LMFDB - Embedded newform 3200.2.d.a.1601.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3200,2,Mod(1601,3200)]

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, 0]))

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

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

chi := DirichletCharacter("3200.1601");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3200 = 2^{7} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3200.d (of order $2$, degree $1$, 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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1601.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3200.1601
Dual form			3200.2.d.a.1601.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -4.00000 q^{7} +2.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -4.00000 q^{7} +2.00000 q^{9} +3.00000i q^{11} -1.00000 q^{17} -7.00000i q^{19} +4.00000i q^{21} -4.00000 q^{23} -5.00000i q^{27} +8.00000i q^{29} +4.00000 q^{31} +3.00000 q^{33} -4.00000i q^{37} +3.00000 q^{41} +8.00000i q^{43} +9.00000 q^{49} +1.00000i q^{51} +12.0000i q^{53} -7.00000 q^{57} +8.00000i q^{59} -4.00000i q^{61} -8.00000 q^{63} +9.00000i q^{67} +4.00000i q^{69} -16.0000 q^{71} +11.0000 q^{73} -12.0000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +1.00000i q^{83} +8.00000 q^{87} +13.0000 q^{89} -4.00000i q^{93} -14.0000 q^{97} +6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{7} + 4 q^{9}+O(q^{10}) $ 2 * q - 8 * q^7 + 4 * q^9	$ 2 q - 8 q^{7} + 4 q^{9} - 2 q^{17} - 8 q^{23} + 8 q^{31} + 6 q^{33} + 6 q^{41} + 18 q^{49} - 14 q^{57} - 16 q^{63} - 32 q^{71} + 22 q^{73} + 8 q^{79} + 2 q^{81} + 16 q^{87} + 26 q^{89} - 28 q^{97}+O(q^{100}) $ 2 * q - 8 * q^7 + 4 * q^9 - 2 * q^17 - 8 * q^23 + 8 * q^31 + 6 * q^33 + 6 * q^41 + 18 * q^49 - 14 * q^57 - 16 * q^63 - 32 * q^71 + 22 * q^73 + 8 * q^79 + 2 * q^81 + 16 * q^87 + 26 * q^89 - 28 * q^97

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
$3$	− 1.00000i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	3.00000i	0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$	3.00000i	0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	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
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.00000i	− 0.962250i
$28$	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
$33$	3.00000	0.522233
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	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
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	1.00000i	0.140028i
$52$	0	0
$53$	12.0000i	1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$	12.0000i	1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−7.00000	−0.927173
$58$	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	$-0.917571\pi$
$61$	− 4.00000i	− 0.512148i	0.966657	−	0.256074i	$-0.0824290\pi$
$62$	0	0
$63$	−8.00000	−1.00791
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.00000i	1.09952i	0.835321	+	0.549762i	$0.185282\pi$
$67$	9.00000i	1.09952i	−0.835321	+	0.549762i	$0.814718\pi$
$68$	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
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 12.0000i	− 1.36753i
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.00000i	0.109764i	0.998493	+	0.0548821i	$0.0174783\pi$
$83$	1.00000i	0.109764i	−0.998493	+	0.0548821i	$0.982522\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.00000	0.857690
$88$	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
$93$	− 4.00000i	− 0.414781i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	6.00000i	0.603023i
$100$	0	0
$101$	12.0000i	1.19404i	0.802225	+	0.597022i	$0.203650\pi$
$101$	12.0000i	1.19404i	−0.802225	+	0.597022i	$0.796350\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.0000i	1.25676i	0.777908	+	0.628379i	$0.216281\pi$
