LMFDB - Embedded newform 1280.2.d.l.641.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1280,2,Mod(641,1280)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [4,0,0,0,0,0,0,0,-20,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$10.2208514587$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			641.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1280.641
Dual form			1280.2.d.l.641.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{3} -1.00000i q^{5} -2.82843 q^{7} -5.00000 q^{9} -5.65685i q^{11} -2.00000i q^{13} -2.82843 q^{15} +2.00000 q^{17} +8.00000i q^{21} +2.82843 q^{23} -1.00000 q^{25} +5.65685i q^{27} +6.00000i q^{29} -5.65685 q^{31} -16.0000 q^{33} +2.82843i q^{35} +10.0000i q^{37} -5.65685 q^{39} -2.00000 q^{41} -8.48528i q^{43} +5.00000i q^{45} +2.82843 q^{47} +1.00000 q^{49} -5.65685i q^{51} -6.00000i q^{53} -5.65685 q^{55} +11.3137i q^{59} -2.00000i q^{61} +14.1421 q^{63} -2.00000 q^{65} +2.82843i q^{67} -8.00000i q^{69} +5.65685 q^{71} +6.00000 q^{73} +2.82843i q^{75} +16.0000i q^{77} -11.3137 q^{79} +1.00000 q^{81} -2.82843i q^{83} -2.00000i q^{85} +16.9706 q^{87} -10.0000 q^{89} +5.65685i q^{91} +16.0000i q^{93} +2.00000 q^{97} +28.2843i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{9} + 8 q^{17} - 4 q^{25} - 64 q^{33} - 8 q^{41} + 4 q^{49} - 8 q^{65} + 24 q^{73} + 4 q^{81} - 40 q^{89} + 8 q^{97}+O(q^{100}) $ 4 * q - 20 * q^9 + 8 * q^17 - 4 * q^25 - 64 * q^33 - 8 * q^41 + 4 * q^49 - 8 * q^65 + 24 * q^73 + 4 * q^81 - 40 * q^89 + 8 * 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
$3$	− 2.82843i	− 1.63299i	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	− 2.82843i	− 1.63299i	0.577350	−	0.816497i	$-0.304087\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	−5.00000	−1.66667
$10$	0	0
$11$	− 5.65685i	− 1.70561i	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	− 5.65685i	− 1.70561i	0.522233	−	0.852803i	$-0.325099\pi$
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	−2.82843	−0.730297
$16$	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
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	8.00000i	1.74574i
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	5.65685i	1.08866i
$28$	0	0
$29$	6.00000i	1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$	6.00000i	1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	0	0
$31$	−5.65685	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	0	0
$33$	−16.0000	−2.78524
$34$	0	0
$35$	2.82843i	0.478091i
$36$	0	0
$37$	10.0000i	1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$	10.0000i	1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$	0	0
$39$	−5.65685	−0.905822
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	− 8.48528i	− 1.29399i	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	− 8.48528i	− 1.29399i	0.762493	−	0.646997i	$-0.223975\pi$
$44$	0	0
$45$	5.00000i	0.745356i
$46$	0	0
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 5.65685i	− 0.792118i
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	−5.65685	−0.762770
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.3137i	1.47292i	0.676481	+	0.736460i	$0.263504\pi$
$59$	11.3137i	1.47292i	−0.676481	+	0.736460i	$0.736496\pi$
$60$	0	0
$61$	− 2.00000i	− 0.256074i	−0.991769	−	0.128037i	$-0.959132\pi$
$61$	− 2.00000i	− 0.256074i	0.991769	−	0.128037i	$-0.0408676\pi$
$62$	0	0
$63$	14.1421	1.78174
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	2.82843i	0.345547i	0.984962	+	0.172774i	$0.0552729\pi$
$67$	2.82843i	0.345547i	−0.984962	+	0.172774i	$0.944727\pi$
$68$	0	0
$69$	− 8.00000i	− 0.963087i
$70$	0	0
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	2.82843i	0.326599i
$76$	0	0
$77$	16.0000i	1.82337i
$78$	0	0
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 2.82843i	− 0.310460i	−0.987878	−	0.155230i	$-0.950388\pi$
$83$	− 2.82843i	− 0.310460i	0.987878	−	0.155230i	$-0.0496119\pi$
$84$	0	0
$85$	− 2.00000i	− 0.216930i
$86$	0	0
$87$	16.9706	1.81944
$88$	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
$91$	5.65685i	0.592999i
$92$	0	0
$93$	16.0000i	1.65912i
$94$	0	0
$95$	0	0
$96$	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
$99$	28.2843i	2.84268i
$100$	0	0
$101$	2.00000i	0.199007i	0.995037	+	0.0995037i	$0.0317255\pi$
$101$	2.00000i	0.199007i	−0.995037	+	0.0995037i	$0.968274\pi$
$102$	0	0
