.0168191	+	0.999859i	$0.494646\pi$
$198$	0	0
$199$	−0.944272	−0.0669377	−0.0334688	−	0.999440i	$-0.510655\pi$
$199$	−0.944272	−0.0669377	−0.0334688	+	0.999440i	$0.510655\pi$
$200$	0	0
$201$	9.52786	0.672044
$202$	0	0
$203$	6.47214	0.454255
$204$	0	0
$205$	−12.4721	−0.871092
$206$	0	0
$207$	7.01316	0.487448
$208$	0	0
$209$	8.94427	0.618688
$210$	0	0
$211$	−1.05573	−0.0726793	−0.0363397	−	0.999339i	$-0.511570\pi$
$211$	−1.05573	−0.0726793	−0.0363397	+	0.999339i	$0.511570\pi$
$212$	0	0
$213$	−3.05573	−0.209375
$214$	0	0
$215$	7.70820	0.525695
$216$	0	0
$217$	−20.9443	−1.42179
$218$	0	0
$219$	5.52786	0.373538
$220$	0	0
$221$	−20.0000	−1.34535
$222$	0	0
$223$	−8.76393	−0.586876	−0.293438	−	0.955978i	$-0.594799\pi$
$223$	−8.76393	−0.586876	−0.293438	+	0.955978i	$0.594799\pi$
$224$	0	0
$225$	−1.47214	−0.0981424
$226$	0	0
$227$	−10.1803	−0.675693	−0.337846	−	0.941201i	$-0.609698\pi$
$227$	−10.1803	−0.675693	−0.337846	+	0.941201i	$0.609698\pi$
$228$	0	0
$229$	2.94427	0.194563	0.0972815	−	0.995257i	$-0.468985\pi$
$229$	2.94427	0.194563	0.0972815	+	0.995257i	$0.468985\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	15.5279	1.01726	0.508632	−	0.860984i	$-0.330151\pi$
$233$	15.5279	1.01726	0.508632	+	0.860984i	$0.330151\pi$
$234$	0	0
$235$	7.23607	0.472029
$236$	0	0
$237$	16.0000	1.03931
$238$	0	0
$239$	−12.9443	−0.837295	−0.418648	−	0.908149i	$-0.637496\pi$
$239$	−12.9443	−0.837295	−0.418648	+	0.908149i	$0.637496\pi$
$240$	0	0
$241$	26.3607	1.69804	0.849020	−	0.528360i	$-0.177193\pi$
$241$	26.3607	1.69804	0.849020	+	0.528360i	$0.177193\pi$
$242$	0	0
$243$	13.5967	0.872232
$244$	0	0
$245$	−3.47214	−0.221827
$246$	0	0
$247$	20.0000	1.27257
$248$	0	0
$249$	−4.58359	−0.290473
$250$	0	0
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	0	0
$253$	9.52786	0.599012
$254$	0	0
$255$	−5.52786	−0.346168
$256$	0	0
$257$	−2.94427	−0.183659	−0.0918293	−	0.995775i	$-0.529271\pi$
$257$	−2.94427	−0.183659	−0.0918293	+	0.995775i	$0.529271\pi$
$258$	0	0
$259$	−22.4721	−1.39635
$260$	0	0
$261$	2.94427	0.182246
$262$	0	0
$263$	−17.7082	−1.09193	−0.545967	−	0.837806i	$-0.683838\pi$
$263$	−17.7082	−1.09193	−0.545967	+	0.837806i	$0.683838\pi$
$264$	0	0
$265$	−0.472136	−0.0290031
$266$	0	0
$267$	−18.4721	−1.13048
$268$	0	0
$269$	−23.8885	−1.45651	−0.728255	−	0.685306i	$-0.759668\pi$
$269$	−23.8885	−1.45651	−0.728255	+	0.685306i	$0.759668\pi$
$270$	0	0
$271$	−24.3607	−1.47981	−0.739903	−	0.672714i	$-0.765129\pi$
$271$	−24.3607	−1.47981	−0.739903	+	0.672714i	$0.765129\pi$
$272$	0	0
$273$	17.8885	1.08266
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	−10.9443	−0.657578	−0.328789	−	0.944403i	$-0.606640\pi$
$277$	−10.9443	−0.657578	−0.328789	+	0.944403i	$0.606640\pi$
$278$	0	0
$279$	−9.52786	−0.570418
$280$	0	0
$281$	3.52786	0.210455	0.105227	−	0.994448i	$-0.466443\pi$
$281$	3.52786	0.210455	0.105227	+	0.994448i	$0.466443\pi$
$282$	0	0
$283$	8.29180	0.492896	0.246448	−	0.969156i	$-0.420737\pi$
$283$	8.29180	0.492896	0.246448	+	0.969156i	$0.420737\pi$
$284$	0	0
$285$	5.52786	0.327442
$286$	0	0
$287$	−40.3607	−2.38242
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	−20.3607	−1.19356
$292$	0	0
$293$	−23.8885	−1.39558	−0.697792	−	0.716301i	$-0.745834\pi$
$293$	−23.8885	−1.39558	−0.697792	+	0.716301i	$0.745834\pi$
$294$	0	0
$295$	8.47214	0.493267
$296$	0	0
$297$	11.0557	0.641518
$298$	0	0
$299$	21.3050	1.23210
$300$	0	0
$301$	24.9443	1.43776
$302$	0	0
$303$	−12.3607	−0.710102
$304$	0	0
$305$	6.00000	0.343559
$306$	0	0
$307$	−0.291796	−0.0166537	−0.00832684	−	0.999965i	$-0.502651\pi$
