LMFDB - Embedded newform 3200.2.c.l.2049.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3200,2,Mod(2049,3200)] 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("3200.2049"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3200 = 2^{7} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3200.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-2,0,4,0,0,0,0,0,0,0,4,0,16,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	no
Analytic conductor:	$25.5521286468$
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			2049.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3200.2049
Dual form			3200.2.c.l.2049.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$.

$n$	$901$	$1151$	$2177$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.00000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	2.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	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
$15$	0	0
$16$	0	0
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	8.00000	1.74574
$22$	0	0
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.00000i	0.769800i
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	4.00000i	0.696311i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	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
$43$	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
$45$	0	0
$46$	0	0
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	4.00000	0.560112
$52$	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
$55$	0	0
$56$	0	0
$57$	4.00000i	0.529813i
$58$	0	0
$59$	14.0000	1.82264	0.911322	−	0.411693i	$-0.135063\pi$
$59$	14.0000	1.82264	0.911322	+	0.411693i	$0.135063\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	4.00000i	0.503953i
$64$	0	0
$65$	0	0
$66$	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
$69$	8.00000	0.963087
$70$	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
$73$	− 14.0000i	− 1.63858i	−0.573382	−	0.819288i	$-0.694369\pi$
$73$	− 14.0000i	− 1.63858i	0.573382	−	0.819288i	$-0.305631\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 8.00000i	− 0.911685i
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 12.0000i	− 1.28654i
$88$	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
$91$	8.00000	0.838628
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.00000i	0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$	2.00000i	0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	20.0000	1.89832
$112$	0	0
$113$	− 2.00000i	− 0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$	− 2.00000i	− 0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$	0	0
$115$	0	0
$116$	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
$125$	0	0
$126$	0	0
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	0	0
$129$	−12.0000	−1.05654
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	− 8.00000i	− 0.693688i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.0000i	0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$	10.0000i	0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	16.0000	1.34744
$142$	0	0
$143$	4.00000i	0.334497i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 18.0000i	− 1.48461i
$148$	0	0
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	2.00000i	0.161690i
$154$	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	$0.0249575\pi$
$163$	2.00000i	0.156652i	−0.996928	+	0.0783260i	$0.975042\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 20.0000i	− 1.54765i	−0.633402	−	0.773823i	$-0.718342\pi$
$167$	− 20.0000i	− 1.54765i	0.633402	−	0.773823i	$-0.281658\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	28.0000i	2.10461i
$178$	0	0
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	− 4.00000i	− 0.295689i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 4.00000i	− 0.292509i
$188$	0	0
$189$	16.0000	1.16383
$190$	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
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.0000i	0.997459i	0.866758	+	0.498729i	$0.166200\pi$
$197$	14.0000i	0.997459i	−0.866758	+	0.498729i	$0.833800\pi$
$198$	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