$103$	14.1421	1.39347	0.696733	−	0.717331i	$-0.254636\pi$
$103$	14.1421	1.39347	0.696733	+	0.717331i	$0.254636\pi$
$104$	0	0
$105$	8.00000	0.780720
$106$	0	0
$107$	− 14.1421i	− 1.36717i	−0.729870	−	0.683586i	$-0.760419\pi$
$107$	− 14.1421i	− 1.36717i	0.729870	−	0.683586i	$-0.239581\pi$
$108$	0	0
$109$	− 18.0000i	− 1.72409i	−0.506834	−	0.862044i	$-0.669184\pi$
$109$	− 18.0000i	− 1.72409i	0.506834	−	0.862044i	$-0.330816\pi$
$110$	0	0
$111$	28.2843	2.68462
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	− 2.82843i	− 0.263752i
$116$	0	0
$117$	10.0000i	0.924500i
$118$	0	0
$119$	−5.65685	−0.518563
$120$	0	0
$121$	−21.0000	−1.90909
$122$	0	0
$123$	5.65685i	0.510061i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	2.82843	0.250982	0.125491	−	0.992095i	$-0.459949\pi$
$127$	2.82843	0.250982	0.125491	+	0.992095i	$0.459949\pi$
$128$	0	0
$129$	−24.0000	−2.11308
$130$	0	0
$131$	− 5.65685i	− 0.494242i	−0.968985	−	0.247121i	$-0.920516\pi$
$131$	− 5.65685i	− 0.494242i	0.968985	−	0.247121i	$-0.0794845\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	5.65685	0.486864
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	11.3137i	0.959616i	0.877373	+	0.479808i	$0.159294\pi$
$139$	11.3137i	0.959616i	−0.877373	+	0.479808i	$0.840706\pi$
$140$	0	0
$141$	− 8.00000i	− 0.673722i
$142$	0	0
$143$	−11.3137	−0.946100
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	− 2.82843i	− 0.233285i
$148$	0	0
$149$	10.0000i	0.819232i	0.912258	+	0.409616i	$0.134337\pi$
$149$	10.0000i	0.819232i	−0.912258	+	0.409616i	$0.865663\pi$
$150$	0	0
$151$	−16.9706	−1.38104	−0.690522	−	0.723311i	$-0.742619\pi$
$151$	−16.9706	−1.38104	−0.690522	+	0.723311i	$0.742619\pi$
$152$	0	0
$153$	−10.0000	−0.808452
$154$	0	0
$155$	5.65685i	0.454369i
$156$	0	0
$157$	− 18.0000i	− 1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$	− 18.0000i	− 1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	0	0
$159$	−16.9706	−1.34585
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	− 14.1421i	− 1.10770i	−0.832617	−	0.553849i	$-0.813159\pi$
$163$	− 14.1421i	− 1.10770i	0.832617	−	0.553849i	$-0.186841\pi$
$164$	0	0
$165$	16.0000i	1.24560i
$166$	0	0
$167$	−14.1421	−1.09435	−0.547176	−	0.837018i	$-0.684297\pi$
$167$	−14.1421	−1.09435	−0.547176	+	0.837018i	$0.684297\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 2.00000i	− 0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$	− 2.00000i	− 0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$	0	0
$175$	2.82843	0.213809
$176$	0	0
$177$	32.0000	2.40527
$178$	0	0
$179$	− 11.3137i	− 0.845626i	−0.906217	−	0.422813i	$-0.861043\pi$
$179$	− 11.3137i	− 0.845626i	0.906217	−	0.422813i	$-0.138957\pi$
$180$	0	0
$181$	− 14.0000i	− 1.04061i	−0.853980	−	0.520306i	$-0.825818\pi$
$181$	− 14.0000i	− 1.04061i	0.853980	−	0.520306i	$-0.174182\pi$
$182$	0	0
$183$	−5.65685	−0.418167
$184$	0	0
$185$	10.0000	0.735215
$186$	0	0
$187$	− 11.3137i	− 0.827340i
$188$	0	0
$189$	− 16.0000i	− 1.16383i
$190$	0	0
$191$	−16.9706	−1.22795	−0.613973	−	0.789327i	$-0.710430\pi$
$191$	−16.9706	−1.22795	−0.613973	+	0.789327i	$0.710430\pi$
$192$	0	0
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	0	0
$195$	5.65685i	0.405096i
$196$	0	0
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	−22.6274	−1.60402	−0.802008	−	0.597314i	$-0.796235\pi$
$199$	−22.6274	−1.60402	−0.802008	+	0.597314i	$0.796235\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	− 16.9706i	− 1.19110i
$204$	0	0
$205$	2.00000i	0.139686i
$206$	0	0
$207$	−14.1421	−0.982946
$208$	0	0
$209$	0	0
$210$	0	0
$211$	16.9706i	1.16830i	0.811645	+	0.584151i	$0.198572\pi$
$211$	16.9706i	1.16830i	−0.811645	+	0.584151i	$0.801428\pi$
$212$	0	0
$213$	− 16.0000i	− 1.09630i
$214$	0	0
$215$	−8.48528	−0.578691
$216$	0	0
$217$	16.0000	1.08615
$218$	0	0
$219$	− 16.9706i	− 1.14676i
$220$	0	0
$221$	− 4.00000i	− 0.269069i
$222$	0	0
$223$	8.48528	0.568216	0.284108	−	0.958792i	$-0.408302\pi$
$223$	8.48528	0.568216	0.284108	+	0.958792i	$0.408302\pi$
$224$	0	0
$225$	5.00000	0.333333
$226$	0	0
$227$	− 19.7990i	− 1.31411i	−0.753845	−	0.657053i	$-0.771803\pi$
$227$	− 19.7990i	− 1.31411i	0.753845	−	0.657053i	$-0.228197\pi$
$228$	0	0
$229$	− 14.0000i	− 0.925146i	−0.886581	−	0.462573i	$-0.846926\pi$
$229$	− 14.0000i	− 0.925146i	0.886581	−	0.462573i	$-0.153074\pi$
$230$	0	0
$231$	45.2548	2.97755
