LMFDB - Embedded newform 1280.2.d.c.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 = [2,0,0,0,0,0,-4,0,-2,0,0,0,0,0,4] 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:	$1$
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 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} +1.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} +2.00000i q^{13} +2.00000 q^{15} -6.00000 q^{17} -4.00000i q^{19} +4.00000i q^{21} -6.00000 q^{23} -1.00000 q^{25} -4.00000i q^{27} +6.00000i q^{29} -4.00000 q^{31} -2.00000i q^{35} -2.00000i q^{37} +4.00000 q^{39} -6.00000 q^{41} +10.0000i q^{43} -1.00000i q^{45} -6.00000 q^{47} -3.00000 q^{49} +12.0000i q^{51} +6.00000i q^{53} -8.00000 q^{57} -12.0000i q^{59} +2.00000i q^{61} +2.00000 q^{63} -2.00000 q^{65} +2.00000i q^{67} +12.0000i q^{69} +12.0000 q^{71} -2.00000 q^{73} +2.00000i q^{75} +8.00000 q^{79} -11.0000 q^{81} +6.00000i q^{83} -6.00000i q^{85} +12.0000 q^{87} +6.00000 q^{89} -4.00000i q^{91} +8.00000i q^{93} +4.00000 q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{7} - 2 q^{9} + 4 q^{15} - 12 q^{17} - 12 q^{23} - 2 q^{25} - 8 q^{31} + 8 q^{39} - 12 q^{41} - 12 q^{47} - 6 q^{49} - 16 q^{57} + 4 q^{63} - 4 q^{65} + 24 q^{71} - 4 q^{73} + 16 q^{79} - 22 q^{81}+ \cdots + 4 q^{97}+O(q^{100}) $ 2 * q - 4 * q^7 - 2 * q^9 + 4 * q^15 - 12 * q^17 - 12 * q^23 - 2 * q^25 - 8 * q^31 + 8 * q^39 - 12 * q^41 - 12 * q^47 - 6 * q^49 - 16 * q^57 + 4 * q^63 - 4 * q^65 + 24 * q^71 - 4 * q^73 + 16 * q^79 - 22 * q^81 + 24 * q^87 + 12 * q^89 + 8 * q^95 + 4 * 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	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	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
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	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
$29$	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
$31$	−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
$33$	0	0
$34$	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
$39$	4.00000	0.640513
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	10.0000i	1.52499i	0.646997	+	0.762493i	$0.276025\pi$
$43$	10.0000i	1.52499i	−0.646997	+	0.762493i	$0.723975\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	12.0000i	1.68034i
$52$	0	0
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−8.00000	−1.05963
$58$	0	0
$59$	− 12.0000i	− 1.56227i	−0.624364	−	0.781133i	$-0.714642\pi$
$59$	− 12.0000i	− 1.56227i	0.624364	−	0.781133i	$-0.285358\pi$
$60$	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
$65$	−2.00000	−0.248069
$66$	0	0
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	0	0
$69$	12.0000i	1.44463i
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	2.00000i	0.230940i
$76$	0	0
$77$	0	0
$78$	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
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	− 6.00000i	− 0.650791i
$86$	0	0
$87$	12.0000	1.28654
$88$	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
$91$	− 4.00000i	− 0.419314i
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 6.00000i	− 0.597022i	−0.954406	−	0.298511i	$-0.903510\pi$
$101$	− 6.00000i	− 0.597022i	0.954406	−	0.298511i	$-0.0964900\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	6.00000i	0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$	6.00000i	0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$	0	0
$109$	2.00000i	0.191565i	0.995402	+	0.0957826i	$0.0305354\pi$
$109$	2.00000i	0.191565i	−0.995402	+	0.0957826i	$0.969465\pi$
$110$	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
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	12.0000i	1.08200i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	20.0000	1.76090
