LMFDB - Embedded newform 5400.2.f.m.649.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5400,2,Mod(649,5400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5400.649"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5400 = 2^{3} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5400.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,0,0,-2,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	no
Analytic conductor:	$43.1192170915$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	5400.649
Dual form			5400.2.f.m.649.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{7} -1.00000 q^{11} +1.00000i q^{13} -1.00000i q^{17} -4.00000 q^{19} -1.00000i q^{23} -5.00000 q^{29} +1.00000 q^{31} -6.00000i q^{37} +7.00000i q^{43} +7.00000i q^{47} +3.00000 q^{49} +12.0000i q^{53} -4.00000 q^{59} +10.0000 q^{61} +4.00000i q^{67} -12.0000 q^{71} +6.00000i q^{73} +2.00000i q^{77} -15.0000 q^{79} -2.00000i q^{83} -12.0000 q^{89} +2.00000 q^{91} -10.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{11} - 8 q^{19} - 10 q^{29} + 2 q^{31} + 6 q^{49} - 8 q^{59} + 20 q^{61} - 24 q^{71} - 30 q^{79} - 24 q^{89} + 4 q^{91}+O(q^{100}) $ 2 * q - 2 * q^11 - 8 * q^19 - 10 * q^29 + 2 * q^31 + 6 * q^49 - 8 * q^59 + 20 * q^61 - 24 * q^71 - 30 * q^79 - 24 * q^89 + 4 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5400\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$	$2701$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.00000i	− 0.242536i	−0.992620	−	0.121268i	$-0.961304\pi$
$17$	− 1.00000i	− 0.242536i	0.992620	−	0.121268i	$-0.0386960\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 1.00000i	− 0.208514i	−0.994550	−	0.104257i	$-0.966753\pi$
$23$	− 1.00000i	− 0.208514i	0.994550	−	0.104257i	$-0.0332465\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	7.00000i	1.06749i	0.845645	+	0.533745i	$0.179216\pi$
$43$	7.00000i	1.06749i	−0.845645	+	0.533745i	$0.820784\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.00000i	1.02105i	0.859861	+	0.510527i	$0.170550\pi$
$47$	7.00000i	1.02105i	−0.859861	+	0.510527i	$0.829450\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	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
$57$	0	0
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00000i	0.227921i
$78$	0	0
$79$	−15.0000	−1.68763	−0.843816	−	0.536633i	$-0.819696\pi$
$79$	−15.0000	−1.68763	−0.843816	+	0.536633i	$0.819696\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 2.00000i	− 0.219529i	−0.993958	−	0.109764i	$-0.964990\pi$
$83$	− 2.00000i	− 0.219529i	0.993958	−	0.109764i	$-0.0350096\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	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
$99$	0	0
$100$	0	0
$101$	15.0000	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$102$	0	0
$103$	− 14.0000i	− 1.37946i	−0.724066	−	0.689730i	$-0.757729\pi$
$103$	− 14.0000i	− 1.37946i	0.724066	−	0.689730i	$-0.242271\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.0000i	1.74013i	0.492941	+	0.870063i	$0.335922\pi$
$107$	18.0000i	1.74013i	−0.492941	+	0.870063i	$0.664078\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 3.00000i	− 0.282216i	−0.989994	−	0.141108i	$-0.954933\pi$
$113$	− 3.00000i	− 0.282216i	0.989994	−	0.141108i	$-0.0450665\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.00000i	0.532414i	0.963916	+	0.266207i	$0.0857705\pi$
$127$	6.00000i	0.532414i	−0.963916	+	0.266207i	$0.914230\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.00000	0.786334	0.393167	−	0.919467i	$-0.371379\pi$
$131$	9.00000	0.786334	0.393167	+	0.919467i	$0.371379\pi$
$132$	0	0
$133$	8.00000i	0.693688i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	22.0000i	1.87959i	0.341743	+	0.939793i	$0.388983\pi$
