LMFDB - Embedded newform 1280.2.d.b.641.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,2,Mod(641,1280)]

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, 1, 0]))

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

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

chi := DirichletCharacter("1280.641");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1280.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$10.2208514587$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			641.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1280.641
Dual form			1280.2.d.b.641.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} +1.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} -4.00000i q^{11} -6.00000i q^{13} -2.00000 q^{15} +2.00000 q^{17} -8.00000i q^{19} -4.00000i q^{21} -6.00000 q^{23} -1.00000 q^{25} +4.00000i q^{27} -2.00000i q^{29} -4.00000 q^{31} +8.00000 q^{33} -2.00000i q^{35} -2.00000i q^{37} +12.0000 q^{39} +10.0000 q^{41} -2.00000i q^{43} -1.00000i q^{45} +2.00000 q^{47} -3.00000 q^{49} +4.00000i q^{51} -2.00000i q^{53} +4.00000 q^{55} +16.0000 q^{57} +2.00000i q^{61} +2.00000 q^{63} +6.00000 q^{65} +6.00000i q^{67} -12.0000i q^{69} -12.0000 q^{71} -10.0000 q^{73} -2.00000i q^{75} +8.00000i q^{77} +8.00000 q^{79} -11.0000 q^{81} +10.0000i q^{83} +2.00000i q^{85} +4.00000 q^{87} +6.00000 q^{89} +12.0000i q^{91} -8.00000i q^{93} +8.00000 q^{95} +10.0000 q^{97} +4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{7} - 2 q^{9} - 4 q^{15} + 4 q^{17} - 12 q^{23} - 2 q^{25} - 8 q^{31} + 16 q^{33} + 24 q^{39} + 20 q^{41} + 4 q^{47} - 6 q^{49} + 8 q^{55} + 32 q^{57} + 4 q^{63} + 12 q^{65} - 24 q^{71} - 20 q^{73}+ \cdots + 20 q^{97}+O(q^{100}) $ 2 * q - 4 * q^7 - 2 * q^9 - 4 * q^15 + 4 * q^17 - 12 * q^23 - 2 * q^25 - 8 * q^31 + 16 * q^33 + 24 * q^39 + 20 * q^41 + 4 * q^47 - 6 * q^49 + 8 * q^55 + 32 * q^57 + 4 * q^63 + 12 * q^65 - 24 * q^71 - 20 * q^73 + 16 * q^79 - 22 * q^81 + 8 * q^87 + 12 * q^89 + 16 * q^95 + 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	2.00000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	2.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 4.00000i	− 1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$	− 4.00000i	− 1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$	0	0
$13$	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 6.00000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	− 8.00000i	− 1.83533i	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	− 8.00000i	− 1.83533i	0.397360	−	0.917663i	$-0.369927\pi$
$20$	0	0
$21$	− 4.00000i	− 0.872872i
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	4.00000i	0.769800i
$28$	0	0
$29$	− 2.00000i	− 0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$	− 2.00000i	− 0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	8.00000	1.39262
$34$	0	0
$35$	− 2.00000i	− 0.338062i
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	12.0000	1.92154
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	− 2.00000i	− 0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$	− 2.00000i	− 0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	4.00000i	0.560112i
$52$	0	0
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	16.0000	2.11925
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	2.00000i	0.256074i	0.991769	+	0.128037i	$0.0408676\pi$
$61$	2.00000i	0.256074i	−0.991769	+	0.128037i	$0.959132\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	6.00000	0.744208
$66$	0	0
$67$	6.00000i	0.733017i	0.930415	+	0.366508i	$0.119447\pi$
$67$	6.00000i	0.733017i	−0.930415	+	0.366508i	$0.880553\pi$
$68$	0	0
$69$	− 12.0000i	− 1.44463i
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	− 2.00000i	− 0.230940i
$76$	0	0
$77$	8.00000i	0.911685i
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	10.0000i	1.09764i	0.835940	+	0.548821i	$0.184923\pi$
$83$	10.0000i	1.09764i	−0.835940	+	0.548821i	$0.815077\pi$
$84$	0	0
$85$	2.00000i	0.216930i
$86$	0	0
$87$	4.00000	0.428845