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	8.00000i	0.693688i
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	4.00000i	0.339276i	0.985506	+	0.169638i	$0.0542598\pi$
$139$	4.00000i	0.339276i	−0.985506	+	0.169638i	$0.945740\pi$
$140$	0	0
$141$	12.0000i	1.01058i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	6.00000i	0.494872i
$148$	0	0
$149$	6.00000i	0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$	6.00000i	0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	− 4.00000i	− 0.321288i
$156$	0	0
$157$	− 22.0000i	− 1.75579i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	− 22.0000i	− 1.75579i	0.478852	−	0.877896i	$-0.341053\pi$
$158$	0	0
$159$	12.0000	0.951662
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	− 10.0000i	− 0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$	− 10.0000i	− 0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	4.00000i	0.305888i
$172$	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
$177$	−24.0000	−1.80395
$178$	0	0
$179$	− 12.0000i	− 0.896922i	−0.893802	−	0.448461i	$-0.851972\pi$
$179$	− 12.0000i	− 0.896922i	0.893802	−	0.448461i	$-0.148028\pi$
$180$	0	0
$181$	10.0000i	0.743294i	0.928374	+	0.371647i	$0.121207\pi$
$181$	10.0000i	0.743294i	−0.928374	+	0.371647i	$0.878793\pi$
$182$	0	0
$183$	4.00000	0.295689
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	0	0
$188$	0	0
$189$	8.00000i	0.581914i
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	26.0000	1.87152	0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000	1.87152	0.935760	+	0.352636i	$0.114715\pi$
$194$	0	0
$195$	4.00000i	0.286446i
$196$	0	0
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	− 6.00000i	− 0.419058i
$206$	0	0
$207$	6.00000	0.417029
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 16.0000i	− 1.10149i	−0.834675	−	0.550743i	$-0.814345\pi$
$211$	− 16.0000i	− 1.10149i	0.834675	−	0.550743i	$-0.185655\pi$
$212$	0	0
$213$	− 24.0000i	− 1.64445i
$214$	0	0
$215$	−10.0000	−0.681994
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	4.00000i	0.270295i
$220$	0	0
$221$	− 12.0000i	− 0.807207i
$222$	0	0
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	− 6.00000i	− 0.398234i	−0.979976	−	0.199117i	$-0.936193\pi$
$227$	− 6.00000i	− 0.398234i	0.979976	−	0.199117i	$-0.0638074\pi$
$228$	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
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	− 6.00000i	− 0.391397i
$236$	0	0
$237$	− 16.0000i	− 1.03931i
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	10.0000i	0.641500i
$244$	0	0
$245$	− 3.00000i	− 0.191663i
$246$	0	0
$247$	8.00000	0.509028
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−12.0000	−0.751469
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	4.00000i	0.248548i
$260$	0	0
$261$	− 6.00000i	− 0.371391i
$262$	0	0
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	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
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 26.0000i	− 1.56219i	−0.624413	−	0.781094i	$-0.714662\pi$
$277$	− 26.0000i	− 1.56219i	0.624413	−	0.781094i	$-0.285338\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	− 14.0000i	− 0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$	− 14.0000i	− 0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$	0	0
$285$	− 8.00000i	− 0.473879i
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	− 4.00000i	− 0.234484i
$292$	0	0
$293$	30.0000i	1.75262i	0.481749	+	0.876309i	$0.340002\pi$
$293$	30.0000i	1.75262i	−0.481749	+	0.876309i	$0.659998\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 12.0000i	− 0.693978i
$300$	0	0
$301$	− 20.0000i	− 1.15278i
