LMFDB - Embedded newform 2448.2.c.i.577.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2448,2,Mod(577,2448)] 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("2448.577"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2448 = 2^{4} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2448.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,-6,0,-4,0,0,0,0,0,6,0,0,0,0, 0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(35)] == traces)

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

Self dual:	no
Analytic conductor:	$19.5473784148$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 306)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	2448.577
Dual form			2448.2.c.i.577.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{5} +4.24264i q^{7} -2.82843i q^{11} +2.00000 q^{13} +(-3.00000 - 2.82843i) q^{17} -2.00000 q^{19} -7.07107i q^{23} +3.00000 q^{25} +7.07107i q^{29} -4.24264i q^{31} +6.00000 q^{35} -4.24264i q^{37} -5.65685i q^{41} +4.00000 q^{43} +12.0000 q^{47} -11.0000 q^{49} +6.00000 q^{53} -4.00000 q^{55} -6.00000 q^{59} +12.7279i q^{61} -2.82843i q^{65} +10.0000 q^{67} +1.41421i q^{71} +12.0000 q^{77} -12.7279i q^{79} +6.00000 q^{83} +(-4.00000 + 4.24264i) q^{85} +6.00000 q^{89} +8.48528i q^{91} +2.82843i q^{95} -8.48528i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{13} - 6 q^{17} - 4 q^{19} + 6 q^{25} + 12 q^{35} + 8 q^{43} + 24 q^{47} - 22 q^{49} + 12 q^{53} - 8 q^{55} - 12 q^{59} + 20 q^{67} + 24 q^{77} + 12 q^{83} - 8 q^{85} + 12 q^{89}+O(q^{100}) $ 2 * q + 4 * q^13 - 6 * q^17 - 4 * q^19 + 6 * q^25 + 12 * q^35 + 8 * q^43 + 24 * q^47 - 22 * q^49 + 12 * q^53 - 8 * q^55 - 12 * q^59 + 20 * q^67 + 24 * q^77 + 12 * q^83 - 8 * q^85 + 12 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2448\mathbb{Z}\right)^\times$.

$n$	$613$	$1361$	$1873$	$2143$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$5$		−	1.41421i		−	0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$		−	1.41421i		−	0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$		0			0
$7$			4.24264i			1.60357i	0.597614	+	0.801784i	$0.296115\pi$
$7$			4.24264i			1.60357i	−0.597614	+	0.801784i	$0.703885\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	2.82843i		−	0.852803i	−0.904534	−	0.426401i	$-0.859781\pi$
$11$		−	2.82843i		−	0.852803i	0.904534	−	0.426401i	$-0.140219\pi$
$12$		0			0
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.00000	−	2.82843i	−0.727607	−	0.685994i
$17$	−3.00000	−	2.82843i	−0.727607	−	0.685994i
$18$		0			0
$19$	−2.00000			−0.458831			−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000			−0.458831			−0.229416	+	0.973329i	$0.573682\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	7.07107i		−	1.47442i	−0.675664	−	0.737210i	$-0.736143\pi$
$23$		−	7.07107i		−	1.47442i	0.675664	−	0.737210i	$-0.263857\pi$
$24$		0			0
$25$	3.00000			0.600000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			7.07107i			1.31306i	0.754298	+	0.656532i	$0.227977\pi$
$29$			7.07107i			1.31306i	−0.754298	+	0.656532i	$0.772023\pi$
$30$		0			0
$31$		−	4.24264i		−	0.762001i	−0.924575	−	0.381000i	$-0.875580\pi$
$31$		−	4.24264i		−	0.762001i	0.924575	−	0.381000i	$-0.124420\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.00000			1.01419
$36$		0			0
$37$		−	4.24264i		−	0.697486i	−0.937218	−	0.348743i	$-0.886609\pi$
$37$		−	4.24264i		−	0.697486i	0.937218	−	0.348743i	$-0.113391\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	5.65685i		−	0.883452i	−0.897150	−	0.441726i	$-0.854366\pi$
$41$		−	5.65685i		−	0.883452i	0.897150	−	0.441726i	$-0.145634\pi$
$42$		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
$45$		0			0
$46$		0			0
$47$	12.0000			1.75038			0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000			1.75038			0.875190	+	0.483779i	$0.160736\pi$
$48$		0			0
$49$	−11.0000			−1.57143
$50$		0			0
$51$		0			0
$52$		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
$55$	−4.00000			−0.539360
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.00000			−0.781133			−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000			−0.781133			−0.390567	+	0.920575i	$0.627721\pi$
$60$		0			0
$61$			12.7279i			1.62964i	0.579712	+	0.814822i	$0.303165\pi$
$61$			12.7279i			1.62964i	−0.579712	+	0.814822i	$0.696835\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	2.82843i		−	0.350823i
$66$		0			0
$67$	10.0000			1.22169			0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000			1.22169			0.610847	+	0.791748i	$0.290829\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			1.41421i			0.167836i	0.996473	+	0.0839181i	$0.0267434\pi$
$71$			1.41421i			0.167836i	−0.996473	+	0.0839181i	$0.973257\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	12.0000			1.36753
