LMFDB - Embedded newform 4056.2.c.j.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(337,4056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4056.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.c (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:	$32.3873230598$
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:	$ 2 $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4056.337
Dual form			4056.2.c.j.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -4.00000i q^{5} +4.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -4.00000i q^{5} +4.00000i q^{7} +1.00000 q^{9} +2.00000i q^{11} -4.00000i q^{15} +6.00000 q^{17} +4.00000i q^{19} +4.00000i q^{21} -4.00000 q^{23} -11.0000 q^{25} +1.00000 q^{27} -6.00000 q^{29} +8.00000i q^{31} +2.00000i q^{33} +16.0000 q^{35} +10.0000i q^{37} -4.00000i q^{41} +4.00000 q^{43} -4.00000i q^{45} +6.00000i q^{47} -9.00000 q^{49} +6.00000 q^{51} +6.00000 q^{53} +8.00000 q^{55} +4.00000i q^{57} +6.00000i q^{59} -6.00000 q^{61} +4.00000i q^{63} -4.00000 q^{69} +10.0000i q^{71} +2.00000i q^{73} -11.0000 q^{75} -8.00000 q^{77} +1.00000 q^{81} -10.0000i q^{83} -24.0000i q^{85} -6.00000 q^{87} -8.00000i q^{89} +8.00000i q^{93} +16.0000 q^{95} -10.0000i q^{97} +2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{9} + 12 q^{17} - 8 q^{23} - 22 q^{25} + 2 q^{27} - 12 q^{29} + 32 q^{35} + 8 q^{43} - 18 q^{49} + 12 q^{51} + 12 q^{53} + 16 q^{55} - 12 q^{61} - 8 q^{69} - 22 q^{75} - 16 q^{77} + 2 q^{81} - 12 q^{87} + 32 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^9 + 12 * q^17 - 8 * q^23 - 22 * q^25 + 2 * q^27 - 12 * q^29 + 32 * q^35 + 8 * q^43 - 18 * q^49 + 12 * q^51 + 12 * q^53 + 16 * q^55 - 12 * q^61 - 8 * q^69 - 22 * q^75 - 16 * q^77 + 2 * q^81 - 12 * q^87 + 32 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times$.

$n$	$1015$	$2029$	$2705$	$3889$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.00000	0.577350
$4$	0	0
$5$	− 4.00000i	− 1.78885i	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	− 4.00000i	− 1.78885i	0.447214	−	0.894427i	$-0.352416\pi$
$6$	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
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000i	0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$	2.00000i	0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 4.00000i	− 1.03280i
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	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
$25$	−11.0000	−2.20000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	8.00000i	1.43684i	0.695608	+	0.718421i	$0.255135\pi$
$31$	8.00000i	1.43684i	−0.695608	+	0.718421i	$0.744865\pi$
$32$	0	0
$33$	2.00000i	0.348155i
$34$	0	0
$35$	16.0000	2.70449
$36$	0	0
$37$	10.0000i	1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$	10.0000i	1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 4.00000i	− 0.624695i	−0.949968	−	0.312348i	$-0.898885\pi$
$41$	− 4.00000i	− 0.624695i	0.949968	−	0.312348i	$-0.101115\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	− 4.00000i	− 0.596285i
$46$	0	0
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	6.00000	0.840168
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	0	0
$57$	4.00000i	0.529813i
$58$	0	0
$59$	6.00000i	0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$	6.00000i	0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	4.00000i	0.503953i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	10.0000i	1.18678i	0.804914	+	0.593391i	$0.202211\pi$
$71$	10.0000i	1.18678i	−0.804914	+	0.593391i	$0.797789\pi$
$72$	0	0
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	0	0
$75$	−11.0000	−1.27017
$76$	0	0
$77$	−8.00000	−0.911685