$302$	0	0
$303$	−12.0000	−0.689382
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	2.00000i	0.114146i	0.998370	+	0.0570730i	$0.0181768\pi$
$307$	2.00000i	0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$	0	0
$309$	28.0000i	1.59286i
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	0	0
$315$	2.00000i	0.112687i
$316$	0	0
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	24.0000i	1.33540i
$324$	0	0
$325$	− 2.00000i	− 0.110940i
$326$	0	0
$327$	4.00000	0.221201
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	− 8.00000i	− 0.439720i	−0.975531	−	0.219860i	$-0.929440\pi$
$331$	− 8.00000i	− 0.439720i	0.975531	−	0.219860i	$-0.0705600\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	−2.00000	−0.109272
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	12.0000i	0.651751i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	−12.0000	−0.646058
$346$	0	0
$347$	30.0000i	1.61048i	0.592946	+	0.805242i	$0.297965\pi$
$347$	30.0000i	1.61048i	−0.592946	+	0.805242i	$0.702035\pi$
$348$	0	0
$349$	− 10.0000i	− 0.535288i	−0.963518	−	0.267644i	$-0.913755\pi$
$349$	− 10.0000i	− 0.535288i	0.963518	−	0.267644i	$-0.0862451\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	12.0000i	0.636894i
$356$	0	0
$357$	− 24.0000i	− 1.27021i
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	− 22.0000i	− 1.15470i
$364$	0	0
$365$	− 2.00000i	− 0.104685i
$366$	0	0
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	− 12.0000i	− 0.623009i
$372$	0	0
$373$	− 26.0000i	− 1.34623i	−0.739538	−	0.673114i	$-0.764956\pi$
$373$	− 26.0000i	− 1.34623i	0.739538	−	0.673114i	$-0.235044\pi$
$374$	0	0
$375$	−2.00000	−0.103280
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	28.0000i	1.43826i	0.694874	+	0.719132i	$0.255460\pi$
$379$	28.0000i	1.43826i	−0.694874	+	0.719132i	$0.744540\pi$
$380$	0	0
$381$	− 4.00000i	− 0.204926i
$382$	0	0
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 10.0000i	− 0.508329i
$388$	0	0
$389$	6.00000i	0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$	6.00000i	0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$	0	0
$391$	36.0000	1.82060
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.00000i	0.402524i
$396$	0	0
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	0	0
$399$	16.0000	0.801002
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	− 8.00000i	− 0.398508i
$404$	0	0
$405$	− 11.0000i	− 0.546594i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	34.0000	1.68119	0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000	1.68119	0.840596	+	0.541663i	$0.182205\pi$
$410$	0	0
$411$	36.0000i	1.77575i
$412$	0	0
$413$	24.0000i	1.18096i
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	8.00000	0.391762
$418$	0	0
$419$	36.0000i	1.75872i	0.476162	+	0.879358i	$0.342028\pi$
$419$	36.0000i	1.75872i	−0.476162	+	0.879358i	$0.657972\pi$
$420$	0	0
$421$	− 26.0000i	− 1.26716i	−0.773676	−	0.633581i	$-0.781584\pi$
$421$	− 26.0000i	− 1.26716i	0.773676	−	0.633581i	$-0.218416\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	− 4.00000i	− 0.193574i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	12.0000i	0.575356i
$436$	0	0
$437$	24.0000i	1.14808i
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	− 6.00000i	− 0.285069i	−0.989790	−	0.142534i	$-0.954475\pi$
$443$	− 6.00000i	− 0.285069i	0.989790	−	0.142534i	$-0.0455251\pi$
$444$	0	0
$445$	6.00000i	0.284427i
$446$	0	0
$447$	12.0000	0.567581
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	40.0000i	1.87936i
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	0	0
$459$	24.0000i	1.12022i
$460$	0	0
$461$	30.0000i	1.39724i	0.715493	+	0.698620i	$0.246202\pi$
