LMFDB - Embedded newform 256.2.b.c.129.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [256,2,Mod(129,256)] 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("256.129"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	256.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$2.04417029174$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	256.129
Dual form			256.2.b.c.129.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} +2.00000i q^{5} +4.00000 q^{7} -1.00000 q^{9} -2.00000i q^{11} -2.00000i q^{13} +4.00000 q^{15} -2.00000 q^{17} -2.00000i q^{19} -8.00000i q^{21} -4.00000 q^{23} +1.00000 q^{25} -4.00000i q^{27} +6.00000i q^{29} -4.00000 q^{33} +8.00000i q^{35} +10.0000i q^{37} -4.00000 q^{39} +6.00000 q^{41} +6.00000i q^{43} -2.00000i q^{45} -8.00000 q^{47} +9.00000 q^{49} +4.00000i q^{51} -6.00000i q^{53} +4.00000 q^{55} -4.00000 q^{57} +14.0000i q^{59} -2.00000i q^{61} -4.00000 q^{63} +4.00000 q^{65} -10.0000i q^{67} +8.00000i q^{69} -12.0000 q^{71} -14.0000 q^{73} -2.00000i q^{75} -8.00000i q^{77} -8.00000 q^{79} -11.0000 q^{81} +6.00000i q^{83} -4.00000i q^{85} +12.0000 q^{87} +2.00000 q^{89} -8.00000i q^{91} +4.00000 q^{95} -2.00000 q^{97} +2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{7} - 2 q^{9} + 8 q^{15} - 4 q^{17} - 8 q^{23} + 2 q^{25} - 8 q^{33} - 8 q^{39} + 12 q^{41} - 16 q^{47} + 18 q^{49} + 8 q^{55} - 8 q^{57} - 8 q^{63} + 8 q^{65} - 24 q^{71} - 28 q^{73} - 16 q^{79}+ \cdots - 4 q^{97}+O(q^{100}) $ 2 * q + 8 * q^7 - 2 * q^9 + 8 * q^15 - 4 * q^17 - 8 * q^23 + 2 * q^25 - 8 * q^33 - 8 * q^39 + 12 * q^41 - 16 * q^47 + 18 * q^49 + 8 * q^55 - 8 * q^57 - 8 * q^63 + 8 * q^65 - 24 * q^71 - 28 * q^73 - 16 * q^79 - 22 * q^81 + 24 * q^87 + 4 * q^89 + 8 * q^95 - 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$-1$	$1$

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
$5$	2.00000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	2.00000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 2.00000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 2.00000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	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
$15$	4.00000	1.03280
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	− 2.00000i	− 0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$	− 2.00000i	− 0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$	0	0
$21$	− 8.00000i	− 1.74574i
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	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
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	8.00000i	1.35225i
$36$	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
$39$	−4.00000	−0.640513
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	6.00000i	0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$	6.00000i	0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	0	0
$45$	− 2.00000i	− 0.298142i
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	4.00000i	0.560112i
$52$	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
$55$	4.00000	0.539360
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	14.0000i	1.82264i	0.411693	+	0.911322i	$0.364937\pi$
$59$	14.0000i	1.82264i	−0.411693	+	0.911322i	$0.635063\pi$
$60$	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
$63$	−4.00000	−0.503953
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	− 10.0000i	− 1.22169i	−0.791748	−	0.610847i	$-0.790829\pi$
$67$	− 10.0000i	− 1.22169i	0.791748	−	0.610847i	$-0.209171\pi$
$68$	0	0
$69$	8.00000i	0.963087i
$70$	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
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	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	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	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
$85$	− 4.00000i	− 0.433861i
$86$	0	0
$87$	12.0000	1.28654