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 10.0000i	− 1.09764i	−0.835940	−	0.548821i	$-0.815077\pi$
$83$	− 10.0000i	− 1.09764i	0.835940	−	0.548821i	$-0.184923\pi$
$84$	0	0
$85$	− 24.0000i	− 2.60317i
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	− 8.00000i	− 0.847998i	−0.905663	−	0.423999i	$-0.860626\pi$
$89$	− 8.00000i	− 0.847998i	0.905663	−	0.423999i	$-0.139374\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	16.0000	1.64157
$96$	0	0
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	0	0
$99$	2.00000i	0.201008i
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	16.0000	1.56144
$106$	0	0
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	10.0000i	0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$	10.0000i	0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$	0	0
$111$	10.0000i	0.949158i
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	16.0000i	1.49201i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	24.0000i	2.20008i
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	− 4.00000i	− 0.360668i
$124$	0	0
$125$	24.0000i	2.14663i
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−16.0000	−1.38738
$134$	0	0
$135$	− 4.00000i	− 0.344265i
$136$	0	0
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	6.00000i	0.505291i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	24.0000i	1.99309i
$146$	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
$151$	8.00000i	0.651031i	0.945537	+	0.325515i	$0.105538\pi$
$151$	8.00000i	0.651031i	−0.945537	+	0.325515i	$0.894462\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	32.0000	2.57030
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	− 16.0000i	− 1.26098i
$162$	0	0
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	0	0
$165$	8.00000	0.622799
$166$	0	0
$167$	− 2.00000i	− 0.154765i	−0.997001	−	0.0773823i	$-0.975344\pi$
$167$	− 2.00000i	− 0.154765i	0.997001	−	0.0773823i	$-0.0246562\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	4.00000i	0.305888i
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	− 44.0000i	− 3.32609i
$176$	0	0
$177$	6.00000i	0.450988i
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	26.0000	1.93256	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000	1.93256	0.966282	+	0.257485i	$0.0828937\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	0	0
$185$	40.0000	2.94086
$186$	0	0
$187$	12.0000i	0.877527i
$188$	0	0
$189$	4.00000i	0.290957i
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	0	0
$193$	− 10.0000i	− 0.719816i	−0.932988	−	0.359908i	$-0.882808\pi$
$193$	− 10.0000i	− 0.719816i	0.932988	−	0.359908i	$-0.117192\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.0000i	1.42494i	0.701702	+	0.712470i	$0.252424\pi$
$197$	20.0000i	1.42494i	−0.701702	+	0.712470i	$0.747576\pi$
$198$	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
$201$	0	0
$202$	0	0
$203$	− 24.0000i	− 1.68447i
$204$	0	0
$205$	−16.0000	−1.11749
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	−8.00000	−0.553372
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	10.0000i	0.685189i
$214$	0	0
$215$	− 16.0000i	− 1.09119i
$216$	0	0
$217$	−32.0000	−2.17230
$218$	0	0
$219$	2.00000i	0.135147i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	0	0
$225$	−11.0000	−0.733333
$226$	0	0
$227$	− 22.0000i	− 1.46019i	−0.683345	−	0.730096i	$-0.739475\pi$
$227$	− 22.0000i	− 1.46019i	0.683345	−	0.730096i	$-0.260525\pi$
$228$	0	0
$229$	6.00000i	0.396491i	0.980152	+	0.198246i	$0.0635244\pi$
$229$	6.00000i	0.396491i	−0.980152	+	0.198246i	$0.936476\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	0	0
$233$	2.00000	0.131024	0.0655122	−	0.997852i	$-0.479132\pi$
$233$	2.00000	0.131024	0.0655122	+	0.997852i	$0.479132\pi$