$107$	13.0000i	1.25676i	−0.777908	+	0.628379i	$0.783719\pi$
$108$	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
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	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
$123$	− 3.00000i	− 0.270501i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	− 8.00000i	− 0.698963i	−0.936943	−	0.349482i	$-0.886358\pi$
$131$	− 8.00000i	− 0.698963i	0.936943	−	0.349482i	$-0.113642\pi$
$132$	0	0
$133$	28.0000i	2.42791i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	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
$147$	− 9.00000i	− 0.742307i
$148$	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
$153$	−2.00000	−0.161690
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 20.0000i	− 1.59617i	−0.602542	−	0.798087i	$-0.705846\pi$
$157$	− 20.0000i	− 1.59617i	0.602542	−	0.798087i	$-0.294154\pi$
$158$	0	0
$159$	12.0000	0.951662
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	9.00000i	0.704934i	0.935824	+	0.352467i	$0.114657\pi$
$163$	9.00000i	0.704934i	−0.935824	+	0.352467i	$0.885343\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	− 14.0000i	− 1.07061i
$172$	0	0
$173$	− 16.0000i	− 1.21646i	−0.793762	−	0.608229i	$-0.791880\pi$
$173$	− 16.0000i	− 1.21646i	0.793762	−	0.608229i	$-0.208120\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.00000	0.601317
$178$	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	$0.147144\pi$
$181$	12.0000i	0.891953i	−0.895045	+	0.445976i	$0.852856\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 3.00000i	− 0.219382i
$188$	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
$193$	7.00000	0.503871	0.251936	−	0.967744i	$-0.418933\pi$
$193$	7.00000	0.503871	0.251936	+	0.967744i	$0.418933\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	9.00000	0.634811
$202$	0	0
$203$	− 32.0000i	− 2.24596i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−8.00000	−0.556038
$208$	0	0
$209$	21.0000	1.45260
$210$	0	0
$211$	15.0000i	1.03264i	0.856395	+	0.516321i	$0.172699\pi$
$211$	15.0000i	1.03264i	−0.856395	+	0.516321i	$0.827301\pi$
$212$	0	0
$213$	16.0000i	1.09630i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−16.0000	−1.08615
$218$	0	0
$219$	− 11.0000i	− 0.743311i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 24.0000i	− 1.59294i	−0.604681	−	0.796468i	$-0.706699\pi$
$227$	− 24.0000i	− 1.59294i	0.604681	−	0.796468i	$-0.293301\pi$
$228$	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
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 4.00000i	− 0.259828i
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	−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
$243$	− 16.0000i	− 1.02640i
$244$	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	$0.134568\pi$
$251$	13.0000i	0.820553i	−0.911961	+	0.410276i	$0.865432\pi$
$252$	0	0
$253$	− 12.0000i	− 0.754434i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	16.0000i	0.994192i
$260$	0	0
$261$	16.0000i	0.990375i
$262$	0	0
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 13.0000i	− 0.795587i
$268$	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
$277$	− 8.00000i	− 0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$	− 8.00000i	− 0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$	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
$283$	19.0000i	1.12943i	0.825285	+	0.564716i	$0.191014\pi$
$283$	19.0000i	1.12943i	−0.825285	+	0.564716i	$0.808986\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	14.0000i	0.820695i
$292$	0	0
$293$	20.0000i	1.16841i	0.811605	+	0.584206i	$0.198594\pi$
$293$	20.0000i	1.16841i	−0.811605	+	0.584206i	$0.801406\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	15.0000	0.870388
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 32.0000i	− 1.84445i
$302$	0	0
$303$	12.0000	0.689382
$304$	0	0
$305$	0	0
$306$	0	0
