LMFDB - Embedded newform 3200.2.f.b.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3200,2,Mod(449,3200)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3200.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3200.449
Dual form			3200.2.f.b.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +4.00000i q^{7} -2.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.00000i q^{7} -2.00000 q^{9} -3.00000i q^{11} +1.00000i q^{17} -7.00000i q^{19} -4.00000i q^{21} -4.00000i q^{23} +5.00000 q^{27} +8.00000i q^{29} +4.00000 q^{31} +3.00000i q^{33} +4.00000 q^{37} +3.00000 q^{41} +8.00000 q^{43} -9.00000 q^{49} -1.00000i q^{51} +12.0000 q^{53} +7.00000i q^{57} +8.00000i q^{59} +4.00000i q^{61} -8.00000i q^{63} -9.00000 q^{67} +4.00000i q^{69} -16.0000 q^{71} +11.0000i q^{73} +12.0000 q^{77} -4.00000 q^{79} +1.00000 q^{81} +1.00000 q^{83} -8.00000i q^{87} -13.0000 q^{89} -4.00000 q^{93} +14.0000i q^{97} +6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 4 * q^9	$ 2 q - 2 q^{3} - 4 q^{9} + 10 q^{27} + 8 q^{31} + 8 q^{37} + 6 q^{41} + 16 q^{43} - 18 q^{49} + 24 q^{53} - 18 q^{67} - 32 q^{71} + 24 q^{77} - 8 q^{79} + 2 q^{81} + 2 q^{83} - 26 q^{89} - 8 q^{93}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^9 + 10 * q^27 + 8 * q^31 + 8 * q^37 + 6 * q^41 + 16 * q^43 - 18 * q^49 + 24 * q^53 - 18 * q^67 - 32 * q^71 + 24 * q^77 - 8 * q^79 + 2 * q^81 + 2 * q^83 - 26 * q^89 - 8 * q^93

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	− 3.00000i	− 0.904534i	−0.891883	−	0.452267i	$-0.850615\pi$
$11$	− 3.00000i	− 0.904534i	0.891883	−	0.452267i	$-0.149385\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000i	0.242536i	0.992620	+	0.121268i	$0.0386960\pi$
$17$	1.00000i	0.242536i	−0.992620	+	0.121268i	$0.961304\pi$
$18$	0	0
$19$	− 7.00000i	− 1.60591i	−0.596040	−	0.802955i	$-0.703260\pi$
$19$	− 7.00000i	− 1.60591i	0.596040	−	0.802955i	$-0.296740\pi$
$20$	0	0
$21$	− 4.00000i	− 0.872872i
$22$	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
$27$	5.00000	0.962250
$28$	0	0
$29$	8.00000i	1.48556i	0.669534	+	0.742781i	$0.266494\pi$
$29$	8.00000i	1.48556i	−0.669534	+	0.742781i	$0.733506\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	3.00000i	0.522233i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	− 1.00000i	− 0.140028i
$52$	0	0
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	7.00000i	0.927173i
$58$	0	0
$59$	8.00000i	1.04151i	0.853706	+	0.520756i	$0.174350\pi$
$59$	8.00000i	1.04151i	−0.853706	+	0.520756i	$0.825650\pi$
$60$	0	0
$61$	4.00000i	0.512148i	0.966657	+	0.256074i	$0.0824290\pi$
$61$	4.00000i	0.512148i	−0.966657	+	0.256074i	$0.917571\pi$
$62$	0	0
$63$	− 8.00000i	− 1.00791i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.00000	−1.09952	−0.549762	−	0.835321i	$-0.685282\pi$
$67$	−9.00000	−1.09952	−0.549762	+	0.835321i	$0.685282\pi$
$68$	0	0
$69$	4.00000i	0.481543i
$70$	0	0
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	0	0
$73$	11.0000i	1.28745i	0.765256	+	0.643726i	$0.222612\pi$
$73$	11.0000i	1.28745i	−0.765256	+	0.643726i	$0.777388\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	12.0000	1.36753
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.00000	0.109764	0.0548821	−	0.998493i	$-0.482522\pi$
$83$	1.00000	0.109764	0.0548821	+	0.998493i	$0.482522\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 8.00000i	− 0.857690i
$88$	0	0
$89$	−13.0000	−1.37800	−0.688999	−	0.724763i	$-0.741949\pi$
$89$	−13.0000	−1.37800	−0.688999	+	0.724763i	$0.741949\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	0	0
$99$	6.00000i	0.603023i
$100$	0	0
$101$	− 12.0000i	− 1.19404i	−0.802225	−	0.597022i	$-0.796350\pi$
$101$	− 12.0000i	− 1.19404i	0.802225	−	0.597022i	$-0.203650\pi$
$102$	0	0
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.0000	−1.25676	−0.628379	−	0.777908i	$-0.716281\pi$
$107$	−13.0000	−1.25676	−0.628379	+	0.777908i	$0.716281\pi$
$108$	0	0