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	− 2.82843i	− 0.184506i
$236$	0	0
$237$	32.0000i	2.07862i
$238$	0	0
$239$	11.3137	0.731823	0.365911	−	0.930650i	$-0.380757\pi$
$239$	11.3137	0.731823	0.365911	+	0.930650i	$0.380757\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	0	0
$243$	14.1421i	0.907218i
$244$	0	0
$245$	− 1.00000i	− 0.0638877i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	− 5.65685i	− 0.357057i	−0.983935	−	0.178529i	$-0.942866\pi$
$251$	− 5.65685i	− 0.357057i	0.983935	−	0.178529i	$-0.0571337\pi$
$252$	0	0
$253$	− 16.0000i	− 1.00591i
$254$	0	0
$255$	−5.65685	−0.354246
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	− 28.2843i	− 1.75750i
$260$	0	0
$261$	− 30.0000i	− 1.85695i
$262$	0	0
$263$	−19.7990	−1.22086	−0.610429	−	0.792071i	$-0.709003\pi$
$263$	−19.7990	−1.22086	−0.610429	+	0.792071i	$0.709003\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	28.2843i	1.73097i
$268$	0	0
$269$	− 18.0000i	− 1.09748i	−0.835993	−	0.548740i	$-0.815108\pi$
$269$	− 18.0000i	− 1.09748i	0.835993	−	0.548740i	$-0.184892\pi$
$270$	0	0
$271$	−16.9706	−1.03089	−0.515444	−	0.856923i	$-0.672373\pi$
$271$	−16.9706	−1.03089	−0.515444	+	0.856923i	$0.672373\pi$
$272$	0	0
$273$	16.0000	0.968364
$274$	0	0
$275$	5.65685i	0.341121i
$276$	0	0
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	0	0
$279$	28.2843	1.69334
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	− 8.48528i	− 0.504398i	−0.967675	−	0.252199i	$-0.918846\pi$
$283$	− 8.48528i	− 0.504398i	0.967675	−	0.252199i	$-0.0811537\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.65685	0.333914
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	− 5.65685i	− 0.331611i
$292$	0	0
$293$	10.0000i	0.584206i	0.956387	+	0.292103i	$0.0943550\pi$
$293$	10.0000i	0.584206i	−0.956387	+	0.292103i	$0.905645\pi$
$294$	0	0
$295$	11.3137	0.658710
$296$	0	0
$297$	32.0000	1.85683
$298$	0	0
$299$	− 5.65685i	− 0.327144i
$300$	0	0
$301$	24.0000i	1.38334i
$302$	0	0
$303$	5.65685	0.324978
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	2.82843i	0.161427i	0.996737	+	0.0807134i	$0.0257199\pi$
$307$	2.82843i	0.161427i	−0.996737	+	0.0807134i	$0.974280\pi$
$308$	0	0
$309$	− 40.0000i	− 2.27552i
$310$	0	0
$311$	−28.2843	−1.60385	−0.801927	−	0.597422i	$-0.796192\pi$
$311$	−28.2843	−1.60385	−0.801927	+	0.597422i	$0.796192\pi$
$312$	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
$315$	− 14.1421i	− 0.796819i
$316$	0	0
$317$	30.0000i	1.68497i	0.538721	+	0.842484i	$0.318908\pi$
$317$	30.0000i	1.68497i	−0.538721	+	0.842484i	$0.681092\pi$
$318$	0	0
$319$	33.9411	1.90034
$320$	0	0
$321$	−40.0000	−2.23258
$322$	0	0
$323$	0	0
$324$	0	0
$325$	2.00000i	0.110940i
$326$	0	0
$327$	−50.9117	−2.81542
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	5.65685i	0.310929i	0.987841	+	0.155464i	$0.0496874\pi$
$331$	5.65685i	0.310929i	−0.987841	+	0.155464i	$0.950313\pi$
$332$	0	0
$333$	− 50.0000i	− 2.73998i
$334$	0	0
$335$	2.82843	0.154533
$336$	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
$339$	− 5.65685i	− 0.307238i
$340$	0	0
$341$	32.0000i	1.73290i
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	−8.00000	−0.430706
$346$	0	0
$347$	8.48528i	0.455514i	0.973718	+	0.227757i	$0.0731391\pi$
$347$	8.48528i	0.455514i	−0.973718	+	0.227757i	$0.926861\pi$
$348$	0	0
$349$	6.00000i	0.321173i	0.987022	+	0.160586i	$0.0513385\pi$
$349$	6.00000i	0.321173i	−0.987022	+	0.160586i	$0.948662\pi$
$350$	0	0
$351$	11.3137	0.603881
$352$	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
$355$	− 5.65685i	− 0.300235i
$356$	0	0
$357$	16.0000i	0.846810i
$358$	0	0
$359$	22.6274	1.19423	0.597115	−	0.802156i	$-0.296314\pi$
$359$	22.6274	1.19423	0.597115	+	0.802156i	$0.296314\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	59.3970i	3.11753i
$364$	0	0
$365$	− 6.00000i	− 0.314054i
$366$	0	0
$367$	−8.48528	−0.442928	−0.221464	−	0.975169i	$-0.571084\pi$
$367$	−8.48528	−0.442928	−0.221464	+	0.975169i	$0.571084\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	16.9706i	0.881068i
$372$	0	0
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	0	0
$375$	2.82843	0.146059
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	− 22.6274i	− 1.16229i	−0.813799	−	0.581146i	$-0.802604\pi$
$379$	− 22.6274i	− 1.16229i	0.813799	−	0.581146i	$-0.197396\pi$
$380$	0	0
$381$	− 8.00000i	− 0.409852i