$307$	7.00000i	0.399511i	0.979846	+	0.199756i	$0.0640148\pi$
$307$	7.00000i	0.399511i	−0.979846	+	0.199756i	$0.935985\pi$
$308$	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
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 8.00000i	− 0.449325i	−0.974437	−	0.224662i	$-0.927872\pi$
$317$	− 8.00000i	− 0.449325i	0.974437	−	0.224662i	$-0.0721279\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	0	0
$321$	13.0000	0.725589
$322$	0	0
$323$	7.00000i	0.389490i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	20.0000	1.10600
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 13.0000i	− 0.714545i	−0.934000	−	0.357272i	$-0.883707\pi$
$331$	− 13.0000i	− 0.714545i	0.934000	−	0.357272i	$-0.116293\pi$
$332$	0	0
$333$	− 8.00000i	− 0.438397i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9.00000	0.490261	0.245131	−	0.969490i	$-0.421169\pi$
$337$	9.00000	0.490261	0.245131	+	0.969490i	$0.421169\pi$
$338$	0	0
$339$	− 9.00000i	− 0.488813i
$340$	0	0
$341$	12.0000i	0.649836i
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.00000i	0.268414i	0.990953	+	0.134207i	$0.0428487\pi$
$347$	5.00000i	0.268414i	−0.990953	+	0.134207i	$0.957151\pi$
$348$	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
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 4.00000i	− 0.211702i
$358$	0	0
$359$	28.0000	1.47778	0.738892	−	0.673824i	$-0.235349\pi$
$359$	28.0000	1.47778	0.738892	+	0.673824i	$0.235349\pi$
$360$	0	0
$361$	−30.0000	−1.57895
$362$	0	0
$363$	− 2.00000i	− 0.104973i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	− 48.0000i	− 2.49204i
$372$	0	0
$373$	20.0000i	1.03556i	0.855514	+	0.517780i	$0.173242\pi$
$373$	20.0000i	1.03556i	−0.855514	+	0.517780i	$0.826758\pi$
$374$	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
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.0000i	0.813326i
$388$	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
$393$	−8.00000	−0.403547
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8.00000i	0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$	8.00000i	0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$	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
$407$	12.0000	0.594818
$408$	0	0
$409$	−21.0000	−1.03838	−0.519192	−	0.854658i	$-0.673767\pi$
$409$	−21.0000	−1.03838	−0.519192	+	0.854658i	$0.673767\pi$
$410$	0	0
$411$	3.00000i	0.147979i
$412$	0	0
$413$	− 32.0000i	− 1.57462i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−13.0000	−0.636613
$418$	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	$-0.937546\pi$
$421$	− 8.00000i	− 0.389896i	0.980814	−	0.194948i	$-0.0624538\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	16.0000i	0.774294i
$428$	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
$433$	31.0000	1.48976	0.744882	−	0.667196i	$-0.232506\pi$
$433$	31.0000	1.48976	0.744882	+	0.667196i	$0.232506\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	28.0000i	1.33942i
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	0	0
$443$	19.0000i	0.902717i	0.892343	+	0.451359i	$0.149060\pi$
$443$	19.0000i	0.902717i	−0.892343	+	0.451359i	$0.850940\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	8.00000	0.378387
$448$	0	0
$449$	−23.0000	−1.08544	−0.542719	−	0.839915i	$-0.682605\pi$
$449$	−23.0000	−1.08544	−0.542719	+	0.839915i	$0.682605\pi$
$450$	0	0
$451$	9.00000i	0.423793i
$452$	0	0
$453$	− 20.0000i	− 0.939682i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.00000	−0.140334	−0.0701670	−	0.997535i	$-0.522353\pi$
$457$	−3.00000	−0.140334	−0.0701670	+	0.997535i	$0.522353\pi$
$458$	0	0
$459$	5.00000i	0.233380i
$460$	0	0
$461$	8.00000i	0.372597i	0.982493	+	0.186299i	$0.0596492\pi$
$461$	8.00000i	0.372597i	−0.982493	+	0.186299i	$0.940351\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	40.0000i	1.85098i	0.378773	+	0.925490i	$0.376346\pi$
$467$	40.0000i	1.85098i	−0.378773	+	0.925490i	$0.623654\pi$