$205$	0	0
$206$	0	0
$207$	4.00000i	0.278019i
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	22.0000	1.51454	0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000	1.51454	0.757271	+	0.653101i	$0.226532\pi$
$212$	0	0
$213$	24.0000i	1.64445i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	28.0000	1.89206
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	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
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	0	0
$233$	18.0000i	1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$	18.0000i	1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	16.0000i	1.03931i
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	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
$245$	0	0
$246$	0	0
$247$	4.00000i	0.254514i
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	− 8.00000i	− 0.502956i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	0	0
$259$	−40.0000	−2.48548
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	− 12.0000i	− 0.739952i	−0.929041	−	0.369976i	$-0.879366\pi$
$263$	− 12.0000i	− 0.739952i	0.929041	−	0.369976i	$-0.120634\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.00000i	0.244796i
$268$	0	0
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	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
$273$	16.0000i	0.968364i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.00000i	0.360505i	0.983620	+	0.180253i	$0.0576915\pi$
$277$	6.00000i	0.360505i	−0.983620	+	0.180253i	$0.942309\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	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
$285$	0	0
$286$	0	0
$287$	24.0000i	1.41668i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	4.00000	0.234484
$292$	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
$295$	0	0
$296$	0	0
$297$	8.00000i	0.464207i
$298$	0	0
$299$	8.00000	0.462652
$300$	0	0
$301$	24.0000	1.38334
$302$	0	0
$303$	12.0000i	0.689382i
$304$	0	0
$305$	0	0
$306$	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
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−28.0000	−1.58773	−0.793867	−	0.608091i	$-0.791935\pi$
$311$	−28.0000	−1.58773	−0.793867	+	0.608091i	$0.791935\pi$
$312$	0	0
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	0	0
$315$	0	0
$316$	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
$325$	0	0
$326$	0	0
$327$	− 12.0000i	− 0.663602i
$328$	0	0
$329$	−32.0000	−1.76422
$330$	0	0
$331$	−14.0000	−0.769510	−0.384755	−	0.923019i	$-0.625714\pi$
$331$	−14.0000	−0.769510	−0.384755	+	0.923019i	$0.625714\pi$
$332$	0	0
$333$	10.0000i	0.547997i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 14.0000i	− 0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$	− 14.0000i	− 0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$	0	0
$339$	4.00000	0.217250
$340$	0	0
$341$	0	0
$342$	0	0
$343$	8.00000i	0.431959i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.0000i	0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$	18.0000i	0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	−8.00000	−0.427008
$352$	0	0
$353$	− 18.0000i	− 0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$	− 18.0000i	− 0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$	0	0
$355$	0	0
$356$	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
$365$	0	0
$366$	0	0
$367$	− 8.00000i	− 0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$	− 8.00000i	− 0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	−24.0000	−1.24602
$372$	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
$375$	0	0
$376$	0	0
$377$	− 12.0000i	− 0.618031i
$378$	0	0
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	0	0
$381$	−32.0000	−1.63941
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 6.00000i	− 0.304997i
$388$	0	0
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	0	0
$393$	12.0000i	0.605320i
$394$	0	0
$395$	0	0
$396$	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
$405$	0	0
$406$	0	0
$407$	− 20.0000i	− 0.991363i
$408$	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
$411$	−20.0000	−0.986527