$461$	30.0000i	1.39724i	−0.715493	+	0.698620i	$0.753798\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	− 30.0000i	− 1.38823i	−0.719862	−	0.694117i	$-0.755795\pi$
$467$	− 30.0000i	− 1.38823i	0.719862	−	0.694117i	$-0.244205\pi$
$468$	0	0
$469$	− 4.00000i	− 0.184703i
$470$	0	0
$471$	−44.0000	−2.02741
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.00000i	0.183533i
$476$	0	0
$477$	− 6.00000i	− 0.274721i
$478$	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
$481$	4.00000	0.182384
$482$	0	0
$483$	− 24.0000i	− 1.09204i
$484$	0	0
$485$	2.00000i	0.0908153i
$486$	0	0
$487$	−26.0000	−1.17817	−0.589086	−	0.808070i	$-0.700512\pi$
$487$	−26.0000	−1.17817	−0.589086	+	0.808070i	$0.700512\pi$
$488$	0	0
$489$	−20.0000	−0.904431
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	− 36.0000i	− 1.62136i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−24.0000	−1.07655
$498$	0	0
$499$	− 4.00000i	− 0.179065i	−0.995984	−	0.0895323i	$-0.971463\pi$
$499$	− 4.00000i	− 0.179065i	0.995984	−	0.0895323i	$-0.0285372\pi$
$500$	0	0
$501$	36.0000i	1.60836i
$502$	0	0
$503$	18.0000	0.802580	0.401290	−	0.915951i	$-0.368562\pi$
$503$	18.0000	0.802580	0.401290	+	0.915951i	$0.368562\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	− 18.0000i	− 0.799408i
$508$	0	0
$509$	6.00000i	0.265945i	0.991120	+	0.132973i	$0.0424523\pi$
$509$	6.00000i	0.265945i	−0.991120	+	0.132973i	$0.957548\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	0	0
$513$	−16.0000	−0.706417
$514$	0	0
$515$	− 14.0000i	− 0.616914i
$516$	0	0
$517$	0	0
$518$	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
$523$	− 14.0000i	− 0.612177i	−0.952003	−	0.306089i	$-0.900980\pi$
$523$	− 14.0000i	− 0.612177i	0.952003	−	0.306089i	$-0.0990204\pi$
$524$	0	0
$525$	− 4.00000i	− 0.174574i
$526$	0	0
$527$	24.0000	1.04546
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	12.0000i	0.520756i
$532$	0	0
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	−6.00000	−0.259403
$536$	0	0
$537$	−24.0000	−1.03568
$538$	0	0
$539$	0	0
$540$	0	0
$541$	14.0000i	0.601907i	0.953639	+	0.300954i	$0.0973049\pi$
$541$	14.0000i	0.601907i	−0.953639	+	0.300954i	$0.902695\pi$
$542$	0	0
$543$	20.0000	0.858282
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	26.0000i	1.11168i	0.831289	+	0.555840i	$0.187603\pi$
$547$	26.0000i	1.11168i	−0.831289	+	0.555840i	$0.812397\pi$
$548$	0	0
$549$	− 2.00000i	− 0.0853579i
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	−16.0000	−0.680389
$554$	0	0
$555$	− 4.00000i	− 0.169791i
$556$	0	0
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	0	0
$559$	−20.0000	−0.845910
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 18.0000i	− 0.758610i	−0.925272	−	0.379305i	$-0.876163\pi$
$563$	− 18.0000i	− 0.758610i	0.925272	−	0.379305i	$-0.123837\pi$
$564$	0	0
$565$	− 6.00000i	− 0.252422i
$566$	0	0
$567$	22.0000	0.923913
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	− 8.00000i	− 0.334790i	−0.985890	−	0.167395i	$-0.946465\pi$
$571$	− 8.00000i	− 0.334790i	0.985890	−	0.167395i	$-0.0535355\pi$
$572$	0	0
$573$	24.0000i	1.00261i
$574$	0	0
$575$	6.00000	0.250217
$576$	0	0
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	0	0
$579$	− 52.0000i	− 2.16105i
$580$	0	0
$581$	− 12.0000i	− 0.497844i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	6.00000i	0.247647i	0.992304	+	0.123823i	$0.0395156\pi$
$587$	6.00000i	0.247647i	−0.992304	+	0.123823i	$0.960484\pi$
$588$	0	0
$589$	16.0000i	0.659269i
$590$	0	0
$591$	−36.0000	−1.48084
$592$	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
$595$	12.0000i	0.491952i
$596$	0	0
$597$	16.0000i	0.654836i
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	− 2.00000i	− 0.0814463i