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	12.0000i	1.25794i
$92$	0	0
$93$	− 8.00000i	− 0.829561i
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	4.00000i	0.402015i
$100$	0	0
$101$	− 14.0000i	− 1.39305i	−0.717532	−	0.696526i	$-0.754728\pi$
$101$	− 14.0000i	− 1.39305i	0.717532	−	0.696526i	$-0.245272\pi$
$102$	0	0
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	− 6.00000i	− 0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	− 6.00000i	− 0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$	0	0
$109$	− 14.0000i	− 1.34096i	−0.741929	−	0.670478i	$-0.766089\pi$
$109$	− 14.0000i	− 1.34096i	0.741929	−	0.670478i	$-0.233911\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	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
$115$	− 6.00000i	− 0.559503i
$116$	0	0
$117$	6.00000i	0.554700i
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	20.0000i	1.80334i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	− 4.00000i	− 0.349482i	−0.984614	−	0.174741i	$-0.944091\pi$
$131$	− 4.00000i	− 0.349482i	0.984614	−	0.174741i	$-0.0559088\pi$
$132$	0	0
$133$	16.0000i	1.38738i
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	− 16.0000i	− 1.35710i	−0.734553	−	0.678551i	$-0.762608\pi$
$139$	− 16.0000i	− 1.35710i	0.734553	−	0.678551i	$-0.237392\pi$
$140$	0	0
$141$	4.00000i	0.336861i
$142$	0	0
$143$	−24.0000	−2.00698
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	− 6.00000i	− 0.494872i
$148$	0	0
$149$	− 10.0000i	− 0.819232i	−0.912258	−	0.409616i	$-0.865663\pi$
$149$	− 10.0000i	− 0.819232i	0.912258	−	0.409616i	$-0.134337\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	− 4.00000i	− 0.321288i
$156$	0	0
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	− 6.00000i	− 0.469956i	−0.972001	−	0.234978i	$-0.924498\pi$
$163$	− 6.00000i	− 0.469956i	0.972001	−	0.234978i	$-0.0755019\pi$
$164$	0	0
$165$	8.00000i	0.622799i
$166$	0	0
$167$	14.0000	1.08335	0.541676	−	0.840587i	$-0.317790\pi$
$167$	14.0000	1.08335	0.541676	+	0.840587i	$0.317790\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	8.00000i	0.611775i
$172$	0	0
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	18.0000i	1.33793i	0.743294	+	0.668965i	$0.233262\pi$
$181$	18.0000i	1.33793i	−0.743294	+	0.668965i	$0.766738\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	− 8.00000i	− 0.585018i
$188$	0	0
$189$	− 8.00000i	− 0.581914i
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	12.0000i	0.859338i
$196$	0	0
$197$	22.0000i	1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$	22.0000i	1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	4.00000i	0.280745i
$204$	0	0
$205$	10.0000i	0.698430i
$206$	0	0
$207$	6.00000	0.417029
$208$	0	0
$209$	−32.0000	−2.21349
$210$	0	0
$211$	− 4.00000i	− 0.275371i	−0.990476	−	0.137686i	$-0.956034\pi$
$211$	− 4.00000i	− 0.275371i	0.990476	−	0.137686i	$-0.0439664\pi$
$212$	0	0
$213$	− 24.0000i	− 1.64445i
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	− 20.0000i	− 1.35147i
$220$	0	0
$221$	− 12.0000i	− 0.807207i
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	− 18.0000i	− 1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$	− 18.0000i	− 1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$	0	0
$229$	10.0000i	0.660819i	0.943838	+	0.330409i	$0.107187\pi$
$229$	10.0000i	0.660819i	−0.943838	+	0.330409i	$0.892813\pi$
$230$	0	0
$231$	−16.0000	−1.05272
$232$	0	0
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	0	0
$235$	2.00000i	0.130466i
$236$	0	0
$237$	16.0000i	1.03931i
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	0	0
$243$	− 10.0000i	− 0.641500i
$244$	0	0
$245$	− 3.00000i	− 0.191663i
$246$	0	0
$247$	−48.0000	−3.05417
$248$	0	0
$249$	−20.0000	−1.26745
$250$	0	0
$251$	− 20.0000i	− 1.26239i	−0.775625	−	0.631194i	$-0.782565\pi$
$251$	− 20.0000i	− 1.26239i	0.775625	−	0.631194i	$-0.217435\pi$
$252$	0	0
$253$	24.0000i	1.50887i
$254$	0	0
$255$	−4.00000	−0.250490
$256$	0	0
$257$	26.0000	1.62184	0.810918	−	0.585160i	$-0.198968\pi$
$257$	26.0000	1.62184	0.810918	+	0.585160i	$0.198968\pi$
