LMFDB - Embedded newform 3200.2.c.p.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 = [2,0,0,0,0,0,0,0,4,0,2,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,4] 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:	$ 1 $
Twist minimal:	yes
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.p.2049.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +2.00000 q^{9} +1.00000 q^{11} +2.00000i q^{13} +3.00000i q^{17} +3.00000 q^{19} -6.00000i q^{23} +5.00000i q^{27} +2.00000 q^{29} +2.00000 q^{31} +1.00000i q^{33} +4.00000i q^{37} -2.00000 q^{39} -3.00000 q^{41} -4.00000i q^{43} -6.00000i q^{47} +7.00000 q^{49} -3.00000 q^{51} +10.0000i q^{53} +3.00000i q^{57} +12.0000 q^{59} -12.0000 q^{61} +1.00000i q^{67} +6.00000 q^{69} +10.0000 q^{71} -1.00000i q^{73} -16.0000 q^{79} +1.00000 q^{81} +11.0000i q^{83} +2.00000i q^{87} +13.0000 q^{89} +2.00000i q^{93} +6.00000i q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{9} + 2 q^{11} + 6 q^{19} + 4 q^{29} + 4 q^{31} - 4 q^{39} - 6 q^{41} + 14 q^{49} - 6 q^{51} + 24 q^{59} - 24 q^{61} + 12 q^{69} + 20 q^{71} - 32 q^{79} + 2 q^{81} + 26 q^{89} + 4 q^{99}+O(q^{100}) $ 2 * q + 4 * q^9 + 2 * q^11 + 6 * q^19 + 4 * q^29 + 4 * q^31 - 4 * q^39 - 6 * q^41 + 14 * q^49 - 6 * q^51 + 24 * q^59 - 24 * q^61 + 12 * q^69 + 20 * q^71 - 32 * q^79 + 2 * q^81 + 26 * q^89 + 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
$3$	1.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	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
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.00000i	0.962250i
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	1.00000i	0.174078i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6.00000i	− 0.875190i	−0.899172	−	0.437595i	$-0.855830\pi$
$47$	− 6.00000i	− 0.875190i	0.899172	−	0.437595i	$-0.144170\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	−3.00000	−0.420084
$52$	0	0
$53$	10.0000i	1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$	10.0000i	1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.00000i	0.397360i
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−12.0000	−1.53644	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000	−1.53644	−0.768221	+	0.640184i	$0.778858\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.00000i	0.122169i	0.998133	+	0.0610847i	$0.0194560\pi$
$67$	1.00000i	0.122169i	−0.998133	+	0.0610847i	$0.980544\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	0	0
$73$	− 1.00000i	− 0.117041i	−0.998286	−	0.0585206i	$-0.981362\pi$
$73$	− 1.00000i	− 0.117041i	0.998286	−	0.0585206i	$-0.0186383\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.0000i	1.20741i	0.797209	+	0.603703i	$0.206309\pi$
$83$	11.0000i	1.20741i	−0.797209	+	0.603703i	$0.793691\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.00000i	0.214423i
$88$	0	0
$89$	13.0000	1.37800	0.688999	−	0.724763i	$-0.258051\pi$
$89$	13.0000	1.37800	0.688999	+	0.724763i	$0.258051\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.00000i	0.207390i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.00000i	0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$	6.00000i	0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	− 14.0000i	− 1.37946i	−0.724066	−	0.689730i	$-0.757729\pi$
$103$	− 14.0000i	− 1.37946i	0.724066	−	0.689730i	$-0.242271\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.00000i	0.676716i	0.941018	+	0.338358i	$0.109871\pi$
$107$	7.00000i	0.676716i	−0.941018	+	0.338358i	$0.890129\pi$
$108$	0	0
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	11.0000i	1.03479i	0.855746	+	0.517396i	$0.173099\pi$