$78$		0			0
$79$		−	12.7279i		−	1.43200i	−0.698099	−	0.716002i	$-0.745970\pi$
$79$		−	12.7279i		−	1.43200i	0.698099	−	0.716002i	$-0.254030\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		0			0
$85$	−4.00000	+	4.24264i	−0.433861	+	0.460179i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		0			0
$91$			8.48528i			0.889499i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			2.82843i			0.290191i
$96$		0			0
$97$		−	8.48528i		−	0.861550i	−0.902459	−	0.430775i	$-0.858240\pi$
$97$		−	8.48528i		−	0.861550i	0.902459	−	0.430775i	$-0.141760\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	18.0000			1.79107			0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000			1.79107			0.895533	+	0.444994i	$0.146794\pi$
$102$		0			0
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			0.197066	+	0.980390i	$0.436859\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	2.82843i		−	0.273434i	−0.990610	−	0.136717i	$-0.956345\pi$
$107$		−	2.82843i		−	0.273434i	0.990610	−	0.136717i	$-0.0436552\pi$
$108$		0			0
$109$		−	4.24264i		−	0.406371i	−0.979140	−	0.203186i	$-0.934871\pi$
$109$		−	4.24264i		−	0.406371i	0.979140	−	0.203186i	$-0.0651295\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	14.1421i		−	1.33038i	−0.746674	−	0.665190i	$-0.768350\pi$
$113$		−	14.1421i		−	1.33038i	0.746674	−	0.665190i	$-0.231650\pi$
$114$		0			0
$115$	−10.0000			−0.932505
$116$		0			0
$117$		0			0
$118$		0			0
$119$	12.0000	−	12.7279i	1.10004	−	1.16677i
$120$		0			0
$121$	3.00000			0.272727
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	11.3137i		−	1.01193i
$126$		0			0
$127$	−20.0000			−1.77471			−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000			−1.77471			−0.887357	+	0.461084i	$0.847461\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			14.1421i			1.23560i	0.786334	+	0.617802i	$0.211977\pi$
$131$			14.1421i			1.23560i	−0.786334	+	0.617802i	$0.788023\pi$
$132$		0			0
$133$		−	8.48528i		−	0.735767i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−12.0000			−1.02523			−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000			−1.02523			−0.512615	+	0.858619i	$0.671323\pi$
$138$		0			0
$139$		−	8.48528i		−	0.719712i	−0.933008	−	0.359856i	$-0.882826\pi$
$139$		−	8.48528i		−	0.719712i	0.933008	−	0.359856i	$-0.117174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	5.65685i		−	0.473050i
$144$		0			0
$145$	10.0000			0.830455
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−18.0000			−1.47462			−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000			−1.47462			−0.737309	+	0.675556i	$0.763904\pi$
$150$		0			0
$151$	16.0000			1.30206			0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000			1.30206			0.651031	+	0.759051i	$0.274337\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.00000			−0.481932
$156$		0			0
$157$	14.0000			1.11732			0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000			1.11732			0.558661	+	0.829396i	$0.311315\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	30.0000			2.36433
$162$		0			0
$163$			16.9706i			1.32924i	0.747183	+	0.664619i	$0.231406\pi$
$163$			16.9706i			1.32924i	−0.747183	+	0.664619i	$0.768594\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	24.0416i		−	1.86040i	−0.367057	−	0.930199i	$-0.619634\pi$
$167$		−	24.0416i		−	1.86040i	0.367057	−	0.930199i	$-0.380366\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	9.89949i		−	0.752645i	−0.926489	−	0.376322i	$-0.877189\pi$
$173$		−	9.89949i		−	0.752645i	0.926489	−	0.376322i	$-0.122811\pi$
$174$		0			0
$175$			12.7279i			0.962140i
$176$		0			0
$177$		0			0
$178$		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
$181$		−	21.2132i		−	1.57676i	−0.615185	−	0.788382i	$-0.710919\pi$
$181$		−	21.2132i		−	1.57676i	0.615185	−	0.788382i	$-0.289081\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.00000			−0.441129
$186$		0			0
$187$	−8.00000	+	8.48528i	−0.585018	+	0.620505i
$188$		0			0
$189$		0			0
$190$		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
$193$			25.4558i			1.83235i	0.400776	+	0.916176i	$0.368740\pi$
$193$			25.4558i			1.83235i	−0.400776	+	0.916176i	$0.631260\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	9.89949i		−	0.705310i	−0.935753	−	0.352655i	$-0.885279\pi$
$197$		−	9.89949i		−	0.705310i	0.935753	−	0.352655i	$-0.114721\pi$
$198$		0			0
$199$		−	4.24264i		−	0.300753i	−0.988629	−	0.150376i	$-0.951951\pi$
$199$		−	4.24264i		−	0.300753i	0.988629	−	0.150376i	$-0.0480486\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−30.0000			−2.10559
$204$		0			0
$205$	−8.00000			−0.558744
$206$		0			0
$207$		0			0
$208$		0			0
$209$			5.65685i			0.391293i
$210$		0			0
$211$		−	8.48528i		−	0.584151i	−0.956395	−	0.292075i	$-0.905654\pi$