$234$	0	0
$235$	24.0000	1.56559
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 6.00000i	− 0.388108i	−0.980991	−	0.194054i	$-0.937836\pi$
$239$	− 6.00000i	− 0.388108i	0.980991	−	0.194054i	$-0.0621637\pi$
$240$	0	0
$241$	− 14.0000i	− 0.901819i	−0.892570	−	0.450910i	$-0.851100\pi$
$241$	− 14.0000i	− 0.901819i	0.892570	−	0.450910i	$-0.148900\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	36.0000i	2.29996i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 10.0000i	− 0.633724i
$250$	0	0
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	0	0
$253$	− 8.00000i	− 0.502956i
$254$	0	0
$255$	− 24.0000i	− 1.50294i
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	−40.0000	−2.48548
$260$	0	0
$261$	−6.00000	−0.371391
$262$	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
$265$	− 24.0000i	− 1.47431i
$266$	0	0
$267$	− 8.00000i	− 0.489592i
$268$	0	0
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	− 20.0000i	− 1.21491i	−0.794353	−	0.607457i	$-0.792190\pi$
$271$	− 20.0000i	− 1.21491i	0.794353	−	0.607457i	$-0.207810\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 22.0000i	− 1.32665i
$276$	0	0
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	0	0
$279$	8.00000i	0.478947i
$280$	0	0
$281$	4.00000i	0.238620i	0.992857	+	0.119310i	$0.0380682\pi$
$281$	4.00000i	0.238620i	−0.992857	+	0.119310i	$0.961932\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	16.0000	0.947758
$286$	0	0
$287$	16.0000	0.944450
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	− 10.0000i	− 0.586210i
$292$	0	0
$293$	24.0000i	1.40209i	0.713115	+	0.701047i	$0.247284\pi$
$293$	24.0000i	1.40209i	−0.713115	+	0.701047i	$0.752716\pi$
$294$	0	0
$295$	24.0000	1.39733
$296$	0	0
$297$	2.00000i	0.116052i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	16.0000i	0.922225i
$302$	0	0
$303$	14.0000	0.804279
$304$	0	0
$305$	24.0000i	1.37424i
$306$	0	0
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	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
$315$	16.0000	0.901498
$316$	0	0
$317$	28.0000i	1.57264i	0.617822	+	0.786318i	$0.288015\pi$
$317$	28.0000i	1.57264i	−0.617822	+	0.786318i	$0.711985\pi$
$318$	0	0
$319$	− 12.0000i	− 0.671871i
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	24.0000i	1.33540i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.0000i	0.553001i
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	24.0000i	1.31916i	0.751635	+	0.659580i	$0.229266\pi$
$331$	24.0000i	1.31916i	−0.751635	+	0.659580i	$0.770734\pi$
$332$	0	0
$333$	10.0000i	0.547997i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	−10.0000	−0.543125
$340$	0	0
$341$	−16.0000	−0.866449
$342$	0	0
$343$	− 8.00000i	− 0.431959i
$344$	0	0
$345$	16.0000i	0.861411i
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	− 6.00000i	− 0.321173i	−0.987022	−	0.160586i	$-0.948662\pi$
$349$	− 6.00000i	− 0.321173i	0.987022	−	0.160586i	$-0.0513385\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 4.00000i	− 0.212899i	−0.994318	−	0.106449i	$-0.966052\pi$
$353$	− 4.00000i	− 0.212899i	0.994318	−	0.106449i	$-0.0339482\pi$
$354$	0	0
$355$	40.0000	2.12298
$356$	0	0
$357$	24.0000i	1.27021i
$358$	0	0
$359$	− 18.0000i	− 0.950004i	−0.879985	−	0.475002i	$-0.842447\pi$
$359$	− 18.0000i	− 0.950004i	0.879985	−	0.475002i	$-0.157553\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	8.00000	0.418739
$366$	0	0
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	0	0
$369$	− 4.00000i	− 0.208232i
$370$	0	0
$371$	24.0000i	1.24602i
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	24.0000i	1.23935i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	12.0000i	0.616399i	0.951322	+	0.308199i	$0.0997264\pi$