$258$	0	0
$259$	4.00000i	0.248548i
$260$	0	0
$261$	2.00000i	0.123797i
$262$	0	0
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	12.0000i	0.734388i
$268$	0	0
$269$	18.0000i	1.09748i	0.835993	+	0.548740i	$0.184892\pi$
$269$	18.0000i	1.09748i	−0.835993	+	0.548740i	$0.815108\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	0	0
$273$	−24.0000	−1.45255
$274$	0	0
$275$	4.00000i	0.241209i
$276$	0	0
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	− 26.0000i	− 1.54554i	−0.634686	−	0.772770i	$-0.718871\pi$
$283$	− 26.0000i	− 1.54554i	0.634686	−	0.772770i	$-0.281129\pi$
$284$	0	0
$285$	16.0000i	0.947758i
$286$	0	0
$287$	−20.0000	−1.18056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	20.0000i	1.17242i
$292$	0	0
$293$	− 18.0000i	− 1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$	− 18.0000i	− 1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	16.0000	0.928414
$298$	0	0
$299$	36.0000i	2.08193i
$300$	0	0
$301$	4.00000i	0.230556i
$302$	0	0
$303$	28.0000	1.60856
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	22.0000i	1.25561i	0.778372	+	0.627803i	$0.216046\pi$
$307$	22.0000i	1.25561i	−0.778372	+	0.627803i	$0.783954\pi$
$308$	0	0
$309$	4.00000i	0.227552i
$310$	0	0
$311$	28.0000	1.58773	0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000	1.58773	0.793867	+	0.608091i	$0.208065\pi$
$312$	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
$315$	2.00000i	0.112687i
$316$	0	0
$317$	2.00000i	0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$	2.00000i	0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	− 16.0000i	− 0.890264i
$324$	0	0
$325$	6.00000i	0.332820i
$326$	0	0
$327$	28.0000	1.54840
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	4.00000i	0.219860i	0.993939	+	0.109930i	$0.0350627\pi$
$331$	4.00000i	0.219860i	−0.993939	+	0.109930i	$0.964937\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	−6.00000	−0.327815
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	− 12.0000i	− 0.651751i
$340$	0	0
$341$	16.0000i	0.866449i
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	12.0000	0.646058
$346$	0	0
$347$	18.0000i	0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$	18.0000i	0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$	0	0
$349$	− 2.00000i	− 0.107058i	−0.998566	−	0.0535288i	$-0.982953\pi$
$349$	− 2.00000i	− 0.107058i	0.998566	−	0.0535288i	$-0.0170469\pi$
$350$	0	0
$351$	24.0000	1.28103
$352$	0	0
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	0	0
$355$	− 12.0000i	− 0.636894i
$356$	0	0
$357$	− 8.00000i	− 0.423405i
$358$	0	0
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	−45.0000	−2.36842
$362$	0	0
$363$	− 10.0000i	− 0.524864i
$364$	0	0
$365$	− 10.0000i	− 0.523424i
$366$	0	0
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	4.00000i	0.207670i
$372$	0	0
$373$	− 10.0000i	− 0.517780i	−0.965907	−	0.258890i	$-0.916643\pi$
$373$	− 10.0000i	− 0.517780i	0.965907	−	0.258890i	$-0.0833568\pi$
$374$	0	0
$375$	2.00000	0.103280
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	24.0000i	1.23280i	0.787434	+	0.616399i	$0.211409\pi$
$379$	24.0000i	1.23280i	−0.787434	+	0.616399i	$0.788591\pi$
$380$	0	0
$381$	− 12.0000i	− 0.614779i
$382$	0	0
$383$	14.0000	0.715367	0.357683	−	0.933843i	$-0.383567\pi$
$383$	14.0000	0.715367	0.357683	+	0.933843i	$0.383567\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	0	0
$387$	2.00000i	0.101666i
$388$	0	0
$389$	− 26.0000i	− 1.31825i	−0.752032	−	0.659126i	$-0.770926\pi$
$389$	− 26.0000i	− 1.31825i	0.752032	−	0.659126i	$-0.229074\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	8.00000	0.403547
$394$	0	0
$395$	8.00000i	0.402524i
$396$	0	0
$397$	26.0000i	1.30490i	0.757831	+	0.652451i	$0.226259\pi$
$397$	26.0000i	1.30490i	−0.757831	+	0.652451i	$0.773741\pi$
$398$	0	0
$399$	−32.0000	−1.60200
$400$	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
$403$	24.0000i	1.19553i
$404$	0	0
$405$	− 11.0000i	− 0.546594i