$211$		−	8.48528i		−	0.584151i	0.956395	−	0.292075i	$-0.0943458\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	5.65685i		−	0.385794i
$216$		0			0
$217$	18.0000			1.22192
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.00000	−	5.65685i	−0.403604	−	0.380521i
$222$		0			0
$223$	−8.00000			−0.535720			−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000			−0.535720			−0.267860	+	0.963458i	$0.586316\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			22.6274i			1.50183i	0.660396	+	0.750917i	$0.270388\pi$
$227$			22.6274i			1.50183i	−0.660396	+	0.750917i	$0.729612\pi$
$228$		0			0
$229$	−10.0000			−0.660819			−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000			−0.660819			−0.330409	+	0.943838i	$0.607187\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			2.82843i			0.185296i	0.995699	+	0.0926482i	$0.0295332\pi$
$233$			2.82843i			0.185296i	−0.995699	+	0.0926482i	$0.970467\pi$
$234$		0			0
$235$		−	16.9706i		−	1.10704i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	24.0000			1.55243			0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000			1.55243			0.776215	+	0.630468i	$0.217137\pi$
$240$		0			0
$241$		−	8.48528i		−	0.546585i	−0.961931	−	0.273293i	$-0.911887\pi$
$241$		−	8.48528i		−	0.546585i	0.961931	−	0.273293i	$-0.0881127\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			15.5563i			0.993859i
$246$		0			0
$247$	−4.00000			−0.254514
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$	−20.0000			−1.25739
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$	18.0000			1.11847
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.0000			−0.739952			−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000			−0.739952			−0.369976	+	0.929041i	$0.620634\pi$
$264$		0			0
$265$		−	8.48528i		−	0.521247i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	18.3848i		−	1.12094i	−0.828175	−	0.560470i	$-0.810621\pi$
$269$		−	18.3848i		−	1.12094i	0.828175	−	0.560470i	$-0.189379\pi$
$270$		0			0
$271$	16.0000			0.971931			0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000			0.971931			0.485965	+	0.873978i	$0.338468\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	8.48528i		−	0.511682i
$276$		0			0
$277$			21.2132i			1.27458i	0.770625	+	0.637289i	$0.219944\pi$
$277$			21.2132i			1.27458i	−0.770625	+	0.637289i	$0.780056\pi$
$278$		0			0
$279$		0			0
$280$		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
$283$		−	8.48528i		−	0.504398i	−0.967675	−	0.252199i	$-0.918846\pi$
$283$		−	8.48528i		−	0.504398i	0.967675	−	0.252199i	$-0.0811537\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	24.0000			1.41668
$288$		0			0
$289$	1.00000	+	16.9706i	0.0588235	+	0.998268i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−18.0000			−1.05157			−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000			−1.05157			−0.525786	+	0.850617i	$0.676229\pi$
$294$		0			0
$295$			8.48528i			0.494032i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	14.1421i		−	0.817861i
$300$		0			0
$301$			16.9706i			0.978167i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	18.0000			1.03068
$306$		0			0
$307$	4.00000			0.228292			0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000			0.228292			0.114146	+	0.993464i	$0.463587\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			1.41421i			0.0801927i	0.999196	+	0.0400963i	$0.0127665\pi$
$311$			1.41421i			0.0801927i	−0.999196	+	0.0400963i	$0.987234\pi$
$312$		0			0
$313$			25.4558i			1.43885i	0.694570	+	0.719425i	$0.255594\pi$
$313$			25.4558i			1.43885i	−0.694570	+	0.719425i	$0.744406\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			32.5269i			1.82689i	0.406958	+	0.913447i	$0.366589\pi$
$317$			32.5269i			1.82689i	−0.406958	+	0.913447i	$0.633411\pi$
$318$		0			0
$319$	20.0000			1.11979
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6.00000	+	5.65685i	0.333849	+	0.314756i
$324$		0			0
$325$	6.00000			0.332820
$326$		0			0
$327$		0			0
$328$		0			0
$329$			50.9117i			2.80685i
$330$		0			0
$331$	4.00000			0.219860			0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000			0.219860			0.109930	+	0.993939i	$0.464937\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	14.1421i		−	0.772667i
$336$		0			0
$337$		−	33.9411i		−	1.84889i	−0.381314	−	0.924445i	$-0.624528\pi$
$337$		−	33.9411i		−	1.84889i	0.381314	−	0.924445i	$-0.375472\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−12.0000			−0.649836
$342$		0			0
$343$		−	16.9706i		−	0.916324i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			22.6274i			1.21470i	0.794433	+	0.607352i	$0.207768\pi$
$347$			22.6274i			1.21470i	−0.794433	+	0.607352i	$0.792232\pi$
$348$		0			0
$349$	26.0000			1.39175			0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000			1.39175			0.695874	+	0.718164i	$0.255017\pi$