$137$	22.0000i	1.87959i	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	6.00000	0.508913	0.254457	−	0.967084i	$-0.418103\pi$
$139$	6.00000	0.508913	0.254457	+	0.967084i	$0.418103\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 1.00000i	− 0.0836242i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.0000	−1.06500	−0.532501	−	0.846430i	$-0.678748\pi$
$149$	−13.0000	−1.06500	−0.532501	+	0.846430i	$0.678748\pi$
$150$	0	0
$151$	−1.00000	−0.0813788	−0.0406894	−	0.999172i	$-0.512955\pi$
$151$	−1.00000	−0.0813788	−0.0406894	+	0.999172i	$0.512955\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	17.0000i	1.35675i	0.734717	+	0.678374i	$0.237315\pi$
$157$	17.0000i	1.35675i	−0.734717	+	0.678374i	$0.762685\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	− 5.00000i	− 0.391630i	−0.980641	−	0.195815i	$-0.937265\pi$
$163$	− 5.00000i	− 0.391630i	0.980641	−	0.195815i	$-0.0627352\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.0000i	0.760286i	0.924928	+	0.380143i	$0.124125\pi$
$173$	10.0000i	0.760286i	−0.924928	+	0.380143i	$0.875875\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−4.00000	−0.297318	−0.148659	−	0.988889i	$-0.547496\pi$
$181$	−4.00000	−0.297318	−0.148659	+	0.988889i	$0.547496\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.00000i	0.0731272i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.0000	1.59186	0.795932	−	0.605386i	$-0.206981\pi$
$191$	22.0000	1.59186	0.795932	+	0.605386i	$0.206981\pi$
$192$	0	0
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 8.00000i	− 0.569976i	−0.958531	−	0.284988i	$-0.908010\pi$
$197$	− 8.00000i	− 0.569976i	0.958531	−	0.284988i	$-0.0919897\pi$
$198$	0	0
$199$	−5.00000	−0.354441	−0.177220	−	0.984171i	$-0.556711\pi$
$199$	−5.00000	−0.354441	−0.177220	+	0.984171i	$0.556711\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	10.0000i	0.701862i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.00000i	− 0.135769i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.00000	0.0672673
$222$	0	0
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.0000i	1.59294i	0.604681	+	0.796468i	$0.293301\pi$
$227$	24.0000i	1.59294i	−0.604681	+	0.796468i	$0.706699\pi$
$228$	0	0
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−18.0000	−1.16432	−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000	−1.16432	−0.582162	+	0.813073i	$0.697793\pi$
$240$	0	0
$241$	5.00000	0.322078	0.161039	−	0.986948i	$-0.448515\pi$
$241$	5.00000	0.322078	0.161039	+	0.986948i	$0.448515\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 4.00000i	− 0.254514i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	0	0
$253$	1.00000i	0.0628695i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	27.0000i	1.68421i	0.539311	+	0.842107i	$0.318685\pi$
$257$	27.0000i	1.68421i	−0.539311	+	0.842107i	$0.681315\pi$
$258$	0	0
$259$	−12.0000	−0.745644
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.0000i	0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$	12.0000i	0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−31.0000	−1.89010	−0.945052	−	0.326921i	$-0.893989\pi$
$269$	−31.0000	−1.89010	−0.945052	+	0.326921i	$0.893989\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	− 16.0000i	− 0.951101i	−0.879688	−	0.475551i	$-0.842249\pi$
$283$	− 16.0000i	− 0.951101i	0.879688	−	0.475551i	$-0.157751\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	16.0000	0.941176
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.00000i	0.116841i	0.998292	+	0.0584206i	$0.0186065\pi$
$293$	2.00000i	0.116841i	−0.998292	+	0.0584206i	$0.981394\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.00000	0.0578315
$300$	0	0
$301$	14.0000	0.806947