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	− 4.00000i	− 0.197305i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−10.0000	−0.490881
$416$	0	0
$417$	32.0000	1.56705
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	− 10.0000i	− 0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$	− 10.0000i	− 0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$	0	0
$423$	−2.00000	−0.0972433
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	− 4.00000i	− 0.193574i
$428$	0	0
$429$	− 48.0000i	− 2.31746i
$430$	0	0
$431$	20.0000	0.963366	0.481683	−	0.876346i	$-0.340026\pi$
$431$	20.0000	0.963366	0.481683	+	0.876346i	$0.340026\pi$
$432$	0	0
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	0	0
$435$	4.00000i	0.191785i
$436$	0	0
$437$	48.0000i	2.29615i
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	− 18.0000i	− 0.855206i	−0.903967	−	0.427603i	$-0.859358\pi$
$443$	− 18.0000i	− 0.855206i	0.903967	−	0.427603i	$-0.140642\pi$
$444$	0	0
$445$	6.00000i	0.284427i
$446$	0	0
$447$	20.0000	0.945968
$448$	0	0
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	0	0
$451$	− 40.0000i	− 1.88353i
$452$	0	0
$453$	40.0000i	1.87936i
$454$	0	0
$455$	−12.0000	−0.562569
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	0	0
$459$	8.00000i	0.373408i
$460$	0	0
$461$	6.00000i	0.279448i	0.990190	+	0.139724i	$0.0446215\pi$
$461$	6.00000i	0.279448i	−0.990190	+	0.139724i	$0.955378\pi$
$462$	0	0
$463$	38.0000	1.76601	0.883005	−	0.469364i	$-0.155517\pi$
$463$	38.0000	1.76601	0.883005	+	0.469364i	$0.155517\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	0	0
$467$	6.00000i	0.277647i	0.990317	+	0.138823i	$0.0443321\pi$
$467$	6.00000i	0.277647i	−0.990317	+	0.138823i	$0.955668\pi$
$468$	0	0
$469$	− 12.0000i	− 0.554109i
$470$	0	0
$471$	−20.0000	−0.921551
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	8.00000i	0.367065i
$476$	0	0
$477$	2.00000i	0.0915737i
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	0	0
$483$	24.0000i	1.09204i
$484$	0	0
$485$	10.0000i	0.454077i
$486$	0	0
$487$	38.0000	1.72194	0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000	1.72194	0.860972	+	0.508652i	$0.169856\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	− 20.0000i	− 0.902587i	−0.892375	−	0.451294i	$-0.850963\pi$
$491$	− 20.0000i	− 0.902587i	0.892375	−	0.451294i	$-0.149037\pi$
$492$	0	0
$493$	− 4.00000i	− 0.180151i
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	24.0000	1.07655
$498$	0	0
$499$	40.0000i	1.79065i	0.445418	+	0.895323i	$0.353055\pi$
$499$	40.0000i	1.79065i	−0.445418	+	0.895323i	$0.646945\pi$
$500$	0	0
$501$	28.0000i	1.25095i
$502$	0	0
$503$	−14.0000	−0.624229	−0.312115	−	0.950044i	$-0.601037\pi$
$503$	−14.0000	−0.624229	−0.312115	+	0.950044i	$0.601037\pi$
$504$	0	0
$505$	14.0000	0.622992
$506$	0	0
$507$	− 46.0000i	− 2.04293i
$508$	0	0
$509$	− 18.0000i	− 0.797836i	−0.916987	−	0.398918i	$-0.869386\pi$
$509$	− 18.0000i	− 0.797836i	0.916987	−	0.398918i	$-0.130614\pi$
$510$	0	0
$511$	20.0000	0.884748
$512$	0	0
$513$	32.0000	1.41283
$514$	0	0
$515$	2.00000i	0.0881305i
$516$	0	0
$517$	− 8.00000i	− 0.351840i
$518$	0	0
$519$	12.0000	0.526742
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	22.0000i	0.961993i	0.876723	+	0.480996i	$0.159725\pi$
$523$	22.0000i	0.961993i	−0.876723	+	0.480996i	$0.840275\pi$
$524$	0	0
$525$	4.00000i	0.174574i
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 60.0000i	− 2.59889i
$534$	0	0
$535$	6.00000	0.259403
$536$	0	0
$537$	0	0
$538$	0	0
$539$	12.0000i	0.516877i
$540$	0	0
$541$	− 10.0000i	− 0.429934i	−0.976621	−	0.214967i	$-0.931036\pi$
$541$	− 10.0000i	− 0.429934i	0.976621	−	0.214967i	$-0.0689643\pi$
$542$	0	0
$543$	−36.0000	−1.54491
$544$	0	0
$545$	14.0000	0.599694
$546$	0	0
$547$	14.0000i	0.598597i	0.954160	+	0.299298i	$0.0967526\pi$
$547$	14.0000i	0.598597i	−0.954160	+	0.299298i	$0.903247\pi$
$548$	0	0
$549$	− 2.00000i	− 0.0853579i
$550$	0	0
$551$	−16.0000	−0.681623
$552$	0	0
$553$	−16.0000	−0.680389
$554$	0	0
$555$	4.00000i	0.169791i
$556$	0	0
$557$	− 46.0000i	− 1.94908i	−0.224208	−	0.974541i	$-0.571980\pi$