$350$		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
$355$	2.00000			0.106149
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12.0000			0.633336			0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000			0.633336			0.316668	+	0.948536i	$0.397436\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$			21.2132i			1.10732i	0.832743	+	0.553660i	$0.186769\pi$
$367$			21.2132i			1.10732i	−0.832743	+	0.553660i	$0.813231\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			25.4558i			1.32160i
$372$		0			0
$373$	−10.0000			−0.517780			−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000			−0.517780			−0.258890	+	0.965907i	$0.583357\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			14.1421i			0.728357i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−12.0000			−0.613171			−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000			−0.613171			−0.306586	+	0.951843i	$0.599187\pi$
$384$		0			0
$385$		−	16.9706i		−	0.864900i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−18.0000			−0.912636			−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000			−0.912636			−0.456318	+	0.889817i	$0.650832\pi$
$390$		0			0
$391$	−20.0000	+	21.2132i	−1.01144	+	1.07280i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−18.0000			−0.905678
$396$		0			0
$397$		−	21.2132i		−	1.06466i	−0.846537	−	0.532330i	$-0.821317\pi$
$397$		−	21.2132i		−	1.06466i	0.846537	−	0.532330i	$-0.178683\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			11.3137i			0.564980i	0.959270	+	0.282490i	$0.0911603\pi$
$401$			11.3137i			0.564980i	−0.959270	+	0.282490i	$0.908840\pi$
$402$		0			0
$403$		−	8.48528i		−	0.422682i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−12.0000			−0.594818
$408$		0			0
$409$	−16.0000			−0.791149			−0.395575	−	0.918434i	$-0.629455\pi$
$409$	−16.0000			−0.791149			−0.395575	+	0.918434i	$0.629455\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	25.4558i		−	1.25260i
$414$		0			0
$415$		−	8.48528i		−	0.416526i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			22.6274i			1.10542i	0.833373	+	0.552711i	$0.186407\pi$
$419$			22.6274i			1.10542i	−0.833373	+	0.552711i	$0.813593\pi$
$420$		0			0
$421$	26.0000			1.26716			0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000			1.26716			0.633581	+	0.773676i	$0.281584\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−9.00000	−	8.48528i	−0.436564	−	0.411597i
$426$		0			0
$427$	−54.0000			−2.61324
$428$		0			0
$429$		0			0
$430$		0			0
$431$			1.41421i			0.0681203i	0.999420	+	0.0340601i	$0.0108438\pi$
$431$			1.41421i			0.0681203i	−0.999420	+	0.0340601i	$0.989156\pi$
$432$		0			0
$433$	−16.0000			−0.768911			−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000			−0.768911			−0.384455	+	0.923144i	$0.625611\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			14.1421i			0.676510i
$438$		0			0
$439$		−	4.24264i		−	0.202490i	−0.994862	−	0.101245i	$-0.967717\pi$
$439$		−	4.24264i		−	0.202490i	0.994862	−	0.101245i	$-0.0322826\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.0000			−0.570137			−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000			−0.570137			−0.285069	+	0.958507i	$0.592016\pi$
$444$		0			0
$445$		−	8.48528i		−	0.402241i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	5.65685i		−	0.266963i	−0.991051	−	0.133482i	$-0.957384\pi$
$449$		−	5.65685i		−	0.266963i	0.991051	−	0.133482i	$-0.0426157\pi$
$450$		0			0
$451$	−16.0000			−0.753411
$452$		0			0
$453$		0			0
$454$		0			0
$455$	12.0000			0.562569
$456$		0			0
$457$	8.00000			0.374224			0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000			0.374224			0.187112	+	0.982339i	$0.440087\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.00000			0.279448			0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000			0.279448			0.139724	+	0.990190i	$0.455378\pi$
$462$		0			0
$463$	4.00000			0.185896			0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000			0.185896			0.0929479	+	0.995671i	$0.470371\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−36.0000			−1.66588			−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000			−1.66588			−0.832941	+	0.553362i	$0.813345\pi$
$468$		0			0
$469$			42.4264i			1.95907i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	11.3137i		−	0.520205i
$474$		0			0
$475$	−6.00000			−0.275299
$476$		0			0
$477$		0			0
$478$		0			0
$479$			9.89949i			0.452319i	0.974090	+	0.226160i	$0.0726171\pi$
$479$			9.89949i			0.452319i	−0.974090	+	0.226160i	$0.927383\pi$
$480$		0			0
$481$		−	8.48528i		−	0.386896i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−12.0000			−0.544892
$486$		0			0
$487$			4.24264i			0.192252i	0.995369	+	0.0961262i	$0.0306452\pi$