$113$	11.0000i	1.03479i	−0.855746	+	0.517396i	$0.826901\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.00000i	0.369800i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	− 3.00000i	− 0.270501i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 6.00000i	− 0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$	− 6.00000i	− 0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.00000i	0.256307i	0.991754	+	0.128154i	$0.0409051\pi$
$137$	3.00000i	0.256307i	−0.991754	+	0.128154i	$0.959095\pi$
$138$	0	0
$139$	19.0000	1.61156	0.805779	−	0.592216i	$-0.201747\pi$
$139$	19.0000	1.61156	0.805779	+	0.592216i	$0.201747\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	2.00000i	0.167248i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	7.00000i	0.577350i
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	6.00000i	0.485071i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	0	0
$162$	0	0
$163$	23.0000i	1.80150i	0.434339	+	0.900750i	$0.356982\pi$
$163$	23.0000i	1.80150i	−0.434339	+	0.900750i	$0.643018\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 14.0000i	− 1.08335i	−0.840587	−	0.541676i	$-0.817790\pi$
$167$	− 14.0000i	− 1.08335i	0.840587	−	0.541676i	$-0.182210\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	6.00000	0.458831
$172$	0	0
$173$	4.00000i	0.304114i	0.988372	+	0.152057i	$0.0485898\pi$
$173$	4.00000i	0.304114i	−0.988372	+	0.152057i	$0.951410\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.0000i	0.901975i
$178$	0	0
$179$	−5.00000	−0.373718	−0.186859	−	0.982387i	$-0.559831\pi$
$179$	−5.00000	−0.373718	−0.186859	+	0.982387i	$0.559831\pi$
$180$	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
$183$	− 12.0000i	− 0.887066i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.00000i	0.219382i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	9.00000i	0.647834i	0.946085	+	0.323917i	$0.105000\pi$
$193$	9.00000i	0.647834i	−0.946085	+	0.323917i	$0.895000\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 12.0000i	− 0.834058i
$208$	0	0
$209$	3.00000	0.207514
$210$	0	0
$211$	−17.0000	−1.17033	−0.585164	−	0.810915i	$-0.698970\pi$
$211$	−17.0000	−1.17033	−0.585164	+	0.810915i	$0.698970\pi$
$212$	0	0
$213$	10.0000i	0.685189i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.00000	0.0675737
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	28.0000i	1.87502i	0.347960	+	0.937509i	$0.386874\pi$
$223$	28.0000i	1.87502i	−0.347960	+	0.937509i	$0.613126\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	8.00000	0.528655	0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000	0.528655	0.264327	+	0.964433i	$0.414850\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.0000i	0.917170i	0.888650	+	0.458585i	$0.151644\pi$
$233$	14.0000i	0.917170i	−0.888650	+	0.458585i	$0.848356\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 16.0000i	− 1.03931i
$238$	0	0
$239$	−26.0000	−1.68180	−0.840900	−	0.541190i	$-0.817974\pi$
$239$	−26.0000	−1.68180	−0.840900	+	0.541190i	$0.817974\pi$
$240$	0	0
$241$	−11.0000	−0.708572	−0.354286	−	0.935137i	$-0.615276\pi$
$241$	−11.0000	−0.708572	−0.354286	+	0.935137i	$0.615276\pi$
$242$	0	0
$243$	16.0000i	1.02640i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.00000i	0.381771i
$248$	0	0
$249$	−11.0000	−0.697097
$250$	0	0
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	0	0
$253$	− 6.00000i	− 0.377217i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6.00000i	− 0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$	− 6.00000i	− 0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	4.00000	0.247594
$262$	0	0
$263$	− 18.0000i	− 1.10993i	−0.831875	−	0.554964i	$-0.812732\pi$