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	− 8.00000i	− 0.838628i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	2.00000i	0.201008i
$100$	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
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	16.0000	1.56144
$106$	0	0
$107$	− 2.00000i	− 0.193347i	−0.995316	−	0.0966736i	$-0.969180\pi$
$107$	− 2.00000i	− 0.193347i	0.995316	−	0.0966736i	$-0.0308203\pi$
$108$	0	0
$109$	6.00000i	0.574696i	0.957826	+	0.287348i	$0.0927736\pi$
$109$	6.00000i	0.574696i	−0.957826	+	0.287348i	$0.907226\pi$
$110$	0	0
$111$	20.0000	1.89832
$112$	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
$115$	− 8.00000i	− 0.746004i
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	−8.00000	−0.733359
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	− 12.0000i	− 1.08200i
$124$	0	0
$125$	12.0000i	1.07331i
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	6.00000i	0.524222i	0.965038	+	0.262111i	$0.0844187\pi$
$131$	6.00000i	0.524222i	−0.965038	+	0.262111i	$0.915581\pi$
$132$	0	0
$133$	− 8.00000i	− 0.693688i
$134$	0	0
$135$	8.00000	0.688530
$136$	0	0
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	− 10.0000i	− 0.848189i	−0.905618	−	0.424094i	$-0.860592\pi$
$139$	− 10.0000i	− 0.848189i	0.905618	−	0.424094i	$-0.139408\pi$
$140$	0	0
$141$	16.0000i	1.34744i
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	−12.0000	−0.996546
$146$	0	0
$147$	− 18.0000i	− 1.48461i
$148$	0	0
$149$	18.0000i	1.47462i	0.675556	+	0.737309i	$0.263904\pi$
$149$	18.0000i	1.47462i	−0.675556	+	0.737309i	$0.736096\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	0	0
$156$	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
$159$	−12.0000	−0.951662
$160$	0	0
$161$	−16.0000	−1.26098
$162$	0	0
$163$	− 2.00000i	− 0.156652i	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$	− 2.00000i	− 0.156652i	0.996928	−	0.0783260i	$-0.0249575\pi$
$164$	0	0
$165$	− 8.00000i	− 0.622799i
$166$	0	0
$167$	20.0000	1.54765	0.773823	−	0.633402i	$-0.218342\pi$
$167$	20.0000	1.54765	0.773823	+	0.633402i	$0.218342\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	2.00000i	0.152944i
$172$	0	0
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	28.0000	2.10461
$178$	0	0
$179$	6.00000i	0.448461i	0.974536	+	0.224231i	$0.0719869\pi$
$179$	6.00000i	0.448461i	−0.974536	+	0.224231i	$0.928013\pi$
$180$	0	0
$181$	2.00000i	0.148659i	0.997234	+	0.0743294i	$0.0236816\pi$
$181$	2.00000i	0.148659i	−0.997234	+	0.0743294i	$0.976318\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	0	0
$185$	−20.0000	−1.47043
$186$	0	0
$187$	4.00000i	0.292509i
$188$	0	0
$189$	− 16.0000i	− 1.16383i
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	− 8.00000i	− 0.572892i
$196$	0	0
$197$	− 14.0000i	− 0.997459i	−0.866758	−	0.498729i	$-0.833800\pi$
$197$	− 14.0000i	− 0.997459i	0.866758	−	0.498729i	$-0.166200\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	−20.0000	−1.41069
$202$	0	0
$203$	24.0000i	1.68447i
$204$	0	0
$205$	12.0000i	0.838116i
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	22.0000i	1.51454i	0.653101	+	0.757271i	$0.273468\pi$
$211$	22.0000i	1.51454i	−0.653101	+	0.757271i	$0.726532\pi$
$212$	0	0
$213$	24.0000i	1.64445i
$214$	0	0
$215$	−12.0000	−0.818393
$216$	0	0
$217$	0	0
$218$	0	0
$219$	28.0000i	1.89206i
$220$	0	0
$221$	4.00000i	0.269069i
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	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
$229$	− 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
$231$	−16.0000	−1.05272
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	− 16.0000i	− 1.04372i
$236$	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
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	10.0000i	0.641500i
$244$	0	0
$245$	18.0000i	1.14998i
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	− 18.0000i	− 1.13615i	−0.822977	−	0.568075i	$-0.807688\pi$