$557$	− 46.0000i	− 1.94908i	0.224208	−	0.974541i	$-0.428020\pi$
$558$	0	0
$559$	−12.0000	−0.507546
$560$	0	0
$561$	16.0000	0.675521
$562$	0	0
$563$	− 30.0000i	− 1.26435i	−0.774826	−	0.632175i	$-0.782163\pi$
$563$	− 30.0000i	− 1.26435i	0.774826	−	0.632175i	$-0.217837\pi$
$564$	0	0
$565$	− 6.00000i	− 0.252422i
$566$	0	0
$567$	22.0000	0.923913
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	4.00000i	0.167395i	0.996491	+	0.0836974i	$0.0266729\pi$
$571$	4.00000i	0.167395i	−0.996491	+	0.0836974i	$0.973327\pi$
$572$	0	0
$573$	8.00000i	0.334205i
$574$	0	0
$575$	6.00000	0.250217
$576$	0	0
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	0	0
$579$	4.00000i	0.166234i
$580$	0	0
$581$	− 20.0000i	− 0.829740i
$582$	0	0
$583$	−8.00000	−0.331326
$584$	0	0
$585$	−6.00000	−0.248069
$586$	0	0
$587$	10.0000i	0.412744i	0.978474	+	0.206372i	$0.0661657\pi$
$587$	10.0000i	0.412744i	−0.978474	+	0.206372i	$0.933834\pi$
$588$	0	0
$589$	32.0000i	1.31854i
$590$	0	0
$591$	−44.0000	−1.80992
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	− 4.00000i	− 0.163984i
$596$	0	0
$597$	− 32.0000i	− 1.30967i
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	− 6.00000i	− 0.244339i
$604$	0	0
$605$	− 5.00000i	− 0.203279i
$606$	0	0
$607$	2.00000	0.0811775	0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000	0.0811775	0.0405887	+	0.999176i	$0.487077\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	− 12.0000i	− 0.485468i
$612$	0	0
$613$	22.0000i	0.888572i	0.895885	+	0.444286i	$0.146543\pi$
$613$	22.0000i	0.888572i	−0.895885	+	0.444286i	$0.853457\pi$
$614$	0	0
$615$	−20.0000	−0.806478
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	0	0
$619$	8.00000i	0.321547i	0.986991	+	0.160774i	$0.0513989\pi$
$619$	8.00000i	0.321547i	−0.986991	+	0.160774i	$0.948601\pi$
$620$	0	0
$621$	− 24.0000i	− 0.963087i
$622$	0	0
$623$	−12.0000	−0.480770
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 64.0000i	− 2.55591i
$628$	0	0
$629$	− 4.00000i	− 0.159490i
$630$	0	0
$631$	36.0000	1.43314	0.716569	−	0.697517i	$-0.245712\pi$
$631$	36.0000	1.43314	0.716569	+	0.697517i	$0.245712\pi$
$632$	0	0
$633$	8.00000	0.317971
$634$	0	0
$635$	− 6.00000i	− 0.238103i
$636$	0	0
$637$	18.0000i	0.713186i
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	− 30.0000i	− 1.18308i	−0.806274	−	0.591542i	$-0.798519\pi$
$643$	− 30.0000i	− 1.18308i	0.806274	−	0.591542i	$-0.201481\pi$
$644$	0	0
$645$	4.00000i	0.157500i
$646$	0	0
$647$	−42.0000	−1.65119	−0.825595	−	0.564263i	$-0.809160\pi$
$647$	−42.0000	−1.65119	−0.825595	+	0.564263i	$0.809160\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	16.0000i	0.627089i
$652$	0	0
$653$	− 30.0000i	− 1.17399i	−0.809590	−	0.586995i	$-0.800311\pi$
$653$	− 30.0000i	− 1.17399i	0.809590	−	0.586995i	$-0.199689\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	16.0000i	0.623272i	0.950202	+	0.311636i	$0.100877\pi$
$659$	16.0000i	0.623272i	−0.950202	+	0.311636i	$0.899123\pi$
$660$	0	0
$661$	− 10.0000i	− 0.388955i	−0.980907	−	0.194477i	$-0.937699\pi$
$661$	− 10.0000i	− 0.388955i	0.980907	−	0.194477i	$-0.0623011\pi$
$662$	0	0
$663$	24.0000	0.932083
$664$	0	0
$665$	−16.0000	−0.620453
$666$	0	0
$667$	12.0000i	0.464642i
$668$	0	0
$669$	− 4.00000i	− 0.154649i
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	0	0
$675$	− 4.00000i	− 0.153960i
$676$	0	0
$677$	− 26.0000i	− 0.999261i	−0.866239	−	0.499631i	$-0.833469\pi$
$677$	− 26.0000i	− 0.999261i	0.866239	−	0.499631i	$-0.166531\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	0	0
$681$	36.0000	1.37952
$682$	0	0
$683$	− 34.0000i	− 1.30097i	−0.759517	−	0.650487i	$-0.774565\pi$
$683$	− 34.0000i	− 1.30097i	0.759517	−	0.650487i	$-0.225435\pi$
$684$	0	0
$685$	− 2.00000i	− 0.0764161i
$686$	0	0
$687$	−20.0000	−0.763048
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	− 12.0000i	− 0.456502i	−0.973602	−	0.228251i	$-0.926699\pi$
$691$	− 12.0000i	− 0.456502i	0.973602	−	0.228251i	$-0.0733006\pi$
$692$	0	0
$693$	− 8.00000i	− 0.303895i
$694$	0	0
$695$	16.0000	0.606915
$696$	0	0