$109$	20.0000i	1.91565i	0.287348	+	0.957826i	$0.407226\pi$
$109$	20.0000i	1.91565i	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	9.00000i	0.846649i	0.905978	+	0.423324i	$0.139137\pi$
$113$	9.00000i	0.846649i	−0.905978	+	0.423324i	$0.860863\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	2.00000	0.181818
$122$	0	0
$123$	−3.00000	−0.270501
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 12.0000i	− 1.06483i	−0.846484	−	0.532414i	$-0.821285\pi$
$127$	− 12.0000i	− 1.06483i	0.846484	−	0.532414i	$-0.178715\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	8.00000i	0.698963i	0.936943	+	0.349482i	$0.113642\pi$
$131$	8.00000i	0.698963i	−0.936943	+	0.349482i	$0.886358\pi$
$132$	0	0
$133$	28.0000	2.42791
$134$	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
$139$	− 13.0000i	− 1.10265i	−0.834292	−	0.551323i	$-0.814123\pi$
$139$	− 13.0000i	− 1.10265i	0.834292	−	0.551323i	$-0.185877\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	9.00000	0.742307
$148$	0	0
$149$	8.00000i	0.655386i	0.944784	+	0.327693i	$0.106271\pi$
$149$	8.00000i	0.655386i	−0.944784	+	0.327693i	$0.893729\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	0	0
$153$	− 2.00000i	− 0.161690i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	20.0000	1.59617	0.798087	−	0.602542i	$-0.205846\pi$
$157$	20.0000	1.59617	0.798087	+	0.602542i	$0.205846\pi$
$158$	0	0
$159$	−12.0000	−0.951662
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	9.00000	0.704934	0.352467	−	0.935824i	$-0.385343\pi$
$163$	9.00000	0.704934	0.352467	+	0.935824i	$0.385343\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.0000i	1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$	16.0000i	1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	14.0000i	1.07061i
$172$	0	0
$173$	−16.0000	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 8.00000i	− 0.601317i
$178$	0	0
$179$	9.00000i	0.672692i	0.941739	+	0.336346i	$0.109191\pi$
$179$	9.00000i	0.672692i	−0.941739	+	0.336346i	$0.890809\pi$
$180$	0	0
$181$	− 12.0000i	− 0.891953i	−0.895045	−	0.445976i	$-0.852856\pi$
$181$	− 12.0000i	− 0.891953i	0.895045	−	0.445976i	$-0.147144\pi$
$182$	0	0
$183$	− 4.00000i	− 0.295689i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.00000	0.219382
$188$	0	0
$189$	20.0000i	1.45479i
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	7.00000i	0.503871i	0.967744	+	0.251936i	$0.0810671\pi$
$193$	7.00000i	0.503871i	−0.967744	+	0.251936i	$0.918933\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	0	0
$201$	9.00000	0.634811
$202$	0	0
$203$	−32.0000	−2.24596
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.00000i	0.556038i
$208$	0	0
$209$	−21.0000	−1.45260
$210$	0	0
$211$	− 15.0000i	− 1.03264i	−0.856395	−	0.516321i	$-0.827301\pi$
$211$	− 15.0000i	− 1.03264i	0.856395	−	0.516321i	$-0.172699\pi$
$212$	0	0
$213$	16.0000	1.09630
$214$	0	0
$215$	0	0
$216$	0	0
$217$	16.0000i	1.08615i
$218$	0	0
$219$	− 11.0000i	− 0.743311i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	0	0
$229$	− 8.00000i	− 0.528655i	−0.964433	−	0.264327i	$-0.914850\pi$
$229$	− 8.00000i	− 0.528655i	0.964433	−	0.264327i	$-0.0851500\pi$
$230$	0	0
$231$	−12.0000	−0.789542
$232$	0	0
$233$	10.0000i	0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$	10.0000i	0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.00000	0.259828
$238$	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
$241$	−15.0000	−0.966235	−0.483117	−	0.875556i	$-0.660496\pi$
$241$	−15.0000	−0.966235	−0.483117	+	0.875556i	$0.660496\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−1.00000	−0.0633724
$250$	0	0
$251$	− 13.0000i	− 0.820553i	−0.911961	−	0.410276i	$-0.865432\pi$
$251$	− 13.0000i	− 0.820553i	0.911961	−	0.410276i	$-0.134568\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	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
$259$	16.0000i	0.994192i
$260$	0	0
$261$	− 16.0000i	− 0.990375i
$262$	0	0
$263$	− 4.00000i	− 0.246651i	−0.992366	−	0.123325i	$-0.960644\pi$