$379$	12.0000i	0.616399i	−0.951322	+	0.308199i	$0.900274\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	− 18.0000i	− 0.919757i	−0.887982	−	0.459879i	$-0.847893\pi$
$383$	− 18.0000i	− 0.919757i	0.887982	−	0.459879i	$-0.152107\pi$
$384$	0	0
$385$	32.0000i	1.63087i
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 22.0000i	− 1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$	− 22.0000i	− 1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$	0	0
$399$	−16.0000	−0.801002
$400$	0	0
$401$	24.0000i	1.19850i	0.800561	+	0.599251i	$0.204535\pi$
$401$	24.0000i	1.19850i	−0.800561	+	0.599251i	$0.795465\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 4.00000i	− 0.198762i
$406$	0	0
$407$	−20.0000	−0.991363
$408$	0	0
$409$	− 10.0000i	− 0.494468i	−0.968956	−	0.247234i	$-0.920478\pi$
$409$	− 10.0000i	− 0.494468i	0.968956	−	0.247234i	$-0.0795217\pi$
$410$	0	0
$411$	− 12.0000i	− 0.591916i
$412$	0	0
$413$	−24.0000	−1.18096
$414$	0	0
$415$	−40.0000	−1.96352
$416$	0	0
$417$	−12.0000	−0.587643
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	− 22.0000i	− 1.07221i	−0.844150	−	0.536107i	$-0.819894\pi$
$421$	− 22.0000i	− 1.07221i	0.844150	−	0.536107i	$-0.180106\pi$
$422$	0	0
$423$	6.00000i	0.291730i
$424$	0	0
$425$	−66.0000	−3.20147
$426$	0	0
$427$	− 24.0000i	− 1.16144i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 2.00000i	− 0.0963366i	−0.998839	−	0.0481683i	$-0.984662\pi$
$431$	− 2.00000i	− 0.0963366i	0.998839	−	0.0481683i	$-0.0153384\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	24.0000i	1.15071i
$436$	0	0
$437$	− 16.0000i	− 0.765384i
$438$	0	0
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	0	0
$441$	−9.00000	−0.428571
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−32.0000	−1.51695
$446$	0	0
$447$	8.00000i	0.378387i
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	8.00000i	0.375873i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 26.0000i	− 1.21623i	−0.793849	−	0.608114i	$-0.791926\pi$
$457$	− 26.0000i	− 1.21623i	0.793849	−	0.608114i	$-0.208074\pi$
$458$	0	0
$459$	6.00000	0.280056
$460$	0	0
$461$	36.0000i	1.67669i	0.545142	+	0.838344i	$0.316476\pi$
$461$	36.0000i	1.67669i	−0.545142	+	0.838344i	$0.683524\pi$
$462$	0	0
$463$	36.0000i	1.67306i	0.547920	+	0.836531i	$0.315420\pi$
$463$	36.0000i	1.67306i	−0.547920	+	0.836531i	$0.684580\pi$
$464$	0	0
$465$	32.0000	1.48396
$466$	0	0
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	8.00000i	0.367840i
$474$	0	0
$475$	− 44.0000i	− 2.01886i
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	− 6.00000i	− 0.274147i	−0.990561	−	0.137073i	$-0.956230\pi$
$479$	− 6.00000i	− 0.274147i	0.990561	−	0.137073i	$-0.0437697\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 16.0000i	− 0.728025i
$484$	0	0
$485$	−40.0000	−1.81631
$486$	0	0
$487$	− 40.0000i	− 1.81257i	−0.422664	−	0.906287i	$-0.638905\pi$
$487$	− 40.0000i	− 1.81257i	0.422664	−	0.906287i	$-0.361095\pi$
$488$	0	0
$489$	20.0000i	0.904431i
$490$	0	0
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	0	0
$493$	−36.0000	−1.62136
$494$	0	0
$495$	8.00000	0.359573
$496$	0	0
$497$	−40.0000	−1.79425
$498$	0	0
$499$	24.0000i	1.07439i	0.843459	+	0.537194i	$0.180516\pi$
$499$	24.0000i	1.07439i	−0.843459	+	0.537194i	$0.819484\pi$
$500$	0	0
$501$	− 2.00000i	− 0.0893534i
$502$	0	0
$503$	−28.0000	−1.24846	−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000	−1.24846	−0.624229	+	0.781241i	$0.714587\pi$
$504$	0	0
$505$	− 56.0000i	− 2.49197i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 28.0000i	− 1.24108i	−0.784176	−	0.620539i	$-0.786914\pi$
$509$	− 28.0000i	− 1.24108i	0.784176	−	0.620539i	$-0.213086\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	0	0