$382$	0	0
$383$	−36.7696	−1.87884	−0.939418	−	0.342773i	$-0.888634\pi$
$383$	−36.7696	−1.87884	−0.939418	+	0.342773i	$0.888634\pi$
$384$	0	0
$385$	16.0000	0.815436
$386$	0	0
$387$	42.4264i	2.15666i
$388$	0	0
$389$	− 6.00000i	− 0.304212i	−0.988364	−	0.152106i	$-0.951394\pi$
$389$	− 6.00000i	− 0.304212i	0.988364	−	0.152106i	$-0.0486055\pi$
$390$	0	0
$391$	5.65685	0.286079
$392$	0	0
$393$	−16.0000	−0.807093
$394$	0	0
$395$	11.3137i	0.569254i
$396$	0	0
$397$	− 34.0000i	− 1.70641i	−0.521575	−	0.853206i	$-0.674655\pi$
$397$	− 34.0000i	− 1.70641i	0.521575	−	0.853206i	$-0.325345\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	11.3137i	0.563576i
$404$	0	0
$405$	− 1.00000i	− 0.0496904i
$406$	0	0
$407$	56.5685	2.80400
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	− 16.9706i	− 0.837096i
$412$	0	0
$413$	− 32.0000i	− 1.57462i
$414$	0	0
$415$	−2.82843	−0.138842
$416$	0	0
$417$	32.0000	1.56705
$418$	0	0
$419$	11.3137i	0.552711i	0.961056	+	0.276355i	$0.0891267\pi$
$419$	11.3137i	0.552711i	−0.961056	+	0.276355i	$0.910873\pi$
$420$	0	0
$421$	− 38.0000i	− 1.85201i	−0.377515	−	0.926003i	$-0.623221\pi$
$421$	− 38.0000i	− 1.85201i	0.377515	−	0.926003i	$-0.376779\pi$
$422$	0	0
$423$	−14.1421	−0.687614
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	5.65685i	0.273754i
$428$	0	0
$429$	32.0000i	1.54497i
$430$	0	0
$431$	5.65685	0.272481	0.136241	−	0.990676i	$-0.456498\pi$
$431$	5.65685	0.272481	0.136241	+	0.990676i	$0.456498\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	− 16.9706i	− 0.813676i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	22.6274	1.07995	0.539974	−	0.841682i	$-0.318434\pi$
$439$	22.6274	1.07995	0.539974	+	0.841682i	$0.318434\pi$
$440$	0	0
$441$	−5.00000	−0.238095
$442$	0	0
$443$	2.82843i	0.134383i	0.997740	+	0.0671913i	$0.0214038\pi$
$443$	2.82843i	0.134383i	−0.997740	+	0.0671913i	$0.978596\pi$
$444$	0	0
$445$	10.0000i	0.474045i
$446$	0	0
$447$	28.2843	1.33780
$448$	0	0
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	11.3137i	0.532742i
$452$	0	0
$453$	48.0000i	2.25524i
$454$	0	0
$455$	5.65685	0.265197
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	0	0
$459$	11.3137i	0.528079i
$460$	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
$463$	8.48528	0.394344	0.197172	−	0.980369i	$-0.436824\pi$
$463$	8.48528	0.394344	0.197172	+	0.980369i	$0.436824\pi$
$464$	0	0
$465$	16.0000	0.741982
$466$	0	0
$467$	14.1421i	0.654420i	0.944952	+	0.327210i	$0.106108\pi$
$467$	14.1421i	0.654420i	−0.944952	+	0.327210i	$0.893892\pi$
$468$	0	0
$469$	− 8.00000i	− 0.369406i
$470$	0	0
$471$	−50.9117	−2.34589
$472$	0	0
$473$	−48.0000	−2.20704
$474$	0	0
$475$	0	0
$476$	0	0
$477$	30.0000i	1.37361i
$478$	0	0
$479$	33.9411	1.55081	0.775405	−	0.631464i	$-0.217546\pi$
$479$	33.9411	1.55081	0.775405	+	0.631464i	$0.217546\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	0	0
$483$	22.6274i	1.02958i
$484$	0	0
$485$	− 2.00000i	− 0.0908153i
$486$	0	0
$487$	31.1127	1.40985	0.704925	−	0.709281i	$-0.250980\pi$
$487$	31.1127	1.40985	0.704925	+	0.709281i	$0.250980\pi$
$488$	0	0
$489$	−40.0000	−1.80886
$490$	0	0
$491$	39.5980i	1.78703i	0.449032	+	0.893516i	$0.351769\pi$
$491$	39.5980i	1.78703i	−0.449032	+	0.893516i	$0.648231\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	28.2843	1.27128
$496$	0	0
$497$	−16.0000	−0.717698
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	40.0000i	1.78707i
$502$	0	0
$503$	−8.48528	−0.378340	−0.189170	−	0.981944i	$-0.560580\pi$
$503$	−8.48528	−0.378340	−0.189170	+	0.981944i	$0.560580\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	0	0
$507$	− 25.4558i	− 1.13053i
$508$	0	0
$509$	− 26.0000i	− 1.15243i	−0.817298	−	0.576215i	$-0.804529\pi$
$509$	− 26.0000i	− 1.15243i	0.817298	−	0.576215i	$-0.195471\pi$
$510$	0	0
$511$	−16.9706	−0.750733
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 14.1421i	− 0.623177i
$516$	0	0
$517$	− 16.0000i	− 0.703679i
$518$	0	0
$519$	−5.65685	−0.248308
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	− 8.48528i	− 0.371035i	−0.982641	−	0.185518i	$-0.940604\pi$
$523$	− 8.48528i	− 0.371035i	0.982641	−	0.185518i	$-0.0593962\pi$
$524$	0	0
$525$	− 8.00000i	− 0.349149i
$526$	0	0
$527$	−11.3137	−0.492833
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	− 56.5685i	− 2.45487i
$532$	0	0
$533$	4.00000i	0.173259i
$534$	0	0