$604$	0	0
$605$	11.0000i	0.447214i
$606$	0	0
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	0	0
$611$	− 12.0000i	− 0.485468i
$612$	0	0
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	0	0
$615$	−12.0000	−0.483887
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	− 20.0000i	− 0.803868i	−0.915669	−	0.401934i	$-0.868338\pi$
$619$	− 20.0000i	− 0.803868i	0.915669	−	0.401934i	$-0.131662\pi$
$620$	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
$627$	0	0
$628$	0	0
$629$	12.0000i	0.478471i
$630$	0	0
$631$	28.0000	1.11466	0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000	1.11466	0.557331	+	0.830290i	$0.311825\pi$
$632$	0	0
$633$	−32.0000	−1.27189
$634$	0	0
$635$	2.00000i	0.0793676i
$636$	0	0
$637$	− 6.00000i	− 0.237729i
$638$	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
$643$	14.0000i	0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$	14.0000i	0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	0	0
$645$	20.0000i	0.787499i
$646$	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
$653$	42.0000i	1.64359i	0.569785	+	0.821794i	$0.307026\pi$
$653$	42.0000i	1.64359i	−0.569785	+	0.821794i	$0.692974\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	36.0000i	1.40236i	0.712984	+	0.701180i	$0.247343\pi$
$659$	36.0000i	1.40236i	−0.712984	+	0.701180i	$0.752657\pi$
$660$	0	0
$661$	22.0000i	0.855701i	0.903850	+	0.427850i	$0.140729\pi$
$661$	22.0000i	0.855701i	−0.903850	+	0.427850i	$0.859271\pi$
$662$	0	0
$663$	−24.0000	−0.932083
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	− 36.0000i	− 1.39393i
$668$	0	0
$669$	20.0000i	0.773245i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	0	0
$675$	4.00000i	0.153960i
$676$	0	0
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	42.0000i	1.60709i	0.595247	+	0.803543i	$0.297054\pi$
$683$	42.0000i	1.60709i	−0.595247	+	0.803543i	$0.702946\pi$
$684$	0	0
$685$	− 18.0000i	− 0.687745i
$686$	0	0
$687$	−28.0000	−1.06827
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	8.00000i	0.304334i	0.988355	+	0.152167i	$0.0486252\pi$
$691$	8.00000i	0.304334i	−0.988355	+	0.152167i	$0.951375\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.00000	−0.151729
$696$	0	0
$697$	36.0000	1.36360
$698$	0	0
$699$	− 12.0000i	− 0.453882i
$700$	0	0
$701$	− 30.0000i	− 1.13308i	−0.824033	−	0.566542i	$-0.808281\pi$
$701$	− 30.0000i	− 1.13308i	0.824033	−	0.566542i	$-0.191719\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	0	0
$705$	−12.0000	−0.451946
$706$	0	0
$707$	12.0000i	0.451306i
$708$	0	0
$709$	34.0000i	1.27690i	0.769665	+	0.638448i	$0.220423\pi$
$709$	34.0000i	1.27690i	−0.769665	+	0.638448i	$0.779577\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	24.0000	0.898807
$714$	0	0
$715$	0	0
$716$	0	0
$717$	48.0000i	1.79259i
$718$	0	0
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	28.0000	1.04277
$722$	0	0
$723$	− 28.0000i	− 1.04133i
$724$	0	0
$725$	− 6.00000i	− 0.222834i
$726$	0	0
$727$	46.0000	1.70605	0.853023	−	0.521874i	$-0.174767\pi$
$727$	46.0000	1.70605	0.853023	+	0.521874i	$0.174767\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	− 60.0000i	− 2.21918i
$732$	0	0
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	0	0
$735$	−6.00000	−0.221313
$736$	0	0
$737$	0	0
$738$	0	0
$739$	20.0000i	0.735712i	0.929883	+	0.367856i	$0.119908\pi$
$739$	20.0000i	0.735712i	−0.929883	+	0.367856i	$0.880092\pi$
$740$	0	0
$741$	− 16.0000i	− 0.587775i
$742$	0	0
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	− 6.00000i	− 0.219529i
$748$	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
$753$	0	0
$754$	0	0
$755$	− 20.0000i	− 0.727875i
$756$	0	0
$757$	22.0000i	0.799604i	0.916602	+	0.399802i	$0.130921\pi$