$468$	0	0
$469$	− 36.0000i	− 1.66233i
$470$	0	0
$471$	−20.0000	−0.921551
$472$	0	0
$473$	−24.0000	−1.10352
$474$	0	0
$475$	0	0
$476$	0	0
$477$	24.0000i	1.09888i
$478$	0	0
$479$	28.0000	1.27935	0.639676	−	0.768644i	$-0.279068\pi$
$479$	28.0000	1.27935	0.639676	+	0.768644i	$0.279068\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 16.0000i	− 0.728025i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	0	0
$489$	9.00000	0.406994
$490$	0	0
$491$	24.0000i	1.08310i	0.840667	+	0.541552i	$0.182163\pi$
$491$	24.0000i	1.08310i	−0.840667	+	0.541552i	$0.817837\pi$
$492$	0	0
$493$	− 8.00000i	− 0.360302i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	64.0000	2.87079
$498$	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
$503$	8.00000	0.356702	0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000	0.356702	0.178351	+	0.983967i	$0.442924\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 13.0000i	− 0.577350i
$508$	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
$513$	−35.0000	−1.54529
$514$	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
$523$	13.0000i	0.568450i	0.958758	+	0.284225i	$0.0917363\pi$
$523$	13.0000i	0.568450i	−0.958758	+	0.284225i	$0.908264\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.00000	−0.174243
$528$	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
$537$	9.00000	0.388379
$538$	0	0
$539$	27.0000i	1.16297i
$540$	0	0
$541$	− 32.0000i	− 1.37579i	−0.725811	−	0.687894i	$-0.758536\pi$
$541$	− 32.0000i	− 1.37579i	0.725811	−	0.687894i	$-0.241464\pi$
$542$	0	0
$543$	12.0000	0.514969
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 1.00000i	− 0.0427569i	−0.999771	−	0.0213785i	$-0.993195\pi$
$547$	− 1.00000i	− 0.0427569i	0.999771	−	0.0213785i	$-0.00680549\pi$
$548$	0	0
$549$	− 8.00000i	− 0.341432i
$550$	0	0
$551$	56.0000	2.38568
$552$	0	0
$553$	−16.0000	−0.680389
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 16.0000i	− 0.677942i	−0.940797	−	0.338971i	$-0.889921\pi$
$557$	− 16.0000i	− 0.677942i	0.940797	−	0.338971i	$-0.110079\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−3.00000	−0.126660
$562$	0	0
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−4.00000	−0.167984
$568$	0	0
$569$	−37.0000	−1.55112	−0.775560	−	0.631273i	$-0.782533\pi$
$569$	−37.0000	−1.55112	−0.775560	+	0.631273i	$0.782533\pi$
$570$	0	0
$571$	− 8.00000i	− 0.334790i	−0.985890	−	0.167395i	$-0.946465\pi$
$571$	− 8.00000i	− 0.334790i	0.985890	−	0.167395i	$-0.0535355\pi$
$572$	0	0
$573$	− 12.0000i	− 0.501307i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−31.0000	−1.29055	−0.645273	−	0.763952i	$-0.723257\pi$
$577$	−31.0000	−1.29055	−0.645273	+	0.763952i	$0.723257\pi$
$578$	0	0
$579$	− 7.00000i	− 0.290910i
$580$	0	0
$581$	− 4.00000i	− 0.165948i
$582$	0	0
$583$	−36.0000	−1.49097
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 5.00000i	− 0.206372i	−0.994662	−	0.103186i	$-0.967096\pi$
$587$	− 5.00000i	− 0.206372i	0.994662	−	0.103186i	$-0.0329037\pi$
$588$	0	0
$589$	− 28.0000i	− 1.15372i
$590$	0	0
$591$	12.0000	0.493614
$592$	0	0
$593$	−23.0000	−0.944497	−0.472248	−	0.881466i	$-0.656557\pi$
$593$	−23.0000	−0.944497	−0.472248	+	0.881466i	$0.656557\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	24.0000i	0.982255i
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	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
$603$	18.0000i	0.733017i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	0	0
$609$	−32.0000	−1.29671
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 4.00000i	− 0.161558i	−0.996732	−	0.0807792i	$-0.974259\pi$
$613$	− 4.00000i	− 0.161558i	0.996732	−	0.0807792i	$-0.0257409\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	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
$623$	−52.0000	−2.08334
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 21.0000i	− 0.838659i