$535$	−14.1421	−0.611418
$536$	0	0
$537$	−32.0000	−1.38090
$538$	0	0
$539$	− 5.65685i	− 0.243658i
$540$	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
$543$	−39.5980	−1.69931
$544$	0	0
$545$	−18.0000	−0.771035
$546$	0	0
$547$	− 42.4264i	− 1.81402i	−0.421107	−	0.907011i	$-0.638358\pi$
$547$	− 42.4264i	− 1.81402i	0.421107	−	0.907011i	$-0.361642\pi$
$548$	0	0
$549$	10.0000i	0.426790i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	32.0000	1.36078
$554$	0	0
$555$	− 28.2843i	− 1.20060i
$556$	0	0
$557$	14.0000i	0.593199i	0.955002	+	0.296600i	$0.0958526\pi$
$557$	14.0000i	0.593199i	−0.955002	+	0.296600i	$0.904147\pi$
$558$	0	0
$559$	−16.9706	−0.717778
$560$	0	0
$561$	−32.0000	−1.35104
$562$	0	0
$563$	19.7990i	0.834428i	0.908808	+	0.417214i	$0.136993\pi$
$563$	19.7990i	0.834428i	−0.908808	+	0.417214i	$0.863007\pi$
$564$	0	0
$565$	− 2.00000i	− 0.0841406i
$566$	0	0
$567$	−2.82843	−0.118783
$568$	0	0
$569$	−34.0000	−1.42535	−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000	−1.42535	−0.712677	+	0.701492i	$0.752517\pi$
$570$	0	0
$571$	28.2843i	1.18366i	0.806063	+	0.591830i	$0.201594\pi$
$571$	28.2843i	1.18366i	−0.806063	+	0.591830i	$0.798406\pi$
$572$	0	0
$573$	48.0000i	2.00523i
$574$	0	0
$575$	−2.82843	−0.117954
$576$	0	0
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	0	0
$579$	− 50.9117i	− 2.11582i
$580$	0	0
$581$	8.00000i	0.331896i
$582$	0	0
$583$	−33.9411	−1.40570
$584$	0	0
$585$	10.0000	0.413449
$586$	0	0
$587$	− 25.4558i	− 1.05068i	−0.850894	−	0.525338i	$-0.823939\pi$
$587$	− 25.4558i	− 1.05068i	0.850894	−	0.525338i	$-0.176061\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−16.9706	−0.698076
$592$	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
$595$	5.65685i	0.231908i
$596$	0	0
$597$	64.0000i	2.61935i
$598$	0	0
$599$	11.3137	0.462266	0.231133	−	0.972922i	$-0.425757\pi$
$599$	11.3137	0.462266	0.231133	+	0.972922i	$0.425757\pi$
$600$	0	0
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	− 14.1421i	− 0.575912i
$604$	0	0
$605$	21.0000i	0.853771i
$606$	0	0
$607$	2.82843	0.114802	0.0574012	−	0.998351i	$-0.481719\pi$
$607$	2.82843	0.114802	0.0574012	+	0.998351i	$0.481719\pi$
$608$	0	0
$609$	−48.0000	−1.94506
$610$	0	0
$611$	− 5.65685i	− 0.228852i
$612$	0	0
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	0	0
$615$	5.65685	0.228106
$616$	0	0
$617$	−10.0000	−0.402585	−0.201292	−	0.979531i	$-0.564514\pi$
$617$	−10.0000	−0.402585	−0.201292	+	0.979531i	$0.564514\pi$
$618$	0	0
$619$	− 45.2548i	− 1.81895i	−0.415764	−	0.909473i	$-0.636486\pi$
$619$	− 45.2548i	− 1.81895i	0.415764	−	0.909473i	$-0.363514\pi$
$620$	0	0
$621$	16.0000i	0.642058i
$622$	0	0
$623$	28.2843	1.13319
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	20.0000i	0.797452i
$630$	0	0
$631$	−16.9706	−0.675587	−0.337794	−	0.941220i	$-0.609681\pi$
$631$	−16.9706	−0.675587	−0.337794	+	0.941220i	$0.609681\pi$
$632$	0	0
$633$	48.0000	1.90783
$634$	0	0
$635$	− 2.82843i	− 0.112243i
$636$	0	0
$637$	− 2.00000i	− 0.0792429i
$638$	0	0
$639$	−28.2843	−1.11891
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	0	0
$643$	31.1127i	1.22697i	0.789708	+	0.613483i	$0.210232\pi$
$643$	31.1127i	1.22697i	−0.789708	+	0.613483i	$0.789768\pi$
$644$	0	0
$645$	24.0000i	0.944999i
$646$	0	0
$647$	−14.1421	−0.555985	−0.277992	−	0.960583i	$-0.589669\pi$
$647$	−14.1421	−0.555985	−0.277992	+	0.960583i	$0.589669\pi$
$648$	0	0
$649$	64.0000	2.51222
$650$	0	0
$651$	− 45.2548i	− 1.77368i
$652$	0	0
$653$	14.0000i	0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$	14.0000i	0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$	0	0
$655$	−5.65685	−0.221032
$656$	0	0
$657$	−30.0000	−1.17041
$658$	0	0
$659$	33.9411i	1.32216i	0.750316	+	0.661079i	$0.229901\pi$
$659$	33.9411i	1.32216i	−0.750316	+	0.661079i	$0.770099\pi$
$660$	0	0
$661$	− 22.0000i	− 0.855701i	−0.903850	−	0.427850i	$-0.859271\pi$
$661$	− 22.0000i	− 0.855701i	0.903850	−	0.427850i	$-0.140729\pi$
$662$	0	0
$663$	−11.3137	−0.439388
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.9706i	0.657103i
$668$	0	0
$669$	− 24.0000i	− 0.927894i
$670$	0	0
$671$	−11.3137	−0.436761
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	0	0
$675$	− 5.65685i	− 0.217732i
$676$	0	0
$677$	− 38.0000i	− 1.46046i	−0.683202	−	0.730229i	$-0.739413\pi$
$677$	− 38.0000i	− 1.46046i	0.683202	−	0.730229i	$-0.260587\pi$