$307$	−0.291796	−0.0166537	−0.00832684	+	0.999965i	$0.502651\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	−15.4164	−0.874184	−0.437092	−	0.899417i	$-0.643992\pi$
$311$	−15.4164	−0.874184	−0.437092	+	0.899417i	$0.643992\pi$
$312$	0	0
$313$	−28.8328	−1.62973	−0.814864	−	0.579653i	$-0.803188\pi$
$313$	−28.8328	−1.62973	−0.814864	+	0.579653i	$0.803188\pi$
$314$	0	0
$315$	−4.76393	−0.268417
$316$	0	0
$317$	−20.4721	−1.14983	−0.574915	−	0.818213i	$-0.694965\pi$
$317$	−20.4721	−1.14983	−0.574915	+	0.818213i	$0.694965\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	−21.3050	−1.18913
$322$	0	0
$323$	−20.0000	−1.11283
$324$	0	0
$325$	−4.47214	−0.248069
$326$	0	0
$327$	18.4721	1.02151
$328$	0	0
$329$	23.4164	1.29099
$330$	0	0
$331$	15.8885	0.873313	0.436657	−	0.899628i	$-0.356162\pi$
$331$	15.8885	0.873313	0.436657	+	0.899628i	$0.356162\pi$
$332$	0	0
$333$	−10.2229	−0.560212
$334$	0	0
$335$	−7.70820	−0.421144
$336$	0	0
$337$	5.05573	0.275403	0.137702	−	0.990474i	$-0.456029\pi$
$337$	5.05573	0.275403	0.137702	+	0.990474i	$0.456029\pi$
$338$	0	0
$339$	−3.63932	−0.197661
$340$	0	0
$341$	−12.9443	−0.700972
$342$	0	0
$343$	11.4164	0.616428
$344$	0	0
$345$	5.88854	0.317029
$346$	0	0
$347$	−17.2361	−0.925281	−0.462640	−	0.886546i	$-0.653098\pi$
$347$	−17.2361	−0.925281	−0.462640	+	0.886546i	$0.653098\pi$
$348$	0	0
$349$	−14.9443	−0.799949	−0.399974	−	0.916526i	$-0.630981\pi$
$349$	−14.9443	−0.799949	−0.399974	+	0.916526i	$0.630981\pi$
$350$	0	0
$351$	24.7214	1.31953
$352$	0	0
$353$	5.05573	0.269089	0.134545	−	0.990908i	$-0.457043\pi$
$353$	5.05573	0.269089	0.134545	+	0.990908i	$0.457043\pi$
$354$	0	0
$355$	2.47214	0.131207
$356$	0	0
$357$	−17.8885	−0.946762
$358$	0	0
$359$	−15.0557	−0.794611	−0.397305	−	0.917686i	$-0.630055\pi$
$359$	−15.0557	−0.794611	−0.397305	+	0.917686i	$0.630055\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−8.65248	−0.454137
$364$	0	0
$365$	−4.47214	−0.234082
$366$	0	0
$367$	−18.2918	−0.954824	−0.477412	−	0.878680i	$-0.658425\pi$
$367$	−18.2918	−0.954824	−0.477412	+	0.878680i	$0.658425\pi$
$368$	0	0
$369$	−18.3607	−0.955819
$370$	0	0
$371$	−1.52786	−0.0793227
$372$	0	0
$373$	5.05573	0.261776	0.130888	−	0.991397i	$-0.458217\pi$
$373$	5.05573	0.261776	0.130888	+	0.991397i	$0.458217\pi$
$374$	0	0
$375$	−1.23607	−0.0638303
$376$	0	0
$377$	8.94427	0.460653
$378$	0	0
$379$	15.5279	0.797613	0.398806	−	0.917035i	$-0.369425\pi$
$379$	15.5279	0.797613	0.398806	+	0.917035i	$0.369425\pi$
$380$	0	0
$381$	−24.9443	−1.27793
$382$	0	0
$383$	10.2918	0.525886	0.262943	−	0.964811i	$-0.415307\pi$
$383$	10.2918	0.525886	0.262943	+	0.964811i	$0.415307\pi$
$384$	0	0
$385$	−6.47214	−0.329851
$386$	0	0
$387$	11.3475	0.576827
$388$	0	0
$389$	11.8885	0.602773	0.301387	−	0.953502i	$-0.402551\pi$
$389$	11.8885	0.602773	0.301387	+	0.953502i	$0.402551\pi$
$390$	0	0
$391$	−21.3050	−1.07744
$392$	0	0
$393$	18.4721	0.931796
$394$	0	0
$395$	−12.9443	−0.651297
$396$	0	0
$397$	−12.4721	−0.625959	−0.312979	−	0.949760i	$-0.601327\pi$
$397$	−12.4721	−0.625959	−0.312979	+	0.949760i	$0.601327\pi$
$398$	0	0
$399$	17.8885	0.895547
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	−28.9443	−1.44182
$404$	0	0
$405$	2.41641	0.120072
$406$	0	0
$407$	−13.8885	−0.688430
$408$	0	0
$409$	17.4164	0.861186	0.430593	−	0.902546i	$-0.358304\pi$
$409$	17.4164	0.861186	0.430593	+	0.902546i	$0.358304\pi$
$410$	0	0
$411$	2.47214	0.121941
$412$	0	0
$413$	27.4164	1.34907
$414$	0	0
$415$	3.70820	0.182029
$416$	0	0
$417$	25.3050	1.23919
$418$	0	0
$419$	−4.47214	−0.218478	−0.109239	−	0.994016i	$-0.534841\pi$