$412$	0	0
$413$	− 56.0000i	− 2.75558i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 20.0000i	− 0.979404i
$418$	0	0
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	0	0
$423$	8.00000i	0.388973i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.00000i	0.387147i
$428$	0	0
$429$	−8.00000	−0.386244
$430$	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
$433$	− 30.0000i	− 1.44171i	−0.693087	−	0.720854i	$-0.743750\pi$
$433$	− 30.0000i	− 1.44171i	0.693087	−	0.720854i	$-0.256250\pi$
$434$	0	0
$435$	0	0
$436$	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
$445$	0	0
$446$	0	0
$447$	36.0000i	1.70274i
$448$	0	0
$449$	34.0000	1.60456	0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000	1.60456	0.802280	+	0.596948i	$0.203620\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	0	0
$453$	8.00000i	0.375873i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 6.00000i	− 0.280668i	−0.990104	−	0.140334i	$-0.955182\pi$
$457$	− 6.00000i	− 0.280668i	0.990104	−	0.140334i	$-0.0448177\pi$
$458$	0	0
$459$	8.00000	0.373408
$460$	0	0
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	0	0
$463$	8.00000i	0.371792i	0.982569	+	0.185896i	$0.0595187\pi$
$463$	8.00000i	0.371792i	−0.982569	+	0.185896i	$0.940481\pi$
$464$	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
$469$	−40.0000	−1.84703
$470$	0	0
$471$	36.0000	1.65879
$472$	0	0
$473$	12.0000i	0.551761i
$474$	0	0
$475$	0	0
$476$	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
$485$	0	0
$486$	0	0
$487$	− 20.0000i	− 0.906287i	−0.891438	−	0.453143i	$-0.850303\pi$
$487$	− 20.0000i	− 0.906287i	0.891438	−	0.453143i	$-0.149697\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	0	0
$491$	10.0000	0.451294	0.225647	−	0.974209i	$-0.427550\pi$
$491$	10.0000	0.451294	0.225647	+	0.974209i	$0.427550\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 48.0000i	− 2.15309i
$498$	0	0
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	0	0
$501$	40.0000	1.78707
$502$	0	0
$503$	− 20.0000i	− 0.891756i	−0.895094	−	0.445878i	$-0.852892\pi$
$503$	− 20.0000i	− 0.891756i	0.895094	−	0.445878i	$-0.147108\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	18.0000i	0.799408i
$508$	0	0
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	0	0
$511$	−56.0000	−2.47729
$512$	0	0
$513$	8.00000i	0.353209i
$514$	0	0
$515$	0	0
$516$	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	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	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
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	−14.0000	−0.607548
$532$	0	0
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 12.0000i	− 0.517838i
$538$	0	0
$539$	−18.0000	−0.775315
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	0	0
$543$	− 4.00000i	− 0.171656i
$544$	0	0
$545$	0	0
$546$	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
$549$	2.00000	0.0853579
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	− 32.0000i	− 1.36078i
$554$	0	0
$555$	0	0
$556$	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	$0.123837\pi$
$563$	18.0000i	0.758610i	−0.925272	+	0.379305i	$0.876163\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	44.0000i	1.84783i
$568$	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
$571$	−38.0000	−1.59025	−0.795125	−	0.606445i	$-0.792595\pi$
$571$	−38.0000	−1.59025	−0.795125	+	0.606445i	$0.792595\pi$
$572$	0	0
$573$	− 32.0000i	− 1.33682i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	0	0
$579$	−4.00000	−0.166234
$580$	0	0
$581$	−24.0000	−0.995688
$582$	0	0
$583$	− 12.0000i	− 0.496989i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	34.0000i	1.40333i	0.712507	+	0.701665i	$0.247560\pi$
$587$	34.0000i	1.40333i	−0.712507	+	0.701665i	$0.752440\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−28.0000	−1.15177
$592$	0	0
$593$	− 18.0000i	− 0.739171i	−0.929197	−	0.369586i	$-0.879500\pi$
$593$	− 18.0000i	− 0.739171i	0.929197	−	0.369586i	$-0.120500\pi$
$594$	0	0
$595$	0	0
$596$	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	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	10.0000i	0.407231i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 16.0000i	− 0.649420i	−0.945814	−	0.324710i	$-0.894733\pi$