$487$			4.24264i			0.192252i	−0.995369	+	0.0961262i	$0.969355\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−36.0000			−1.62466			−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000			−1.62466			−0.812329	+	0.583200i	$0.801800\pi$
$492$		0			0
$493$	20.0000	−	21.2132i	0.900755	−	0.955395i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.00000			−0.269137
$498$		0			0
$499$			8.48528i			0.379853i	0.981798	+	0.189927i	$0.0608250\pi$
$499$			8.48528i			0.379853i	−0.981798	+	0.189927i	$0.939175\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	15.5563i		−	0.693623i	−0.937935	−	0.346812i	$-0.887264\pi$
$503$		−	15.5563i		−	0.693623i	0.937935	−	0.346812i	$-0.112736\pi$
$504$		0			0
$505$		−	25.4558i		−	1.13277i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6.00000			−0.265945			−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000			−0.265945			−0.132973	+	0.991120i	$0.542452\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	5.65685i		−	0.249271i
$516$		0			0
$517$		−	33.9411i		−	1.49273i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	5.65685i		−	0.247831i	−0.992293	−	0.123916i	$-0.960455\pi$
$521$		−	5.65685i		−	0.247831i	0.992293	−	0.123916i	$-0.0395452\pi$
$522$		0			0
$523$	34.0000			1.48672			0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000			1.48672			0.743358	+	0.668894i	$0.233232\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12.0000	+	12.7279i	−0.522728	+	0.554437i
$528$		0			0
$529$	−27.0000			−1.17391
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	11.3137i		−	0.490051i
$534$		0			0
$535$	−4.00000			−0.172935
$536$		0			0
$537$		0			0
$538$		0			0
$539$			31.1127i			1.34012i
$540$		0			0
$541$			21.2132i			0.912027i	0.889973	+	0.456013i	$0.150723\pi$
$541$			21.2132i			0.912027i	−0.889973	+	0.456013i	$0.849277\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−6.00000			−0.257012
$546$		0			0
$547$			8.48528i			0.362804i	0.983409	+	0.181402i	$0.0580636\pi$
$547$			8.48528i			0.362804i	−0.983409	+	0.181402i	$0.941936\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	14.1421i		−	0.602475i
$552$		0			0
$553$	54.0000			2.29631
$554$		0			0
$555$		0			0
$556$		0			0
$557$	18.0000			0.762684			0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000			0.762684			0.381342	+	0.924434i	$0.375462\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−30.0000			−1.26435			−0.632175	−	0.774826i	$-0.717837\pi$
$563$	−30.0000			−1.26435			−0.632175	+	0.774826i	$0.717837\pi$
$564$		0			0
$565$	−20.0000			−0.841406
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−36.0000			−1.50920			−0.754599	−	0.656186i	$-0.772169\pi$
$569$	−36.0000			−1.50920			−0.754599	+	0.656186i	$0.772169\pi$
$570$		0			0
$571$		−	16.9706i		−	0.710196i	−0.934829	−	0.355098i	$-0.884448\pi$
$571$		−	16.9706i		−	0.710196i	0.934829	−	0.355098i	$-0.115552\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	21.2132i		−	0.884652i
$576$		0			0
$577$	2.00000			0.0832611			0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000			0.0832611			0.0416305	+	0.999133i	$0.486745\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			25.4558i			1.05609i
$582$		0			0
$583$		−	16.9706i		−	0.702849i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12.0000			−0.495293			−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000			−0.495293			−0.247647	+	0.968850i	$0.579657\pi$
$588$		0			0
$589$			8.48528i			0.349630i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$	−18.0000	−	16.9706i	−0.737928	−	0.695725i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$			33.9411i			1.38449i	0.721664	+	0.692244i	$0.243378\pi$
$601$			33.9411i			1.38449i	−0.721664	+	0.692244i	$0.756622\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	4.24264i		−	0.172488i
$606$		0			0
$607$		−	12.7279i		−	0.516610i	−0.966063	−	0.258305i	$-0.916836\pi$
$607$		−	12.7279i		−	0.516610i	0.966063	−	0.258305i	$-0.0831640\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	24.0000			0.970936
$612$		0			0
$613$	−46.0000			−1.85792			−0.928961	−	0.370177i	$-0.879297\pi$
$613$	−46.0000			−1.85792			−0.928961	+	0.370177i	$0.879297\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			2.82843i			0.113868i	0.998378	+	0.0569341i	$0.0181325\pi$
$617$			2.82843i			0.113868i	−0.998378	+	0.0569341i	$0.981868\pi$
$618$		0			0
$619$			33.9411i			1.36421i	0.731255	+	0.682105i	$0.238935\pi$
$619$			33.9411i			1.36421i	−0.731255	+	0.682105i	$0.761065\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			25.4558i			1.01987i
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.0000	+	12.7279i	−0.478471	+	0.507495i
$630$		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
$635$			28.2843i			1.12243i
$636$		0			0