$419$	−4.47214	−0.218478	−0.109239	+	0.994016i	$0.534841\pi$
$420$	0	0
$421$	−28.8328	−1.40523	−0.702613	−	0.711572i	$-0.747983\pi$
$421$	−28.8328	−1.40523	−0.702613	+	0.711572i	$0.747983\pi$
$422$	0	0
$423$	10.6525	0.517941
$424$	0	0
$425$	4.47214	0.216930
$426$	0	0
$427$	19.4164	0.939626
$428$	0	0
$429$	11.0557	0.533776
$430$	0	0
$431$	−24.3607	−1.17341	−0.586706	−	0.809800i	$-0.699576\pi$
$431$	−24.3607	−1.17341	−0.586706	+	0.809800i	$0.699576\pi$
$432$	0	0
$433$	−0.472136	−0.0226894	−0.0113447	−	0.999936i	$-0.503611\pi$
$433$	−0.472136	−0.0226894	−0.0113447	+	0.999936i	$0.503611\pi$
$434$	0	0
$435$	2.47214	0.118530
$436$	0	0
$437$	21.3050	1.01915
$438$	0	0
$439$	15.0557	0.718571	0.359285	−	0.933228i	$-0.383020\pi$
$439$	15.0557	0.718571	0.359285	+	0.933228i	$0.383020\pi$
$440$	0	0
$441$	−5.11146	−0.243403
$442$	0	0
$443$	−4.65248	−0.221046	−0.110523	−	0.993874i	$-0.535253\pi$
$443$	−4.65248	−0.221046	−0.110523	+	0.993874i	$0.535253\pi$
$444$	0	0
$445$	14.9443	0.708426
$446$	0	0
$447$	8.58359	0.405990
$448$	0	0
$449$	−1.41641	−0.0668444	−0.0334222	−	0.999441i	$-0.510641\pi$
$449$	−1.41641	−0.0668444	−0.0334222	+	0.999441i	$0.510641\pi$
$450$	0	0
$451$	−24.9443	−1.17458
$452$	0	0
$453$	28.9443	1.35992
$454$	0	0
$455$	−14.4721	−0.678464
$456$	0	0
$457$	3.88854	0.181898	0.0909492	−	0.995856i	$-0.471010\pi$
$457$	3.88854	0.181898	0.0909492	+	0.995856i	$0.471010\pi$
$458$	0	0
$459$	−24.7214	−1.15389
$460$	0	0
$461$	41.7771	1.94575	0.972876	−	0.231325i	$-0.0743061\pi$
$461$	41.7771	1.94575	0.972876	+	0.231325i	$0.0743061\pi$
$462$	0	0
$463$	1.12461	0.0522651	0.0261326	−	0.999658i	$-0.491681\pi$
$463$	1.12461	0.0522651	0.0261326	+	0.999658i	$0.491681\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	41.5967	1.92487	0.962434	−	0.271516i	$-0.0875249\pi$
$467$	41.5967	1.92487	0.962434	+	0.271516i	$0.0875249\pi$
$468$	0	0
$469$	−24.9443	−1.15182
$470$	0	0
$471$	6.24922	0.287949
$472$	0	0
$473$	15.4164	0.708847
$474$	0	0
$475$	−4.47214	−0.205196
$476$	0	0
$477$	−0.695048	−0.0318241
$478$	0	0
$479$	12.9443	0.591439	0.295719	−	0.955275i	$-0.404441\pi$
$479$	12.9443	0.591439	0.295719	+	0.955275i	$0.404441\pi$
$480$	0	0
$481$	−31.0557	−1.41602
$482$	0	0
$483$	19.0557	0.867066
$484$	0	0
$485$	16.4721	0.747961
$486$	0	0
$487$	12.7639	0.578389	0.289194	−	0.957270i	$-0.406613\pi$
$487$	12.7639	0.578389	0.289194	+	0.957270i	$0.406613\pi$
$488$	0	0
$489$	−14.4721	−0.654453
$490$	0	0
$491$	−21.0557	−0.950232	−0.475116	−	0.879923i	$-0.657594\pi$
$491$	−21.0557	−0.950232	−0.475116	+	0.879923i	$0.657594\pi$
$492$	0	0
$493$	−8.94427	−0.402830
$494$	0	0
$495$	−2.94427	−0.132335
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	14.5836	0.652851	0.326426	−	0.945223i	$-0.394156\pi$
$499$	14.5836	0.652851	0.326426	+	0.945223i	$0.394156\pi$
$500$	0	0
$501$	18.1115	0.809160
$502$	0	0
$503$	5.12461	0.228495	0.114248	−	0.993452i	$-0.463554\pi$
$503$	5.12461	0.228495	0.114248	+	0.993452i	$0.463554\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	8.65248	0.384270
$508$	0	0
$509$	−16.8328	−0.746101	−0.373051	−	0.927811i	$-0.621688\pi$
$509$	−16.8328	−0.746101	−0.373051	+	0.927811i	$0.621688\pi$
$510$	0	0
$511$	−14.4721	−0.640210
$512$	0	0
$513$	24.7214	1.09147
$514$	0	0
$515$	−3.23607	−0.142598
$516$	0	0
$517$	14.4721	0.636484
$518$	0	0
$519$	−13.5279	−0.593807
$520$	0	0
$521$	−15.8885	−0.696090	−0.348045	−	0.937478i	$-0.613154\pi$
$521$	−15.8885	−0.696090	−0.348045	+	0.937478i	$0.613154\pi$
$522$	0	0
$523$	20.0689	0.877551	0.438776	−	0.898597i	$-0.355412\pi$
$523$	20.0689	0.877551	0.438776	+	0.898597i	$0.355412\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	28.9443	1.26083