$678$	0	0
$679$	−5.65685	−0.217090
$680$	0	0
$681$	−56.0000	−2.14592
$682$	0	0
$683$	14.1421i	0.541134i	0.962701	+	0.270567i	$0.0872111\pi$
$683$	14.1421i	0.541134i	−0.962701	+	0.270567i	$0.912789\pi$
$684$	0	0
$685$	− 6.00000i	− 0.229248i
$686$	0	0
$687$	−39.5980	−1.51076
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	28.2843i	1.07598i	0.842950	+	0.537992i	$0.180817\pi$
$691$	28.2843i	1.07598i	−0.842950	+	0.537992i	$0.819183\pi$
$692$	0	0
$693$	− 80.0000i	− 3.03895i
$694$	0	0
$695$	11.3137	0.429153
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	28.2843i	1.06981i
$700$	0	0
$701$	− 2.00000i	− 0.0755390i	−0.999286	−	0.0377695i	$-0.987975\pi$
$701$	− 2.00000i	− 0.0755390i	0.999286	−	0.0377695i	$-0.0120253\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	− 5.65685i	− 0.212748i
$708$	0	0
$709$	− 14.0000i	− 0.525781i	−0.964826	−	0.262891i	$-0.915324\pi$
$709$	− 14.0000i	− 0.525781i	0.964826	−	0.262891i	$-0.0846758\pi$
$710$	0	0
$711$	56.5685	2.12149
$712$	0	0
$713$	−16.0000	−0.599205
$714$	0	0
$715$	11.3137i	0.423109i
$716$	0	0
$717$	− 32.0000i	− 1.19506i
$718$	0	0
$719$	11.3137	0.421930	0.210965	−	0.977494i	$-0.432339\pi$
$719$	11.3137	0.421930	0.210965	+	0.977494i	$0.432339\pi$
$720$	0	0
$721$	−40.0000	−1.48968
$722$	0	0
$723$	− 73.5391i	− 2.73495i
$724$	0	0
$725$	− 6.00000i	− 0.222834i
$726$	0	0
$727$	−25.4558	−0.944105	−0.472052	−	0.881570i	$-0.656487\pi$
$727$	−25.4558	−0.944105	−0.472052	+	0.881570i	$0.656487\pi$
$728$	0	0
$729$	43.0000	1.59259
$730$	0	0
$731$	− 16.9706i	− 0.627679i
$732$	0	0
$733$	46.0000i	1.69905i	0.527549	+	0.849524i	$0.323111\pi$
$733$	46.0000i	1.69905i	−0.527549	+	0.849524i	$0.676889\pi$
$734$	0	0
$735$	−2.82843	−0.104328
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	11.3137i	0.416181i	0.978110	+	0.208091i	$0.0667249\pi$
$739$	11.3137i	0.416181i	−0.978110	+	0.208091i	$0.933275\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	14.1421	0.518825	0.259412	−	0.965767i	$-0.416471\pi$
$743$	14.1421	0.518825	0.259412	+	0.965767i	$0.416471\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	0	0
$747$	14.1421i	0.517434i
$748$	0	0
$749$	40.0000i	1.46157i
$750$	0	0
$751$	16.9706	0.619265	0.309632	−	0.950856i	$-0.399794\pi$
$751$	16.9706	0.619265	0.309632	+	0.950856i	$0.399794\pi$
$752$	0	0
$753$	−16.0000	−0.583072
$754$	0	0
$755$	16.9706i	0.617622i
$756$	0	0
$757$	10.0000i	0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$	10.0000i	0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	0	0
$759$	−45.2548	−1.64265
$760$	0	0
$761$	−26.0000	−0.942499	−0.471250	−	0.882000i	$-0.656197\pi$
$761$	−26.0000	−0.942499	−0.471250	+	0.882000i	$0.656197\pi$
$762$	0	0
$763$	50.9117i	1.84313i
$764$	0	0
$765$	10.0000i	0.361551i
$766$	0	0
$767$	22.6274	0.817029
$768$	0	0
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	39.5980i	1.42609i
$772$	0	0
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	0	0
$775$	5.65685	0.203200
$776$	0	0
$777$	−80.0000	−2.86998
$778$	0	0
$779$	0	0
$780$	0	0
$781$	− 32.0000i	− 1.14505i
$782$	0	0
$783$	−33.9411	−1.21296
$784$	0	0
$785$	−18.0000	−0.642448
$786$	0	0
$787$	− 8.48528i	− 0.302468i	−0.988498	−	0.151234i	$-0.951675\pi$
$787$	− 8.48528i	− 0.302468i	0.988498	−	0.151234i	$-0.0483246\pi$
$788$	0	0
$789$	56.0000i	1.99365i
$790$	0	0
$791$	−5.65685	−0.201135
$792$	0	0
$793$	−4.00000	−0.142044
$794$	0	0
$795$	16.9706i	0.601884i
$796$	0	0
$797$	30.0000i	1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$	30.0000i	1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$	0	0
$799$	5.65685	0.200125
$800$	0	0
$801$	50.0000	1.76666
$802$	0	0
$803$	− 33.9411i	− 1.19776i
$804$	0	0
$805$	8.00000i	0.281963i
$806$	0	0
$807$	−50.9117	−1.79218
$808$	0	0
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	0	0
$811$	− 5.65685i	− 0.198639i	−0.995056	−	0.0993195i	$-0.968333\pi$
$811$	− 5.65685i	− 0.198639i	0.995056	−	0.0993195i	$-0.0316666\pi$
$812$	0	0
$813$	48.0000i	1.68343i
$814$	0	0
$815$	−14.1421	−0.495377
$816$	0	0
$817$	0	0
$818$	0	0
$819$	− 28.2843i	− 0.988332i
$820$	0	0
$821$	10.0000i	0.349002i	0.984657	+	0.174501i	$0.0558313\pi$
$821$	10.0000i	0.349002i	−0.984657	+	0.174501i	$0.944169\pi$
$822$	0	0
$823$	25.4558	0.887335	0.443667	−	0.896191i	$-0.353677\pi$
$823$	25.4558	0.887335	0.443667	+	0.896191i	$0.353677\pi$
$824$	0	0