$628$	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
$633$	15.0000	0.596196
$634$	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
$643$	− 24.0000i	− 0.946468i	−0.880937	−	0.473234i	$-0.843087\pi$
$643$	− 24.0000i	− 0.946468i	0.880937	−	0.473234i	$-0.156913\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	16.0000i	0.627089i
$652$	0	0
$653$	− 36.0000i	− 1.40879i	−0.709809	−	0.704394i	$-0.751219\pi$
$653$	− 36.0000i	− 1.40879i	0.709809	−	0.704394i	$-0.248781\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	22.0000	0.858302
$658$	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	$-0.716280\pi$
$661$	− 40.0000i	− 1.55582i	0.628376	−	0.777910i	$-0.283720\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 32.0000i	− 1.23904i
$668$	0	0
$669$	− 8.00000i	− 0.309298i
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	48.0000i	1.84479i	0.386248	+	0.922395i	$0.373771\pi$
$677$	48.0000i	1.84479i	−0.386248	+	0.922395i	$0.626229\pi$
$678$	0	0
$679$	56.0000	2.14908
$680$	0	0
$681$	−24.0000	−0.919682
$682$	0	0
$683$	13.0000i	0.497431i	0.968577	+	0.248716i	$0.0800084\pi$
$683$	13.0000i	0.497431i	−0.968577	+	0.248716i	$0.919992\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−8.00000	−0.305219
$688$	0	0
$689$	0	0
$690$	0	0
$691$	9.00000i	0.342376i	0.985238	+	0.171188i	$0.0547606\pi$
$691$	9.00000i	0.342376i	−0.985238	+	0.171188i	$0.945239\pi$
$692$	0	0
$693$	− 24.0000i	− 0.911685i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−3.00000	−0.113633
$698$	0	0
$699$	− 10.0000i	− 0.378235i
$700$	0	0
$701$	16.0000i	0.604312i	0.953259	+	0.302156i	$0.0977063\pi$
$701$	16.0000i	0.604312i	−0.953259	+	0.302156i	$0.902294\pi$
$702$	0	0
$703$	−28.0000	−1.05604
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 48.0000i	− 1.80523i
$708$	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
$713$	−16.0000	−0.599205
$714$	0	0
$715$	0	0
$716$	0	0
$717$	24.0000i	0.896296i
$718$	0	0
$719$	−20.0000	−0.745874	−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000	−0.745874	−0.372937	+	0.927857i	$0.621649\pi$
$720$	0	0
$721$	−64.0000	−2.38348
$722$	0	0
$723$	15.0000i	0.557856i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	− 8.00000i	− 0.295891i
$732$	0	0
$733$	32.0000i	1.18195i	0.806691	+	0.590973i	$0.201256\pi$
$733$	32.0000i	1.18195i	−0.806691	+	0.590973i	$0.798744\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−27.0000	−0.994558
$738$	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
$743$	12.0000	0.440237	0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000	0.440237	0.220119	+	0.975473i	$0.429356\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.00000i	0.0731762i
$748$	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
$753$	13.0000	0.473746
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 32.0000i	− 1.16306i	−0.813525	−	0.581530i	$-0.802454\pi$
$757$	− 32.0000i	− 1.16306i	0.813525	−	0.581530i	$-0.197546\pi$
$758$	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
$763$	− 80.0000i	− 2.89619i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−39.0000	−1.40638	−0.703188	−	0.711004i	$-0.748241\pi$
$769$	−39.0000	−1.40638	−0.703188	+	0.711004i	$0.748241\pi$
$770$	0	0
$771$	18.0000i	0.648254i
$772$	0	0
$773$	− 8.00000i	− 0.287740i	−0.989597	−	0.143870i	$-0.954045\pi$
$773$	− 8.00000i	− 0.287740i	0.989597	−	0.143870i	$-0.0459547\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	16.0000	0.573997
$778$	0	0
$779$	− 21.0000i	− 0.752403i
$780$	0	0
$781$	− 48.0000i	− 1.71758i
$782$	0	0
$783$	40.0000	1.42948
$784$	0	0
$785$	0	0
$786$	0	0
$787$	8.00000i	0.285169i	0.989783	+	0.142585i	$0.0455413\pi$
$787$	8.00000i	0.285169i	−0.989783	+	0.142585i	$0.954459\pi$
$788$	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