$263$	− 18.0000i	− 1.10993i	0.831875	−	0.554964i	$-0.187268\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	13.0000i	0.795587i
$268$	0	0
$269$	16.0000	0.975537	0.487769	−	0.872973i	$-0.337811\pi$
$269$	16.0000	0.975537	0.487769	+	0.872973i	$0.337811\pi$
$270$	0	0
$271$	10.0000	0.607457	0.303728	−	0.952759i	$-0.401768\pi$
$271$	10.0000	0.607457	0.303728	+	0.952759i	$0.401768\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 14.0000i	− 0.841178i	−0.907251	−	0.420589i	$-0.861823\pi$
$277$	− 14.0000i	− 0.841178i	0.907251	−	0.420589i	$-0.138177\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	− 17.0000i	− 1.01055i	−0.862960	−	0.505273i	$-0.831392\pi$
$283$	− 17.0000i	− 1.01055i	0.862960	−	0.505273i	$-0.168608\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	−6.00000	−0.351726
$292$	0	0
$293$	− 12.0000i	− 0.701047i	−0.936554	−	0.350524i	$-0.886004\pi$
$293$	− 12.0000i	− 0.701047i	0.936554	−	0.350524i	$-0.113996\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.00000i	0.290129i
$298$	0	0
$299$	12.0000	0.693978
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 25.0000i	− 1.42683i	−0.700744	−	0.713413i	$-0.747149\pi$
$307$	− 25.0000i	− 1.42683i	0.700744	−	0.713413i	$-0.252851\pi$
$308$	0	0
$309$	14.0000	0.796432
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	2.00000i	0.113047i	0.998401	+	0.0565233i	$0.0180015\pi$
$313$	2.00000i	0.113047i	−0.998401	+	0.0565233i	$0.981998\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	32.0000i	1.79730i	0.438667	+	0.898650i	$0.355451\pi$
$317$	32.0000i	1.79730i	−0.438667	+	0.898650i	$0.644549\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	0	0
$321$	−7.00000	−0.390702
$322$	0	0
$323$	9.00000i	0.500773i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	16.0000i	0.884802i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−11.0000	−0.604615	−0.302307	−	0.953211i	$-0.597757\pi$
$331$	−11.0000	−0.604615	−0.302307	+	0.953211i	$0.597757\pi$
$332$	0	0
$333$	8.00000i	0.438397i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 23.0000i	− 1.25289i	−0.779466	−	0.626445i	$-0.784509\pi$
$337$	− 23.0000i	− 1.25289i	0.779466	−	0.626445i	$-0.215491\pi$
$338$	0	0
$339$	−11.0000	−0.597438
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.00000i	0.161048i	0.996753	+	0.0805242i	$0.0256594\pi$
$347$	3.00000i	0.161048i	−0.996753	+	0.0805242i	$0.974341\pi$
$348$	0	0
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	0	0
$351$	−10.0000	−0.533761
$352$	0	0
$353$	− 10.0000i	− 0.532246i	−0.963939	−	0.266123i	$-0.914257\pi$
$353$	− 10.0000i	− 0.532246i	0.963939	−	0.266123i	$-0.0857428\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	0	0
$363$	− 10.0000i	− 0.524864i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 16.0000i	− 0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$	− 16.0000i	− 0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 18.0000i	− 0.932005i	−0.884783	−	0.466002i	$-0.845694\pi$
$373$	− 18.0000i	− 0.932005i	0.884783	−	0.466002i	$-0.154306\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.00000i	0.206010i
$378$	0	0
$379$	5.00000	0.256833	0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000	0.256833	0.128416	+	0.991720i	$0.459011\pi$
$380$	0	0
$381$	6.00000	0.307389
$382$	0	0
$383$	20.0000i	1.02195i	0.859595	+	0.510976i	$0.170716\pi$
$383$	20.0000i	1.02195i	−0.859595	+	0.510976i	$0.829284\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 8.00000i	− 0.406663i
$388$	0	0
$389$	−22.0000	−1.11544	−0.557722	−	0.830028i	$-0.688325\pi$
$389$	−22.0000	−1.11544	−0.557722	+	0.830028i	$0.688325\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	0	0
$393$	− 12.0000i	− 0.605320i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 28.0000i	− 1.40528i	−0.711546	−	0.702640i	$-0.752005\pi$