$697$	20.0000	0.757554
$698$	0	0
$699$	− 4.00000i	− 0.151294i
$700$	0	0
$701$	18.0000i	0.679851i	0.940452	+	0.339925i	$0.110402\pi$
$701$	18.0000i	0.679851i	−0.940452	+	0.339925i	$0.889598\pi$
$702$	0	0
$703$	−16.0000	−0.603451
$704$	0	0
$705$	−4.00000	−0.150649
$706$	0	0
$707$	28.0000i	1.05305i
$708$	0	0
$709$	10.0000i	0.375558i	0.982211	+	0.187779i	$0.0601289\pi$
$709$	10.0000i	0.375558i	−0.982211	+	0.187779i	$0.939871\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	24.0000	0.898807
$714$	0	0
$715$	− 24.0000i	− 0.897549i
$716$	0	0
$717$	16.0000i	0.597531i
$718$	0	0
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	− 36.0000i	− 1.33885i
$724$	0	0
$725$	2.00000i	0.0742781i
$726$	0	0
$727$	−2.00000	−0.0741759	−0.0370879	−	0.999312i	$-0.511808\pi$
$727$	−2.00000	−0.0741759	−0.0370879	+	0.999312i	$0.511808\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	− 4.00000i	− 0.147945i
$732$	0	0
$733$	− 6.00000i	− 0.221615i	−0.993842	−	0.110808i	$-0.964656\pi$
$733$	− 6.00000i	− 0.221615i	0.993842	−	0.110808i	$-0.0353437\pi$
$734$	0	0
$735$	6.00000	0.221313
$736$	0	0
$737$	24.0000	0.884051
$738$	0	0
$739$	− 16.0000i	− 0.588570i	−0.955718	−	0.294285i	$-0.904919\pi$
$739$	− 16.0000i	− 0.588570i	0.955718	−	0.294285i	$-0.0950814\pi$
$740$	0	0
$741$	− 96.0000i	− 3.52665i
$742$	0	0
$743$	42.0000	1.54083	0.770415	−	0.637542i	$-0.220049\pi$
$743$	42.0000	1.54083	0.770415	+	0.637542i	$0.220049\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	0	0
$747$	− 10.0000i	− 0.365881i
$748$	0	0
$749$	12.0000i	0.438470i
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	0	0
$753$	40.0000	1.45768
$754$	0	0
$755$	20.0000i	0.727875i
$756$	0	0
$757$	− 10.0000i	− 0.363456i	−0.983349	−	0.181728i	$-0.941831\pi$
$757$	− 10.0000i	− 0.363456i	0.983349	−	0.181728i	$-0.0581691\pi$
$758$	0	0
$759$	−48.0000	−1.74229
$760$	0	0
$761$	−10.0000	−0.362500	−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000	−0.362500	−0.181250	+	0.983437i	$0.558014\pi$
$762$	0	0
$763$	28.0000i	1.01367i
$764$	0	0
$765$	− 2.00000i	− 0.0723102i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	52.0000i	1.87273i
$772$	0	0
$773$	38.0000i	1.36677i	0.730061	+	0.683383i	$0.239492\pi$
$773$	38.0000i	1.36677i	−0.730061	+	0.683383i	$0.760508\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	−8.00000	−0.286998
$778$	0	0
$779$	− 80.0000i	− 2.86630i
$780$	0	0
$781$	48.0000i	1.71758i
$782$	0	0
$783$	8.00000	0.285897
$784$	0	0
$785$	−10.0000	−0.356915
$786$	0	0
$787$	14.0000i	0.499046i	0.968369	+	0.249523i	$0.0802738\pi$
$787$	14.0000i	0.499046i	−0.968369	+	0.249523i	$0.919726\pi$
$788$	0	0
$789$	4.00000i	0.142404i
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	12.0000	0.426132
$794$	0	0
$795$	4.00000i	0.141865i
$796$	0	0
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	0	0
$799$	4.00000	0.141510
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	40.0000i	1.41157i
$804$	0	0
$805$	12.0000i	0.422944i
$806$	0	0
$807$	−36.0000	−1.26726
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	− 20.0000i	− 0.702295i	−0.936320	−	0.351147i	$-0.885792\pi$
$811$	− 20.0000i	− 0.702295i	0.936320	−	0.351147i	$-0.114208\pi$
$812$	0	0
$813$	− 56.0000i	− 1.96401i
$814$	0	0
$815$	6.00000	0.210171
$816$	0	0
$817$	−16.0000	−0.559769
$818$	0	0
$819$	− 12.0000i	− 0.419314i
$820$	0	0
$821$	38.0000i	1.32621i	0.748527	+	0.663105i	$0.230762\pi$
$821$	38.0000i	1.32621i	−0.748527	+	0.663105i	$0.769238\pi$
$822$	0	0
$823$	−6.00000	−0.209147	−0.104573	−	0.994517i	$-0.533348\pi$
$823$	−6.00000	−0.209147	−0.104573	+	0.994517i	$0.533348\pi$
$824$	0	0
$825$	−8.00000	−0.278524
$826$	0	0
$827$	50.0000i	1.73867i	0.494223	+	0.869335i	$0.335453\pi$
$827$	50.0000i	1.73867i	−0.494223	+	0.869335i	$0.664547\pi$
$828$	0	0
$829$	2.00000i	0.0694629i	0.999397	+	0.0347314i	$0.0110576\pi$
$829$	2.00000i	0.0694629i	−0.999397	+	0.0347314i	$0.988942\pi$
$830$	0	0
$831$	−44.0000	−1.52634
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	14.0000i	0.484490i
$836$	0	0