$263$	− 4.00000i	− 0.246651i	0.992366	−	0.123325i	$-0.0393559\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	13.0000	0.795587
$268$	0	0
$269$	16.0000i	0.975537i	0.872973	+	0.487769i	$0.162189\pi$
$269$	16.0000i	0.975537i	−0.872973	+	0.487769i	$0.837811\pi$
$270$	0	0
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	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
$283$	19.0000	1.12943	0.564716	−	0.825285i	$-0.308986\pi$
$283$	19.0000	1.12943	0.564716	+	0.825285i	$0.308986\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.0000i	0.708338i
$288$	0	0
$289$	16.0000	0.941176
$290$	0	0
$291$	− 14.0000i	− 0.820695i
$292$	0	0
$293$	20.0000	1.16841	0.584206	−	0.811605i	$-0.301406\pi$
$293$	20.0000	1.16841	0.584206	+	0.811605i	$0.301406\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 15.0000i	− 0.870388i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	32.0000i	1.84445i
$302$	0	0
$303$	12.0000i	0.689382i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−7.00000	−0.399511	−0.199756	−	0.979846i	$-0.564015\pi$
$307$	−7.00000	−0.399511	−0.199756	+	0.979846i	$0.564015\pi$
$308$	0	0
$309$	− 16.0000i	− 0.910208i
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.00000	0.449325	0.224662	−	0.974437i	$-0.427872\pi$
$317$	8.00000	0.449325	0.224662	+	0.974437i	$0.427872\pi$
$318$	0	0
$319$	24.0000	1.34374
$320$	0	0
$321$	13.0000	0.725589
$322$	0	0
$323$	7.00000	0.389490
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 20.0000i	− 1.10600i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	13.0000i	0.714545i	0.934000	+	0.357272i	$0.116293\pi$
$331$	13.0000i	0.714545i	−0.934000	+	0.357272i	$0.883707\pi$
$332$	0	0
$333$	−8.00000	−0.438397
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 9.00000i	− 0.490261i	−0.969490	−	0.245131i	$-0.921169\pi$
$337$	− 9.00000i	− 0.490261i	0.969490	−	0.245131i	$-0.0788309\pi$
$338$	0	0
$339$	− 9.00000i	− 0.488813i
$340$	0	0
$341$	− 12.0000i	− 0.649836i
$342$	0	0
$343$	− 8.00000i	− 0.431959i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.00000	−0.268414	−0.134207	−	0.990953i	$-0.542849\pi$
$347$	−5.00000	−0.268414	−0.134207	+	0.990953i	$0.542849\pi$
$348$	0	0
$349$	4.00000i	0.214115i	0.994253	+	0.107058i	$0.0341429\pi$
$349$	4.00000i	0.214115i	−0.994253	+	0.107058i	$0.965857\pi$
$350$	0	0
$351$	0	0
$352$	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
$357$	4.00000	0.211702
$358$	0	0
$359$	−28.0000	−1.47778	−0.738892	−	0.673824i	$-0.764651\pi$
$359$	−28.0000	−1.47778	−0.738892	+	0.673824i	$0.764651\pi$
$360$	0	0
$361$	−30.0000	−1.57895
$362$	0	0
$363$	−2.00000	−0.104973
$364$	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
$371$	48.0000i	2.49204i
$372$	0	0
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	29.0000i	1.48963i	0.667271	+	0.744815i	$0.267462\pi$
$379$	29.0000i	1.48963i	−0.667271	+	0.744815i	$0.732538\pi$
$380$	0	0
$381$	12.0000i	0.614779i
$382$	0	0
$383$	− 4.00000i	− 0.204390i	−0.994764	−	0.102195i	$-0.967413\pi$
$383$	− 4.00000i	− 0.204390i	0.994764	−	0.102195i	$-0.0325866\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−16.0000	−0.813326
$388$	0	0
$389$	− 12.0000i	− 0.608424i	−0.952604	−	0.304212i	$-0.901607\pi$
$389$	− 12.0000i	− 0.608424i	0.952604	−	0.304212i	$-0.0983931\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	0	0
$393$	− 8.00000i	− 0.403547i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−8.00000	−0.401508	−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000	−0.401508	−0.200754	+	0.979642i	$0.564339\pi$
$398$	0	0
$399$	−28.0000	−1.40175
$400$	0	0
$401$	−15.0000	−0.749064	−0.374532	−	0.927214i	$-0.622197\pi$
$401$	−15.0000	−0.749064	−0.374532	+	0.927214i	$0.622197\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 12.0000i	− 0.594818i
$408$	0	0
$409$	21.0000	1.03838	0.519192	−	0.854658i	$-0.326233\pi$
$409$	21.0000	1.03838	0.519192	+	0.854658i	$0.326233\pi$