$397$	− 28.0000i	− 1.40528i	0.711546	−	0.702640i	$-0.247995\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.0000	0.649189	0.324595	−	0.945853i	$-0.394772\pi$
$401$	13.0000	0.649189	0.324595	+	0.945853i	$0.394772\pi$
$402$	0	0
$403$	4.00000i	0.199254i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.00000i	0.198273i
$408$	0	0
$409$	7.00000	0.346128	0.173064	−	0.984911i	$-0.444633\pi$
$409$	7.00000	0.346128	0.173064	+	0.984911i	$0.444633\pi$
$410$	0	0
$411$	−3.00000	−0.147979
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	19.0000i	0.930434i
$418$	0	0
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	−32.0000	−1.55958	−0.779792	−	0.626038i	$-0.784675\pi$
$421$	−32.0000	−1.55958	−0.779792	+	0.626038i	$0.784675\pi$
$422$	0	0
$423$	− 12.0000i	− 0.583460i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	− 23.0000i	− 1.10531i	−0.833410	−	0.552655i	$-0.813615\pi$
$433$	− 23.0000i	− 1.10531i	0.833410	−	0.552655i	$-0.186385\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 18.0000i	− 0.861057i
$438$	0	0
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	0	0
$441$	14.0000	0.666667
$442$	0	0
$443$	− 13.0000i	− 0.617649i	−0.951119	−	0.308824i	$-0.900064\pi$
$443$	− 13.0000i	− 0.617649i	0.951119	−	0.308824i	$-0.0999355\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.00000i	0.283790i
$448$	0	0
$449$	−5.00000	−0.235965	−0.117982	−	0.993016i	$-0.537643\pi$
$449$	−5.00000	−0.235965	−0.117982	+	0.993016i	$0.537643\pi$
$450$	0	0
$451$	−3.00000	−0.141264
$452$	0	0
$453$	12.0000i	0.563809i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 9.00000i	− 0.421002i	−0.977594	−	0.210501i	$-0.932490\pi$
$457$	− 9.00000i	− 0.421002i	0.977594	−	0.210501i	$-0.0675096\pi$
$458$	0	0
$459$	−15.0000	−0.700140
$460$	0	0
$461$	32.0000	1.49039	0.745194	−	0.666847i	$-0.232357\pi$
$461$	32.0000	1.49039	0.745194	+	0.666847i	$0.232357\pi$
$462$	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
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−14.0000	−0.645086
$472$	0	0
$473$	− 4.00000i	− 0.183920i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	20.0000i	0.915737i
$478$	0	0
$479$	22.0000	1.00521	0.502603	−	0.864517i	$-0.332376\pi$
$479$	22.0000	1.00521	0.502603	+	0.864517i	$0.332376\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	26.0000i	1.17817i	0.808070	+	0.589086i	$0.200512\pi$
$487$	26.0000i	1.17817i	−0.808070	+	0.589086i	$0.799488\pi$
$488$	0	0
$489$	−23.0000	−1.04010
$490$	0	0
$491$	40.0000	1.80517	0.902587	−	0.430507i	$-0.141665\pi$
$491$	40.0000	1.80517	0.902587	+	0.430507i	$0.141665\pi$
$492$	0	0
$493$	6.00000i	0.270226i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	14.0000	0.625474
$502$	0	0
$503$	− 4.00000i	− 0.178351i	−0.996016	−	0.0891756i	$-0.971577\pi$
$503$	− 4.00000i	− 0.178351i	0.996016	−	0.0891756i	$-0.0284232\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.00000i	0.399704i
$508$	0	0
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	15.0000i	0.662266i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 6.00000i	− 0.263880i
$518$	0	0
$519$	−4.00000	−0.175581
$520$	0	0
$521$	37.0000	1.62100	0.810500	−	0.585739i	$-0.199196\pi$
$521$	37.0000	1.62100	0.810500	+	0.585739i	$0.199196\pi$
$522$	0	0
$523$	− 39.0000i	− 1.70535i	−0.522441	−	0.852675i	$-0.674978\pi$
$523$	− 39.0000i	− 1.70535i	0.522441	−	0.852675i	$-0.325022\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.00000i	0.261364i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	24.0000	1.04151
$532$	0	0
$533$	− 6.00000i	− 0.259889i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 5.00000i	− 0.215766i