$797$	12.0000i	0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$	12.0000i	0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	26.0000	0.918665
$802$	0	0
$803$	33.0000i	1.16454i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	16.0000	0.563227
$808$	0	0
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	− 8.00000i	− 0.280918i	−0.990086	−	0.140459i	$-0.955142\pi$
$811$	− 8.00000i	− 0.280918i	0.990086	−	0.140459i	$-0.0448578\pi$
$812$	0	0
$813$	− 4.00000i	− 0.140286i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	56.0000	1.95919
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 36.0000i	− 1.25641i	−0.778048	−	0.628204i	$-0.783790\pi$
$821$	− 36.0000i	− 1.25641i	0.778048	−	0.628204i	$-0.216210\pi$
$822$	0	0
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	51.0000i	1.77344i	0.462303	+	0.886722i	$0.347023\pi$
$827$	51.0000i	1.77344i	−0.462303	+	0.886722i	$0.652977\pi$
$828$	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
$833$	−9.00000	−0.311832
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 20.0000i	− 0.691301i
$838$	0	0
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	−35.0000	−1.20690
$842$	0	0
$843$	− 6.00000i	− 0.206651i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−8.00000	−0.274883
$848$	0	0
$849$	19.0000	0.652078
$850$	0	0
$851$	16.0000i	0.548473i
$852$	0	0
$853$	− 28.0000i	− 0.958702i	−0.877623	−	0.479351i	$-0.840872\pi$
$853$	− 28.0000i	− 0.958702i	0.877623	−	0.479351i	$-0.159128\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27.0000	−0.922302	−0.461151	−	0.887322i	$-0.652563\pi$
$857$	−27.0000	−0.922302	−0.461151	+	0.887322i	$0.652563\pi$
$858$	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
$863$	56.0000	1.90626	0.953131	−	0.302558i	$-0.0978405\pi$
$863$	56.0000	1.90626	0.953131	+	0.302558i	$0.0978405\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	16.0000i	0.543388i
$868$	0	0
$869$	12.0000i	0.407072i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−28.0000	−0.947656
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.00000i	0.270141i	0.990836	+	0.135070i	$0.0431261\pi$
$877$	8.00000i	0.270141i	−0.990836	+	0.135070i	$0.956874\pi$
$878$	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
$883$	− 25.0000i	− 0.841317i	−0.907219	−	0.420658i	$-0.861799\pi$
$883$	− 25.0000i	− 0.841317i	0.907219	−	0.420658i	$-0.138201\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	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
$903$	−32.0000	−1.06489
$904$	0	0
$905$	0	0
$906$	0	0
$907$	40.0000i	1.32818i	0.747653	+	0.664089i	$0.231180\pi$
$907$	40.0000i	1.32818i	−0.747653	+	0.664089i	$0.768820\pi$
$908$	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
$913$	−3.00000	−0.0992855
$914$	0	0
$915$	0	0
$916$	0	0
$917$	32.0000i	1.05673i
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	0	0
$921$	7.00000	0.230658
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	32.0000	1.05102
$928$	0	0
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	− 63.0000i	− 2.06474i
$932$	0	0
$933$	− 12.0000i	− 0.392862i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−5.00000	−0.163343	−0.0816714	−	0.996659i	$-0.526026\pi$
$937$	−5.00000	−0.163343	−0.0816714	+	0.996659i	$0.526026\pi$
$938$	0	0
$939$	− 6.00000i	− 0.195803i
$940$	0	0
$941$	48.0000i	1.56476i	0.622804	+	0.782378i	$0.285993\pi$
$941$	48.0000i	1.56476i	−0.622804	+	0.782378i	$0.714007\pi$
$942$	0	0
$943$	−12.0000	−0.390774
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 8.00000i	− 0.259965i	−0.991516	−	0.129983i	$-0.958508\pi$
$947$	− 8.00000i	− 0.259965i	0.991516	−	0.129983i	$-0.0414921\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−8.00000	−0.259418
$952$	0	0
$953$	−27.0000	−0.874616	−0.437308	−	0.899312i	$-0.644068\pi$
$953$	−27.0000	−0.874616	−0.437308	+	0.899312i	$0.644068\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	24.0000i	0.775810i