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5.00000i	0.285365i	0.989769	+	0.142683i	$0.0455728\pi$
$307$	5.00000i	0.285365i	−0.989769	+	0.142683i	$0.954427\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	8.00000i	0.452187i	0.974106	+	0.226093i	$0.0725954\pi$
$313$	8.00000i	0.452187i	−0.974106	+	0.226093i	$0.927405\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.0000i	1.34797i	0.738743	+	0.673987i	$0.235420\pi$
$317$	24.0000i	1.34797i	−0.738743	+	0.673987i	$0.764580\pi$
$318$	0	0
$319$	5.00000	0.279946
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.00000i	0.222566i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	14.0000	0.771845
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8.00000i	0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$	8.00000i	0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.00000	−0.0541530
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 16.0000i	− 0.858925i	−0.903085	−	0.429463i	$-0.858703\pi$
$347$	− 16.0000i	− 0.858925i	0.903085	−	0.429463i	$-0.141297\pi$
$348$	0	0
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 17.0000i	− 0.904819i	−0.891810	−	0.452409i	$-0.850565\pi$
$353$	− 17.0000i	− 0.904819i	0.891810	−	0.452409i	$-0.149435\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 2.00000i	− 0.104399i	−0.998637	−	0.0521996i	$-0.983377\pi$
$367$	− 2.00000i	− 0.104399i	0.998637	−	0.0521996i	$-0.0166232\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	24.0000	1.24602
$372$	0	0
$373$	19.0000i	0.983783i	0.870657	+	0.491891i	$0.163694\pi$
$373$	19.0000i	0.983783i	−0.870657	+	0.491891i	$0.836306\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 5.00000i	− 0.257513i
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	15.0000i	0.766464i	0.923652	+	0.383232i	$0.125189\pi$
$383$	15.0000i	0.766464i	−0.923652	+	0.383232i	$0.874811\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9.00000	−0.456318	−0.228159	−	0.973624i	$-0.573271\pi$
$389$	−9.00000	−0.456318	−0.228159	+	0.973624i	$0.573271\pi$
$390$	0	0
$391$	−1.00000	−0.0505722
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	11.0000i	0.552074i	0.961147	+	0.276037i	$0.0890213\pi$
$397$	11.0000i	0.552074i	−0.961147	+	0.276037i	$0.910979\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−36.0000	−1.79775	−0.898877	−	0.438201i	$-0.855616\pi$
$401$	−36.0000	−1.79775	−0.898877	+	0.438201i	$0.855616\pi$
$402$	0	0
$403$	1.00000i	0.0498135i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.00000i	0.297409i
$408$	0	0
$409$	−31.0000	−1.53285	−0.766426	−	0.642333i	$-0.777967\pi$
$409$	−31.0000	−1.53285	−0.766426	+	0.642333i	$0.777967\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.00000i	0.393654i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−33.0000	−1.61216	−0.806078	−	0.591810i	$-0.798414\pi$
$419$	−33.0000	−1.61216	−0.806078	+	0.591810i	$0.798414\pi$
$420$	0	0
$421$	12.0000	0.584844	0.292422	−	0.956289i	$-0.405539\pi$
$421$	12.0000	0.584844	0.292422	+	0.956289i	$0.405539\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 20.0000i	− 0.967868i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	0	0
$433$	26.0000i	1.24948i	0.780833	+	0.624740i	$0.214795\pi$
$433$	26.0000i	1.24948i	−0.780833	+	0.624740i	$0.785205\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.00000i	0.191346i
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 28.0000i	− 1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$457$	− 28.0000i	− 1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	6.00000i	0.278844i	0.990233	+	0.139422i	$0.0445244\pi$