$528$	0	0
$529$	−0.304952	−0.0132588
$530$	0	0
$531$	12.4721	0.541245
$532$	0	0
$533$	−55.7771	−2.41597
$534$	0	0
$535$	17.2361	0.745180
$536$	0	0
$537$	−25.3050	−1.09199
$538$	0	0
$539$	−6.94427	−0.299111
$540$	0	0
$541$	23.8885	1.02705	0.513524	−	0.858075i	$-0.328340\pi$
$541$	23.8885	1.02705	0.513524	+	0.858075i	$0.328340\pi$
$542$	0	0
$543$	13.5279	0.580536
$544$	0	0
$545$	−14.9443	−0.640142
$546$	0	0
$547$	20.6525	0.883036	0.441518	−	0.897252i	$-0.354440\pi$
$547$	20.6525	0.883036	0.441518	+	0.897252i	$0.354440\pi$
$548$	0	0
$549$	8.83282	0.376975
$550$	0	0
$551$	8.94427	0.381039
$552$	0	0
$553$	−41.8885	−1.78128
$554$	0	0
$555$	−8.58359	−0.364353
$556$	0	0
$557$	13.0557	0.553189	0.276594	−	0.960987i	$-0.410794\pi$
$557$	13.0557	0.553189	0.276594	+	0.960987i	$0.410794\pi$
$558$	0	0
$559$	34.4721	1.45802
$560$	0	0
$561$	−11.0557	−0.466773
$562$	0	0
$563$	4.29180	0.180878	0.0904388	−	0.995902i	$-0.471173\pi$
$563$	4.29180	0.180878	0.0904388	+	0.995902i	$0.471173\pi$
$564$	0	0
$565$	2.94427	0.123866
$566$	0	0
$567$	7.81966	0.328395
$568$	0	0
$569$	−34.3607	−1.44047	−0.720237	−	0.693728i	$-0.755967\pi$
$569$	−34.3607	−1.44047	−0.720237	+	0.693728i	$0.755967\pi$
$570$	0	0
$571$	−16.8328	−0.704431	−0.352216	−	0.935919i	$-0.614572\pi$
$571$	−16.8328	−0.704431	−0.352216	+	0.935919i	$0.614572\pi$
$572$	0	0
$573$	14.1115	0.589515
$574$	0	0
$575$	−4.76393	−0.198670
$576$	0	0
$577$	19.8885	0.827971	0.413985	−	0.910283i	$-0.364136\pi$
$577$	19.8885	0.827971	0.413985	+	0.910283i	$0.364136\pi$
$578$	0	0
$579$	−14.2492	−0.592178
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	−0.944272	−0.0391077
$584$	0	0
$585$	−6.58359	−0.272198
$586$	0	0
$587$	8.65248	0.357126	0.178563	−	0.983928i	$-0.442855\pi$
$587$	8.65248	0.357126	0.178563	+	0.983928i	$0.442855\pi$
$588$	0	0
$589$	−28.9443	−1.19263
$590$	0	0
$591$	0.583592	0.0240058
$592$	0	0
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	14.4721	0.593300
$596$	0	0
$597$	−1.16718	−0.0477697
$598$	0	0
$599$	−21.8885	−0.894342	−0.447171	−	0.894449i	$-0.647568\pi$
$599$	−21.8885	−0.894342	−0.447171	+	0.894449i	$0.647568\pi$
$600$	0	0
$601$	−22.3607	−0.912111	−0.456056	−	0.889951i	$-0.650738\pi$
$601$	−22.3607	−0.912111	−0.456056	+	0.889951i	$0.650738\pi$
$602$	0	0
$603$	−11.3475	−0.462107
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	6.87539	0.279063	0.139532	−	0.990218i	$-0.455440\pi$
$607$	6.87539	0.279063	0.139532	+	0.990218i	$0.455440\pi$
$608$	0	0
$609$	8.00000	0.324176
$610$	0	0
$611$	32.3607	1.30917
$612$	0	0
$613$	19.5279	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$613$	19.5279	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$614$	0	0
$615$	−15.4164	−0.621650
$616$	0	0
$617$	25.4164	1.02323	0.511613	−	0.859216i	$-0.329048\pi$
$617$	25.4164	1.02323	0.511613	+	0.859216i	$0.329048\pi$
$618$	0	0
$619$	20.4721	0.822845	0.411422	−	0.911445i	$-0.365032\pi$
$619$	20.4721	0.822845	0.411422	+	0.911445i	$0.365032\pi$
$620$	0	0
$621$	26.3344	1.05676
$622$	0	0
$623$	48.3607	1.93753
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	11.0557	0.441523
$628$	0	0
$629$	31.0557	1.23827
$630$	0	0
$631$	−34.4721	−1.37231	−0.686157	−	0.727453i	$-0.740704\pi$
$631$	−34.4721	−1.37231	−0.686157	+	0.727453i	$0.740704\pi$
$632$	0	0
$633$	−1.30495	−0.0518672
$634$	0	0
$635$	20.1803	0.800832
$636$	0	0
$637$	−15.5279	−0.615236
$638$	0	0
$639$	3.63932	0.143969
$640$	0	0
$641$	0.472136	0.0186482	0.00932412	−	0.999957i	$-0.497032\pi$
$641$	0.472136	0.0186482	0.00932412	+	0.999957i	$0.497032\pi$