$538$	0	0
$539$	7.00000	0.301511
$540$	0	0
$541$	−20.0000	−0.859867	−0.429934	−	0.902861i	$-0.641463\pi$
$541$	−20.0000	−0.859867	−0.429934	+	0.902861i	$0.641463\pi$
$542$	0	0
$543$	− 2.00000i	− 0.0858282i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 13.0000i	− 0.555840i	−0.960604	−	0.277920i	$-0.910355\pi$
$547$	− 13.0000i	− 0.555840i	0.960604	−	0.277920i	$-0.0896450\pi$
$548$	0	0
$549$	−24.0000	−1.02430
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 38.0000i	− 1.61011i	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	− 38.0000i	− 1.61011i	0.593199	−	0.805056i	$-0.297865\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	−3.00000	−0.126660
$562$	0	0
$563$	− 4.00000i	− 0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$	− 4.00000i	− 0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−25.0000	−1.04805	−0.524027	−	0.851701i	$-0.675571\pi$
$569$	−25.0000	−1.04805	−0.524027	+	0.851701i	$0.675571\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	− 18.0000i	− 0.751961i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 39.0000i	− 1.62359i	−0.583942	−	0.811796i	$-0.698490\pi$
$577$	− 39.0000i	− 1.62359i	0.583942	−	0.811796i	$-0.301510\pi$
$578$	0	0
$579$	−9.00000	−0.374027
$580$	0	0
$581$	0	0
$582$	0	0
$583$	10.0000i	0.414158i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 39.0000i	− 1.60970i	−0.593477	−	0.804851i	$-0.702245\pi$
$587$	− 39.0000i	− 1.60970i	0.593477	−	0.804851i	$-0.297755\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	− 17.0000i	− 0.698106i	−0.937103	−	0.349053i	$-0.886503\pi$
$593$	− 17.0000i	− 0.698106i	0.937103	−	0.349053i	$-0.113497\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 4.00000i	− 0.163709i
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	11.0000	0.448699	0.224350	−	0.974509i	$-0.427974\pi$
$601$	11.0000	0.448699	0.224350	+	0.974509i	$0.427974\pi$
$602$	0	0
$603$	2.00000i	0.0814463i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	28.0000i	1.13648i	0.822861	+	0.568242i	$0.192376\pi$
$607$	28.0000i	1.13648i	−0.822861	+	0.568242i	$0.807624\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	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
$615$	0	0
$616$	0	0
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	0	0
$619$	−24.0000	−0.964641	−0.482321	−	0.875995i	$-0.660206\pi$
$619$	−24.0000	−0.964641	−0.482321	+	0.875995i	$0.660206\pi$
$620$	0	0
$621$	30.0000	1.20386
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.00000i	0.119808i
$628$	0	0
$629$	−12.0000	−0.478471
$630$	0	0
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	0	0
$633$	− 17.0000i	− 0.675689i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	14.0000i	0.554700i
$638$	0	0
$639$	20.0000	0.791188
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	0	0
$643$	− 36.0000i	− 1.41970i	−0.704352	−	0.709851i	$-0.748762\pi$
$643$	− 36.0000i	− 1.41970i	0.704352	−	0.709851i	$-0.251238\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 28.0000i	− 1.10079i	−0.834903	−	0.550397i	$-0.814476\pi$
$647$	− 28.0000i	− 1.10079i	0.834903	−	0.550397i	$-0.185524\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	0	0
$653$	28.0000i	1.09572i	0.836569	+	0.547862i	$0.184558\pi$
$653$	28.0000i	1.09572i	−0.836569	+	0.547862i	$0.815442\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.00000i	− 0.0780274i
$658$	0	0
$659$	−27.0000	−1.05177	−0.525885	−	0.850555i	$-0.676266\pi$
$659$	−27.0000	−1.05177	−0.525885	+	0.850555i	$0.676266\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	− 6.00000i	− 0.233021i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 12.0000i	− 0.464642i
$668$	0	0