$757$	22.0000i	0.799604i	−0.916602	+	0.399802i	$0.869079\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	− 4.00000i	− 0.144810i
$764$	0	0
$765$	6.00000i	0.216930i
$766$	0	0
$767$	24.0000	0.866590
$768$	0	0
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	12.0000i	0.432169i
$772$	0	0
$773$	30.0000i	1.07903i	0.841978	+	0.539513i	$0.181391\pi$
$773$	30.0000i	1.07903i	−0.841978	+	0.539513i	$0.818609\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	8.00000	0.286998
$778$	0	0
$779$	24.0000i	0.859889i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	24.0000	0.857690
$784$	0	0
$785$	22.0000	0.785214
$786$	0	0
$787$	26.0000i	0.926800i	0.886149	+	0.463400i	$0.153371\pi$
$787$	26.0000i	0.926800i	−0.886149	+	0.463400i	$0.846629\pi$
$788$	0	0
$789$	− 36.0000i	− 1.28163i
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	−4.00000	−0.142044
$794$	0	0
$795$	12.0000i	0.425596i
$796$	0	0
$797$	42.0000i	1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$797$	42.0000i	1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$798$	0	0
$799$	36.0000	1.27359
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	0	0
$804$	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
$811$	16.0000i	0.561836i	0.959732	+	0.280918i	$0.0906389\pi$
$811$	16.0000i	0.561836i	−0.959732	+	0.280918i	$0.909361\pi$
$812$	0	0
$813$	− 40.0000i	− 1.40286i
$814$	0	0
$815$	10.0000	0.350285
$816$	0	0
$817$	40.0000	1.39942
$818$	0	0
$819$	4.00000i	0.139771i
$820$	0	0
$821$	54.0000i	1.88461i	0.334751	+	0.942306i	$0.391348\pi$
$821$	54.0000i	1.88461i	−0.334751	+	0.942306i	$0.608652\pi$
$822$	0	0
$823$	−38.0000	−1.32460	−0.662298	−	0.749240i	$-0.730419\pi$
$823$	−38.0000	−1.32460	−0.662298	+	0.749240i	$0.730419\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.0000i	1.04320i	0.853189	+	0.521601i	$0.174665\pi$
$827$	30.0000i	1.04320i	−0.853189	+	0.521601i	$0.825335\pi$
$828$	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
$831$	−52.0000	−1.80386
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	− 18.0000i	− 0.622916i
$836$	0	0
$837$	16.0000i	0.553041i
$838$	0	0
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	12.0000i	0.413302i
$844$	0	0
$845$	9.00000i	0.309609i
$846$	0	0
$847$	−22.0000	−0.755929
$848$	0	0
$849$	−28.0000	−0.960958
$850$	0	0
$851$	12.0000i	0.411355i
$852$	0	0
$853$	− 50.0000i	− 1.71197i	−0.517003	−	0.855984i	$-0.672952\pi$
$853$	− 50.0000i	− 1.71197i	0.517003	−	0.855984i	$-0.327048\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	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
$859$	4.00000i	0.136478i	0.997669	+	0.0682391i	$0.0217381\pi$
$859$	4.00000i	0.136478i	−0.997669	+	0.0682391i	$0.978262\pi$
$860$	0	0
$861$	− 24.0000i	− 0.817918i
$862$	0	0
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	0	0
$867$	− 38.0000i	− 1.29055i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−4.00000	−0.135535
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	2.00000i	0.0676123i
$876$	0	0
$877$	26.0000i	0.877958i	0.898497	+	0.438979i	$0.144660\pi$
$877$	26.0000i	0.877958i	−0.898497	+	0.438979i	$0.855340\pi$
$878$	0	0
$879$	60.0000	2.02375
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	14.0000i	0.471138i	0.971858	+	0.235569i	$0.0756953\pi$
$883$	14.0000i	0.471138i	−0.971858	+	0.235569i	$0.924305\pi$
$884$	0	0
$885$	− 24.0000i	− 0.806751i
$886$	0	0
$887$	−18.0000	−0.604381	−0.302190	−	0.953248i	$-0.597718\pi$
$887$	−18.0000	−0.604381	−0.302190	+	0.953248i	$0.597718\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	0	0
$892$	0	0
$893$	24.0000i	0.803129i
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	−24.0000	−0.801337