$837$	− 16.0000i	− 0.553041i
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	20.0000i	0.688837i
$844$	0	0
$845$	− 23.0000i	− 0.791224i
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	52.0000	1.78464
$850$	0	0
$851$	12.0000i	0.411355i
$852$	0	0
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	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
$859$	− 16.0000i	− 0.545913i	−0.962026	−	0.272956i	$-0.911998\pi$
$859$	− 16.0000i	− 0.545913i	0.962026	−	0.272956i	$-0.0880015\pi$
$860$	0	0
$861$	− 40.0000i	− 1.36320i
$862$	0	0
$863$	46.0000	1.56586	0.782929	−	0.622111i	$-0.213725\pi$
$863$	46.0000	1.56586	0.782929	+	0.622111i	$0.213725\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	0	0
$867$	− 26.0000i	− 0.883006i
$868$	0	0
$869$	− 32.0000i	− 1.08553i
$870$	0	0
$871$	36.0000	1.21981
$872$	0	0
$873$	−10.0000	−0.338449
$874$	0	0
$875$	2.00000i	0.0676123i
$876$	0	0
$877$	10.0000i	0.337676i	0.985644	+	0.168838i	$0.0540015\pi$
$877$	10.0000i	0.337676i	−0.985644	+	0.168838i	$0.945999\pi$
$878$	0	0
$879$	36.0000	1.21425
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	2.00000i	0.0673054i	0.999434	+	0.0336527i	$0.0107140\pi$
$883$	2.00000i	0.0673054i	−0.999434	+	0.0336527i	$0.989286\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.00000	−0.0671534	−0.0335767	−	0.999436i	$-0.510690\pi$
$887$	−2.00000	−0.0671534	−0.0335767	+	0.999436i	$0.510690\pi$
$888$	0	0
$889$	12.0000	0.402467
$890$	0	0
$891$	44.0000i	1.47406i
$892$	0	0
$893$	− 16.0000i	− 0.535420i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−72.0000	−2.40401
$898$	0	0
$899$	8.00000i	0.266815i
$900$	0	0
$901$	− 4.00000i	− 0.133259i
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	−18.0000	−0.598340
$906$	0	0
$907$	2.00000i	0.0664089i	0.999449	+	0.0332045i	$0.0105712\pi$
$907$	2.00000i	0.0664089i	−0.999449	+	0.0332045i	$0.989429\pi$
$908$	0	0
$909$	14.0000i	0.464351i
$910$	0	0
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	0	0
$913$	40.0000	1.32381
$914$	0	0
$915$	− 4.00000i	− 0.132236i
$916$	0	0
$917$	8.00000i	0.264183i
$918$	0	0
$919$	56.0000	1.84727	0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000	1.84727	0.923635	+	0.383274i	$0.125203\pi$
$920$	0	0
$921$	−44.0000	−1.44985
$922$	0	0
$923$	72.0000i	2.36991i
$924$	0	0
$925$	2.00000i	0.0657596i
$926$	0	0
$927$	−2.00000	−0.0656886
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	24.0000i	0.786568i
$932$	0	0
$933$	56.0000i	1.83336i
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	0	0
$939$	12.0000i	0.391605i
$940$	0	0
$941$	38.0000i	1.23876i	0.785090	+	0.619382i	$0.212617\pi$
$941$	38.0000i	1.23876i	−0.785090	+	0.619382i	$0.787383\pi$
$942$	0	0
$943$	−60.0000	−1.95387
$944$	0	0
$945$	8.00000	0.260240
$946$	0	0
$947$	38.0000i	1.23483i	0.786636	+	0.617417i	$0.211821\pi$
$947$	38.0000i	1.23483i	−0.786636	+	0.617417i	$0.788179\pi$
$948$	0	0
$949$	60.0000i	1.94768i
$950$	0	0
$951$	−4.00000	−0.129709
$952$	0	0
$953$	−58.0000	−1.87880	−0.939402	−	0.342817i	$-0.888619\pi$
$953$	−58.0000	−1.87880	−0.939402	+	0.342817i	$0.888619\pi$
$954$	0	0
$955$	4.00000i	0.129437i
$956$	0	0
$957$	− 16.0000i	− 0.517207i
$958$	0	0
$959$	4.00000	0.129167
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	6.00000i	0.193347i
$964$	0	0
$965$	2.00000i	0.0643823i
$966$	0	0
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	0	0
$969$	32.0000	1.02799
$970$	0	0
$971$	36.0000i	1.15529i	0.816286	+	0.577647i	$0.196029\pi$
$971$	36.0000i	1.15529i	−0.816286	+	0.577647i	$0.803971\pi$
$972$	0	0
$973$	32.0000i	1.02587i
$974$	0	0
$975$	−12.0000	−0.384308
$976$	0	0
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	− 24.0000i	− 0.767043i
$980$	0	0
$981$	14.0000i	0.446986i
$982$	0	0
$983$	−46.0000	−1.46717	−0.733586	−	0.679597i	$-0.762155\pi$
$983$	−46.0000	−1.46717	−0.733586	+	0.679597i	$0.762155\pi$
$984$	0	0
$985$	−22.0000	−0.700978
$986$	0	0
$987$	− 8.00000i	− 0.254643i
$988$	0	0
$989$	12.0000i	0.381578i
$990$	0	0