$669$	−28.0000	−1.08254
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	− 50.0000i	− 1.92736i	−0.267063	−	0.963679i	$-0.586053\pi$
$673$	− 50.0000i	− 1.92736i	0.267063	−	0.963679i	$-0.413947\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 4.00000i	− 0.153732i	−0.997041	−	0.0768662i	$-0.975509\pi$
$677$	− 4.00000i	− 0.153732i	0.997041	−	0.0768662i	$-0.0244914\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.00000i	0.344375i	0.985064	+	0.172188i	$0.0550836\pi$
$683$	9.00000i	0.344375i	−0.985064	+	0.172188i	$0.944916\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	8.00000i	0.305219i
$688$	0	0
$689$	−20.0000	−0.761939
$690$	0	0
$691$	−35.0000	−1.33146	−0.665731	−	0.746191i	$-0.731880\pi$
$691$	−35.0000	−1.33146	−0.665731	+	0.746191i	$0.731880\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 9.00000i	− 0.340899i
$698$	0	0
$699$	−14.0000	−0.529529
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	12.0000i	0.452589i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−24.0000	−0.901339	−0.450669	−	0.892691i	$-0.648815\pi$
$709$	−24.0000	−0.901339	−0.450669	+	0.892691i	$0.648815\pi$
$710$	0	0
$711$	−32.0000	−1.20009
$712$	0	0
$713$	− 12.0000i	− 0.449404i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 26.0000i	− 0.970988i
$718$	0	0
$719$	−4.00000	−0.149175	−0.0745874	−	0.997214i	$-0.523764\pi$
$719$	−4.00000	−0.149175	−0.0745874	+	0.997214i	$0.523764\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 11.0000i	− 0.409094i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	22.0000i	0.815935i	0.912996	+	0.407967i	$0.133762\pi$
$727$	22.0000i	0.815935i	−0.912996	+	0.407967i	$0.866238\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	12.0000	0.443836
$732$	0	0
$733$	− 54.0000i	− 1.99454i	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	− 54.0000i	− 1.99454i	0.0738717	−	0.997268i	$-0.476464\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.00000i	0.0368355i
$738$	0	0
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	0	0
$743$	26.0000i	0.953847i	0.878945	+	0.476924i	$0.158248\pi$
$743$	26.0000i	0.953847i	−0.878945	+	0.476924i	$0.841752\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	22.0000i	0.804938i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	34.0000	1.24068	0.620339	−	0.784334i	$-0.286995\pi$
$751$	34.0000	1.24068	0.620339	+	0.784334i	$0.286995\pi$
$752$	0	0
$753$	15.0000i	0.546630i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 36.0000i	− 1.30844i	−0.756303	−	0.654221i	$-0.772997\pi$
$757$	− 36.0000i	− 1.30844i	0.756303	−	0.654221i	$-0.227003\pi$
$758$	0	0
$759$	6.00000	0.217786
$760$	0	0
$761$	−23.0000	−0.833749	−0.416875	−	0.908964i	$-0.636875\pi$
$761$	−23.0000	−0.833749	−0.416875	+	0.908964i	$0.636875\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	24.0000i	0.866590i
$768$	0	0
$769$	−25.0000	−0.901523	−0.450762	−	0.892644i	$-0.648848\pi$
$769$	−25.0000	−0.901523	−0.450762	+	0.892644i	$0.648848\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	0	0
$773$	− 26.0000i	− 0.935155i	−0.883952	−	0.467578i	$-0.845127\pi$
$773$	− 26.0000i	− 0.935155i	0.883952	−	0.467578i	$-0.154873\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.00000	−0.322458
$780$	0	0
$781$	10.0000	0.357828
$782$	0	0
$783$	10.0000i	0.357371i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 36.0000i	− 1.28326i	−0.767014	−	0.641631i	$-0.778258\pi$
$787$	− 36.0000i	− 1.28326i	0.767014	−	0.641631i	$-0.221742\pi$
$788$	0	0
$789$	18.0000	0.640817
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 24.0000i	− 0.852265i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	38.0000i	1.34603i	0.739629	+	0.673015i	$0.235001\pi$