$637$	−22.0000			−0.871672
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	22.6274i		−	0.893729i	−0.894602	−	0.446865i	$-0.852541\pi$
$641$		−	22.6274i		−	0.893729i	0.894602	−	0.446865i	$-0.147459\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−12.0000			−0.471769			−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000			−0.471769			−0.235884	+	0.971781i	$0.575799\pi$
$648$		0			0
$649$			16.9706i			0.666153i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	1.41421i		−	0.0553425i	−0.999617	−	0.0276712i	$-0.991191\pi$
$653$		−	1.41421i		−	0.0553425i	0.999617	−	0.0276712i	$-0.00880915\pi$
$654$		0			0
$655$	20.0000			0.781465
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−30.0000			−1.16863			−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000			−1.16863			−0.584317	+	0.811525i	$0.698638\pi$
$660$		0			0
$661$	14.0000			0.544537			0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000			0.544537			0.272268	+	0.962221i	$0.412226\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−12.0000			−0.465340
$666$		0			0
$667$	50.0000			1.93601
$668$		0			0
$669$		0			0
$670$		0			0
$671$	36.0000			1.38976
$672$		0			0
$673$		−	25.4558i		−	0.981251i	−0.871371	−	0.490625i	$-0.836768\pi$
$673$		−	25.4558i		−	0.981251i	0.871371	−	0.490625i	$-0.163232\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			49.4975i			1.90234i	0.308664	+	0.951171i	$0.400118\pi$
$677$			49.4975i			1.90234i	−0.308664	+	0.951171i	$0.599882\pi$
$678$		0			0
$679$	36.0000			1.38155
$680$		0			0
$681$		0			0
$682$		0			0
$683$			5.65685i			0.216454i	0.994126	+	0.108227i	$0.0345173\pi$
$683$			5.65685i			0.216454i	−0.994126	+	0.108227i	$0.965483\pi$
$684$		0			0
$685$			16.9706i			0.648412i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	12.0000			0.457164
$690$		0			0
$691$		−	25.4558i		−	0.968386i	−0.874961	−	0.484193i	$-0.839113\pi$
$691$		−	25.4558i		−	0.968386i	0.874961	−	0.484193i	$-0.160887\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−12.0000			−0.455186
$696$		0			0
$697$	−16.0000	+	16.9706i	−0.606043	+	0.642806i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	42.0000			1.58632			0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000			1.58632			0.793159	+	0.609015i	$0.208435\pi$
$702$		0			0
$703$			8.48528i			0.320028i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			76.3675i			2.87210i
$708$		0			0
$709$			12.7279i			0.478007i	0.971019	+	0.239004i	$0.0768208\pi$
$709$			12.7279i			0.478007i	−0.971019	+	0.239004i	$0.923179\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−30.0000			−1.12351
$714$		0			0
$715$	−8.00000			−0.299183
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	32.5269i		−	1.21305i	−0.795065	−	0.606525i	$-0.792563\pi$
$719$		−	32.5269i		−	1.21305i	0.795065	−	0.606525i	$-0.207437\pi$
$720$		0			0
$721$			16.9706i			0.632017i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			21.2132i			0.787839i
$726$		0			0
$727$	−20.0000			−0.741759			−0.370879	−	0.928681i	$-0.620944\pi$
$727$	−20.0000			−0.741759			−0.370879	+	0.928681i	$0.620944\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12.0000	−	11.3137i	−0.443836	−	0.418453i
$732$		0			0
$733$	−10.0000			−0.369358			−0.184679	−	0.982799i	$-0.559125\pi$
$733$	−10.0000			−0.369358			−0.184679	+	0.982799i	$0.559125\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	28.2843i		−	1.04186i
$738$		0			0
$739$	10.0000			0.367856			0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000			0.367856			0.183928	+	0.982940i	$0.441119\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			26.8701i			0.985767i	0.870095	+	0.492883i	$0.164057\pi$
$743$			26.8701i			0.985767i	−0.870095	+	0.492883i	$0.835943\pi$
$744$		0			0
$745$			25.4558i			0.932630i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	12.0000			0.438470
$750$		0			0
$751$			29.6985i			1.08371i	0.840471	+	0.541857i	$0.182278\pi$
$751$			29.6985i			1.08371i	−0.840471	+	0.541857i	$0.817722\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	22.6274i		−	0.823496i
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−12.0000			−0.435000			−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000			−0.435000			−0.217500	+	0.976060i	$0.569790\pi$
$762$		0			0
$763$	18.0000			0.651644
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−12.0000			−0.433295
$768$		0			0
$769$	50.0000			1.80305			0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000			1.80305			0.901523	+	0.432731i	$0.142450\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	18.0000			0.647415			0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000			0.647415			0.323708	+	0.946157i	$0.395071\pi$
$774$		0			0