$898$	0	0
$899$	− 24.0000i	− 0.800445i
$900$	0	0
$901$	− 36.0000i	− 1.19933i
$902$	0	0
$903$	−40.0000	−1.33112
$904$	0	0
$905$	−10.0000	−0.332411
$906$	0	0
$907$	46.0000i	1.52740i	0.645568	+	0.763702i	$0.276621\pi$
$907$	46.0000i	1.52740i	−0.645568	+	0.763702i	$0.723379\pi$
$908$	0	0
$909$	6.00000i	0.199007i
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	4.00000i	0.132236i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	24.0000i	0.789970i
$924$	0	0
$925$	2.00000i	0.0657596i
$926$	0	0
$927$	14.0000	0.459820
$928$	0	0
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	0	0
$931$	12.0000i	0.393284i
$932$	0	0
$933$	24.0000i	0.785725i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	− 44.0000i	− 1.43589i
$940$	0	0
$941$	− 18.0000i	− 0.586783i	−0.955992	−	0.293392i	$-0.905216\pi$
$941$	− 18.0000i	− 0.586783i	0.955992	−	0.293392i	$-0.0947840\pi$
$942$	0	0
$943$	36.0000	1.17232
$944$	0	0
$945$	−8.00000	−0.260240
$946$	0	0
$947$	18.0000i	0.584921i	0.956278	+	0.292461i	$0.0944741\pi$
$947$	18.0000i	0.584921i	−0.956278	+	0.292461i	$0.905526\pi$
$948$	0	0
$949$	− 4.00000i	− 0.129845i
$950$	0	0
$951$	−12.0000	−0.389127
$952$	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
$955$	− 12.0000i	− 0.388311i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36.0000	1.16250
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	− 6.00000i	− 0.193347i
$964$	0	0
$965$	26.0000i	0.836970i
$966$	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
$969$	48.0000	1.54198
$970$	0	0
$971$	24.0000i	0.770197i	0.922876	+	0.385098i	$0.125832\pi$
$971$	24.0000i	0.770197i	−0.922876	+	0.385098i	$0.874168\pi$
$972$	0	0
$973$	− 8.00000i	− 0.256468i
$974$	0	0
$975$	−4.00000	−0.128103
$976$	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
$979$	0	0
$980$	0	0
$981$	− 2.00000i	− 0.0638551i
$982$	0	0
$983$	18.0000	0.574111	0.287055	−	0.957914i	$-0.407324\pi$
$983$	18.0000	0.574111	0.287055	+	0.957914i	$0.407324\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	− 24.0000i	− 0.763928i
$988$	0	0
$989$	− 60.0000i	− 1.90789i
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	0	0
$993$	−16.0000	−0.507745
$994$	0	0
$995$	− 8.00000i	− 0.253617i
$996$	0	0
$997$	− 26.0000i	− 0.823428i	−0.911313	−	0.411714i	$-0.864930\pi$
$997$	− 26.0000i	− 0.823428i	0.911313	−	0.411714i	$-0.135070\pi$
$998$	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.c.641.1		2
4.3	odd	2		1280.2.d.g.641.2		2
8.3	odd	2		1280.2.d.g.641.1		2
8.5	even	2	inner	1280.2.d.c.641.2		2
16.3	odd	4		320.2.a.a.1.1		1
16.5	even	4		20.2.a.a.1.1	✓	1
16.11	odd	4		80.2.a.b.1.1		1
16.13	even	4		320.2.a.f.1.1		1
48.5	odd	4		180.2.a.a.1.1		1
48.11	even	4		720.2.a.h.1.1		1
48.29	odd	4		2880.2.a.m.1.1		1
48.35	even	4		2880.2.a.f.1.1		1
80.3	even	4		1600.2.c.e.449.1		2
80.13	odd	4		1600.2.c.d.449.2		2
80.19	odd	4		1600.2.a.w.1.1		1
80.27	even	4		400.2.c.b.49.1		2
80.29	even	4		1600.2.a.c.1.1		1
80.37	odd	4		100.2.c.a.49.2		2
80.43	even	4		400.2.c.b.49.2		2
80.53	odd	4		100.2.c.a.49.1		2
80.59	odd	4		400.2.a.c.1.1		1
80.67	even	4		1600.2.c.e.449.2		2
80.69	even	4		100.2.a.a.1.1		1
80.77	odd	4		1600.2.c.d.449.1		2
112.5	odd	12		980.2.i.c.361.1		2
112.27	even	4		3920.2.a.h.1.1		1
112.37	even	12		980.2.i.i.361.1		2
112.53	even	12		980.2.i.i.961.1		2
112.69	odd	4		980.2.a.h.1.1		1
112.101	odd	12		980.2.i.c.961.1		2
144.5	odd	12		1620.2.i.b.541.1		2
144.85	even	12		1620.2.i.h.541.1		2
144.101	odd	12		1620.2.i.b.1081.1		2
144.133	even	12		1620.2.i.h.1081.1		2
176.21	odd	4		2420.2.a.a.1.1		1
176.43	even	4		9680.2.a.ba.1.1		1
208.5	odd	4		3380.2.f.b.3041.2		2
208.21	odd	4		3380.2.f.b.3041.1		2
208.181	even	4		3380.2.a.c.1.1		1