$251$	− 18.0000i	− 1.13615i	0.822977	−	0.568075i	$-0.192312\pi$
$252$	0	0
$253$	8.00000i	0.502956i
$254$	0	0
$255$	−8.00000	−0.500979
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	40.0000i	2.48548i
$260$	0	0
$261$	− 6.00000i	− 0.371391i
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	0	0
$267$	− 4.00000i	− 0.244796i
$268$	0	0
$269$	− 10.0000i	− 0.609711i	−0.952399	−	0.304855i	$-0.901392\pi$
$269$	− 10.0000i	− 0.609711i	0.952399	−	0.304855i	$-0.0986081\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	−16.0000	−0.968364
$274$	0	0
$275$	− 2.00000i	− 0.120605i
$276$	0	0
$277$	− 6.00000i	− 0.360505i	−0.983620	−	0.180253i	$-0.942309\pi$
$277$	− 6.00000i	− 0.360505i	0.983620	−	0.180253i	$-0.0576915\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	6.00000i	0.356663i	0.983970	+	0.178331i	$0.0570699\pi$
$283$	6.00000i	0.356663i	−0.983970	+	0.178331i	$0.942930\pi$
$284$	0	0
$285$	− 8.00000i	− 0.473879i
$286$	0	0
$287$	24.0000	1.41668
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	4.00000i	0.234484i
$292$	0	0
$293$	− 14.0000i	− 0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$	− 14.0000i	− 0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$	0	0
$295$	−28.0000	−1.63022
$296$	0	0
$297$	−8.00000	−0.464207
$298$	0	0
$299$	8.00000i	0.462652i
$300$	0	0
$301$	24.0000i	1.38334i
$302$	0	0
$303$	−12.0000	−0.689382
$304$	0	0
$305$	4.00000	0.229039
$306$	0	0
$307$	− 18.0000i	− 1.02731i	−0.857996	−	0.513657i	$-0.828290\pi$
$307$	− 18.0000i	− 1.02731i	0.857996	−	0.513657i	$-0.171710\pi$
$308$	0	0
$309$	− 8.00000i	− 0.455104i
$310$	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
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	− 8.00000i	− 0.450749i
$316$	0	0
$317$	6.00000i	0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$	6.00000i	0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	4.00000i	0.222566i
$324$	0	0
$325$	− 2.00000i	− 0.110940i
$326$	0	0
$327$	12.0000	0.663602
$328$	0	0
$329$	−32.0000	−1.76422
$330$	0	0
$331$	14.0000i	0.769510i	0.923019	+	0.384755i	$0.125714\pi$
$331$	14.0000i	0.769510i	−0.923019	+	0.384755i	$0.874286\pi$
$332$	0	0
$333$	− 10.0000i	− 0.547997i
$334$	0	0
$335$	20.0000	1.09272
$336$	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
$339$	− 4.00000i	− 0.217250i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	−16.0000	−0.861411
$346$	0	0
$347$	− 18.0000i	− 0.966291i	−0.875540	−	0.483145i	$-0.839494\pi$
$347$	− 18.0000i	− 0.966291i	0.875540	−	0.483145i	$-0.160506\pi$
$348$	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
$355$	− 24.0000i	− 1.27379i
$356$	0	0
$357$	16.0000i	0.846810i
$358$	0	0
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	0	0
$363$	− 14.0000i	− 0.734809i
$364$	0	0
$365$	− 28.0000i	− 1.46559i
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	− 24.0000i	− 1.24602i
$372$	0	0
$373$	10.0000i	0.517780i	0.965907	+	0.258890i	$0.0833568\pi$
$373$	10.0000i	0.517780i	−0.965907	+	0.258890i	$0.916643\pi$
$374$	0	0
$375$	24.0000	1.23935
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	− 2.00000i	− 0.102733i	−0.998680	−	0.0513665i	$-0.983642\pi$
$379$	− 2.00000i	− 0.102733i	0.998680	−	0.0513665i	$-0.0163577\pi$
$380$	0	0
$381$	− 32.0000i	− 1.63941i
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	16.0000	0.815436
$386$	0	0
$387$	− 6.00000i	− 0.304997i
$388$	0	0
$389$	10.0000i	0.507020i	0.967333	+	0.253510i	$0.0815851\pi$
$389$	10.0000i	0.507020i	−0.967333	+	0.253510i	$0.918415\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	− 16.0000i	− 0.805047i
$396$	0	0
$397$	6.00000i	0.301131i	0.988600	+	0.150566i	$0.0481095\pi$
$397$	6.00000i	0.301131i	−0.988600	+	0.150566i	$0.951890\pi$
$398$	0	0
$399$	−16.0000	−0.801002
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 22.0000i	− 1.09319i
$406$	0	0
$407$	20.0000	0.991363