$797$	38.0000i	1.34603i	−0.739629	+	0.673015i	$0.764999\pi$
$798$	0	0
$799$	18.0000	0.636794
$800$	0	0
$801$	26.0000	0.918665
$802$	0	0
$803$	− 1.00000i	− 0.0352892i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	16.0000i	0.563227i
$808$	0	0
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	10.0000i	0.350715i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 12.0000i	− 0.419827i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	40.0000i	1.39431i	0.716919	+	0.697156i	$0.245552\pi$
$823$	40.0000i	1.39431i	−0.716919	+	0.697156i	$0.754448\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 27.0000i	− 0.938882i	−0.882964	−	0.469441i	$-0.844455\pi$
$827$	− 27.0000i	− 0.938882i	0.882964	−	0.469441i	$-0.155545\pi$
$828$	0	0
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	0	0
$833$	21.0000i	0.727607i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	10.0000i	0.345651i
$838$	0	0
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	6.00000i	0.206651i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	17.0000	0.583438
$850$	0	0
$851$	24.0000	0.822709
$852$	0	0
$853$	− 26.0000i	− 0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	− 26.0000i	− 0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 25.0000i	− 0.853984i	−0.904255	−	0.426992i	$-0.859573\pi$
$857$	− 25.0000i	− 0.853984i	0.904255	−	0.426992i	$-0.140427\pi$
$858$	0	0
$859$	−13.0000	−0.443554	−0.221777	−	0.975097i	$-0.571186\pi$
$859$	−13.0000	−0.443554	−0.221777	+	0.975097i	$0.571186\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 34.0000i	− 1.15737i	−0.815550	−	0.578687i	$-0.803565\pi$
$863$	− 34.0000i	− 1.15737i	0.815550	−	0.578687i	$-0.196435\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.00000i	0.271694i
$868$	0	0
$869$	−16.0000	−0.542763
$870$	0	0
$871$	−2.00000	−0.0677674
$872$	0	0
$873$	12.0000i	0.406138i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 48.0000i	− 1.62084i	−0.585846	−	0.810422i	$-0.699238\pi$
$877$	− 48.0000i	− 1.62084i	0.585846	−	0.810422i	$-0.300762\pi$
$878$	0	0
$879$	12.0000	0.404750
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	− 39.0000i	− 1.31245i	−0.754563	−	0.656227i	$-0.772151\pi$
$883$	− 39.0000i	− 1.31245i	0.754563	−	0.656227i	$-0.227849\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 4.00000i	− 0.134307i	−0.997743	−	0.0671534i	$-0.978608\pi$
$887$	− 4.00000i	− 0.134307i	0.997743	−	0.0671534i	$-0.0213917\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	− 18.0000i	− 0.602347i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	12.0000i	0.400668i
$898$	0	0
$899$	4.00000	0.133407
$900$	0	0
$901$	−30.0000	−0.999445
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 20.0000i	− 0.664089i	−0.943264	−	0.332045i	$-0.892262\pi$
$907$	− 20.0000i	− 0.664089i	0.943264	−	0.332045i	$-0.107738\pi$
$908$	0	0
$909$	0	0
$910$	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
$913$	11.0000i	0.364047i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−50.0000	−1.64935	−0.824674	−	0.565608i	$-0.808641\pi$
$919$	−50.0000	−1.64935	−0.824674	+	0.565608i	$0.808641\pi$
$920$	0	0
$921$	25.0000	0.823778
$922$	0	0
$923$	20.0000i	0.658308i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 28.0000i	− 0.919641i
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	21.0000	0.688247
$932$	0	0
$933$	4.00000i	0.130954i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 47.0000i	− 1.53542i	−0.640796	−	0.767712i	$-0.721395\pi$
$937$	− 47.0000i	− 1.53542i	0.640796	−	0.767712i	$-0.278605\pi$
$938$	0	0
$939$	−2.00000	−0.0652675
$940$	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