$463$	6.00000i	0.278844i	−0.990233	+	0.139422i	$0.955476\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 28.0000i	− 1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$	− 28.0000i	− 1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 7.00000i	− 0.321860i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−20.0000	−0.913823	−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000	−0.913823	−0.456912	+	0.889512i	$0.651044\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 4.00000i	− 0.181257i	−0.995885	−	0.0906287i	$-0.971112\pi$
$487$	− 4.00000i	− 0.181257i	0.995885	−	0.0906287i	$-0.0288876\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	5.00000i	0.225189i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	24.0000i	1.07655i
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 9.00000i	− 0.401290i	−0.979664	−	0.200645i	$-0.935696\pi$
$503$	− 9.00000i	− 0.401290i	0.979664	−	0.200645i	$-0.0643038\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7.00000	0.310270	0.155135	−	0.987893i	$-0.450419\pi$
$509$	7.00000	0.310270	0.155135	+	0.987893i	$0.450419\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 7.00000i	− 0.307860i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	0	0
$523$	1.00000i	0.0437269i	0.999761	+	0.0218635i	$0.00695991\pi$
$523$	1.00000i	0.0437269i	−0.999761	+	0.0218635i	$0.993040\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.00000i	− 0.0435607i
$528$	0	0
$529$	22.0000	0.956522
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.00000	−0.129219
$540$	0	0
$541$	34.0000	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	37.0000i	1.58201i	0.611812	+	0.791003i	$0.290441\pi$
$547$	37.0000i	1.58201i	−0.611812	+	0.791003i	$0.709559\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	20.0000	0.852029
$552$	0	0
$553$	30.0000i	1.27573i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 28.0000i	− 1.18640i	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	− 28.0000i	− 1.18640i	0.805056	−	0.593199i	$-0.202135\pi$
$558$	0	0
$559$	−7.00000	−0.296068
$560$	0	0
$561$	0	0
$562$	0	0
$563$	38.0000i	1.60151i	0.598993	+	0.800755i	$0.295568\pi$
$563$	38.0000i	1.60151i	−0.598993	+	0.800755i	$0.704432\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	32.0000	1.34151	0.670755	−	0.741679i	$-0.265970\pi$
$569$	32.0000	1.34151	0.670755	+	0.741679i	$0.265970\pi$
$570$	0	0
$571$	22.0000	0.920671	0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000	0.920671	0.460336	+	0.887745i	$0.347729\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 32.0000i	− 1.33218i	−0.745873	−	0.666089i	$-0.767967\pi$
$577$	− 32.0000i	− 1.33218i	0.745873	−	0.666089i	$-0.232033\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	− 12.0000i	− 0.496989i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	42.0000i	1.73353i	0.498721	+	0.866763i	$0.333803\pi$
$587$	42.0000i	1.73353i	−0.498721	+	0.866763i	$0.666197\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 15.0000i	− 0.615976i	−0.951390	−	0.307988i	$-0.900344\pi$
$593$	− 15.0000i	− 0.615976i	0.951390	−	0.307988i	$-0.0996557\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	0	0
$601$	−35.0000	−1.42768	−0.713840	−	0.700309i	$-0.753046\pi$
$601$	−35.0000	−1.42768	−0.713840	+	0.700309i	$0.753046\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 30.0000i	− 1.21766i	−0.793300	−	0.608831i	$-0.791639\pi$
$607$	− 30.0000i	− 1.21766i	0.793300	−	0.608831i	$-0.208361\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7.00000	−0.283190
$612$	0	0
$613$	− 15.0000i	− 0.605844i	−0.953015	−	0.302922i	$-0.902038\pi$
$613$	− 15.0000i	− 0.605844i	0.953015	−	0.302922i	$-0.0979622\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.00000i	0.362326i	0.983453	+	0.181163i	$0.0579862\pi$
$617$	9.00000i	0.362326i	−0.983453	+	0.181163i	$0.942014\pi$
$618$	0	0
$619$	−38.0000	−1.52735	−0.763674	−	0.645601i	$-0.776607\pi$