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	20.0000i	0.986527i
$412$	0	0
$413$	56.0000i	2.75558i
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	−20.0000	−0.979404
$418$	0	0
$419$	− 26.0000i	− 1.27018i	−0.772437	−	0.635092i	$-0.780962\pi$
$419$	− 26.0000i	− 1.27018i	0.772437	−	0.635092i	$-0.219038\pi$
$420$	0	0
$421$	34.0000i	1.65706i	0.559946	+	0.828529i	$0.310822\pi$
$421$	34.0000i	1.65706i	−0.559946	+	0.828529i	$0.689178\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	− 8.00000i	− 0.387147i
$428$	0	0
$429$	8.00000i	0.386244i
$430$	0	0
$431$	40.0000	1.92673	0.963366	−	0.268190i	$-0.0864254\pi$
$431$	40.0000	1.92673	0.963366	+	0.268190i	$0.0864254\pi$
$432$	0	0
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	0	0
$435$	24.0000i	1.15071i
$436$	0	0
$437$	8.00000i	0.382692i
$438$	0	0
$439$	−36.0000	−1.71819	−0.859093	−	0.511819i	$-0.828972\pi$
$439$	−36.0000	−1.71819	−0.859093	+	0.511819i	$0.828972\pi$
$440$	0	0
$441$	−9.00000	−0.428571
$442$	0	0
$443$	6.00000i	0.285069i	0.989790	+	0.142534i	$0.0455251\pi$
$443$	6.00000i	0.285069i	−0.989790	+	0.142534i	$0.954475\pi$
$444$	0	0
$445$	4.00000i	0.189618i
$446$	0	0
$447$	36.0000	1.70274
$448$	0	0
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	0	0
$451$	− 12.0000i	− 0.565058i
$452$	0	0
$453$	8.00000i	0.375873i
$454$	0	0
$455$	16.0000	0.750092
$456$	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
$461$	− 10.0000i	− 0.465746i	−0.972507	−	0.232873i	$-0.925187\pi$
$461$	− 10.0000i	− 0.465746i	0.972507	−	0.232873i	$-0.0748127\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.0000i	0.647843i	0.946084	+	0.323921i	$0.105001\pi$
$467$	14.0000i	0.647843i	−0.946084	+	0.323921i	$0.894999\pi$
$468$	0	0
$469$	− 40.0000i	− 1.84703i
$470$	0	0
$471$	−36.0000	−1.65879
$472$	0	0
$473$	12.0000	0.551761
$474$	0	0
$475$	− 2.00000i	− 0.0917663i
$476$	0	0
$477$	6.00000i	0.274721i
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	0	0
$483$	32.0000i	1.45605i
$484$	0	0
$485$	− 4.00000i	− 0.181631i
$486$	0	0
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	0	0
$491$	− 10.0000i	− 0.451294i	−0.974209	−	0.225647i	$-0.927550\pi$
$491$	− 10.0000i	− 0.451294i	0.974209	−	0.225647i	$-0.0724495\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	−48.0000	−2.15309
$498$	0	0
$499$	22.0000i	0.984855i	0.870353	+	0.492428i	$0.163890\pi$
$499$	22.0000i	0.984855i	−0.870353	+	0.492428i	$0.836110\pi$
$500$	0	0
$501$	− 40.0000i	− 1.78707i
$502$	0	0
$503$	−20.0000	−0.891756	−0.445878	−	0.895094i	$-0.647108\pi$
$503$	−20.0000	−0.891756	−0.445878	+	0.895094i	$0.647108\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	0	0
$507$	− 18.0000i	− 0.799408i
$508$	0	0
$509$	14.0000i	0.620539i	0.950649	+	0.310270i	$0.100419\pi$
$509$	14.0000i	0.620539i	−0.950649	+	0.310270i	$0.899581\pi$
$510$	0	0
$511$	−56.0000	−2.47729
$512$	0	0
$513$	−8.00000	−0.353209
$514$	0	0
$515$	8.00000i	0.352522i
$516$	0	0
$517$	16.0000i	0.703679i
$518$	0	0
$519$	−36.0000	−1.58022
$520$	0	0
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	0	0
$523$	14.0000i	0.612177i	0.952003	+	0.306089i	$0.0990204\pi$
$523$	14.0000i	0.612177i	−0.952003	+	0.306089i	$0.900980\pi$
$524$	0	0
$525$	− 8.00000i	− 0.349149i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	− 14.0000i	− 0.607548i
$532$	0	0
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	4.00000	0.172935
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	− 18.0000i	− 0.775315i
$540$	0	0
$541$	− 34.0000i	− 1.46177i	−0.682498	−	0.730887i	$-0.739107\pi$
$541$	− 34.0000i	− 1.46177i	0.682498	−	0.730887i	$-0.260893\pi$
$542$	0	0
$543$	4.00000	0.171656
$544$	0	0
$545$	−12.0000	−0.514024
$546$	0	0
$547$	38.0000i	1.62476i	0.583127	+	0.812381i	$0.301829\pi$
$547$	38.0000i	1.62476i	−0.583127	+	0.812381i	$0.698171\pi$
$548$	0	0
$549$	2.00000i	0.0853579i
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	−32.0000	−1.36078