$410$	0	0
$411$	− 3.00000i	− 0.147979i
$412$	0	0
$413$	−32.0000	−1.57462
$414$	0	0
$415$	0	0
$416$	0	0
$417$	13.0000i	0.636613i
$418$	0	0
$419$	− 31.0000i	− 1.51445i	−0.653155	−	0.757225i	$-0.726555\pi$
$419$	− 31.0000i	− 1.51445i	0.653155	−	0.757225i	$-0.273445\pi$
$420$	0	0
$421$	8.00000i	0.389896i	0.980814	+	0.194948i	$0.0624538\pi$
$421$	8.00000i	0.389896i	−0.980814	+	0.194948i	$0.937546\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−16.0000	−0.774294
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−28.0000	−1.34871	−0.674356	−	0.738406i	$-0.735579\pi$
$431$	−28.0000	−1.34871	−0.674356	+	0.738406i	$0.735579\pi$
$432$	0	0
$433$	31.0000i	1.48976i	0.667196	+	0.744882i	$0.267494\pi$
$433$	31.0000i	1.48976i	−0.667196	+	0.744882i	$0.732506\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−28.0000	−1.33942
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	0	0
$443$	19.0000	0.902717	0.451359	−	0.892343i	$-0.350940\pi$
$443$	19.0000	0.902717	0.451359	+	0.892343i	$0.350940\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 8.00000i	− 0.378387i
$448$	0	0
$449$	23.0000	1.08544	0.542719	−	0.839915i	$-0.317395\pi$
$449$	23.0000	1.08544	0.542719	+	0.839915i	$0.317395\pi$
$450$	0	0
$451$	− 9.00000i	− 0.423793i
$452$	0	0
$453$	−20.0000	−0.939682
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.00000i	0.140334i	0.997535	+	0.0701670i	$0.0223532\pi$
$457$	3.00000i	0.140334i	−0.997535	+	0.0701670i	$0.977647\pi$
$458$	0	0
$459$	5.00000i	0.233380i
$460$	0	0
$461$	− 8.00000i	− 0.372597i	−0.982493	−	0.186299i	$-0.940351\pi$
$461$	− 8.00000i	− 0.372597i	0.982493	−	0.186299i	$-0.0596492\pi$
$462$	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
$467$	−40.0000	−1.85098	−0.925490	−	0.378773i	$-0.876346\pi$
$467$	−40.0000	−1.85098	−0.925490	+	0.378773i	$0.876346\pi$
$468$	0	0
$469$	− 36.0000i	− 1.66233i
$470$	0	0
$471$	−20.0000	−0.921551
$472$	0	0
$473$	− 24.0000i	− 1.10352i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−24.0000	−1.09888
$478$	0	0
$479$	−28.0000	−1.27935	−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000	−1.27935	−0.639676	+	0.768644i	$0.720932\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−16.0000	−0.728025
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.0000i	0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$	12.0000i	0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$	0	0
$489$	−9.00000	−0.406994
$490$	0	0
$491$	− 24.0000i	− 1.08310i	−0.840667	−	0.541552i	$-0.817837\pi$
$491$	− 24.0000i	− 1.08310i	0.840667	−	0.541552i	$-0.182163\pi$
$492$	0	0
$493$	−8.00000	−0.360302
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 64.0000i	− 2.87079i
$498$	0	0
$499$	− 8.00000i	− 0.358129i	−0.983837	−	0.179065i	$-0.942693\pi$
$499$	− 8.00000i	− 0.358129i	0.983837	−	0.179065i	$-0.0573071\pi$
$500$	0	0
$501$	− 16.0000i	− 0.714827i
$502$	0	0
$503$	8.00000i	0.356702i	0.983967	+	0.178351i	$0.0570763\pi$
$503$	8.00000i	0.356702i	−0.983967	+	0.178351i	$0.942924\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	13.0000	0.577350
$508$	0	0
$509$	− 12.0000i	− 0.531891i	−0.963988	−	0.265945i	$-0.914316\pi$
$509$	− 12.0000i	− 0.531891i	0.963988	−	0.265945i	$-0.0856841\pi$
$510$	0	0
$511$	−44.0000	−1.94645
$512$	0	0
$513$	− 35.0000i	− 1.54529i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	16.0000	0.702322
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	0	0
$523$	13.0000	0.568450	0.284225	−	0.958758i	$-0.408264\pi$
$523$	13.0000	0.568450	0.284225	+	0.958758i	$0.408264\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.00000i	0.174243i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	− 16.0000i	− 0.694341i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 9.00000i	− 0.388379i
$538$	0	0
$539$	27.0000i	1.16297i
$540$	0	0
$541$	32.0000i	1.37579i	0.725811	+	0.687894i	$0.241464\pi$
$541$	32.0000i	1.37579i	−0.725811	+	0.687894i	$0.758536\pi$
$542$	0	0