$619$	−38.0000	−1.52735	−0.763674	+	0.645601i	$0.776607\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	24.0000i	0.961540i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6.00000	−0.239236
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.00000i	0.118864i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	− 9.00000i	− 0.354925i	−0.984128	−	0.177463i	$-0.943211\pi$
$643$	− 9.00000i	− 0.354925i	0.984128	−	0.177463i	$-0.0567889\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 12.0000i	− 0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$	− 12.0000i	− 0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.00000i	0.156532i	0.996933	+	0.0782660i	$0.0249384\pi$
$653$	4.00000i	0.156532i	−0.996933	+	0.0782660i	$0.975062\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.00000i	0.193601i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.0000	−0.386046
$672$	0	0
$673$	− 6.00000i	− 0.231283i	−0.993291	−	0.115642i	$-0.963108\pi$
$673$	− 6.00000i	− 0.231283i	0.993291	−	0.115642i	$-0.0368924\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.00000i	0.153732i	0.997041	+	0.0768662i	$0.0244914\pi$
$677$	4.00000i	0.153732i	−0.997041	+	0.0768662i	$0.975509\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−11.0000	−0.415464	−0.207732	−	0.978186i	$-0.566608\pi$
$701$	−11.0000	−0.415464	−0.207732	+	0.978186i	$0.566608\pi$
$702$	0	0
$703$	24.0000i	0.905177i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 30.0000i	− 1.12827i
$708$	0	0
$709$	52.0000	1.95290	0.976450	−	0.215742i	$-0.0692169\pi$
$709$	52.0000	1.95290	0.976450	+	0.215742i	$0.0692169\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1.00000i	− 0.0374503i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.00000	−0.0745874	−0.0372937	−	0.999304i	$-0.511874\pi$
$719$	−2.00000	−0.0745874	−0.0372937	+	0.999304i	$0.511874\pi$
$720$	0	0
$721$	−28.0000	−1.04277
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	28.0000i	1.03846i	0.854634	+	0.519231i	$0.173782\pi$
$727$	28.0000i	1.03846i	−0.854634	+	0.519231i	$0.826218\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	7.00000	0.258904
$732$	0	0
$733$	42.0000i	1.55131i	0.631160	+	0.775653i	$0.282579\pi$
$733$	42.0000i	1.55131i	−0.631160	+	0.775653i	$0.717421\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 4.00000i	− 0.147342i
$738$	0	0
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 51.0000i	− 1.87101i	−0.353315	−	0.935504i	$-0.614946\pi$
$743$	− 51.0000i	− 1.87101i	0.353315	−	0.935504i	$-0.385054\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	36.0000	1.31541
$750$	0	0
$751$	3.00000	0.109472	0.0547358	−	0.998501i	$-0.482568\pi$
$751$	3.00000	0.109472	0.0547358	+	0.998501i	$0.482568\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.00000i	0.0363456i	0.999835	+	0.0181728i	$0.00578490\pi$
$757$	1.00000i	0.0363456i	−0.999835	+	0.0181728i	$0.994215\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	− 8.00000i	− 0.289619i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 4.00000i	− 0.144432i
$768$	0	0
$769$	−7.00000	−0.252426	−0.126213	−	0.992003i	$-0.540282\pi$
$769$	−7.00000	−0.252426	−0.126213	+	0.992003i	$0.540282\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	12.0000	0.429394
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 47.0000i	− 1.67537i	−0.546154	−	0.837685i	$-0.683909\pi$
$787$	− 47.0000i	− 1.67537i	0.546154	−	0.837685i	$-0.316091\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	10.0000i	0.355110i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 36.0000i	− 1.27519i	−0.770374	−	0.637593i	$-0.779930\pi$
$797$	− 36.0000i	− 1.27519i	0.770374	−	0.637593i	$-0.220070\pi$
$798$	0	0
$799$	7.00000	0.247642