$554$	0	0
$555$	40.0000i	1.69791i
$556$	0	0
$557$	− 2.00000i	− 0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$	− 2.00000i	− 0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	0	0
$561$	8.00000	0.337760
$562$	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
$565$	4.00000i	0.168281i
$566$	0	0
$567$	−44.0000	−1.84783
$568$	0	0
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	0	0
$571$	38.0000i	1.59025i	0.606445	+	0.795125i	$0.292595\pi$
$571$	38.0000i	1.59025i	−0.606445	+	0.795125i	$0.707405\pi$
$572$	0	0
$573$	32.0000i	1.33682i
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	4.00000i	0.166234i
$580$	0	0
$581$	24.0000i	0.995688i
$582$	0	0
$583$	−12.0000	−0.496989
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	− 34.0000i	− 1.40333i	−0.712507	−	0.701665i	$-0.752440\pi$
$587$	− 34.0000i	− 1.40333i	0.712507	−	0.701665i	$-0.247560\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−28.0000	−1.15177
$592$	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
$595$	− 16.0000i	− 0.655936i
$596$	0	0
$597$	− 8.00000i	− 0.327418i
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	0	0
$603$	10.0000i	0.407231i
$604$	0	0
$605$	14.0000i	0.569181i
$606$	0	0
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	0	0
$609$	48.0000	1.94506
$610$	0	0
$611$	16.0000i	0.647291i
$612$	0	0
$613$	34.0000i	1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$	34.0000i	1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$	0	0
$615$	24.0000	0.967773
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	0	0
$619$	46.0000i	1.84890i	0.381308	+	0.924448i	$0.375474\pi$
$619$	46.0000i	1.84890i	−0.381308	+	0.924448i	$0.624526\pi$
$620$	0	0
$621$	16.0000i	0.642058i
$622$	0	0
$623$	8.00000	0.320513
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	8.00000i	0.319489i
$628$	0	0
$629$	− 20.0000i	− 0.797452i
$630$	0	0
$631$	44.0000	1.75161	0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000	1.75161	0.875806	+	0.482663i	$0.160330\pi$
$632$	0	0
$633$	44.0000	1.74884
$634$	0	0
$635$	32.0000i	1.26988i
$636$	0	0
$637$	− 18.0000i	− 0.713186i
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	− 42.0000i	− 1.65632i	−0.560493	−	0.828159i	$-0.689388\pi$
$643$	− 42.0000i	− 1.65632i	0.560493	−	0.828159i	$-0.310612\pi$
$644$	0	0
$645$	24.0000i	0.944999i
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	28.0000	1.09910
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 42.0000i	− 1.64359i	−0.569785	−	0.821794i	$-0.692974\pi$
$653$	− 42.0000i	− 1.64359i	0.569785	−	0.821794i	$-0.307026\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	0	0
$657$	14.0000	0.546192
$658$	0	0
$659$	6.00000i	0.233727i	0.993148	+	0.116863i	$0.0372840\pi$
$659$	6.00000i	0.233727i	−0.993148	+	0.116863i	$0.962716\pi$
$660$	0	0
$661$	34.0000i	1.32245i	0.750189	+	0.661223i	$0.229962\pi$
$661$	34.0000i	1.32245i	−0.750189	+	0.661223i	$0.770038\pi$
$662$	0	0
$663$	8.00000	0.310694
$664$	0	0
$665$	16.0000	0.620453
$666$	0	0
$667$	− 24.0000i	− 0.929284i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	0	0
$675$	− 4.00000i	− 0.153960i
$676$	0	0
$677$	− 22.0000i	− 0.845529i	−0.906240	−	0.422764i	$-0.861060\pi$
$677$	− 22.0000i	− 0.845529i	0.906240	−	0.422764i	$-0.138940\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	0	0
$681$	−36.0000	−1.37952
$682$	0	0
$683$	− 42.0000i	− 1.60709i	−0.595247	−	0.803543i	$-0.702946\pi$
$683$	− 42.0000i	− 1.60709i	0.595247	−	0.803543i	$-0.297054\pi$
$684$	0	0
$685$	− 20.0000i	− 0.764161i
$686$	0	0
$687$	−28.0000	−1.06827
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	6.00000i	0.228251i	0.993466	+	0.114125i	$0.0364066\pi$
$691$	6.00000i	0.228251i	−0.993466	+	0.114125i	$0.963593\pi$
$692$	0	0
$693$	8.00000i	0.303895i
$694$	0	0
$695$	20.0000	0.758643
$696$	0	0
$697$	−12.0000	−0.454532
$698$	0	0
$699$	− 36.0000i	− 1.36165i