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	26.0000i	0.837838i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	0	0
$969$	7.00000	0.224872
$970$	0	0
$971$	− 45.0000i	− 1.44412i	−0.691831	−	0.722059i	$-0.743196\pi$
$971$	− 45.0000i	− 1.44412i	0.691831	−	0.722059i	$-0.256804\pi$
$972$	0	0
$973$	52.0000i	1.66704i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.0000	0.991778	0.495889	−	0.868386i	$-0.334842\pi$
$977$	31.0000	0.991778	0.495889	+	0.868386i	$0.334842\pi$
$978$	0	0
$979$	39.0000i	1.24645i
$980$	0	0
$981$	40.0000i	1.27710i
$982$	0	0
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	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
$993$	−13.0000	−0.412543
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 24.0000i	− 0.760088i	−0.924968	−	0.380044i	$-0.875909\pi$
$997$	− 24.0000i	− 0.760088i	0.924968	−	0.380044i	$-0.124091\pi$
$998$	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.d.a.1601.1	✓	2
4.3	odd	2		3200.2.d.g.1601.2	yes	2
5.2	odd	4		3200.2.f.b.449.1		2
5.3	odd	4		3200.2.f.f.449.2		2
5.4	even	2		3200.2.d.h.1601.2	yes	2
8.3	odd	2		3200.2.d.g.1601.1	yes	2
8.5	even	2	inner	3200.2.d.a.1601.2	yes	2
16.3	odd	4		6400.2.a.b.1.1		1
16.5	even	4		6400.2.a.h.1.1		1
16.11	odd	4		6400.2.a.q.1.1		1
16.13	even	4		6400.2.a.w.1.1		1
20.3	even	4		3200.2.f.a.449.1		2
20.7	even	4		3200.2.f.e.449.2		2
20.19	odd	2		3200.2.d.b.1601.1	yes	2
40.3	even	4		3200.2.f.e.449.1		2
40.13	odd	4		3200.2.f.b.449.2		2
40.19	odd	2		3200.2.d.b.1601.2	yes	2
40.27	even	4		3200.2.f.a.449.2		2
40.29	even	2		3200.2.d.h.1601.1	yes	2
40.37	odd	4		3200.2.f.f.449.1		2
80.19	odd	4		6400.2.a.v.1.1		1
80.29	even	4		6400.2.a.c.1.1		1
80.59	odd	4		6400.2.a.i.1.1		1
80.69	even	4		6400.2.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3200.2.d.a.1601.1	✓	2	1.1	even	1	trivial
3200.2.d.a.1601.2	yes	2	8.5	even	2	inner
3200.2.d.b.1601.1	yes	2	20.19	odd	2
3200.2.d.b.1601.2	yes	2	40.19	odd	2
3200.2.d.g.1601.1	yes	2	8.3	odd	2
3200.2.d.g.1601.2	yes	2	4.3	odd	2
3200.2.d.h.1601.1	yes	2	40.29	even	2
3200.2.d.h.1601.2	yes	2	5.4	even	2
3200.2.f.a.449.1		2	20.3	even	4
3200.2.f.a.449.2		2	40.27	even	4
3200.2.f.b.449.1		2	5.2	odd	4
3200.2.f.b.449.2		2	40.13	odd	4
3200.2.f.e.449.1		2	40.3	even	4
3200.2.f.e.449.2		2	20.7	even	4
3200.2.f.f.449.1		2	40.37	odd	4
3200.2.f.f.449.2		2	5.3	odd	4
6400.2.a.b.1.1		1	16.3	odd	4
6400.2.a.c.1.1		1	80.29	even	4
6400.2.a.h.1.1		1	16.5	even	4
6400.2.a.i.1.1		1	80.59	odd	4
6400.2.a.p.1.1		1	80.69	even	4
6400.2.a.q.1.1		1	16.11	odd	4
6400.2.a.v.1.1		1	80.19	odd	4
6400.2.a.w.1.1		1	16.13	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -4.00000 q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -4.00000 q^{7} +2.00000 q^{9} +3.00000i q^{11} -1.00000 q^{17} -7.00000i q^{19} +4.00000i q^{21} -4.00000 q^{23} -5.00000i q^{27} +8.00000i q^{29} +4.00000 q^{31} +3.00000 q^{33} -4.00000i q^{37} +3.00000 q^{41} +8.00000i q^{43} +9.00000 q^{49} +1.00000i q^{51} +12.0000i q^{53} -7.00000 q^{57} +8.00000i q^{59} -4.00000i q^{61} -8.00000 q^{63} +9.00000i q^{67} +4.00000i q^{69} -16.0000 q^{71} +11.0000 q^{73} -12.0000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +1.00000i q^{83} +8.00000 q^{87} +13.0000 q^{89} -4.00000i q^{93} -14.0000 q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{7} + 4 q^{9}+O(q^{10}) \) 2 * q - 8 * q^7 + 4 * q^9	\( 2 q - 8 q^{7} + 4 q^{9} - 2 q^{17} - 8 q^{23} + 8 q^{31} + 6 q^{33} + 6 q^{41} + 18 q^{49} - 14 q^{57} - 16 q^{63} - 32 q^{71} + 22 q^{73} + 8 q^{79} + 2 q^{81} + 16 q^{87} + 26 q^{89} - 28 q^{97}+O(q^{100}) \) 2 * q - 8 * q^7 + 4 * q^9 - 2 * q^17 - 8 * q^23 + 8 * q^31 + 6 * q^33 + 6 * q^41 + 18 * q^49 - 14 * q^57 - 16 * q^63 - 32 * q^71 + 22 * q^73 + 8 * q^79 + 2 * q^81 + 16 * q^87 + 26 * q^89 - 28 * q^97

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