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 6.00000i	− 0.211735i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	−10.0000	−0.351147	−0.175574	−	0.984466i	$-0.556178\pi$
$811$	−10.0000	−0.351147	−0.175574	+	0.984466i	$0.556178\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 28.0000i	− 0.979596i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	38.0000	1.32621	0.663105	−	0.748527i	$-0.269238\pi$
$821$	38.0000	1.32621	0.663105	+	0.748527i	$0.269238\pi$
$822$	0	0
$823$	− 20.0000i	− 0.697156i	−0.937280	−	0.348578i	$-0.886665\pi$
$823$	− 20.0000i	− 0.697156i	0.937280	−	0.348578i	$-0.113335\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 30.0000i	− 1.04320i	−0.853189	−	0.521601i	$-0.825335\pi$
$827$	− 30.0000i	− 1.04320i	0.853189	−	0.521601i	$-0.174665\pi$
$828$	0	0
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 3.00000i	− 0.103944i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	26.0000	0.897620	0.448810	−	0.893627i	$-0.351848\pi$
$839$	26.0000	0.897620	0.448810	+	0.893627i	$0.351848\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	20.0000i	0.687208i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−6.00000	−0.205677
$852$	0	0
$853$	− 31.0000i	− 1.06142i	−0.847554	−	0.530710i	$-0.821925\pi$
$853$	− 31.0000i	− 1.06142i	0.847554	−	0.530710i	$-0.178075\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.00000i	0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$	6.00000i	0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$	0	0
$859$	22.0000	0.750630	0.375315	−	0.926897i	$-0.377534\pi$
$859$	22.0000	0.750630	0.375315	+	0.926897i	$0.377534\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	21.0000i	0.714848i	0.933942	+	0.357424i	$0.116345\pi$
$863$	21.0000i	0.714848i	−0.933942	+	0.357424i	$0.883655\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	15.0000	0.508840
$870$	0	0
$871$	−4.00000	−0.135535
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	53.0000i	1.78968i	0.446384	+	0.894841i	$0.352711\pi$
$877$	53.0000i	1.78968i	−0.446384	+	0.894841i	$0.647289\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	− 52.0000i	− 1.74994i	−0.484178	−	0.874970i	$-0.660881\pi$
$883$	− 52.0000i	− 1.74994i	0.484178	−	0.874970i	$-0.339119\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19.0000i	0.637958i	0.947762	+	0.318979i	$0.103340\pi$
$887$	19.0000i	0.637958i	−0.947762	+	0.318979i	$0.896660\pi$
$888$	0	0
$889$	12.0000	0.402467
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 28.0000i	− 0.936984i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−5.00000	−0.166759
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	45.0000i	1.49420i	0.664711	+	0.747100i	$0.268555\pi$
$907$	45.0000i	1.49420i	−0.664711	+	0.747100i	$0.731445\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	56.0000	1.85536	0.927681	−	0.373373i	$-0.121799\pi$
$911$	56.0000	1.85536	0.927681	+	0.373373i	$0.121799\pi$
$912$	0	0
$913$	2.00000i	0.0661903i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 18.0000i	− 0.594412i
$918$	0	0
$919$	−35.0000	−1.15454	−0.577272	−	0.816552i	$-0.695883\pi$
$919$	−35.0000	−1.15454	−0.577272	+	0.816552i	$0.695883\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 12.0000i	− 0.394985i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−20.0000	−0.656179	−0.328089	−	0.944647i	$-0.606405\pi$
$929$	−20.0000	−0.656179	−0.328089	+	0.944647i	$0.606405\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	14.0000i	0.457360i	0.973502	+	0.228680i	$0.0734410\pi$
$937$	14.0000i	0.457360i	−0.973502	+	0.228680i	$0.926559\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	17.0000	0.554184	0.277092	−	0.960843i	$-0.410629\pi$