$825$	16.0000	0.557048
$826$	0	0
$827$	31.1127i	1.08189i	0.841057	+	0.540947i	$0.181934\pi$
$827$	31.1127i	1.08189i	−0.841057	+	0.540947i	$0.818066\pi$
$828$	0	0
$829$	− 2.00000i	− 0.0694629i	−0.999397	−	0.0347314i	$-0.988942\pi$
$829$	− 2.00000i	− 0.0694629i	0.999397	−	0.0347314i	$-0.0110576\pi$
$830$	0	0
$831$	28.2843	0.981170
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	14.1421i	0.489409i
$836$	0	0
$837$	− 32.0000i	− 1.10608i
$838$	0	0
$839$	11.3137	0.390593	0.195296	−	0.980744i	$-0.437433\pi$
$839$	11.3137	0.390593	0.195296	+	0.980744i	$0.437433\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	− 84.8528i	− 2.92249i
$844$	0	0
$845$	− 9.00000i	− 0.309609i
$846$	0	0
$847$	59.3970	2.04090
$848$	0	0
$849$	−24.0000	−0.823678
$850$	0	0
$851$	28.2843i	0.969572i
$852$	0	0
$853$	− 22.0000i	− 0.753266i	−0.926363	−	0.376633i	$-0.877082\pi$
$853$	− 22.0000i	− 0.753266i	0.926363	−	0.376633i	$-0.122918\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	54.0000	1.84460	0.922302	−	0.386469i	$-0.126305\pi$
$857$	54.0000	1.84460	0.922302	+	0.386469i	$0.126305\pi$
$858$	0	0
$859$	33.9411i	1.15806i	0.815308	+	0.579028i	$0.196568\pi$
$859$	33.9411i	1.15806i	−0.815308	+	0.579028i	$0.803432\pi$
$860$	0	0
$861$	− 16.0000i	− 0.545279i
$862$	0	0
$863$	42.4264	1.44421	0.722106	−	0.691783i	$-0.243174\pi$
$863$	42.4264	1.44421	0.722106	+	0.691783i	$0.243174\pi$
$864$	0	0
$865$	−2.00000	−0.0680020
$866$	0	0
$867$	36.7696i	1.24876i
$868$	0	0
$869$	64.0000i	2.17105i
$870$	0	0
$871$	5.65685	0.191675
$872$	0	0
$873$	−10.0000	−0.338449
$874$	0	0
$875$	− 2.82843i	− 0.0956183i
$876$	0	0
$877$	− 2.00000i	− 0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$	− 2.00000i	− 0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$	0	0
$879$	28.2843	0.954005
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	− 2.82843i	− 0.0951842i	−0.998867	−	0.0475921i	$-0.984845\pi$
$883$	− 2.82843i	− 0.0951842i	0.998867	−	0.0475921i	$-0.0151548\pi$
$884$	0	0
$885$	− 32.0000i	− 1.07567i
$886$	0	0
$887$	−2.82843	−0.0949693	−0.0474846	−	0.998872i	$-0.515121\pi$
$887$	−2.82843	−0.0949693	−0.0474846	+	0.998872i	$0.515121\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	− 5.65685i	− 0.189512i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−11.3137	−0.378176
$896$	0	0
$897$	−16.0000	−0.534224
$898$	0	0
$899$	− 33.9411i	− 1.13200i
$900$	0	0
$901$	− 12.0000i	− 0.399778i
$902$	0	0
$903$	67.8823	2.25898
$904$	0	0
$905$	−14.0000	−0.465376
$906$	0	0
$907$	− 25.4558i	− 0.845247i	−0.906305	−	0.422624i	$-0.861109\pi$
$907$	− 25.4558i	− 0.845247i	0.906305	−	0.422624i	$-0.138891\pi$
$908$	0	0
$909$	− 10.0000i	− 0.331679i
$910$	0	0
$911$	−5.65685	−0.187420	−0.0937100	−	0.995600i	$-0.529873\pi$
$911$	−5.65685	−0.187420	−0.0937100	+	0.995600i	$0.529873\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	0	0
$915$	5.65685i	0.187010i
$916$	0	0
$917$	16.0000i	0.528367i
$918$	0	0
$919$	33.9411	1.11961	0.559807	−	0.828623i	$-0.310875\pi$
$919$	33.9411	1.11961	0.559807	+	0.828623i	$0.310875\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	0	0
$923$	− 11.3137i	− 0.372395i
$924$	0	0
$925$	− 10.0000i	− 0.328798i
$926$	0	0
$927$	−70.7107	−2.32244
$928$	0	0
$929$	26.0000	0.853032	0.426516	−	0.904480i	$-0.359741\pi$
$929$	26.0000	0.853032	0.426516	+	0.904480i	$0.359741\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	80.0000i	2.61908i
$934$	0	0
$935$	−11.3137	−0.369998
$936$	0	0
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	0	0
$939$	− 16.9706i	− 0.553813i
$940$	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
$943$	−5.65685	−0.184213
$944$	0	0
$945$	−16.0000	−0.520480
$946$	0	0
$947$	− 42.4264i	− 1.37867i	−0.724441	−	0.689336i	$-0.757902\pi$
$947$	− 42.4264i	− 1.37867i	0.724441	−	0.689336i	$-0.242098\pi$
$948$	0	0
$949$	− 12.0000i	− 0.389536i
$950$	0	0
$951$	84.8528	2.75154
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	16.9706i	0.549155i
$956$	0	0
$957$	− 96.0000i	− 3.10324i
$958$	0	0
$959$	−16.9706	−0.548008
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	70.7107i	2.27862i
$964$	0	0
$965$	− 18.0000i	− 0.579441i
$966$	0	0
$967$	42.4264	1.36434	0.682171	−	0.731193i	$-0.261036\pi$
$967$	42.4264	1.36434	0.682171	+	0.731193i	$0.261036\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	28.2843i	0.907685i	0.891082	+	0.453843i	$0.149947\pi$