$543$	12.0000i	0.514969i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.00000	0.0427569	0.0213785	−	0.999771i	$-0.493195\pi$
$547$	1.00000	0.0427569	0.0213785	+	0.999771i	$0.493195\pi$
$548$	0	0
$549$	− 8.00000i	− 0.341432i
$550$	0	0
$551$	56.0000	2.38568
$552$	0	0
$553$	− 16.0000i	− 0.680389i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.0000	0.677942	0.338971	−	0.940797i	$-0.389921\pi$
$557$	16.0000	0.677942	0.338971	+	0.940797i	$0.389921\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−3.00000	−0.126660
$562$	0	0
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.00000i	0.167984i
$568$	0	0
$569$	37.0000	1.55112	0.775560	−	0.631273i	$-0.217467\pi$
$569$	37.0000	1.55112	0.775560	+	0.631273i	$0.217467\pi$
$570$	0	0
$571$	8.00000i	0.334790i	0.985890	+	0.167395i	$0.0535355\pi$
$571$	8.00000i	0.334790i	−0.985890	+	0.167395i	$0.946465\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	0	0
$575$	0	0
$576$	0	0
$577$	31.0000i	1.29055i	0.763952	+	0.645273i	$0.223257\pi$
$577$	31.0000i	1.29055i	−0.763952	+	0.645273i	$0.776743\pi$
$578$	0	0
$579$	− 7.00000i	− 0.290910i
$580$	0	0
$581$	4.00000i	0.165948i
$582$	0	0
$583$	− 36.0000i	− 1.49097i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.00000	0.206372	0.103186	−	0.994662i	$-0.467096\pi$
$587$	5.00000	0.206372	0.103186	+	0.994662i	$0.467096\pi$
$588$	0	0
$589$	− 28.0000i	− 1.15372i
$590$	0	0
$591$	12.0000	0.493614
$592$	0	0
$593$	− 23.0000i	− 0.944497i	−0.881466	−	0.472248i	$-0.843443\pi$
$593$	− 23.0000i	− 0.944497i	0.881466	−	0.472248i	$-0.156557\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−24.0000	−0.982255
$598$	0	0
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	0	0
$603$	18.0000	0.733017
$604$	0	0
$605$	0	0
$606$	0	0
$607$	32.0000i	1.29884i	0.760430	+	0.649420i	$0.224988\pi$
$607$	32.0000i	1.29884i	−0.760430	+	0.649420i	$0.775012\pi$
$608$	0	0
$609$	32.0000	1.29671
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−4.00000	−0.161558	−0.0807792	−	0.996732i	$-0.525741\pi$
$613$	−4.00000	−0.161558	−0.0807792	+	0.996732i	$0.525741\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 42.0000i	− 1.69086i	−0.534089	−	0.845428i	$-0.679345\pi$
$617$	− 42.0000i	− 1.69086i	0.534089	−	0.845428i	$-0.320655\pi$
$618$	0	0
$619$	− 40.0000i	− 1.60774i	−0.594808	−	0.803868i	$-0.702772\pi$
$619$	− 40.0000i	− 1.60774i	0.594808	−	0.803868i	$-0.297228\pi$
$620$	0	0
$621$	− 20.0000i	− 0.802572i
$622$	0	0
$623$	− 52.0000i	− 2.08334i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	21.0000	0.838659
$628$	0	0
$629$	4.00000i	0.159490i
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	0	0
$633$	15.0000i	0.596196i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	32.0000	1.26590
$640$	0	0
$641$	14.0000	0.552967	0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000	0.552967	0.276483	+	0.961019i	$0.410831\pi$
$642$	0	0
$643$	−24.0000	−0.946468	−0.473234	−	0.880937i	$-0.656913\pi$
$643$	−24.0000	−0.946468	−0.473234	+	0.880937i	$0.656913\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 24.0000i	− 0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$	− 24.0000i	− 0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	0	0
$651$	− 16.0000i	− 0.627089i
$652$	0	0
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 22.0000i	− 0.858302i
$658$	0	0
$659$	− 1.00000i	− 0.0389545i	−0.999810	−	0.0194772i	$-0.993800\pi$
$659$	− 1.00000i	− 0.0389545i	0.999810	−	0.0194772i	$-0.00620019\pi$
$660$	0	0
$661$	40.0000i	1.55582i	0.628376	+	0.777910i	$0.283720\pi$
$661$	40.0000i	1.55582i	−0.628376	+	0.777910i	$0.716280\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	32.0000	1.23904
$668$	0	0
$669$	− 8.00000i	− 0.309298i
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	− 14.0000i	− 0.539660i	−0.962908	−	0.269830i	$-0.913032\pi$
$673$	− 14.0000i	− 0.539660i	0.962908	−	0.269830i	$-0.0869676\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−48.0000	−1.84479	−0.922395	−	0.386248i	$-0.873771\pi$