$941$	17.0000	0.554184	0.277092	+	0.960843i	$0.410629\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 14.0000i	− 0.454939i	−0.973785	−	0.227469i	$-0.926955\pi$
$947$	− 14.0000i	− 0.454939i	0.973785	−	0.227469i	$-0.0730452\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 21.0000i	− 0.680257i	−0.940379	−	0.340128i	$-0.889529\pi$
$953$	− 21.0000i	− 0.680257i	0.940379	−	0.340128i	$-0.110471\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	44.0000	1.42083
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	6.00000i	0.192947i	0.995336	+	0.0964735i	$0.0307563\pi$
$967$	6.00000i	0.192947i	−0.995336	+	0.0964735i	$0.969244\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−53.0000	−1.70085	−0.850425	−	0.526096i	$-0.823655\pi$
$971$	−53.0000	−1.70085	−0.850425	+	0.526096i	$0.823655\pi$
$972$	0	0
$973$	− 12.0000i	− 0.384702i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 61.0000i	− 1.95156i	−0.218748	−	0.975781i	$-0.570197\pi$
$977$	− 61.0000i	− 1.95156i	0.218748	−	0.975781i	$-0.429803\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 33.0000i	− 1.05254i	−0.850319	−	0.526268i	$-0.823591\pi$
$983$	− 33.0000i	− 1.05254i	0.850319	−	0.526268i	$-0.176409\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7.00000	0.222587
$990$	0	0
$991$	−55.0000	−1.74713	−0.873566	−	0.486705i	$-0.838199\pi$
$991$	−55.0000	−1.74713	−0.873566	+	0.486705i	$0.838199\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 39.0000i	− 1.23514i	−0.786515	−	0.617571i	$-0.788117\pi$
$997$	− 39.0000i	− 1.23514i	0.786515	−	0.617571i	$-0.211883\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5400.2.f.m.649.1		2
3.2	odd	2		5400.2.f.p.649.1		2
5.2	odd	4		1080.2.a.f.1.1	✓	1
5.3	odd	4		5400.2.a.m.1.1		1
5.4	even	2	inner	5400.2.f.m.649.2		2
15.2	even	4		1080.2.a.k.1.1	yes	1
15.8	even	4		5400.2.a.n.1.1		1
15.14	odd	2		5400.2.f.p.649.2		2
20.7	even	4		2160.2.a.d.1.1		1
40.27	even	4		8640.2.a.bk.1.1		1
40.37	odd	4		8640.2.a.cb.1.1		1
45.2	even	12		3240.2.q.c.1081.1		2
45.7	odd	12		3240.2.q.o.1081.1		2
45.22	odd	12		3240.2.q.o.2161.1		2
45.32	even	12		3240.2.q.c.2161.1		2
60.47	odd	4		2160.2.a.n.1.1		1
120.77	even	4		8640.2.a.v.1.1		1
120.107	odd	4		8640.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.a.f.1.1	✓	1	5.2	odd	4
1080.2.a.k.1.1	yes	1	15.2	even	4
2160.2.a.d.1.1		1	20.7	even	4
2160.2.a.n.1.1		1	60.47	odd	4
3240.2.q.c.1081.1		2	45.2	even	12
3240.2.q.c.2161.1		2	45.32	even	12
3240.2.q.o.1081.1		2	45.7	odd	12
3240.2.q.o.2161.1		2	45.22	odd	12
5400.2.a.m.1.1		1	5.3	odd	4
5400.2.a.n.1.1		1	15.8	even	4
5400.2.f.m.649.1		2	1.1	even	1	trivial
5400.2.f.m.649.2		2	5.4	even	2	inner
5400.2.f.p.649.1		2	3.2	odd	2
5400.2.f.p.649.2		2	15.14	odd	2
8640.2.a.i.1.1		1	120.107	odd	4
8640.2.a.v.1.1		1	120.77	even	4
8640.2.a.bk.1.1		1	40.27	even	4
8640.2.a.cb.1.1		1	40.37	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5400 = 2^{3} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5400.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{7} -1.00000 q^{11} +1.00000i q^{13} -1.00000i q^{17} -4.00000 q^{19} -1.00000i q^{23} -5.00000 q^{29} +1.00000 q^{31} -6.00000i q^{37} +7.00000i q^{43} +7.00000i q^{47} +3.00000 q^{49} +12.0000i q^{53} -4.00000 q^{59} +10.0000 q^{61} +4.00000i q^{67} -12.0000 q^{71} +6.00000i q^{73} +2.00000i q^{77} -15.0000 q^{79} -2.00000i q^{83} -12.0000 q^{89} +2.00000 q^{91} -10.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{11} - 8 q^{19} - 10 q^{29} + 2 q^{31} + 6 q^{49} - 8 q^{59} + 20 q^{61} - 24 q^{71} - 30 q^{79} - 24 q^{89} + 4 q^{91}+O(q^{100}) \) 2 * q - 2 * q^11 - 8 * q^19 - 10 * q^29 + 2 * q^31 + 6 * q^49 - 8 * q^59 + 20 * q^61 - 24 * q^71 - 30 * q^79 - 24 * q^89 + 4 * q^91

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