$700$	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
$703$	20.0000	0.754314
$704$	0	0
$705$	−32.0000	−1.20519
$706$	0	0
$707$	− 24.0000i	− 0.902613i
$708$	0	0
$709$	− 22.0000i	− 0.826227i	−0.910679	−	0.413114i	$-0.864441\pi$
$709$	− 22.0000i	− 0.826227i	0.910679	−	0.413114i	$-0.135559\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 8.00000i	− 0.299183i
$716$	0	0
$717$	48.0000i	1.79259i
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	0	0
$723$	4.00000i	0.148762i
$724$	0	0
$725$	6.00000i	0.222834i
$726$	0	0
$727$	12.0000	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$727$	12.0000	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	− 12.0000i	− 0.443836i
$732$	0	0
$733$	6.00000i	0.221615i	0.993842	+	0.110808i	$0.0353437\pi$
$733$	6.00000i	0.221615i	−0.993842	+	0.110808i	$0.964656\pi$
$734$	0	0
$735$	36.0000	1.32788
$736$	0	0
$737$	−20.0000	−0.736709
$738$	0	0
$739$	− 18.0000i	− 0.662141i	−0.943606	−	0.331070i	$-0.892590\pi$
$739$	− 18.0000i	− 0.662141i	0.943606	−	0.331070i	$-0.107410\pi$
$740$	0	0
$741$	8.00000i	0.293887i
$742$	0	0
$743$	−44.0000	−1.61420	−0.807102	−	0.590412i	$-0.798965\pi$
$743$	−44.0000	−1.61420	−0.807102	+	0.590412i	$0.798965\pi$
$744$	0	0
$745$	−36.0000	−1.31894
$746$	0	0
$747$	− 6.00000i	− 0.219529i
$748$	0	0
$749$	− 8.00000i	− 0.292314i
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	−36.0000	−1.31191
$754$	0	0
$755$	− 8.00000i	− 0.291150i
$756$	0	0
$757$	− 46.0000i	− 1.67190i	−0.548807	−	0.835949i	$-0.684918\pi$
$757$	− 46.0000i	− 1.67190i	0.548807	−	0.835949i	$-0.315082\pi$
$758$	0	0
$759$	16.0000	0.580763
$760$	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
$763$	24.0000i	0.868858i
$764$	0	0
$765$	4.00000i	0.144620i
$766$	0	0
$767$	28.0000	1.01102
$768$	0	0
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	− 36.0000i	− 1.29651i
$772$	0	0
$773$	− 54.0000i	− 1.94225i	−0.238581	−	0.971123i	$-0.576682\pi$
$773$	− 54.0000i	− 1.94225i	0.238581	−	0.971123i	$-0.423318\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	80.0000	2.86998
$778$	0	0
$779$	− 12.0000i	− 0.429945i
$780$	0	0
$781$	24.0000i	0.858788i
$782$	0	0
$783$	24.0000	0.857690
$784$	0	0
$785$	36.0000	1.28490
$786$	0	0
$787$	22.0000i	0.784215i	0.919919	+	0.392108i	$0.128254\pi$
$787$	22.0000i	0.784215i	−0.919919	+	0.392108i	$0.871746\pi$
$788$	0	0
$789$	24.0000i	0.854423i
$790$	0	0
$791$	8.00000	0.284447
$792$	0	0
$793$	−4.00000	−0.142044
$794$	0	0
$795$	− 24.0000i	− 0.851192i
$796$	0	0
$797$	− 18.0000i	− 0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$	− 18.0000i	− 0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	0	0
$803$	28.0000i	0.988099i
$804$	0	0
$805$	− 32.0000i	− 1.12785i
$806$	0	0
$807$	−20.0000	−0.704033
$808$	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
$811$	− 18.0000i	− 0.632065i	−0.948748	−	0.316033i	$-0.897649\pi$
$811$	− 18.0000i	− 0.632065i	0.948748	−	0.316033i	$-0.102351\pi$
$812$	0	0
$813$	− 16.0000i	− 0.561144i
$814$	0	0
$815$	4.00000	0.140114
$816$	0	0
$817$	12.0000	0.419827
$818$	0	0
$819$	8.00000i	0.279543i
$820$	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
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	22.0000i	0.765015i	0.923952	+	0.382507i	$0.124939\pi$
$827$	22.0000i	0.765015i	−0.923952	+	0.382507i	$0.875061\pi$
$828$	0	0
$829$	14.0000i	0.486240i	0.969996	+	0.243120i	$0.0781709\pi$
$829$	14.0000i	0.486240i	−0.969996	+	0.243120i	$0.921829\pi$
$830$	0	0
$831$	−12.0000	−0.416275
$832$	0	0
$833$	−18.0000	−0.623663
$834$	0	0
$835$	40.0000i	1.38426i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	− 36.0000i	− 1.23991i
$844$	0	0
$845$	18.0000i	0.619219i
$846$	0	0
$847$	28.0000	0.962091
$848$	0	0
$849$	12.0000	0.411839
$850$	0	0
$851$	− 40.0000i	− 1.37118i
$852$	0	0
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	− 50.0000i	− 1.70598i	−0.521929	−	0.852989i	$-0.674787\pi$