$677$	−48.0000	−1.84479	−0.922395	+	0.386248i	$0.873771\pi$
$678$	0	0
$679$	−56.0000	−2.14908
$680$	0	0
$681$	−24.0000	−0.919682
$682$	0	0
$683$	13.0000	0.497431	0.248716	−	0.968577i	$-0.419992\pi$
$683$	13.0000	0.497431	0.248716	+	0.968577i	$0.419992\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	8.00000i	0.305219i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 9.00000i	− 0.342376i	−0.985238	−	0.171188i	$-0.945239\pi$
$691$	− 9.00000i	− 0.342376i	0.985238	−	0.171188i	$-0.0547606\pi$
$692$	0	0
$693$	−24.0000	−0.911685
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.00000i	0.113633i
$698$	0	0
$699$	− 10.0000i	− 0.378235i
$700$	0	0
$701$	− 16.0000i	− 0.604312i	−0.953259	−	0.302156i	$-0.902294\pi$
$701$	− 16.0000i	− 0.604312i	0.953259	−	0.302156i	$-0.0977063\pi$
$702$	0	0
$703$	− 28.0000i	− 1.05604i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	48.0000	1.80523
$708$	0	0
$709$	− 40.0000i	− 1.50223i	−0.660171	−	0.751116i	$-0.729516\pi$
$709$	− 40.0000i	− 1.50223i	0.660171	−	0.751116i	$-0.270484\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	− 16.0000i	− 0.599205i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−24.0000	−0.896296
$718$	0	0
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	0	0
$721$	−64.0000	−2.38348
$722$	0	0
$723$	15.0000	0.557856
$724$	0	0
$725$	0	0
$726$	0	0
$727$	40.0000i	1.48352i	0.670667	+	0.741759i	$0.266008\pi$
$727$	40.0000i	1.48352i	−0.670667	+	0.741759i	$0.733992\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	8.00000i	0.295891i
$732$	0	0
$733$	32.0000	1.18195	0.590973	−	0.806691i	$-0.298744\pi$
$733$	32.0000	1.18195	0.590973	+	0.806691i	$0.298744\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	27.0000i	0.994558i
$738$	0	0
$739$	24.0000i	0.882854i	0.897297	+	0.441427i	$0.145528\pi$
$739$	24.0000i	0.882854i	−0.897297	+	0.441427i	$0.854472\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12.0000i	0.440237i	0.975473	+	0.220119i	$0.0706445\pi$
$743$	12.0000i	0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−2.00000	−0.0731762
$748$	0	0
$749$	− 52.0000i	− 1.90004i
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	0	0
$753$	13.0000i	0.473746i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	32.0000	1.16306	0.581530	−	0.813525i	$-0.302454\pi$
$757$	32.0000	1.16306	0.581530	+	0.813525i	$0.302454\pi$
$758$	0	0
$759$	12.0000	0.435572
$760$	0	0
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	0	0
$763$	−80.0000	−2.89619
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	39.0000	1.40638	0.703188	−	0.711004i	$-0.251759\pi$
$769$	39.0000	1.40638	0.703188	+	0.711004i	$0.251759\pi$
$770$	0	0
$771$	− 18.0000i	− 0.648254i
$772$	0	0
$773$	−8.00000	−0.287740	−0.143870	−	0.989597i	$-0.545955\pi$
$773$	−8.00000	−0.287740	−0.143870	+	0.989597i	$0.545955\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 16.0000i	− 0.573997i
$778$	0	0
$779$	− 21.0000i	− 0.752403i
$780$	0	0
$781$	48.0000i	1.71758i
$782$	0	0
$783$	40.0000i	1.42948i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−8.00000	−0.285169	−0.142585	−	0.989783i	$-0.545541\pi$
$787$	−8.00000	−0.285169	−0.142585	+	0.989783i	$0.545541\pi$
$788$	0	0
$789$	4.00000i	0.142404i
$790$	0	0
$791$	−36.0000	−1.28001
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	26.0000	0.918665
$802$	0	0
$803$	33.0000	1.16454
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 16.0000i	− 0.563227i
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	8.00000i	0.280918i	0.990086	+	0.140459i	$0.0448578\pi$
$811$	8.00000i	0.280918i	−0.990086	+	0.140459i	$0.955142\pi$
$812$	0	0
$813$	−4.00000	−0.140286
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 56.0000i	− 1.95919i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	36.0000i	1.25641i	0.778048	+	0.628204i	$0.216210\pi$
$821$	36.0000i	1.25641i	−0.778048	+	0.628204i	$0.783790\pi$
$822$	0	0