$607$	− 16.0000i	− 0.649420i	0.945814	−	0.324710i	$-0.105267\pi$
$608$	0	0
$609$	−48.0000	−1.94506
$610$	0	0
$611$	16.0000	0.647291
$612$	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
$615$	0	0
$616$	0	0
$617$	− 2.00000i	− 0.0805170i	−0.999189	−	0.0402585i	$-0.987182\pi$
$617$	− 2.00000i	− 0.0805170i	0.999189	−	0.0402585i	$-0.0128181\pi$
$618$	0	0
$619$	46.0000	1.84890	0.924448	−	0.381308i	$-0.124526\pi$
$619$	46.0000	1.84890	0.924448	+	0.381308i	$0.124526\pi$
$620$	0	0
$621$	16.0000	0.642058
$622$	0	0
$623$	− 8.00000i	− 0.320513i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8.00000i	0.319489i
$628$	0	0
$629$	−20.0000	−0.797452
$630$	0	0
$631$	−44.0000	−1.75161	−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−44.0000	−1.75161	−0.875806	+	0.482663i	$0.839670\pi$
$632$	0	0
$633$	44.0000i	1.74884i
$634$	0	0
$635$	0	0
$636$	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	$0.310612\pi$
$643$	42.0000i	1.65632i	−0.560493	+	0.828159i	$0.689388\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.0000i	0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$	12.0000i	0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$	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	$0.307026\pi$
$653$	42.0000i	1.64359i	−0.569785	+	0.821794i	$0.692974\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	14.0000i	0.546192i
$658$	0	0
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	−34.0000	−1.32245	−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000	−1.32245	−0.661223	+	0.750189i	$0.729962\pi$
$662$	0	0
$663$	8.00000i	0.310694i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	24.0000i	0.929284i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	2.00000i	0.0770943i	0.999257	+	0.0385472i	$0.0122730\pi$
$673$	2.00000i	0.0770943i	−0.999257	+	0.0385472i	$0.987727\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.0000i	0.845529i	0.906240	+	0.422764i	$0.138940\pi$
$677$	22.0000i	0.845529i	−0.906240	+	0.422764i	$0.861060\pi$
$678$	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
$685$	0	0
$686$	0	0
$687$	− 28.0000i	− 1.06827i
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	6.00000	0.228251	0.114125	−	0.993466i	$-0.463593\pi$
$691$	6.00000	0.228251	0.114125	+	0.993466i	$0.463593\pi$
$692$	0	0
$693$	8.00000i	0.303895i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	12.0000i	0.454532i
$698$	0	0
$699$	−36.0000	−1.36165
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	− 20.0000i	− 0.754314i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 24.0000i	− 0.902613i
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	0	0
$714$	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
$721$	16.0000	0.595871
$722$	0	0
$723$	− 4.00000i	− 0.148762i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 12.0000i	− 0.445055i	−0.974926	−	0.222528i	$-0.928569\pi$
$727$	− 12.0000i	− 0.445055i	0.974926	−	0.222528i	$-0.0714308\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	12.0000	0.443836
$732$	0	0
$733$	− 6.00000i	− 0.221615i	−0.993842	−	0.110808i	$-0.964656\pi$
$733$	− 6.00000i	− 0.221615i	0.993842	−	0.110808i	$-0.0353437\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 20.0000i	− 0.736709i
$738$	0	0
$739$	18.0000	0.662141	0.331070	−	0.943606i	$-0.392590\pi$
$739$	18.0000	0.662141	0.331070	+	0.943606i	$0.392590\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	− 44.0000i	− 1.61420i	−0.590412	−	0.807102i	$-0.701035\pi$
$743$	− 44.0000i	− 1.61420i	0.590412	−	0.807102i	$-0.298965\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.00000i	0.219529i
$748$	0	0
$749$	8.00000	0.292314
$750$	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
$753$	36.0000i	1.31191i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	46.0000i	1.67190i	0.548807	+	0.835949i	$0.315082\pi$
$757$	46.0000i	1.67190i	−0.548807	+	0.835949i	$0.684918\pi$
$758$	0	0
$759$	16.0000	0.580763