$775$		−	12.7279i		−	0.457200i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			11.3137i			0.405356i
$780$		0			0
$781$	4.00000			0.143131
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	19.7990i		−	0.706656i
$786$		0			0
$787$			25.4558i			0.907403i	0.891154	+	0.453701i	$0.149897\pi$
$787$			25.4558i			0.907403i	−0.891154	+	0.453701i	$0.850103\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	60.0000			2.13335
$792$		0			0
$793$			25.4558i			0.903964i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	30.0000			1.06265			0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000			1.06265			0.531327	+	0.847167i	$0.321693\pi$
$798$		0			0
$799$	−36.0000	−	33.9411i	−1.27359	−	1.20075i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		−	42.4264i		−	1.49533i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			45.2548i			1.59108i	0.605904	+	0.795538i	$0.292811\pi$
$809$			45.2548i			1.59108i	−0.605904	+	0.795538i	$0.707189\pi$
$810$		0			0
$811$			33.9411i			1.19183i	0.803046	+	0.595917i	$0.203211\pi$
$811$			33.9411i			1.19183i	−0.803046	+	0.595917i	$0.796789\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	24.0000			0.840683
$816$		0			0
$817$	−8.00000			−0.279885
$818$		0			0
$819$		0			0
$820$		0			0
$821$			15.5563i			0.542920i	0.962450	+	0.271460i	$0.0875065\pi$
$821$			15.5563i			0.542920i	−0.962450	+	0.271460i	$0.912493\pi$
$822$		0			0
$823$			4.24264i			0.147889i	0.997262	+	0.0739446i	$0.0235588\pi$
$823$			4.24264i			0.147889i	−0.997262	+	0.0739446i	$0.976441\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	19.7990i		−	0.688478i	−0.938882	−	0.344239i	$-0.888137\pi$
$827$		−	19.7990i		−	0.688478i	0.938882	−	0.344239i	$-0.111863\pi$
$828$		0			0
$829$	14.0000			0.486240			0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000			0.486240			0.243120	+	0.969996i	$0.421829\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	33.0000	+	31.1127i	1.14338	+	1.07799i
$834$		0			0
$835$	−34.0000			−1.17662
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	15.5563i		−	0.537065i	−0.963271	−	0.268532i	$-0.913461\pi$
$839$		−	15.5563i		−	0.537065i	0.963271	−	0.268532i	$-0.0865386\pi$
$840$		0			0
$841$	−21.0000			−0.724138
$842$		0			0
$843$		0			0
$844$		0			0
$845$			12.7279i			0.437854i
$846$		0			0
$847$			12.7279i			0.437337i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−30.0000			−1.02839
$852$		0			0
$853$		−	29.6985i		−	1.01686i	−0.861104	−	0.508428i	$-0.830227\pi$
$853$		−	29.6985i		−	1.01686i	0.861104	−	0.508428i	$-0.169773\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	22.6274i		−	0.772938i	−0.922302	−	0.386469i	$-0.873695\pi$
$857$		−	22.6274i		−	0.772938i	0.922302	−	0.386469i	$-0.126305\pi$
$858$		0			0
$859$	4.00000			0.136478			0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000			0.136478			0.0682391	+	0.997669i	$0.478262\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−24.0000			−0.816970			−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000			−0.816970			−0.408485	+	0.912765i	$0.633943\pi$
$864$		0			0
$865$	−14.0000			−0.476014
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−36.0000			−1.22122
$870$		0			0
$871$	20.0000			0.677674
$872$		0			0
$873$		0			0
$874$		0			0
$875$	48.0000			1.62270
$876$		0			0
$877$			12.7279i			0.429791i	0.976637	+	0.214896i	$0.0689412\pi$
$877$			12.7279i			0.429791i	−0.976637	+	0.214896i	$0.931059\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	14.1421i		−	0.476461i	−0.971209	−	0.238230i	$-0.923433\pi$
$881$		−	14.1421i		−	0.476461i	0.971209	−	0.238230i	$-0.0765673\pi$
$882$		0			0
$883$	−26.0000			−0.874970			−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000			−0.874970			−0.437485	+	0.899226i	$0.644131\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1.41421i			0.0474846i	0.999718	+	0.0237423i	$0.00755813\pi$
$887$			1.41421i			0.0474846i	−0.999718	+	0.0237423i	$0.992442\pi$
$888$		0			0
$889$		−	84.8528i		−	2.84587i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−24.0000			−0.803129
$894$		0			0
$895$			16.9706i			0.567263i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	30.0000			1.00056
$900$		0			0
$901$	−18.0000	−	16.9706i	−0.599667	−	0.565371i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−30.0000			−0.997234
$906$		0			0
$907$		−	42.4264i		−	1.40875i	−0.709830	−	0.704373i	$-0.751228\pi$
$907$		−	42.4264i		−	1.40875i	0.709830	−	0.704373i	$-0.248772\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	15.5563i		−	0.515405i	−0.966224	−	0.257702i	$-0.917035\pi$
$911$		−	15.5563i		−	0.515405i	0.966224	−	0.257702i	$-0.0829654\pi$
$912$		0			0
$913$		−	16.9706i		−	0.561644i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−60.0000			−1.98137
$918$		0			0