$823$	24.0000i	0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$	24.0000i	0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−51.0000	−1.77344	−0.886722	−	0.462303i	$-0.847023\pi$
$827$	−51.0000	−1.77344	−0.886722	+	0.462303i	$0.847023\pi$
$828$	0	0
$829$	− 32.0000i	− 1.11141i	−0.831381	−	0.555703i	$-0.812449\pi$
$829$	− 32.0000i	− 1.11141i	0.831381	−	0.555703i	$-0.187551\pi$
$830$	0	0
$831$	−8.00000	−0.277517
$832$	0	0
$833$	− 9.00000i	− 0.311832i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	20.0000	0.691301
$838$	0	0
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	−35.0000	−1.20690
$842$	0	0
$843$	−6.00000	−0.206651
$844$	0	0
$845$	0	0
$846$	0	0
$847$	8.00000i	0.274883i
$848$	0	0
$849$	−19.0000	−0.652078
$850$	0	0
$851$	− 16.0000i	− 0.548473i
$852$	0	0
$853$	−28.0000	−0.958702	−0.479351	−	0.877623i	$-0.659128\pi$
$853$	−28.0000	−0.958702	−0.479351	+	0.877623i	$0.659128\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	27.0000i	0.922302i	0.887322	+	0.461151i	$0.152563\pi$
$857$	27.0000i	0.922302i	−0.887322	+	0.461151i	$0.847437\pi$
$858$	0	0
$859$	− 5.00000i	− 0.170598i	−0.996355	−	0.0852989i	$-0.972815\pi$
$859$	− 5.00000i	− 0.170598i	0.996355	−	0.0852989i	$-0.0271845\pi$
$860$	0	0
$861$	− 12.0000i	− 0.408959i
$862$	0	0
$863$	56.0000i	1.90626i	0.302558	+	0.953131i	$0.402160\pi$
$863$	56.0000i	1.90626i	−0.302558	+	0.953131i	$0.597840\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−16.0000	−0.543388
$868$	0	0
$869$	12.0000i	0.407072i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 28.0000i	− 0.947656i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−8.00000	−0.270141	−0.135070	−	0.990836i	$-0.543126\pi$
$877$	−8.00000	−0.270141	−0.135070	+	0.990836i	$0.543126\pi$
$878$	0	0
$879$	−20.0000	−0.674583
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	0	0
$883$	−25.0000	−0.841317	−0.420658	−	0.907219i	$-0.638201\pi$
$883$	−25.0000	−0.841317	−0.420658	+	0.907219i	$0.638201\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.0000i	0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$	24.0000i	0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$	0	0
$889$	48.0000	1.60987
$890$	0	0
$891$	− 3.00000i	− 0.100504i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	32.0000i	1.06726i
$900$	0	0
$901$	12.0000i	0.399778i
$902$	0	0
$903$	− 32.0000i	− 1.06489i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−40.0000	−1.32818	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	−40.0000	−1.32818	−0.664089	+	0.747653i	$0.731180\pi$
$908$	0	0
$909$	24.0000i	0.796030i
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	− 3.00000i	− 0.0992855i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−32.0000	−1.05673
$918$	0	0
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	0	0
$921$	7.00000	0.230658
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 32.0000i	− 1.05102i
$928$	0	0
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	0	0
$931$	63.0000i	2.06474i
$932$	0	0
$933$	−12.0000	−0.392862
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5.00000i	0.163343i	0.996659	+	0.0816714i	$0.0260258\pi$
$937$	5.00000i	0.163343i	−0.996659	+	0.0816714i	$0.973974\pi$
$938$	0	0
$939$	− 6.00000i	− 0.195803i
$940$	0	0
$941$	− 48.0000i	− 1.56476i	−0.622804	−	0.782378i	$-0.714007\pi$
$941$	− 48.0000i	− 1.56476i	0.622804	−	0.782378i	$-0.285993\pi$
$942$	0	0
$943$	− 12.0000i	− 0.390774i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.00000	0.259965	0.129983	−	0.991516i	$-0.458508\pi$
$947$	8.00000	0.259965	0.129983	+	0.991516i	$0.458508\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−8.00000	−0.259418
$952$	0	0
$953$	− 27.0000i	− 0.874616i	−0.899312	−	0.437308i	$-0.855932\pi$
$953$	− 27.0000i	− 0.874616i	0.899312	−	0.437308i	$-0.144068\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−24.0000	−0.775810
$958$	0	0
$959$	−12.0000	−0.387500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	26.0000	0.837838