$991$	44.0000	1.39771	0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000	1.39771	0.698853	+	0.715265i	$0.253694\pi$
$992$	0	0
$993$	−8.00000	−0.253872
$994$	0	0
$995$	− 16.0000i	− 0.507234i
$996$	0	0
$997$	− 18.0000i	− 0.570066i	−0.958518	−	0.285033i	$-0.907995\pi$
$997$	− 18.0000i	− 0.570066i	0.958518	−	0.285033i	$-0.0920045\pi$
$998$	0	0
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.2.d.b.641.2		2
4.3	odd	2		1280.2.d.h.641.1		2
8.3	odd	2		1280.2.d.h.641.2		2
8.5	even	2	inner	1280.2.d.b.641.1		2
16.3	odd	4		320.2.a.e.1.1		1
16.5	even	4		160.2.a.b.1.1	yes	1
16.11	odd	4		160.2.a.a.1.1	✓	1
16.13	even	4		320.2.a.b.1.1		1
48.5	odd	4		1440.2.a.l.1.1		1
48.11	even	4		1440.2.a.i.1.1		1
48.29	odd	4		2880.2.a.o.1.1		1
48.35	even	4		2880.2.a.d.1.1		1
80.3	even	4		1600.2.c.f.449.2		2
80.13	odd	4		1600.2.c.c.449.1		2
80.19	odd	4		1600.2.a.e.1.1		1
80.27	even	4		800.2.c.a.449.2		2
80.29	even	4		1600.2.a.t.1.1		1
80.37	odd	4		800.2.c.b.449.1		2
80.43	even	4		800.2.c.a.449.1		2
80.53	odd	4		800.2.c.b.449.2		2
80.59	odd	4		800.2.a.i.1.1		1
80.67	even	4		1600.2.c.f.449.1		2
80.69	even	4		800.2.a.a.1.1		1
80.77	odd	4		1600.2.c.c.449.2		2
112.27	even	4		7840.2.a.w.1.1		1
112.69	odd	4		7840.2.a.e.1.1		1
240.53	even	4		7200.2.f.g.6049.1		2
240.59	even	4		7200.2.a.bp.1.1		1
240.107	odd	4		7200.2.f.w.6049.1		2
240.149	odd	4		7200.2.a.l.1.1		1
240.197	even	4		7200.2.f.g.6049.2		2
240.203	odd	4		7200.2.f.w.6049.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.2.a.a.1.1	✓	1	16.11	odd	4
160.2.a.b.1.1	yes	1	16.5	even	4
320.2.a.b.1.1		1	16.13	even	4
320.2.a.e.1.1		1	16.3	odd	4
800.2.a.a.1.1		1	80.69	even	4
800.2.a.i.1.1		1	80.59	odd	4
800.2.c.a.449.1		2	80.43	even	4
800.2.c.a.449.2		2	80.27	even	4
800.2.c.b.449.1		2	80.37	odd	4
800.2.c.b.449.2		2	80.53	odd	4
1280.2.d.b.641.1		2	8.5	even	2	inner
1280.2.d.b.641.2		2	1.1	even	1	trivial
1280.2.d.h.641.1		2	4.3	odd	2
1280.2.d.h.641.2		2	8.3	odd	2
1440.2.a.i.1.1		1	48.11	even	4
1440.2.a.l.1.1		1	48.5	odd	4
1600.2.a.e.1.1		1	80.19	odd	4
1600.2.a.t.1.1		1	80.29	even	4
1600.2.c.c.449.1		2	80.13	odd	4
1600.2.c.c.449.2		2	80.77	odd	4
1600.2.c.f.449.1		2	80.67	even	4
1600.2.c.f.449.2		2	80.3	even	4
2880.2.a.d.1.1		1	48.35	even	4
2880.2.a.o.1.1		1	48.29	odd	4
7200.2.a.l.1.1		1	240.149	odd	4
7200.2.a.bp.1.1		1	240.59	even	4
7200.2.f.g.6049.1		2	240.53	even	4
7200.2.f.g.6049.2		2	240.197	even	4
7200.2.f.w.6049.1		2	240.107	odd	4
7200.2.f.w.6049.2		2	240.203	odd	4
7840.2.a.e.1.1		1	112.69	odd	4
7840.2.a.w.1.1		1	112.27	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1280.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} +1.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} -4.00000i q^{11} -6.00000i q^{13} -2.00000 q^{15} +2.00000 q^{17} -8.00000i q^{19} -4.00000i q^{21} -6.00000 q^{23} -1.00000 q^{25} +4.00000i q^{27} -2.00000i q^{29} -4.00000 q^{31} +8.00000 q^{33} -2.00000i q^{35} -2.00000i q^{37} +12.0000 q^{39} +10.0000 q^{41} -2.00000i q^{43} -1.00000i q^{45} +2.00000 q^{47} -3.00000 q^{49} +4.00000i q^{51} -2.00000i q^{53} +4.00000 q^{55} +16.0000 q^{57} +2.00000i q^{61} +2.00000 q^{63} +6.00000 q^{65} +6.00000i q^{67} -12.0000i q^{69} -12.0000 q^{71} -10.0000 q^{73} -2.00000i q^{75} +8.00000i q^{77} +8.00000 q^{79} -11.0000 q^{81} +10.0000i q^{83} +2.00000i q^{85} +4.00000 q^{87} +6.00000 q^{89} +12.0000i q^{91} -8.00000i q^{93} +8.00000 q^{95} +10.0000 q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{7} - 2 q^{9} - 4 q^{15} + 4 q^{17} - 12 q^{23} - 2 q^{25} - 8 q^{31} + 16 q^{33} + 24 q^{39} + 20 q^{41} + 4 q^{47} - 6 q^{49} + 8 q^{55} + 32 q^{57} + 4 q^{63} + 12 q^{65} - 24 q^{71} - 20 q^{73}+ \cdots + 20 q^{97}+O(q^{100}) \) 2 * q - 4 * q^7 - 2 * q^9 - 4 * q^15 + 4 * q^17 - 12 * q^23 - 2 * q^25 - 8 * q^31 + 16 * q^33 + 24 * q^39 + 20 * q^41 + 4 * q^47 - 6 * q^49 + 8 * q^55 + 32 * q^57 + 4 * q^63 + 12 * q^65 - 24 * q^71 - 20 * q^73 + 16 * q^79 - 22 * q^81 + 8 * q^87 + 12 * q^89 + 16 * q^95 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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