$642$	0	0
$643$	22.1803	0.874707	0.437354	−	0.899290i	$-0.355916\pi$
$643$	22.1803	0.874707	0.437354	+	0.899290i	$0.355916\pi$
$644$	0	0
$645$	9.52786	0.375159
$646$	0	0
$647$	12.7639	0.501802	0.250901	−	0.968013i	$-0.419273\pi$
$647$	12.7639	0.501802	0.250901	+	0.968013i	$0.419273\pi$
$648$	0	0
$649$	16.9443	0.665121
$650$	0	0
$651$	−25.8885	−1.01465
$652$	0	0
$653$	−49.4164	−1.93381	−0.966907	−	0.255130i	$-0.917882\pi$
$653$	−49.4164	−1.93381	−0.966907	+	0.255130i	$0.917882\pi$
$654$	0	0
$655$	−14.9443	−0.583921
$656$	0	0
$657$	−6.58359	−0.256850
$658$	0	0
$659$	21.4164	0.834265	0.417132	−	0.908846i	$-0.363035\pi$
$659$	21.4164	0.834265	0.417132	+	0.908846i	$0.363035\pi$
$660$	0	0
$661$	35.8885	1.39590	0.697951	−	0.716145i	$-0.254095\pi$
$661$	35.8885	1.39590	0.697951	+	0.716145i	$0.254095\pi$
$662$	0	0
$663$	−24.7214	−0.960098
$664$	0	0
$665$	−14.4721	−0.561205
$666$	0	0
$667$	9.52786	0.368920
$668$	0	0
$669$	−10.8328	−0.418821
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	−23.3050	−0.898340	−0.449170	−	0.893446i	$-0.648280\pi$
$673$	−23.3050	−0.898340	−0.449170	+	0.893446i	$0.648280\pi$
$674$	0	0
$675$	−5.52786	−0.212768
$676$	0	0
$677$	−30.3607	−1.16686	−0.583428	−	0.812165i	$-0.698289\pi$
$677$	−30.3607	−1.16686	−0.583428	+	0.812165i	$0.698289\pi$
$678$	0	0
$679$	53.3050	2.04566
$680$	0	0
$681$	−12.5836	−0.482204
$682$	0	0
$683$	24.2918	0.929500	0.464750	−	0.885442i	$-0.346144\pi$
$683$	24.2918	0.929500	0.464750	+	0.885442i	$0.346144\pi$
$684$	0	0
$685$	−2.00000	−0.0764161
$686$	0	0
$687$	3.63932	0.138849
$688$	0	0
$689$	−2.11146	−0.0804401
$690$	0	0
$691$	−30.0000	−1.14125	−0.570627	−	0.821209i	$-0.693300\pi$
$691$	−30.0000	−1.14125	−0.570627	+	0.821209i	$0.693300\pi$
$692$	0	0
$693$	−9.52786	−0.361934
$694$	0	0
$695$	−20.4721	−0.776552
$696$	0	0
$697$	55.7771	2.11271
$698$	0	0
$699$	19.1935	0.725965
$700$	0	0
$701$	−9.05573	−0.342030	−0.171015	−	0.985268i	$-0.554705\pi$
$701$	−9.05573	−0.342030	−0.171015	+	0.985268i	$0.554705\pi$
$702$	0	0
$703$	−31.0557	−1.17129
$704$	0	0
$705$	8.94427	0.336861
$706$	0	0
$707$	32.3607	1.21705
$708$	0	0
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	0	0
$711$	−19.0557	−0.714646
$712$	0	0
$713$	−30.8328	−1.15470
$714$	0	0
$715$	−8.94427	−0.334497
$716$	0	0
$717$	−16.0000	−0.597531
$718$	0	0
$719$	22.8328	0.851520	0.425760	−	0.904836i	$-0.360007\pi$
$719$	22.8328	0.851520	0.425760	+	0.904836i	$0.360007\pi$
$720$	0	0
$721$	−10.4721	−0.390003
$722$	0	0
$723$	32.5836	1.21180
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	1.70820	0.0633538	0.0316769	−	0.999498i	$-0.489915\pi$
$727$	1.70820	0.0633538	0.0316769	+	0.999498i	$0.489915\pi$
$728$	0	0
$729$	24.0557	0.890953
$730$	0	0
$731$	−34.4721	−1.27500
$732$	0	0
$733$	51.8885	1.91655	0.958274	−	0.285853i	$-0.0922768\pi$
$733$	51.8885	1.91655	0.958274	+	0.285853i	$0.0922768\pi$
$734$	0	0
$735$	−4.29180	−0.158305
$736$	0	0
$737$	−15.4164	−0.567871
$738$	0	0
$739$	49.1935	1.80961	0.904806	−	0.425824i	$-0.140016\pi$
$739$	49.1935	1.80961	0.904806	+	0.425824i	$0.140016\pi$
$740$	0	0
$741$	24.7214	0.908162
$742$	0	0
$743$	−30.6525	−1.12453	−0.562265	−	0.826957i	$-0.690070\pi$
$743$	−30.6525	−1.12453	−0.562265	+	0.826957i	$0.690070\pi$
$744$	0	0
$745$	−6.94427	−0.254418
$746$	0	0
$747$	5.45898	0.199734
$748$	0	0
$749$	55.7771	2.03805
$750$	0	0
$751$	16.3607	0.597010	0.298505	−	0.954408i	$-0.403512\pi$
$751$	16.3607	0.597010	0.298505	+	0.954408i	$0.403512\pi$
$752$	0	0
$753$	−2.47214	−0.0900896
$754$	0	0
$755$	−23.4164	−0.852210
$756$	0	0
$757$	19.8885	0.722861	0.361431	−	0.932399i	$-0.382288\pi$