$971$	28.2843i	0.907685i	−0.891082	+	0.453843i	$0.850053\pi$
$972$	0	0
$973$	− 32.0000i	− 1.02587i
$974$	0	0
$975$	5.65685	0.181164
$976$	0	0
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	56.5685i	1.80794i
$980$	0	0
$981$	90.0000i	2.87348i
$982$	0	0
$983$	2.82843	0.0902128	0.0451064	−	0.998982i	$-0.485637\pi$
$983$	2.82843	0.0902128	0.0451064	+	0.998982i	$0.485637\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	22.6274i	0.720239i
$988$	0	0
$989$	− 24.0000i	− 0.763156i
$990$	0	0
$991$	16.9706	0.539088	0.269544	−	0.962988i	$-0.413127\pi$
$991$	16.9706	0.539088	0.269544	+	0.962988i	$0.413127\pi$
$992$	0	0
$993$	16.0000	0.507745
$994$	0	0
$995$	22.6274i	0.717337i
$996$	0	0
$997$	10.0000i	0.316703i	0.987383	+	0.158352i	$0.0506179\pi$
$997$	10.0000i	0.316703i	−0.987383	+	0.158352i	$0.949382\pi$
$998$	0	0
$999$	−56.5685	−1.78975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.2.d.l.641.1		4
4.3	odd	2	inner	1280.2.d.l.641.3		4
8.3	odd	2	inner	1280.2.d.l.641.2		4
8.5	even	2	inner	1280.2.d.l.641.4		4
16.3	odd	4		320.2.a.g.1.1		2
16.5	even	4		160.2.a.c.1.1	✓	2
16.11	odd	4		160.2.a.c.1.2	yes	2
16.13	even	4		320.2.a.g.1.2		2
48.5	odd	4		1440.2.a.o.1.2		2
48.11	even	4		1440.2.a.o.1.1		2
48.29	odd	4		2880.2.a.bk.1.2		2
48.35	even	4		2880.2.a.bk.1.1		2
80.3	even	4		1600.2.c.n.449.2		4
80.13	odd	4		1600.2.c.n.449.3		4
80.19	odd	4		1600.2.a.bc.1.2		2
80.27	even	4		800.2.c.f.449.1		4
80.29	even	4		1600.2.a.bc.1.1		2
80.37	odd	4		800.2.c.f.449.4		4
80.43	even	4		800.2.c.f.449.3		4
80.53	odd	4		800.2.c.f.449.2		4
80.59	odd	4		800.2.a.m.1.1		2
80.67	even	4		1600.2.c.n.449.4		4
80.69	even	4		800.2.a.m.1.2		2
80.77	odd	4		1600.2.c.n.449.1		4
112.27	even	4		7840.2.a.bf.1.1		2
112.69	odd	4		7840.2.a.bf.1.2		2
240.53	even	4		7200.2.f.bh.6049.1		4
240.59	even	4		7200.2.a.cm.1.2		2
240.107	odd	4		7200.2.f.bh.6049.2		4
240.149	odd	4		7200.2.a.cm.1.1		2
240.197	even	4		7200.2.f.bh.6049.3		4
240.203	odd	4		7200.2.f.bh.6049.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.2.a.c.1.1	✓	2	16.5	even	4
160.2.a.c.1.2	yes	2	16.11	odd	4
320.2.a.g.1.1		2	16.3	odd	4
320.2.a.g.1.2		2	16.13	even	4
800.2.a.m.1.1		2	80.59	odd	4
800.2.a.m.1.2		2	80.69	even	4
800.2.c.f.449.1		4	80.27	even	4
800.2.c.f.449.2		4	80.53	odd	4
800.2.c.f.449.3		4	80.43	even	4
800.2.c.f.449.4		4	80.37	odd	4
1280.2.d.l.641.1		4	1.1	even	1	trivial
1280.2.d.l.641.2		4	8.3	odd	2	inner
1280.2.d.l.641.3		4	4.3	odd	2	inner
1280.2.d.l.641.4		4	8.5	even	2	inner
1440.2.a.o.1.1		2	48.11	even	4
1440.2.a.o.1.2		2	48.5	odd	4
1600.2.a.bc.1.1		2	80.29	even	4
1600.2.a.bc.1.2		2	80.19	odd	4
1600.2.c.n.449.1		4	80.77	odd	4
1600.2.c.n.449.2		4	80.3	even	4
1600.2.c.n.449.3		4	80.13	odd	4
1600.2.c.n.449.4		4	80.67	even	4
2880.2.a.bk.1.1		2	48.35	even	4
2880.2.a.bk.1.2		2	48.29	odd	4
7200.2.a.cm.1.1		2	240.149	odd	4
7200.2.a.cm.1.2		2	240.59	even	4
7200.2.f.bh.6049.1		4	240.53	even	4
7200.2.f.bh.6049.2		4	240.107	odd	4
7200.2.f.bh.6049.3		4	240.197	even	4
7200.2.f.bh.6049.4		4	240.203	odd	4
7840.2.a.bf.1.1		2	112.27	even	4
7840.2.a.bf.1.2		2	112.69	odd	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-2.82843i q^{3} -1.00000i q^{5} -2.82843 q^{7} -5.00000 q^{9} -5.65685i q^{11} -2.00000i q^{13} -2.82843 q^{15} +2.00000 q^{17} +8.00000i q^{21} +2.82843 q^{23} -1.00000 q^{25} +5.65685i q^{27} +6.00000i q^{29} -5.65685 q^{31} -16.0000 q^{33} +2.82843i q^{35} +10.0000i q^{37} -5.65685 q^{39} -2.00000 q^{41} -8.48528i q^{43} +5.00000i q^{45} +2.82843 q^{47} +1.00000 q^{49} -5.65685i q^{51} -6.00000i q^{53} -5.65685 q^{55} +11.3137i q^{59} -2.00000i q^{61} +14.1421 q^{63} -2.00000 q^{65} +2.82843i q^{67} -8.00000i q^{69} +5.65685 q^{71} +6.00000 q^{73} +2.82843i q^{75} +16.0000i q^{77} -11.3137 q^{79} +1.00000 q^{81} -2.82843i q^{83} -2.00000i q^{85} +16.9706 q^{87} -10.0000 q^{89} +5.65685i q^{91} +16.0000i q^{93} +2.00000 q^{97} +28.2843i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{9} + 8 q^{17} - 4 q^{25} - 64 q^{33} - 8 q^{41} + 4 q^{49} - 8 q^{65} + 24 q^{73} + 4 q^{81} - 40 q^{89} + 8 q^{97}+O(q^{100}) \) 4 * q - 20 * q^9 + 8 * q^17 - 4 * q^25 - 64 * q^33 - 8 * q^41 + 4 * q^49 - 8 * q^65 + 24 * q^73 + 4 * q^81 - 40 * q^89 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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