$513$	4.00000i	0.176604i
$514$	0	0
$515$	64.0000i	2.82018i
$516$	0	0
$517$	−12.0000	−0.527759
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	− 44.0000i	− 1.92032i
$526$	0	0
$527$	48.0000i	2.09091i
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	6.00000i	0.260378i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 32.0000i	− 1.38348i
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	− 18.0000i	− 0.775315i
$540$	0	0
$541$	6.00000i	0.257960i	0.991647	+	0.128980i	$0.0411703\pi$
$541$	6.00000i	0.257960i	−0.991647	+	0.128980i	$0.958830\pi$
$542$	0	0
$543$	26.0000	1.11577
$544$	0	0
$545$	40.0000	1.71341
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	0	0
$549$	−6.00000	−0.256074
$550$	0	0
$551$	− 24.0000i	− 1.02243i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	40.0000	1.69791
$556$	0	0
$557$	32.0000i	1.35588i	0.735116	+	0.677942i	$0.237128\pi$
$557$	32.0000i	1.35588i	−0.735116	+	0.677942i	$0.762872\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	12.0000i	0.506640i
$562$	0	0
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	40.0000i	1.68281i
$566$	0	0
$567$	4.00000i	0.167984i
$568$	0	0
$569$	2.00000	0.0838444	0.0419222	−	0.999121i	$-0.486652\pi$
$569$	2.00000	0.0838444	0.0419222	+	0.999121i	$0.486652\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	16.0000	0.668410
$574$	0	0
$575$	44.0000	1.83493
$576$	0	0
$577$	34.0000i	1.41544i	0.706494	+	0.707719i	$0.250276\pi$
$577$	34.0000i	1.41544i	−0.706494	+	0.707719i	$0.749724\pi$
$578$	0	0
$579$	− 10.0000i	− 0.415586i
$580$	0	0
$581$	40.0000	1.65948
$582$	0	0
$583$	12.0000i	0.496989i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 18.0000i	− 0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$	− 18.0000i	− 0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	20.0000i	0.822690i
$592$	0	0
$593$	− 32.0000i	− 1.31408i	−0.753855	−	0.657041i	$-0.771808\pi$
$593$	− 32.0000i	− 1.31408i	0.753855	−	0.657041i	$-0.228192\pi$
$594$	0	0
$595$	96.0000	3.93562
$596$	0	0
$597$	24.0000	0.982255
$598$	0	0
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	−42.0000	−1.71322	−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000	−1.71322	−0.856608	+	0.515968i	$0.827432\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 28.0000i	− 1.13836i
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	− 24.0000i	− 0.972529i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	30.0000i	1.21169i	0.795583	+	0.605844i	$0.207165\pi$
$613$	30.0000i	1.21169i	−0.795583	+	0.605844i	$0.792835\pi$
$614$	0	0
$615$	−16.0000	−0.645182
$616$	0	0
$617$	− 28.0000i	− 1.12724i	−0.826035	−	0.563619i	$-0.809409\pi$
$617$	− 28.0000i	− 1.12724i	0.826035	−	0.563619i	$-0.190591\pi$
$618$	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
$621$	−4.00000	−0.160514
$622$	0	0
$623$	32.0000	1.28205
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	−8.00000	−0.319489
$628$	0	0
$629$	60.0000i	2.39236i
$630$	0	0
$631$	− 48.0000i	− 1.91085i	−0.295234	−	0.955425i	$-0.595398\pi$
$631$	− 48.0000i	− 1.91085i	0.295234	−	0.955425i	$-0.404602\pi$
$632$	0	0
$633$	4.00000	0.158986
$634$	0	0
$635$	− 32.0000i	− 1.26988i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	10.0000i	0.395594i
$640$	0	0
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	16.0000i	0.630978i	0.948929	+	0.315489i	$0.102169\pi$
$643$	16.0000i	0.630978i	−0.948929	+	0.315489i	$0.897831\pi$
$644$	0	0
$645$	− 16.0000i	− 0.629999i
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	−32.0000	−1.25418
$652$	0	0
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.00000i	0.0780274i
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	0	0