$757$	19.8885	0.722861	0.361431	+	0.932399i	$0.382288\pi$
$758$	0	0
$759$	11.7771	0.427481
$760$	0	0
$761$	3.88854	0.140960	0.0704798	−	0.997513i	$-0.477547\pi$
$761$	3.88854	0.140960	0.0704798	+	0.997513i	$0.477547\pi$
$762$	0	0
$763$	−48.3607	−1.75077
$764$	0	0
$765$	6.58359	0.238030
$766$	0	0
$767$	37.8885	1.36808
$768$	0	0
$769$	−14.9443	−0.538904	−0.269452	−	0.963014i	$-0.586843\pi$
$769$	−14.9443	−0.538904	−0.269452	+	0.963014i	$0.586843\pi$
$770$	0	0
$771$	−3.63932	−0.131067
$772$	0	0
$773$	18.3607	0.660388	0.330194	−	0.943913i	$-0.392886\pi$
$773$	18.3607	0.660388	0.330194	+	0.943913i	$0.392886\pi$
$774$	0	0
$775$	6.47214	0.232486
$776$	0	0
$777$	−27.7771	−0.996497
$778$	0	0
$779$	−55.7771	−1.99842
$780$	0	0
$781$	4.94427	0.176920
$782$	0	0
$783$	11.0557	0.395099
$784$	0	0
$785$	−5.05573	−0.180447
$786$	0	0
$787$	−44.0689	−1.57089	−0.785443	−	0.618934i	$-0.787565\pi$
$787$	−44.0689	−1.57089	−0.785443	+	0.618934i	$0.787565\pi$
$788$	0	0
$789$	−21.8885	−0.779253
$790$	0	0
$791$	9.52786	0.338772
$792$	0	0
$793$	26.8328	0.952861
$794$	0	0
$795$	−0.583592	−0.0206979
$796$	0	0
$797$	−25.4164	−0.900295	−0.450148	−	0.892954i	$-0.648629\pi$
$797$	−25.4164	−0.900295	−0.450148	+	0.892954i	$0.648629\pi$
$798$	0	0
$799$	−32.3607	−1.14484
$800$	0	0
$801$	22.0000	0.777332
$802$	0	0
$803$	−8.94427	−0.315637
$804$	0	0
$805$	−15.4164	−0.543357
$806$	0	0
$807$	−29.5279	−1.03943
$808$	0	0
$809$	−12.1115	−0.425816	−0.212908	−	0.977072i	$-0.568293\pi$
$809$	−12.1115	−0.425816	−0.212908	+	0.977072i	$0.568293\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	0	0
$813$	−30.1115	−1.05605
$814$	0	0
$815$	11.7082	0.410120
$816$	0	0
$817$	34.4721	1.20603
$818$	0	0
$819$	−21.3050	−0.744455
$820$	0	0
$821$	21.0557	0.734850	0.367425	−	0.930053i	$-0.380239\pi$
$821$	21.0557	0.734850	0.367425	+	0.930053i	$0.380239\pi$
$822$	0	0
$823$	−1.70820	−0.0595442	−0.0297721	−	0.999557i	$-0.509478\pi$
$823$	−1.70820	−0.0595442	−0.0297721	+	0.999557i	$0.509478\pi$
$824$	0	0
$825$	−2.47214	−0.0860687
$826$	0	0
$827$	−6.18034	−0.214911	−0.107456	−	0.994210i	$-0.534270\pi$
$827$	−6.18034	−0.214911	−0.107456	+	0.994210i	$0.534270\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	0	0
$831$	−13.5279	−0.469276
$832$	0	0
$833$	15.5279	0.538009
$834$	0	0
$835$	−14.6525	−0.507070
$836$	0	0
$837$	−35.7771	−1.23664
$838$	0	0
$839$	4.00000	0.138095	0.0690477	−	0.997613i	$-0.478004\pi$
$839$	4.00000	0.138095	0.0690477	+	0.997613i	$0.478004\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	4.36068	0.150190
$844$	0	0
$845$	−7.00000	−0.240807
$846$	0	0
$847$	22.6525	0.778348
$848$	0	0
$849$	10.2492	0.351752
$850$	0	0
$851$	−33.0820	−1.13404
$852$	0	0
$853$	−7.52786	−0.257749	−0.128875	−	0.991661i	$-0.541136\pi$
$853$	−7.52786	−0.257749	−0.128875	+	0.991661i	$0.541136\pi$
$854$	0	0
$855$	−6.58359	−0.225154
$856$	0	0
$857$	24.8328	0.848273	0.424136	−	0.905598i	$-0.360578\pi$
$857$	24.8328	0.848273	0.424136	+	0.905598i	$0.360578\pi$
$858$	0	0
$859$	49.4164	1.68607	0.843033	−	0.537862i	$-0.180768\pi$
$859$	49.4164	1.68607	0.843033	+	0.537862i	$0.180768\pi$
$860$	0	0
$861$	−49.8885	−1.70020
$862$	0	0
$863$	18.2918	0.622660	0.311330	−	0.950302i	$-0.399226\pi$
$863$	18.2918	0.622660	0.311330	+	0.950302i	$0.399226\pi$
$864$	0	0
$865$	10.9443	0.372116
$866$	0	0
$867$	3.70820	0.125937
$868$	0	0
$869$	−25.8885	−0.878209
$870$	0	0
$871$	−34.4721	−1.16804
$872$	0	0
$873$	24.2492	0.820712
$874$	0	0
$875$	3.23607	0.109399
$876$	0	0
$877$	51.8885	1.75215	0.876076	−	0.482173i	$-0.160152\pi$