240.53	even	4		900.2.d.c.649.1		2
240.59	even	4		3600.2.a.be.1.1		1
240.107	odd	4		3600.2.f.j.2449.1		2
240.149	odd	4		900.2.a.b.1.1		1
240.197	even	4		900.2.d.c.649.2		2
240.203	odd	4		3600.2.f.j.2449.2		2
272.21	even	4		5780.2.c.a.5201.2		2
272.101	even	4		5780.2.a.f.1.1		1
272.149	even	4		5780.2.c.a.5201.1		2
304.37	odd	4		7220.2.a.f.1.1		1
336.293	even	4		8820.2.a.g.1.1		1
560.69	odd	4		4900.2.a.e.1.1		1
560.293	even	4		4900.2.e.f.2549.2		2
560.517	even	4		4900.2.e.f.2549.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.2.a.a.1.1	✓	1	16.5	even	4
80.2.a.b.1.1		1	16.11	odd	4
100.2.a.a.1.1		1	80.69	even	4
100.2.c.a.49.1		2	80.53	odd	4
100.2.c.a.49.2		2	80.37	odd	4
180.2.a.a.1.1		1	48.5	odd	4
320.2.a.a.1.1		1	16.3	odd	4
320.2.a.f.1.1		1	16.13	even	4
400.2.a.c.1.1		1	80.59	odd	4
400.2.c.b.49.1		2	80.27	even	4
400.2.c.b.49.2		2	80.43	even	4
720.2.a.h.1.1		1	48.11	even	4
900.2.a.b.1.1		1	240.149	odd	4
900.2.d.c.649.1		2	240.53	even	4
900.2.d.c.649.2		2	240.197	even	4
980.2.a.h.1.1		1	112.69	odd	4
980.2.i.c.361.1		2	112.5	odd	12
980.2.i.c.961.1		2	112.101	odd	12
980.2.i.i.361.1		2	112.37	even	12
980.2.i.i.961.1		2	112.53	even	12
1280.2.d.c.641.1		2	1.1	even	1	trivial
1280.2.d.c.641.2		2	8.5	even	2	inner
1280.2.d.g.641.1		2	8.3	odd	2
1280.2.d.g.641.2		2	4.3	odd	2
1600.2.a.c.1.1		1	80.29	even	4
1600.2.a.w.1.1		1	80.19	odd	4
1600.2.c.d.449.1		2	80.77	odd	4
1600.2.c.d.449.2		2	80.13	odd	4
1600.2.c.e.449.1		2	80.3	even	4
1600.2.c.e.449.2		2	80.67	even	4
1620.2.i.b.541.1		2	144.5	odd	12
1620.2.i.b.1081.1		2	144.101	odd	12
1620.2.i.h.541.1		2	144.85	even	12
1620.2.i.h.1081.1		2	144.133	even	12
2420.2.a.a.1.1		1	176.21	odd	4
2880.2.a.f.1.1		1	48.35	even	4
2880.2.a.m.1.1		1	48.29	odd	4
3380.2.a.c.1.1		1	208.181	even	4
3380.2.f.b.3041.1		2	208.21	odd	4
3380.2.f.b.3041.2		2	208.5	odd	4
3600.2.a.be.1.1		1	240.59	even	4
3600.2.f.j.2449.1		2	240.107	odd	4
3600.2.f.j.2449.2		2	240.203	odd	4
3920.2.a.h.1.1		1	112.27	even	4
4900.2.a.e.1.1		1	560.69	odd	4
4900.2.e.f.2549.1		2	560.517	even	4
4900.2.e.f.2549.2		2	560.293	even	4
5780.2.a.f.1.1		1	272.101	even	4
5780.2.c.a.5201.1		2	272.149	even	4
5780.2.c.a.5201.2		2	272.21	even	4
7220.2.a.f.1.1		1	304.37	odd	4
8820.2.a.g.1.1		1	336.293	even	4
9680.2.a.ba.1.1		1	176.43	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} +1.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} +2.00000i q^{13} +2.00000 q^{15} -6.00000 q^{17} -4.00000i q^{19} +4.00000i q^{21} -6.00000 q^{23} -1.00000 q^{25} -4.00000i q^{27} +6.00000i q^{29} -4.00000 q^{31} -2.00000i q^{35} -2.00000i q^{37} +4.00000 q^{39} -6.00000 q^{41} +10.0000i q^{43} -1.00000i q^{45} -6.00000 q^{47} -3.00000 q^{49} +12.0000i q^{51} +6.00000i q^{53} -8.00000 q^{57} -12.0000i q^{59} +2.00000i q^{61} +2.00000 q^{63} -2.00000 q^{65} +2.00000i q^{67} +12.0000i q^{69} +12.0000 q^{71} -2.00000 q^{73} +2.00000i q^{75} +8.00000 q^{79} -11.0000 q^{81} +6.00000i q^{83} -6.00000i q^{85} +12.0000 q^{87} +6.00000 q^{89} -4.00000i q^{91} +8.00000i q^{93} +4.00000 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{7} - 2 q^{9} + 4 q^{15} - 12 q^{17} - 12 q^{23} - 2 q^{25} - 8 q^{31} + 8 q^{39} - 12 q^{41} - 12 q^{47} - 6 q^{49} - 16 q^{57} + 4 q^{63} - 4 q^{65} + 24 q^{71} - 4 q^{73} + 16 q^{79} - 22 q^{81}+ \cdots + 4 q^{97}+O(q^{100}) \) 2 * q - 4 * q^7 - 2 * q^9 + 4 * q^15 - 12 * q^17 - 12 * q^23 - 2 * q^25 - 8 * q^31 + 8 * q^39 - 12 * q^41 - 12 * q^47 - 6 * q^49 - 16 * q^57 + 4 * q^63 - 4 * q^65 + 24 * q^71 - 4 * q^73 + 16 * q^79 - 22 * q^81 + 24 * q^87 + 12 * q^89 + 8 * q^95 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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