$943$	18.0000i	0.586161i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.0000i	1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$	36.0000i	1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$	0	0
$949$	2.00000	0.0649227
$950$	0	0
$951$	−32.0000	−1.03767
$952$	0	0
$953$	45.0000i	1.45769i	0.684677	+	0.728846i	$0.259943\pi$
$953$	45.0000i	1.45769i	−0.684677	+	0.728846i	$0.740057\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.00000i	0.0646508i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	14.0000i	0.451144i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.00000i	0.128631i	0.997930	+	0.0643157i	$0.0204865\pi$
$967$	4.00000i	0.128631i	−0.997930	+	0.0643157i	$0.979514\pi$
$968$	0	0
$969$	−9.00000	−0.289122
$970$	0	0
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.00000i	0.223950i	0.993711	+	0.111975i	$0.0357176\pi$
$977$	7.00000i	0.223950i	−0.993711	+	0.111975i	$0.964282\pi$
$978$	0	0
$979$	13.0000	0.415482
$980$	0	0
$981$	32.0000	1.02168
$982$	0	0
$983$	36.0000i	1.14822i	0.818778	+	0.574111i	$0.194652\pi$
$983$	36.0000i	1.14822i	−0.818778	+	0.574111i	$0.805348\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−24.0000	−0.763156
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	− 11.0000i	− 0.349074i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	36.0000i	1.14013i	0.821599	+	0.570066i	$0.193082\pi$
$997$	36.0000i	1.14013i	−0.821599	+	0.570066i	$0.806918\pi$
$998$	0	0
$999$	−20.0000	−0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3200.2.c.p.2049.2		2
4.3	odd	2		3200.2.c.n.2049.1		2
5.2	odd	4		3200.2.a.t.1.1	yes	1
5.3	odd	4		3200.2.a.k.1.1	yes	1
5.4	even	2	inner	3200.2.c.p.2049.1		2
8.3	odd	2		3200.2.c.o.2049.2		2
8.5	even	2		3200.2.c.m.2049.1		2
20.3	even	4		3200.2.a.q.1.1	yes	1
20.7	even	4		3200.2.a.j.1.1	yes	1
20.19	odd	2		3200.2.c.n.2049.2		2
40.3	even	4		3200.2.a.l.1.1	yes	1
40.13	odd	4		3200.2.a.r.1.1	yes	1
40.19	odd	2		3200.2.c.o.2049.1		2
40.27	even	4		3200.2.a.s.1.1	yes	1
40.29	even	2		3200.2.c.m.2049.2		2
40.37	odd	4		3200.2.a.i.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3200.2.a.i.1.1	✓	1	40.37	odd	4
3200.2.a.j.1.1	yes	1	20.7	even	4
3200.2.a.k.1.1	yes	1	5.3	odd	4
3200.2.a.l.1.1	yes	1	40.3	even	4
3200.2.a.q.1.1	yes	1	20.3	even	4
3200.2.a.r.1.1	yes	1	40.13	odd	4
3200.2.a.s.1.1	yes	1	40.27	even	4
3200.2.a.t.1.1	yes	1	5.2	odd	4
3200.2.c.m.2049.1		2	8.5	even	2
3200.2.c.m.2049.2		2	40.29	even	2
3200.2.c.n.2049.1		2	4.3	odd	2
3200.2.c.n.2049.2		2	20.19	odd	2
3200.2.c.o.2049.1		2	40.19	odd	2
3200.2.c.o.2049.2		2	8.3	odd	2
3200.2.c.p.2049.1		2	5.4	even	2	inner
3200.2.c.p.2049.2		2	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +2.00000 q^{9} +1.00000 q^{11} +2.00000i q^{13} +3.00000i q^{17} +3.00000 q^{19} -6.00000i q^{23} +5.00000i q^{27} +2.00000 q^{29} +2.00000 q^{31} +1.00000i q^{33} +4.00000i q^{37} -2.00000 q^{39} -3.00000 q^{41} -4.00000i q^{43} -6.00000i q^{47} +7.00000 q^{49} -3.00000 q^{51} +10.0000i q^{53} +3.00000i q^{57} +12.0000 q^{59} -12.0000 q^{61} +1.00000i q^{67} +6.00000 q^{69} +10.0000 q^{71} -1.00000i q^{73} -16.0000 q^{79} +1.00000 q^{81} +11.0000i q^{83} +2.00000i q^{87} +13.0000 q^{89} +2.00000i q^{93} +6.00000i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9} + 2 q^{11} + 6 q^{19} + 4 q^{29} + 4 q^{31} - 4 q^{39} - 6 q^{41} + 14 q^{49} - 6 q^{51} + 24 q^{59} - 24 q^{61} + 12 q^{69} + 20 q^{71} - 32 q^{79} + 2 q^{81} + 26 q^{89} + 4 q^{99}+O(q^{100}) \) 2 * q + 4 * q^9 + 2 * q^11 + 6 * q^19 + 4 * q^29 + 4 * q^31 - 4 * q^39 - 6 * q^41 + 14 * q^49 - 6 * q^51 + 24 * q^59 - 24 * q^61 + 12 * q^69 + 20 * q^71 - 32 * q^79 + 2 * q^81 + 26 * q^89 + 4 * q^99

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