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	24.0000i	0.868858i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	28.0000i	1.01102i
$768$	0	0
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	−36.0000	−1.29651
$772$	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
$777$	− 80.0000i	− 2.86998i
$778$	0	0
$779$	−12.0000	−0.429945
$780$	0	0
$781$	24.0000	0.858788
$782$	0	0
$783$	− 24.0000i	− 0.857690i
$784$	0	0
$785$	0	0
$786$	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
$789$	24.0000	0.854423
$790$	0	0
$791$	−8.00000	−0.284447
$792$	0	0
$793$	− 4.00000i	− 0.142044i
$794$	0	0
$795$	0	0
$796$	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
$805$	0	0
$806$	0	0
$807$	20.0000i	0.704033i
$808$	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
$811$	18.0000	0.632065	0.316033	−	0.948748i	$-0.397649\pi$
$811$	18.0000	0.632065	0.316033	+	0.948748i	$0.397649\pi$
$812$	0	0
$813$	16.0000i	0.561144i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	12.0000i	0.419827i
$818$	0	0
$819$	−8.00000	−0.279543
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	0	0
$823$	28.0000i	0.976019i	0.872838	+	0.488009i	$0.162277\pi$
$823$	28.0000i	0.976019i	−0.872838	+	0.488009i	$0.837723\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 22.0000i	− 0.765015i	−0.923952	−	0.382507i	$-0.875061\pi$
$827$	− 22.0000i	− 0.765015i	0.923952	−	0.382507i	$-0.124939\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	−12.0000	−0.416275
$832$	0	0
$833$	18.0000i	0.623663i
$834$	0	0
$835$	0	0
$836$	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
$845$	0	0
$846$	0	0
$847$	28.0000i	0.962091i
$848$	0	0
$849$	−12.0000	−0.411839
$850$	0	0
$851$	−40.0000	−1.37118
$852$	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
$855$	0	0
$856$	0	0
$857$	− 6.00000i	− 0.204956i	−0.994735	−	0.102478i	$-0.967323\pi$
$857$	− 6.00000i	− 0.204956i	0.994735	−	0.102478i	$-0.0326771\pi$
$858$	0	0
$859$	−50.0000	−1.70598	−0.852989	−	0.521929i	$-0.825213\pi$
$859$	−50.0000	−1.70598	−0.852989	+	0.521929i	$0.825213\pi$
$860$	0	0
$861$	−48.0000	−1.63584
$862$	0	0
$863$	32.0000i	1.08929i	0.838666	+	0.544646i	$0.183336\pi$
$863$	32.0000i	1.08929i	−0.838666	+	0.544646i	$0.816664\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	26.0000i	0.883006i
$868$	0	0
$869$	16.0000	0.542763
$870$	0	0
$871$	20.0000	0.677674
$872$	0	0
$873$	2.00000i	0.0676897i
$874$	0	0
$875$	0	0
$876$	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	$0.193869\pi$
$883$	34.0000i	1.14419i	−0.820187	+	0.572096i	$0.806131\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36.0000i	1.20876i	0.796696	+	0.604381i	$0.206579\pi$
$887$	36.0000i	1.20876i	−0.796696	+	0.604381i	$0.793421\pi$
$888$	0	0
$889$	64.0000	2.14649
$890$	0	0
$891$	−22.0000	−0.737028
$892$	0	0
$893$	− 16.0000i	− 0.535420i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	16.0000i	0.534224i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	48.0000i	1.59734i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 38.0000i	− 1.26177i	−0.775877	−	0.630885i	$-0.782692\pi$
$907$	− 38.0000i	− 1.26177i	0.775877	−	0.630885i	$-0.217308\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	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
$913$	− 12.0000i	− 0.397142i
$914$	0	0
$915$	0	0
$916$	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
$925$	0	0
$926$	0	0
$927$	− 4.00000i	− 0.131377i
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	0	0
$933$	− 56.0000i	− 1.83336i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	46.0000i	1.50275i	0.659873	+	0.751377i	$0.270610\pi$
$937$	46.0000i	1.50275i	−0.659873	+	0.751377i	$0.729390\pi$
$938$	0	0
$939$	20.0000	0.652675
$940$	0	0
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	0	0
$943$	24.0000i	0.781548i
$944$	0	0
$945$	0	0
$946$	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
$949$	28.0000	0.908918
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	− 58.0000i	− 1.87880i	−0.342817	−	0.939402i	$-0.611381\pi$
$953$	− 58.0000i	− 1.87880i	0.342817	−	0.939402i	$-0.388619\pi$
$954$	0	0
$955$	0	0
$956$	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
$965$	0	0