$859$	− 50.0000i	− 1.70598i	0.521929	−	0.852989i	$-0.325213\pi$
$860$	0	0
$861$	− 48.0000i	− 1.63584i
$862$	0	0
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	0	0
$865$	36.0000	1.22404
$866$	0	0
$867$	26.0000i	0.883006i
$868$	0	0
$869$	16.0000i	0.542763i
$870$	0	0
$871$	−20.0000	−0.677674
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	48.0000i	1.62270i
$876$	0	0
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	0	0
$879$	−28.0000	−0.944417
$880$	0	0
$881$	−46.0000	−1.54978	−0.774890	−	0.632096i	$-0.782195\pi$
$881$	−46.0000	−1.54978	−0.774890	+	0.632096i	$0.782195\pi$
$882$	0	0
$883$	− 34.0000i	− 1.14419i	−0.820187	−	0.572096i	$-0.806131\pi$
$883$	− 34.0000i	− 1.14419i	0.820187	−	0.572096i	$-0.193869\pi$
$884$	0	0
$885$	56.0000i	1.88242i
$886$	0	0
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	0	0
$889$	64.0000	2.14649
$890$	0	0
$891$	22.0000i	0.737028i
$892$	0	0
$893$	16.0000i	0.535420i
$894$	0	0
$895$	−12.0000	−0.401116
$896$	0	0
$897$	16.0000	0.534224
$898$	0	0
$899$	0	0
$900$	0	0
$901$	12.0000i	0.399778i
$902$	0	0
$903$	48.0000	1.59734
$904$	0	0
$905$	−4.00000	−0.132964
$906$	0	0
$907$	38.0000i	1.26177i	0.775877	+	0.630885i	$0.217308\pi$
$907$	38.0000i	1.26177i	−0.775877	+	0.630885i	$0.782692\pi$
$908$	0	0
$909$	6.00000i	0.199007i
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	− 8.00000i	− 0.264472i
$916$	0	0
$917$	24.0000i	0.792550i
$918$	0	0
$919$	−36.0000	−1.18753	−0.593765	−	0.804638i	$-0.702359\pi$
$919$	−36.0000	−1.18753	−0.593765	+	0.804638i	$0.702359\pi$
$920$	0	0
$921$	−36.0000	−1.18624
$922$	0	0
$923$	24.0000i	0.789970i
$924$	0	0
$925$	10.0000i	0.328798i
$926$	0	0
$927$	−4.00000	−0.131377
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	− 18.0000i	− 0.589926i
$932$	0	0
$933$	− 56.0000i	− 1.83336i
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	−46.0000	−1.50275	−0.751377	−	0.659873i	$-0.770610\pi$
$937$	−46.0000	−1.50275	−0.751377	+	0.659873i	$0.770610\pi$
$938$	0	0
$939$	20.0000i	0.652675i
$940$	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
$943$	−24.0000	−0.781548
$944$	0	0
$945$	32.0000	1.04096
$946$	0	0
$947$	14.0000i	0.454939i	0.973785	+	0.227469i	$0.0730452\pi$
$947$	14.0000i	0.454939i	−0.973785	+	0.227469i	$0.926955\pi$
$948$	0	0
$949$	28.0000i	0.908918i
$950$	0	0
$951$	12.0000	0.389127
$952$	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
$955$	− 32.0000i	− 1.03550i
$956$	0	0
$957$	− 24.0000i	− 0.775810i
$958$	0	0
$959$	−40.0000	−1.29167
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	2.00000i	0.0644491i
$964$	0	0
$965$	− 4.00000i	− 0.128765i
$966$	0	0
$967$	−28.0000	−0.900419	−0.450210	−	0.892923i	$-0.648651\pi$
$967$	−28.0000	−0.900419	−0.450210	+	0.892923i	$0.648651\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	38.0000i	1.21948i	0.792602	+	0.609739i	$0.208726\pi$
$971$	38.0000i	1.21948i	−0.792602	+	0.609739i	$0.791274\pi$
$972$	0	0
$973$	− 40.0000i	− 1.28234i
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	− 4.00000i	− 0.127841i
$980$	0	0
$981$	− 6.00000i	− 0.191565i
$982$	0	0
$983$	−20.0000	−0.637901	−0.318950	−	0.947771i	$-0.603330\pi$
$983$	−20.0000	−0.637901	−0.318950	+	0.947771i	$0.603330\pi$
$984$	0	0
$985$	28.0000	0.892154
$986$	0	0
$987$	64.0000i	2.03714i
$988$	0	0
$989$	− 24.0000i	− 0.763156i
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	0	0
$995$	8.00000i	0.253617i
$996$	0	0
$997$	− 54.0000i	− 1.71020i	−0.518465	−	0.855099i	$-0.673497\pi$
$997$	− 54.0000i	− 1.71020i	0.518465	−	0.855099i	$-0.326503\pi$
$998$	0	0
$999$	40.0000	1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.2.b.c.129.1		2
3.2	odd	2		2304.2.d.r.1153.1		2
4.3	odd	2		256.2.b.a.129.2		2
8.3	odd	2		256.2.b.a.129.1		2
8.5	even	2	inner	256.2.b.c.129.2		2
12.11	even	2		2304.2.d.b.1153.1		2
16.3	odd	4		128.2.a.b.1.1	yes	1
16.5	even	4		128.2.a.a.1.1	✓	1
16.11	odd	4		128.2.a.c.1.1	yes	1