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 32.0000i	− 1.02905i	−0.857475	−	0.514525i	$-0.827968\pi$
$967$	− 32.0000i	− 1.02905i	0.857475	−	0.514525i	$-0.172032\pi$
$968$	0	0
$969$	−7.00000	−0.224872
$970$	0	0
$971$	45.0000i	1.44412i	0.691831	+	0.722059i	$0.256804\pi$
$971$	45.0000i	1.44412i	−0.691831	+	0.722059i	$0.743196\pi$
$972$	0	0
$973$	52.0000	1.66704
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 31.0000i	− 0.991778i	−0.868386	−	0.495889i	$-0.834842\pi$
$977$	− 31.0000i	− 0.991778i	0.868386	−	0.495889i	$-0.165158\pi$
$978$	0	0
$979$	39.0000i	1.24645i
$980$	0	0
$981$	− 40.0000i	− 1.27710i
$982$	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
$989$	− 32.0000i	− 1.01754i
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	0	0
$993$	− 13.0000i	− 0.412543i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	24.0000	0.760088	0.380044	−	0.924968i	$-0.375909\pi$
$997$	24.0000	0.760088	0.380044	+	0.924968i	$0.375909\pi$
$998$	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.f.b.449.2		2
4.3	odd	2		3200.2.f.e.449.1		2
5.2	odd	4		3200.2.d.a.1601.2	yes	2
5.3	odd	4		3200.2.d.h.1601.1	yes	2
5.4	even	2		3200.2.f.f.449.1		2
8.3	odd	2		3200.2.f.a.449.1		2
8.5	even	2		3200.2.f.f.449.2		2
20.3	even	4		3200.2.d.b.1601.2	yes	2
20.7	even	4		3200.2.d.g.1601.1	yes	2
20.19	odd	2		3200.2.f.a.449.2		2
40.3	even	4		3200.2.d.b.1601.1	yes	2
40.13	odd	4		3200.2.d.h.1601.2	yes	2
40.19	odd	2		3200.2.f.e.449.2		2
40.27	even	4		3200.2.d.g.1601.2	yes	2
40.29	even	2	inner	3200.2.f.b.449.1		2
40.37	odd	4		3200.2.d.a.1601.1	✓	2
80.3	even	4		6400.2.a.i.1.1		1
80.13	odd	4		6400.2.a.p.1.1		1
80.27	even	4		6400.2.a.b.1.1		1
80.37	odd	4		6400.2.a.w.1.1		1
80.43	even	4		6400.2.a.v.1.1		1
80.53	odd	4		6400.2.a.c.1.1		1
80.67	even	4		6400.2.a.q.1.1		1
80.77	odd	4		6400.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3200.2.d.a.1601.1	✓	2	40.37	odd	4
3200.2.d.a.1601.2	yes	2	5.2	odd	4
3200.2.d.b.1601.1	yes	2	40.3	even	4
3200.2.d.b.1601.2	yes	2	20.3	even	4
3200.2.d.g.1601.1	yes	2	20.7	even	4
3200.2.d.g.1601.2	yes	2	40.27	even	4
3200.2.d.h.1601.1	yes	2	5.3	odd	4
3200.2.d.h.1601.2	yes	2	40.13	odd	4
3200.2.f.a.449.1		2	8.3	odd	2
3200.2.f.a.449.2		2	20.19	odd	2
3200.2.f.b.449.1		2	40.29	even	2	inner
3200.2.f.b.449.2		2	1.1	even	1	trivial
3200.2.f.e.449.1		2	4.3	odd	2
3200.2.f.e.449.2		2	40.19	odd	2
3200.2.f.f.449.1		2	5.4	even	2
3200.2.f.f.449.2		2	8.5	even	2
6400.2.a.b.1.1		1	80.27	even	4
6400.2.a.c.1.1		1	80.53	odd	4
6400.2.a.h.1.1		1	80.77	odd	4
6400.2.a.i.1.1		1	80.3	even	4
6400.2.a.p.1.1		1	80.13	odd	4
6400.2.a.q.1.1		1	80.67	even	4
6400.2.a.v.1.1		1	80.43	even	4
6400.2.a.w.1.1		1	80.37	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.00000i q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.00000i q^{7} -2.00000 q^{9} -3.00000i q^{11} +1.00000i q^{17} -7.00000i q^{19} -4.00000i q^{21} -4.00000i q^{23} +5.00000 q^{27} +8.00000i q^{29} +4.00000 q^{31} +3.00000i q^{33} +4.00000 q^{37} +3.00000 q^{41} +8.00000 q^{43} -9.00000 q^{49} -1.00000i q^{51} +12.0000 q^{53} +7.00000i q^{57} +8.00000i q^{59} +4.00000i q^{61} -8.00000i q^{63} -9.00000 q^{67} +4.00000i q^{69} -16.0000 q^{71} +11.0000i q^{73} +12.0000 q^{77} -4.00000 q^{79} +1.00000 q^{81} +1.00000 q^{83} -8.00000i q^{87} -13.0000 q^{89} -4.00000 q^{93} +14.0000i q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 4 * q^9	\( 2 q - 2 q^{3} - 4 q^{9} + 10 q^{27} + 8 q^{31} + 8 q^{37} + 6 q^{41} + 16 q^{43} - 18 q^{49} + 24 q^{53} - 18 q^{67} - 32 q^{71} + 24 q^{77} - 8 q^{79} + 2 q^{81} + 2 q^{83} - 26 q^{89} - 8 q^{93}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^9 + 10 * q^27 + 8 * q^31 + 8 * q^37 + 6 * q^41 + 16 * q^43 - 18 * q^49 + 24 * q^53 - 18 * q^67 - 32 * q^71 + 24 * q^77 - 8 * q^79 + 2 * q^81 + 2 * q^83 - 26 * q^89 - 8 * q^93

Introduction

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