$966$	0	0
$967$	28.0000i	0.900419i	0.892923	+	0.450210i	$0.148651\pi$
$967$	28.0000i	0.900419i	−0.892923	+	0.450210i	$0.851349\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	−38.0000	−1.21948	−0.609739	−	0.792602i	$-0.708726\pi$
$971$	−38.0000	−1.21948	−0.609739	+	0.792602i	$0.708726\pi$
$972$	0	0
$973$	40.0000i	1.28234i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	30.0000i	0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$	30.0000i	0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$	0	0
$979$	4.00000	0.127841
$980$	0	0
$981$	6.00000	0.191565
$982$	0	0
$983$	− 20.0000i	− 0.637901i	−0.947771	−	0.318950i	$-0.896670\pi$
$983$	− 20.0000i	− 0.637901i	0.947771	−	0.318950i	$-0.103330\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 64.0000i	− 2.03714i
$988$	0	0
$989$	24.0000	0.763156
$990$	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
$993$	− 28.0000i	− 0.888553i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	54.0000i	1.71020i	0.518465	+	0.855099i	$0.326503\pi$
$997$	54.0000i	1.71020i	−0.518465	+	0.855099i	$0.673497\pi$
$998$	0	0
$999$	40.0000	1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3200.2.c.l.2049.2		2
4.3	odd	2		3200.2.c.f.2049.1		2
5.2	odd	4		3200.2.a.x.1.1		1
5.3	odd	4		128.2.a.a.1.1	✓	1
5.4	even	2	inner	3200.2.c.l.2049.1		2
8.3	odd	2		3200.2.c.k.2049.2		2
8.5	even	2		3200.2.c.e.2049.1		2
15.8	even	4		1152.2.a.m.1.1		1
20.3	even	4		128.2.a.c.1.1	yes	1
20.7	even	4		3200.2.a.e.1.1		1
20.19	odd	2		3200.2.c.f.2049.2		2
35.13	even	4		6272.2.a.h.1.1		1
40.3	even	4		128.2.a.b.1.1	yes	1
40.13	odd	4		128.2.a.d.1.1	yes	1
40.19	odd	2		3200.2.c.k.2049.1		2
40.27	even	4		3200.2.a.u.1.1		1
40.29	even	2		3200.2.c.e.2049.2		2
40.37	odd	4		3200.2.a.h.1.1		1
60.23	odd	4		1152.2.a.r.1.1		1
80.3	even	4		256.2.b.a.129.2		2
80.13	odd	4		256.2.b.c.129.1		2
80.43	even	4		256.2.b.a.129.1		2
80.53	odd	4		256.2.b.c.129.2		2
120.53	even	4		1152.2.a.c.1.1		1
120.83	odd	4		1152.2.a.h.1.1		1
140.83	odd	4		6272.2.a.b.1.1		1
160.3	even	8		1024.2.e.m.257.2		4
160.13	odd	8		1024.2.e.i.257.2		4
160.43	even	8		1024.2.e.m.769.2		4
160.53	odd	8		1024.2.e.i.769.1		4
160.83	even	8		1024.2.e.m.257.1		4
160.93	odd	8		1024.2.e.i.257.1		4
160.123	even	8		1024.2.e.m.769.1		4
160.133	odd	8		1024.2.e.i.769.2		4
240.53	even	4		2304.2.d.r.1153.2		2
240.83	odd	4		2304.2.d.b.1153.1		2
240.173	even	4		2304.2.d.r.1153.1		2
240.203	odd	4		2304.2.d.b.1153.2		2
280.13	even	4		6272.2.a.a.1.1		1
280.83	odd	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	5.3	odd	4
128.2.a.b.1.1	yes	1	40.3	even	4
128.2.a.c.1.1	yes	1	20.3	even	4
128.2.a.d.1.1	yes	1	40.13	odd	4
256.2.b.a.129.1		2	80.43	even	4
256.2.b.a.129.2		2	80.3	even	4
256.2.b.c.129.1		2	80.13	odd	4
256.2.b.c.129.2		2	80.53	odd	4
1024.2.e.i.257.1		4	160.93	odd	8
1024.2.e.i.257.2		4	160.13	odd	8
1024.2.e.i.769.1		4	160.53	odd	8
1024.2.e.i.769.2		4	160.133	odd	8
1024.2.e.m.257.1		4	160.83	even	8
1024.2.e.m.257.2		4	160.3	even	8
1024.2.e.m.769.1		4	160.123	even	8
1024.2.e.m.769.2		4	160.43	even	8
1152.2.a.c.1.1		1	120.53	even	4
1152.2.a.h.1.1		1	120.83	odd	4
1152.2.a.m.1.1		1	15.8	even	4
1152.2.a.r.1.1		1	60.23	odd	4
2304.2.d.b.1153.1		2	240.83	odd	4
2304.2.d.b.1153.2		2	240.203	odd	4
2304.2.d.r.1153.1		2	240.173	even	4
2304.2.d.r.1153.2		2	240.53	even	4
3200.2.a.e.1.1		1	20.7	even	4
3200.2.a.h.1.1		1	40.37	odd	4
3200.2.a.u.1.1		1	40.27	even	4
3200.2.a.x.1.1		1	5.2	odd	4
3200.2.c.e.2049.1		2	8.5	even	2
3200.2.c.e.2049.2		2	40.29	even	2
3200.2.c.f.2049.1		2	4.3	odd	2
3200.2.c.f.2049.2		2	20.19	odd	2
3200.2.c.k.2049.1		2	40.19	odd	2
3200.2.c.k.2049.2		2	8.3	odd	2
3200.2.c.l.2049.1		2	5.4	even	2	inner
3200.2.c.l.2049.2		2	1.1	even	1	trivial
6272.2.a.a.1.1		1	280.13	even	4
6272.2.a.b.1.1		1	140.83	odd	4
6272.2.a.g.1.1		1	280.83	odd	4
6272.2.a.h.1.1		1	35.13	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:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.c (of order \(2\), degree \(1\), not minimal)

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

Introduction

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