$661$	− 38.0000i	− 1.47803i	−0.673690	−	0.739014i	$-0.735292\pi$
$661$	− 38.0000i	− 1.47803i	0.673690	−	0.739014i	$-0.264708\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	64.0000i	2.48181i
$666$	0	0
$667$	24.0000	0.929284
$668$	0	0
$669$	− 8.00000i	− 0.309298i
$670$	0	0
$671$	− 12.0000i	− 0.463255i
$672$	0	0
$673$	30.0000	1.15642	0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000	1.15642	0.578208	+	0.815890i	$0.303752\pi$
$674$	0	0
$675$	−11.0000	−0.423390
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	40.0000	1.53506
$680$	0	0
$681$	− 22.0000i	− 0.843042i
$682$	0	0
$683$	18.0000i	0.688751i	0.938832	+	0.344375i	$0.111909\pi$
$683$	18.0000i	0.688751i	−0.938832	+	0.344375i	$0.888091\pi$
$684$	0	0
$685$	−48.0000	−1.83399
$686$	0	0
$687$	6.00000i	0.228914i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 8.00000i	− 0.304334i	−0.988355	−	0.152167i	$-0.951375\pi$
$691$	− 8.00000i	− 0.304334i	0.988355	−	0.152167i	$-0.0486252\pi$
$692$	0	0
$693$	−8.00000	−0.303895
$694$	0	0
$695$	48.0000i	1.82074i
$696$	0	0
$697$	− 24.0000i	− 0.909065i
$698$	0	0
$699$	2.00000	0.0756469
$700$	0	0
$701$	−38.0000	−1.43524	−0.717620	−	0.696435i	$-0.754769\pi$
$701$	−38.0000	−1.43524	−0.717620	+	0.696435i	$0.754769\pi$
$702$	0	0
$703$	−40.0000	−1.50863
$704$	0	0
$705$	24.0000	0.903892
$706$	0	0
$707$	56.0000i	2.10610i
$708$	0	0
$709$	14.0000i	0.525781i	0.964826	+	0.262891i	$0.0846758\pi$
$709$	14.0000i	0.525781i	−0.964826	+	0.262891i	$0.915324\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 32.0000i	− 1.19841i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 6.00000i	− 0.224074i
$718$	0	0
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	0	0
$721$	− 64.0000i	− 2.38348i
$722$	0	0
$723$	− 14.0000i	− 0.520666i
$724$	0	0
$725$	66.0000	2.45118
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	− 14.0000i	− 0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$	− 14.0000i	− 0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	0	0
$735$	36.0000i	1.32788i
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 16.0000i	− 0.588570i	−0.955718	−	0.294285i	$-0.904919\pi$
$739$	− 16.0000i	− 0.588570i	0.955718	−	0.294285i	$-0.0950814\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 22.0000i	− 0.807102i	−0.914957	−	0.403551i	$-0.867776\pi$
$743$	− 22.0000i	− 0.807102i	0.914957	−	0.403551i	$-0.132224\pi$
$744$	0	0
$745$	32.0000	1.17239
$746$	0	0
$747$	− 10.0000i	− 0.365881i
$748$	0	0
$749$	32.0000i	1.16925i
$750$	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
$753$	16.0000	0.583072
$754$	0	0
$755$	32.0000	1.16460
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	0	0
$759$	− 8.00000i	− 0.290382i
$760$	0	0
$761$	36.0000i	1.30500i	0.757789	+	0.652499i	$0.226280\pi$
$761$	36.0000i	1.30500i	−0.757789	+	0.652499i	$0.773720\pi$
$762$	0	0
$763$	−40.0000	−1.44810
$764$	0	0
$765$	− 24.0000i	− 0.867722i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	10.0000i	0.360609i	0.983611	+	0.180305i	$0.0577084\pi$
$769$	10.0000i	0.360609i	−0.983611	+	0.180305i	$0.942292\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	0	0
$773$	− 24.0000i	− 0.863220i	−0.902060	−	0.431610i	$-0.857946\pi$
$773$	− 24.0000i	− 0.863220i	0.902060	−	0.431610i	$-0.142054\pi$
$774$	0	0
$775$	− 88.0000i	− 3.16105i
$776$	0	0
$777$	−40.0000	−1.43499
$778$	0	0
$779$	16.0000	0.573259
$780$	0	0
$781$	−20.0000	−0.715656
$782$	0	0
$783$	−6.00000	−0.214423
$784$	0	0
$785$	8.00000i	0.285532i
$786$	0	0
$787$	48.0000i	1.71102i	0.517790	+	0.855508i	$0.326755\pi$
$787$	48.0000i	1.71102i	−0.517790	+	0.855508i	$0.673245\pi$
$788$	0	0
$789$	−4.00000	−0.142404
$790$	0	0
$791$	− 40.0000i	− 1.42224i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 24.0000i	− 0.851192i