$877$	51.8885	1.75215	0.876076	+	0.482173i	$0.160152\pi$
$878$	0	0
$879$	−29.5279	−0.995950
$880$	0	0
$881$	−24.4721	−0.824487	−0.412244	−	0.911074i	$-0.635255\pi$
$881$	−24.4721	−0.824487	−0.412244	+	0.911074i	$0.635255\pi$
$882$	0	0
$883$	−27.7082	−0.932455	−0.466228	−	0.884665i	$-0.654387\pi$
$883$	−27.7082	−0.932455	−0.466228	+	0.884665i	$0.654387\pi$
$884$	0	0
$885$	10.4721	0.352017
$886$	0	0
$887$	11.5967	0.389381	0.194690	−	0.980865i	$-0.437630\pi$
$887$	11.5967	0.389381	0.194690	+	0.980865i	$0.437630\pi$
$888$	0	0
$889$	65.3050	2.19026
$890$	0	0
$891$	4.83282	0.161905
$892$	0	0
$893$	32.3607	1.08291
$894$	0	0
$895$	20.4721	0.684308
$896$	0	0
$897$	26.3344	0.879279
$898$	0	0
$899$	−12.9443	−0.431716
$900$	0	0
$901$	2.11146	0.0703428
$902$	0	0
$903$	30.8328	1.02605
$904$	0	0
$905$	−10.9443	−0.363800
$906$	0	0
$907$	−9.23607	−0.306679	−0.153339	−	0.988174i	$-0.549003\pi$
$907$	−9.23607	−0.306679	−0.153339	+	0.988174i	$0.549003\pi$
$908$	0	0
$909$	14.7214	0.488277
$910$	0	0
$911$	−53.3050	−1.76607	−0.883036	−	0.469305i	$-0.844504\pi$
$911$	−53.3050	−1.76607	−0.883036	+	0.469305i	$0.844504\pi$
$912$	0	0
$913$	7.41641	0.245447
$914$	0	0
$915$	7.41641	0.245179
$916$	0	0
$917$	−48.3607	−1.59701
$918$	0	0
$919$	−5.88854	−0.194245	−0.0971226	−	0.995272i	$-0.530964\pi$
$919$	−5.88854	−0.194245	−0.0971226	+	0.995272i	$0.530964\pi$
$920$	0	0
$921$	−0.360680	−0.0118848
$922$	0	0
$923$	11.0557	0.363904
$924$	0	0
$925$	6.94427	0.228326
$926$	0	0
$927$	−4.76393	−0.156468
$928$	0	0
$929$	24.4721	0.802905	0.401452	−	0.915880i	$-0.368506\pi$
$929$	24.4721	0.802905	0.401452	+	0.915880i	$0.368506\pi$
$930$	0	0
$931$	−15.5279	−0.508905
$932$	0	0
$933$	−19.0557	−0.623857
$934$	0	0
$935$	8.94427	0.292509
$936$	0	0
$937$	−3.52786	−0.115250	−0.0576251	−	0.998338i	$-0.518353\pi$
$937$	−3.52786	−0.115250	−0.0576251	+	0.998338i	$0.518353\pi$
$938$	0	0
$939$	−35.6393	−1.16305
$940$	0	0
$941$	44.8328	1.46151	0.730754	−	0.682641i	$-0.239169\pi$
$941$	44.8328	1.46151	0.730754	+	0.682641i	$0.239169\pi$
$942$	0	0
$943$	−59.4164	−1.93486
$944$	0	0
$945$	−17.8885	−0.581914
$946$	0	0
$947$	21.8197	0.709044	0.354522	−	0.935048i	$-0.384644\pi$
$947$	21.8197	0.709044	0.354522	+	0.935048i	$0.384644\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	−25.3050	−0.820569
$952$	0	0
$953$	45.7771	1.48287	0.741433	−	0.671027i	$-0.234147\pi$
$953$	45.7771	1.48287	0.741433	+	0.671027i	$0.234147\pi$
$954$	0	0
$955$	−11.4164	−0.369426
$956$	0	0
$957$	4.94427	0.159826
$958$	0	0
$959$	−6.47214	−0.208996
$960$	0	0
$961$	10.8885	0.351243
$962$	0	0
$963$	25.3738	0.817660
$964$	0	0
$965$	11.5279	0.371095
$966$	0	0
$967$	−1.34752	−0.0433335	−0.0216667	−	0.999765i	$-0.506897\pi$
$967$	−1.34752	−0.0433335	−0.0216667	+	0.999765i	$0.506897\pi$
$968$	0	0
$969$	−24.7214	−0.794164
$970$	0	0
$971$	23.8885	0.766620	0.383310	−	0.923620i	$-0.374784\pi$
$971$	23.8885	0.766620	0.383310	+	0.923620i	$0.374784\pi$
$972$	0	0
$973$	−66.2492	−2.12385
$974$	0	0
$975$	−5.52786	−0.177033
$976$	0	0
$977$	−39.3050	−1.25748	−0.628738	−	0.777617i	$-0.716428\pi$
$977$	−39.3050	−1.25748	−0.628738	+	0.777617i	$0.716428\pi$
$978$	0	0
$979$	29.8885	0.955242
$980$	0	0
$981$	−22.0000	−0.702406
$982$	0	0
$983$	39.0132	1.24433	0.622163	−	0.782888i	$-0.286254\pi$
$983$	39.0132	1.24433	0.622163	+	0.782888i	$0.286254\pi$
$984$	0	0
$985$	−0.472136	−0.0150435
$986$	0	0
$987$	28.9443	0.921306
$988$	0	0
$989$	36.7214	1.16767
$990$	0	0
$991$	−17.5279	−0.556791	−0.278395	−	0.960467i	$-0.589803\pi$
$991$	−17.5279	−0.556791	−0.278395	+	0.960467i	$0.589803\pi$
$992$	0	0
$993$	19.6393	0.623235
$994$	0	0