$919$	−20.0000			−0.659739			−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000			−0.659739			−0.329870	+	0.944027i	$0.607005\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			2.82843i			0.0930988i
$924$		0			0
$925$		−	12.7279i		−	0.418491i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	22.6274i		−	0.742381i	−0.928557	−	0.371191i	$-0.878950\pi$
$929$		−	22.6274i		−	0.742381i	0.928557	−	0.371191i	$-0.121050\pi$
$930$		0			0
$931$	22.0000			0.721021
$932$		0			0
$933$		0			0
$934$		0			0
$935$	12.0000	+	11.3137i	0.392442	+	0.369998i
$936$		0			0
$937$	−4.00000			−0.130674			−0.0653372	−	0.997863i	$-0.520812\pi$
$937$	−4.00000			−0.130674			−0.0653372	+	0.997863i	$0.520812\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1.41421i		−	0.0461020i	−0.999734	−	0.0230510i	$-0.992662\pi$
$941$		−	1.41421i		−	0.0461020i	0.999734	−	0.0230510i	$-0.00733802\pi$
$942$		0			0
$943$	−40.0000			−1.30258
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	53.7401i		−	1.74632i	−0.487435	−	0.873160i	$-0.662067\pi$
$947$		−	53.7401i		−	1.74632i	0.487435	−	0.873160i	$-0.337933\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		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
$955$		−	16.9706i		−	0.549155i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	50.9117i		−	1.64402i
$960$		0			0
$961$	13.0000			0.419355
$962$		0			0
$963$		0			0
$964$		0			0
$965$	36.0000			1.15888
$966$		0			0
$967$	−32.0000			−1.02905			−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000			−1.02905			−0.514525	+	0.857475i	$0.672032\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	42.0000			1.34784			0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000			1.34784			0.673922	+	0.738802i	$0.264608\pi$
$972$		0			0
$973$	36.0000			1.15411
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−60.0000			−1.91957			−0.959785	−	0.280736i	$-0.909421\pi$
$977$	−60.0000			−1.91957			−0.959785	+	0.280736i	$0.909421\pi$
$978$		0			0
$979$		−	16.9706i		−	0.542382i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			9.89949i			0.315745i	0.987460	+	0.157872i	$0.0504635\pi$
$983$			9.89949i			0.315745i	−0.987460	+	0.157872i	$0.949537\pi$
$984$		0			0
$985$	−14.0000			−0.446077
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	28.2843i		−	0.899388i
$990$		0			0
$991$		−	12.7279i		−	0.404316i	−0.979353	−	0.202158i	$-0.935205\pi$
$991$		−	12.7279i		−	0.404316i	0.979353	−	0.202158i	$-0.0647954\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−6.00000			−0.190213
$996$		0			0
$997$			29.6985i			0.940560i	0.882517	+	0.470280i	$0.155847\pi$
$997$			29.6985i			0.940560i	−0.882517	+	0.470280i	$0.844153\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2448.2.c.i.577.1		2
3.2	odd	2		2448.2.c.k.577.2		2
4.3	odd	2		306.2.b.c.271.1	yes	2
12.11	even	2		306.2.b.b.271.2	yes	2
17.16	even	2	inner	2448.2.c.i.577.2		2
51.50	odd	2		2448.2.c.k.577.1		2
68.47	odd	4		5202.2.a.t.1.2		2
68.55	odd	4		5202.2.a.t.1.1		2
68.67	odd	2		306.2.b.c.271.2	yes	2
204.47	even	4		5202.2.a.bd.1.1		2
204.191	even	4		5202.2.a.bd.1.2		2
204.203	even	2		306.2.b.b.271.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
306.2.b.b.271.1	✓	2	204.203	even	2
306.2.b.b.271.2	yes	2	12.11	even	2
306.2.b.c.271.1	yes	2	4.3	odd	2
306.2.b.c.271.2	yes	2	68.67	odd	2
2448.2.c.i.577.1		2	1.1	even	1	trivial
2448.2.c.i.577.2		2	17.16	even	2	inner
2448.2.c.k.577.1		2	51.50	odd	2
2448.2.c.k.577.2		2	3.2	odd	2
5202.2.a.t.1.1		2	68.55	odd	4
5202.2.a.t.1.2		2	68.47	odd	4
5202.2.a.bd.1.1		2	204.47	even	4
5202.2.a.bd.1.2		2	204.191	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2448.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{5} +4.24264i q^{7} -2.82843i q^{11} +2.00000 q^{13} +(-3.00000 - 2.82843i) q^{17} -2.00000 q^{19} -7.07107i q^{23} +3.00000 q^{25} +7.07107i q^{29} -4.24264i q^{31} +6.00000 q^{35} -4.24264i q^{37} -5.65685i q^{41} +4.00000 q^{43} +12.0000 q^{47} -11.0000 q^{49} +6.00000 q^{53} -4.00000 q^{55} -6.00000 q^{59} +12.7279i q^{61} -2.82843i q^{65} +10.0000 q^{67} +1.41421i q^{71} +12.0000 q^{77} -12.7279i q^{79} +6.00000 q^{83} +(-4.00000 + 4.24264i) q^{85} +6.00000 q^{89} +8.48528i q^{91} +2.82843i q^{95} -8.48528i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{13} - 6 q^{17} - 4 q^{19} + 6 q^{25} + 12 q^{35} + 8 q^{43} + 24 q^{47} - 22 q^{49} + 12 q^{53} - 8 q^{55} - 12 q^{59} + 20 q^{67} + 24 q^{77} + 12 q^{83} - 8 q^{85} + 12 q^{89}+O(q^{100}) \) 2 * q + 4 * q^13 - 6 * q^17 - 4 * q^19 + 6 * q^25 + 12 * q^35 + 8 * q^43 + 24 * q^47 - 22 * q^49 + 12 * q^53 - 8 * q^55 - 12 * q^59 + 20 * q^67 + 24 * q^77 + 12 * q^83 - 8 * q^85 + 12 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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