16.13	even	4		128.2.a.d.1.1	yes	1
24.5	odd	2		2304.2.d.r.1153.2		2
24.11	even	2		2304.2.d.b.1153.2		2
32.3	odd	8		1024.2.e.m.769.1		4
32.5	even	8		1024.2.e.i.257.2		4
32.11	odd	8		1024.2.e.m.257.2		4
32.13	even	8		1024.2.e.i.769.1		4
32.19	odd	8		1024.2.e.m.769.2		4
32.21	even	8		1024.2.e.i.257.1		4
32.27	odd	8		1024.2.e.m.257.1		4
32.29	even	8		1024.2.e.i.769.2		4
48.5	odd	4		1152.2.a.m.1.1		1
48.11	even	4		1152.2.a.r.1.1		1
48.29	odd	4		1152.2.a.c.1.1		1
48.35	even	4		1152.2.a.h.1.1		1
80.3	even	4		3200.2.c.k.2049.1		2
80.13	odd	4		3200.2.c.e.2049.2		2
80.19	odd	4		3200.2.a.u.1.1		1
80.27	even	4		3200.2.c.f.2049.1		2
80.29	even	4		3200.2.a.h.1.1		1
80.37	odd	4		3200.2.c.l.2049.2		2
80.43	even	4		3200.2.c.f.2049.2		2
80.53	odd	4		3200.2.c.l.2049.1		2
80.59	odd	4		3200.2.a.e.1.1		1
80.67	even	4		3200.2.c.k.2049.2		2
80.69	even	4		3200.2.a.x.1.1		1
80.77	odd	4		3200.2.c.e.2049.1		2
112.13	odd	4		6272.2.a.a.1.1		1
112.27	even	4		6272.2.a.b.1.1		1
112.69	odd	4		6272.2.a.h.1.1		1
112.83	even	4		6272.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.2.a.a.1.1	✓	1	16.5	even	4
128.2.a.b.1.1	yes	1	16.3	odd	4
128.2.a.c.1.1	yes	1	16.11	odd	4
128.2.a.d.1.1	yes	1	16.13	even	4
256.2.b.a.129.1		2	8.3	odd	2
256.2.b.a.129.2		2	4.3	odd	2
256.2.b.c.129.1		2	1.1	even	1	trivial
256.2.b.c.129.2		2	8.5	even	2	inner
1024.2.e.i.257.1		4	32.21	even	8
1024.2.e.i.257.2		4	32.5	even	8
1024.2.e.i.769.1		4	32.13	even	8
1024.2.e.i.769.2		4	32.29	even	8
1024.2.e.m.257.1		4	32.27	odd	8
1024.2.e.m.257.2		4	32.11	odd	8
1024.2.e.m.769.1		4	32.3	odd	8
1024.2.e.m.769.2		4	32.19	odd	8
1152.2.a.c.1.1		1	48.29	odd	4
1152.2.a.h.1.1		1	48.35	even	4
1152.2.a.m.1.1		1	48.5	odd	4
1152.2.a.r.1.1		1	48.11	even	4
2304.2.d.b.1153.1		2	12.11	even	2
2304.2.d.b.1153.2		2	24.11	even	2
2304.2.d.r.1153.1		2	3.2	odd	2
2304.2.d.r.1153.2		2	24.5	odd	2
3200.2.a.e.1.1		1	80.59	odd	4
3200.2.a.h.1.1		1	80.29	even	4
3200.2.a.u.1.1		1	80.19	odd	4
3200.2.a.x.1.1		1	80.69	even	4
3200.2.c.e.2049.1		2	80.77	odd	4
3200.2.c.e.2049.2		2	80.13	odd	4
3200.2.c.f.2049.1		2	80.27	even	4
3200.2.c.f.2049.2		2	80.43	even	4
3200.2.c.k.2049.1		2	80.3	even	4
3200.2.c.k.2049.2		2	80.67	even	4
3200.2.c.l.2049.1		2	80.53	odd	4
3200.2.c.l.2049.2		2	80.37	odd	4
6272.2.a.a.1.1		1	112.13	odd	4
6272.2.a.b.1.1		1	112.27	even	4
6272.2.a.g.1.1		1	112.83	even	4
6272.2.a.h.1.1		1	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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} +2.00000i q^{5} +4.00000 q^{7} -1.00000 q^{9} -2.00000i q^{11} -2.00000i q^{13} +4.00000 q^{15} -2.00000 q^{17} -2.00000i q^{19} -8.00000i q^{21} -4.00000 q^{23} +1.00000 q^{25} -4.00000i q^{27} +6.00000i q^{29} -4.00000 q^{33} +8.00000i q^{35} +10.0000i q^{37} -4.00000 q^{39} +6.00000 q^{41} +6.00000i q^{43} -2.00000i q^{45} -8.00000 q^{47} +9.00000 q^{49} +4.00000i q^{51} -6.00000i q^{53} +4.00000 q^{55} -4.00000 q^{57} +14.0000i q^{59} -2.00000i q^{61} -4.00000 q^{63} +4.00000 q^{65} -10.0000i q^{67} +8.00000i q^{69} -12.0000 q^{71} -14.0000 q^{73} -2.00000i q^{75} -8.00000i q^{77} -8.00000 q^{79} -11.0000 q^{81} +6.00000i q^{83} -4.00000i q^{85} +12.0000 q^{87} +2.00000 q^{89} -8.00000i q^{91} +4.00000 q^{95} -2.00000 q^{97} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{7} - 2 q^{9} + 8 q^{15} - 4 q^{17} - 8 q^{23} + 2 q^{25} - 8 q^{33} - 8 q^{39} + 12 q^{41} - 16 q^{47} + 18 q^{49} + 8 q^{55} - 8 q^{57} - 8 q^{63} + 8 q^{65} - 24 q^{71} - 28 q^{73} - 16 q^{79}+ \cdots - 4 q^{97}+O(q^{100}) \) 2 * q + 8 * q^7 - 2 * q^9 + 8 * q^15 - 4 * q^17 - 8 * q^23 + 2 * q^25 - 8 * q^33 - 8 * q^39 + 12 * q^41 - 16 * q^47 + 18 * q^49 + 8 * q^55 - 8 * q^57 - 8 * q^63 + 8 * q^65 - 24 * q^71 - 28 * q^73 - 16 * q^79 - 22 * q^81 + 24 * q^87 + 4 * q^89 + 8 * q^95 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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