$796$	0	0
$797$	−14.0000	−0.495905	−0.247953	−	0.968772i	$-0.579758\pi$
$797$	−14.0000	−0.495905	−0.247953	+	0.968772i	$0.579758\pi$
$798$	0	0
$799$	36.0000i	1.27359i
$800$	0	0
$801$	− 8.00000i	− 0.282666i
$802$	0	0
$803$	−4.00000	−0.141157
$804$	0	0
$805$	−64.0000	−2.25570
$806$	0	0
$807$	10.0000	0.352017
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	− 28.0000i	− 0.983213i	−0.870817	−	0.491606i	$-0.836410\pi$
$811$	− 28.0000i	− 0.983213i	0.870817	−	0.491606i	$-0.163590\pi$
$812$	0	0
$813$	− 20.0000i	− 0.701431i
$814$	0	0
$815$	80.0000	2.80228
$816$	0	0
$817$	16.0000i	0.559769i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 32.0000i	− 1.11681i	−0.829569	−	0.558404i	$-0.811414\pi$
$821$	− 32.0000i	− 1.11681i	0.829569	−	0.558404i	$-0.188586\pi$
$822$	0	0
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	0	0
$825$	− 22.0000i	− 0.765942i
$826$	0	0
$827$	10.0000i	0.347734i	0.984769	+	0.173867i	$0.0556263\pi$
$827$	10.0000i	0.347734i	−0.984769	+	0.173867i	$0.944374\pi$
$828$	0	0
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	0	0
$833$	−54.0000	−1.87099
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	8.00000i	0.276520i
$838$	0	0
$839$	42.0000i	1.45000i	0.688748	+	0.725001i	$0.258161\pi$
$839$	42.0000i	1.45000i	−0.688748	+	0.725001i	$0.741839\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	4.00000i	0.137767i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	28.0000i	0.962091i
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	− 40.0000i	− 1.37118i
$852$	0	0
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	0	0
$855$	16.0000	0.547188
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	16.0000	0.545279
$862$	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
$865$	− 24.0000i	− 0.816024i
$866$	0	0
$867$	19.0000	0.645274
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 10.0000i	− 0.338449i
$874$	0	0
$875$	−96.0000	−3.24539
$876$	0	0
$877$	− 10.0000i	− 0.337676i	−0.985644	−	0.168838i	$-0.945999\pi$
$877$	− 10.0000i	− 0.337676i	0.985644	−	0.168838i	$-0.0540015\pi$
$878$	0	0
$879$	24.0000i	0.809500i
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	0	0
$885$	24.0000	0.806751
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	32.0000i	1.07325i
$890$	0	0
$891$	2.00000i	0.0670025i
$892$	0	0
$893$	−24.0000	−0.803129
$894$	0	0
$895$	48.0000i	1.60446i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 48.0000i	− 1.60089i
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	16.0000i	0.532447i
$904$	0	0
$905$	− 104.000i	− 3.45708i
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	14.0000	0.464351
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	0	0
$913$	20.0000	0.661903
$914$	0	0
$915$	24.0000i	0.793416i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	− 16.0000i	− 0.527218i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 110.000i	− 3.61678i
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	− 20.0000i	− 0.656179i	−0.944647	−	0.328089i	$-0.893595\pi$
$929$	− 20.0000i	− 0.656179i	0.944647	−	0.328089i	$-0.106405\pi$
$930$	0	0
$931$	− 36.0000i	− 1.17985i
$932$	0	0
$933$	−20.0000	−0.654771
$934$	0	0
$935$	48.0000	1.56977
$936$	0	0
$937$	−50.0000	−1.63343	−0.816714	−	0.577042i	$-0.804207\pi$
$937$	−50.0000	−1.63343	−0.816714	+	0.577042i	$0.804207\pi$
$938$	0	0
$939$	6.00000	0.195803
$940$	0	0
$941$	− 40.0000i	− 1.30396i	−0.758235	−	0.651981i	$-0.773938\pi$
$941$	− 40.0000i	− 1.30396i	0.758235	−	0.651981i	$-0.226062\pi$
$942$	0	0
$943$	16.0000i	0.521032i
$944$	0	0
$945$	16.0000	0.520480
$946$	0	0
$947$	− 18.0000i	− 0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$	− 18.0000i	− 0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	28.0000i	0.907962i