$995$	0.944272	0.0299354
$996$	0	0
$997$	−34.5836	−1.09527	−0.547637	−	0.836716i	$-0.684472\pi$
$997$	−34.5836	−1.09527	−0.547637	+	0.836716i	$0.684472\pi$
$998$	0	0
$999$	−38.3870	−1.21451

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	640.2.a.i.1.2	✓	2
3.2	odd	2		5760.2.a.ch.1.1		2
4.3	odd	2		640.2.a.k.1.1	yes	2
5.2	odd	4		3200.2.c.u.2049.2		4
5.3	odd	4		3200.2.c.u.2049.3		4
5.4	even	2		3200.2.a.bl.1.1		2
8.3	odd	2		640.2.a.j.1.2	yes	2
8.5	even	2		640.2.a.l.1.1	yes	2
12.11	even	2		5760.2.a.ci.1.2		2
16.3	odd	4		1280.2.d.k.641.2		4
16.5	even	4		1280.2.d.m.641.2		4
16.11	odd	4		1280.2.d.k.641.3		4
16.13	even	4		1280.2.d.m.641.3		4
20.3	even	4		3200.2.c.w.2049.2		4
20.7	even	4		3200.2.c.w.2049.3		4
20.19	odd	2		3200.2.a.be.1.2		2
24.5	odd	2		5760.2.a.bw.1.1		2
24.11	even	2		5760.2.a.cd.1.2		2
40.3	even	4		3200.2.c.v.2049.3		4
40.13	odd	4		3200.2.c.x.2049.2		4
40.19	odd	2		3200.2.a.bk.1.1		2
40.27	even	4		3200.2.c.v.2049.2		4
40.29	even	2		3200.2.a.bf.1.2		2
40.37	odd	4		3200.2.c.x.2049.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.2.a.i.1.2	✓	2	1.1	even	1	trivial
640.2.a.j.1.2	yes	2	8.3	odd	2
640.2.a.k.1.1	yes	2	4.3	odd	2
640.2.a.l.1.1	yes	2	8.5	even	2
1280.2.d.k.641.2		4	16.3	odd	4
1280.2.d.k.641.3		4	16.11	odd	4
1280.2.d.m.641.2		4	16.5	even	4
1280.2.d.m.641.3		4	16.13	even	4
3200.2.a.be.1.2		2	20.19	odd	2
3200.2.a.bf.1.2		2	40.29	even	2
3200.2.a.bk.1.1		2	40.19	odd	2
3200.2.a.bl.1.1		2	5.4	even	2
3200.2.c.u.2049.2		4	5.2	odd	4
3200.2.c.u.2049.3		4	5.3	odd	4
3200.2.c.v.2049.2		4	40.27	even	4
3200.2.c.v.2049.3		4	40.3	even	4
3200.2.c.w.2049.2		4	20.3	even	4
3200.2.c.w.2049.3		4	20.7	even	4
3200.2.c.x.2049.2		4	40.13	odd	4
3200.2.c.x.2049.3		4	40.37	odd	4
5760.2.a.bw.1.1		2	24.5	odd	2
5760.2.a.cd.1.2		2	24.11	even	2
5760.2.a.ch.1.1		2	3.2	odd	2
5760.2.a.ci.1.2		2	12.11	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:	\( N \)	\(=\)	\( 640 = 2^{7} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	640.a (trivial)

\(f(q)\)	\(=\)	\(q+1.23607 q^{3} -1.00000 q^{5} -3.23607 q^{7} -1.47214 q^{9} -2.00000 q^{11} -4.47214 q^{13} -1.23607 q^{15} +4.47214 q^{17} -4.47214 q^{19} -4.00000 q^{21} -4.76393 q^{23} +1.00000 q^{25} -5.52786 q^{27} -2.00000 q^{29} +6.47214 q^{31} -2.47214 q^{33} +3.23607 q^{35} +6.94427 q^{37} -5.52786 q^{39} +12.4721 q^{41} -7.70820 q^{43} +1.47214 q^{45} -7.23607 q^{47} +3.47214 q^{49} +5.52786 q^{51} +0.472136 q^{53} +2.00000 q^{55} -5.52786 q^{57} -8.47214 q^{59} -6.00000 q^{61} +4.76393 q^{63} +4.47214 q^{65} +7.70820 q^{67} -5.88854 q^{69} -2.47214 q^{71} +4.47214 q^{73} +1.23607 q^{75} +6.47214 q^{77} +12.9443 q^{79} -2.41641 q^{81} -3.70820 q^{83} -4.47214 q^{85} -2.47214 q^{87} -14.9443 q^{89} +14.4721 q^{91} +8.00000 q^{93} +4.47214 q^{95} -16.4721 q^{97} +2.94427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 6 q^{9} - 4 q^{11} + 2 q^{15} - 8 q^{21} - 14 q^{23} + 2 q^{25} - 20 q^{27} - 4 q^{29} + 4 q^{31} + 4 q^{33} + 2 q^{35} - 4 q^{37} - 20 q^{39} + 16 q^{41} - 2 q^{43}+ \cdots - 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 6 * q^9 - 4 * q^11 + 2 * q^15 - 8 * q^21 - 14 * q^23 + 2 * q^25 - 20 * q^27 - 4 * q^29 + 4 * q^31 + 4 * q^33 + 2 * q^35 - 4 * q^37 - 20 * q^39 + 16 * q^41 - 2 * q^43 - 6 * q^45 - 10 * q^47 - 2 * q^49 + 20 * q^51 - 8 * q^53 + 4 * q^55 - 20 * q^57 - 8 * q^59 - 12 * q^61 + 14 * q^63 + 2 * q^67 + 24 * q^69 + 4 * q^71 - 2 * q^75 + 4 * q^77 + 8 * q^79 + 22 * q^81 + 6 * q^83 + 4 * q^87 - 12 * q^89 + 20 * q^91 + 16 * q^93 - 24 * q^97 - 12 * q^99

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