$952$	0	0
$953$	42.0000	1.36051	0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000	1.36051	0.680257	+	0.732974i	$0.261868\pi$
$954$	0	0
$955$	− 64.0000i	− 2.07099i
$956$	0	0
$957$	− 12.0000i	− 0.387905i
$958$	0	0
$959$	48.0000	1.55000
$960$	0	0
$961$	−33.0000	−1.06452
$962$	0	0
$963$	8.00000	0.257796
$964$	0	0
$965$	−40.0000	−1.28765
$966$	0	0
$967$	32.0000i	1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$	32.0000i	1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$	0	0
$969$	24.0000i	0.770991i
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	− 48.0000i	− 1.53881i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.0000i	0.767828i	0.923369	+	0.383914i	$0.125424\pi$
$977$	24.0000i	0.767828i	−0.923369	+	0.383914i	$0.874576\pi$
$978$	0	0
$979$	16.0000	0.511362
$980$	0	0
$981$	10.0000i	0.319275i
$982$	0	0
$983$	− 38.0000i	− 1.21201i	−0.795460	−	0.606006i	$-0.792771\pi$
$983$	− 38.0000i	− 1.21201i	0.795460	−	0.606006i	$-0.207229\pi$
$984$	0	0
$985$	80.0000	2.54901
$986$	0	0
$987$	−24.0000	−0.763928
$988$	0	0
$989$	−16.0000	−0.508770
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	24.0000i	0.761617i
$994$	0	0
$995$	− 96.0000i	− 3.04340i
$996$	0	0
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	0	0
$999$	10.0000i	0.316386i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.c.j.337.1		2
13.5	odd	4		312.2.a.d.1.1	✓	1
13.8	odd	4		4056.2.a.s.1.1		1
13.12	even	2	inner	4056.2.c.j.337.2		2
39.5	even	4		936.2.a.i.1.1		1
52.31	even	4		624.2.a.a.1.1		1
52.47	even	4		8112.2.a.o.1.1		1
65.44	odd	4		7800.2.a.j.1.1		1
104.5	odd	4		2496.2.a.n.1.1		1
104.83	even	4		2496.2.a.bd.1.1		1
156.83	odd	4		1872.2.a.t.1.1		1
312.5	even	4		7488.2.a.b.1.1		1
312.83	odd	4		7488.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.a.d.1.1	✓	1	13.5	odd	4
624.2.a.a.1.1		1	52.31	even	4
936.2.a.i.1.1		1	39.5	even	4
1872.2.a.t.1.1		1	156.83	odd	4
2496.2.a.n.1.1		1	104.5	odd	4
2496.2.a.bd.1.1		1	104.83	even	4
4056.2.a.s.1.1		1	13.8	odd	4
4056.2.c.j.337.1		2	1.1	even	1	trivial
4056.2.c.j.337.2		2	13.12	even	2	inner
7488.2.a.b.1.1		1	312.5	even	4
7488.2.a.e.1.1		1	312.83	odd	4
7800.2.a.j.1.1		1	65.44	odd	4
8112.2.a.o.1.1		1	52.47	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:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.00000i q^{5} +4.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -4.00000i q^{5} +4.00000i q^{7} +1.00000 q^{9} +2.00000i q^{11} -4.00000i q^{15} +6.00000 q^{17} +4.00000i q^{19} +4.00000i q^{21} -4.00000 q^{23} -11.0000 q^{25} +1.00000 q^{27} -6.00000 q^{29} +8.00000i q^{31} +2.00000i q^{33} +16.0000 q^{35} +10.0000i q^{37} -4.00000i q^{41} +4.00000 q^{43} -4.00000i q^{45} +6.00000i q^{47} -9.00000 q^{49} +6.00000 q^{51} +6.00000 q^{53} +8.00000 q^{55} +4.00000i q^{57} +6.00000i q^{59} -6.00000 q^{61} +4.00000i q^{63} -4.00000 q^{69} +10.0000i q^{71} +2.00000i q^{73} -11.0000 q^{75} -8.00000 q^{77} +1.00000 q^{81} -10.0000i q^{83} -24.0000i q^{85} -6.00000 q^{87} -8.00000i q^{89} +8.00000i q^{93} +16.0000 q^{95} -10.0000i q^{97} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{9} + 12 q^{17} - 8 q^{23} - 22 q^{25} + 2 q^{27} - 12 q^{29} + 32 q^{35} + 8 q^{43} - 18 q^{49} + 12 q^{51} + 12 q^{53} + 16 q^{55} - 12 q^{61} - 8 q^{69} - 22 q^{75} - 16 q^{77} + 2 q^{81} - 12 q^{87} + 32 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^9 + 12 * q^17 - 8 * q^23 - 22 * q^25 + 2 * q^27 - 12 * q^29 + 32 * q^35 + 8 * q^43 - 18 * q^49 + 12 * q^51 + 12 * q^53 + 16 * q^55 - 12 * q^61 - 8 * q^69 - 22 * q^75 - 16 * q^77 + 2 * q^81 - 12 * q^87 + 32 * q^95

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