LMFDB - Embedded newform 576.3.e.f.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,3,Mod(449,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("576.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	576.e (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:	$15.6948632272$
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:	$ 3 $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	576.449
Dual form			576.3.e.f.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.24264i q^{5} +4.00000 q^{7} +O(q^{10})$	$q-4.24264i q^{5} +4.00000 q^{7} -16.9706i q^{11} -8.00000 q^{13} +12.7279i q^{17} -16.0000 q^{19} -16.9706i q^{23} +7.00000 q^{25} +4.24264i q^{29} -44.0000 q^{31} -16.9706i q^{35} +34.0000 q^{37} -46.6690i q^{41} -40.0000 q^{43} -84.8528i q^{47} -33.0000 q^{49} +38.1838i q^{53} -72.0000 q^{55} -33.9411i q^{59} -50.0000 q^{61} +33.9411i q^{65} +8.00000 q^{67} -50.9117i q^{71} -16.0000 q^{73} -67.8823i q^{77} +76.0000 q^{79} -118.794i q^{83} +54.0000 q^{85} -12.7279i q^{89} -32.0000 q^{91} +67.8823i q^{95} +176.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{7}+O(q^{10}) $ 2 * q + 8 * q^7	$ 2 q + 8 q^{7} - 16 q^{13} - 32 q^{19} + 14 q^{25} - 88 q^{31} + 68 q^{37} - 80 q^{43} - 66 q^{49} - 144 q^{55} - 100 q^{61} + 16 q^{67} - 32 q^{73} + 152 q^{79} + 108 q^{85} - 64 q^{91} + 352 q^{97}+O(q^{100}) $ 2 * q + 8 * q^7 - 16 * q^13 - 32 * q^19 + 14 * q^25 - 88 * q^31 + 68 * q^37 - 80 * q^43 - 66 * q^49 - 144 * q^55 - 100 * q^61 + 16 * q^67 - 32 * q^73 + 152 * q^79 + 108 * q^85 - 64 * q^91 + 352 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 4.24264i	− 0.848528i	−0.905539	−	0.424264i	$-0.860533\pi$
$5$	− 4.24264i	− 0.848528i	0.905539	−	0.424264i	$-0.139467\pi$
$6$	0	0
$7$	4.00000	0.571429	0.285714	−	0.958315i	$-0.407769\pi$
$7$	4.00000	0.571429	0.285714	+	0.958315i	$0.407769\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 16.9706i	− 1.54278i	−0.636364	−	0.771389i	$-0.719562\pi$
$11$	− 16.9706i	− 1.54278i	0.636364	−	0.771389i	$-0.280438\pi$
$12$	0	0
$13$	−8.00000	−0.615385	−0.307692	−	0.951486i	$-0.599557\pi$
$13$	−8.00000	−0.615385	−0.307692	+	0.951486i	$0.599557\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	12.7279i	0.748701i	0.927287	+	0.374351i	$0.122134\pi$
$17$	12.7279i	0.748701i	−0.927287	+	0.374351i	$0.877866\pi$
$18$	0	0
$19$	−16.0000	−0.842105	−0.421053	−	0.907036i	$-0.638339\pi$
$19$	−16.0000	−0.842105	−0.421053	+	0.907036i	$0.638339\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 16.9706i	− 0.737851i	−0.929459	−	0.368925i	$-0.879726\pi$
$23$	− 16.9706i	− 0.737851i	0.929459	−	0.368925i	$-0.120274\pi$
$24$	0	0
$25$	7.00000	0.280000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.24264i	0.146298i	0.997321	+	0.0731490i	$0.0233049\pi$
$29$	4.24264i	0.146298i	−0.997321	+	0.0731490i	$0.976695\pi$
$30$	0	0
$31$	−44.0000	−1.41935	−0.709677	−	0.704527i	$-0.751159\pi$
$31$	−44.0000	−1.41935	−0.709677	+	0.704527i	$0.751159\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 16.9706i	− 0.484873i
$36$	0	0
$37$	34.0000	0.918919	0.459459	−	0.888199i	$-0.348043\pi$
$37$	34.0000	0.918919	0.459459	+	0.888199i	$0.348043\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 46.6690i	− 1.13827i	−0.822244	−	0.569135i	$-0.807278\pi$
$41$	− 46.6690i	− 1.13827i	0.822244	−	0.569135i	$-0.192722\pi$
$42$	0	0
$43$	−40.0000	−0.930233	−0.465116	−	0.885250i	$-0.653987\pi$
$43$	−40.0000	−0.930233	−0.465116	+	0.885250i	$0.653987\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 84.8528i	− 1.80538i	−0.430293	−	0.902690i	$-0.641590\pi$
$47$	− 84.8528i	− 1.80538i	0.430293	−	0.902690i	$-0.358410\pi$
$48$	0	0
$49$	−33.0000	−0.673469
$50$	0	0
$51$	0	0
$52$	0	0
$53$	38.1838i	0.720448i	0.932866	+	0.360224i	$0.117300\pi$
$53$	38.1838i	0.720448i	−0.932866	+	0.360224i	$0.882700\pi$
$54$	0	0
$55$	−72.0000	−1.30909
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 33.9411i	− 0.575273i	−0.957740	−	0.287637i	$-0.907130\pi$
$59$	− 33.9411i	− 0.575273i	0.957740	−	0.287637i	$-0.0928695\pi$
$60$	0	0
$61$	−50.0000	−0.819672	−0.409836	−	0.912159i	$-0.634414\pi$
$61$	−50.0000	−0.819672	−0.409836	+	0.912159i	$0.634414\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	33.9411i	0.522171i
$66$	0	0
$67$	8.00000	0.119403	0.0597015	−	0.998216i	$-0.480985\pi$
$67$	8.00000	0.119403	0.0597015	+	0.998216i	$0.480985\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 50.9117i	− 0.717066i	−0.933517	−	0.358533i	$-0.883277\pi$
$71$	− 50.9117i	− 0.717066i	0.933517	−	0.358533i	$-0.116723\pi$
$72$	0	0
$73$	−16.0000	−0.219178	−0.109589	−	0.993977i	$-0.534953\pi$
$73$	−16.0000	−0.219178	−0.109589	+	0.993977i	$0.534953\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 67.8823i	− 0.881588i
$78$	0	0
$79$	76.0000	0.962025	0.481013	−	0.876714i	$-0.340269\pi$
$79$	76.0000	0.962025	0.481013	+	0.876714i	$0.340269\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 118.794i	− 1.43125i	−0.698484	−	0.715626i	$-0.746141\pi$
$83$	− 118.794i	− 1.43125i	0.698484	−	0.715626i	$-0.253859\pi$
$84$	0	0
$85$	54.0000	0.635294
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 12.7279i	− 0.143010i	−0.997440	−	0.0715052i	$-0.977220\pi$
$89$	− 12.7279i	− 0.143010i	0.997440	−	0.0715052i	$-0.0227802\pi$
$90$	0	0
$91$	−32.0000	−0.351648
$92$	0	0
$93$	0	0
$94$	0	0
$95$	67.8823i	0.714550i
$96$	0	0
$97$	176.000	1.81443	0.907216	−	0.420664i	$-0.138203\pi$
$97$	176.000	1.81443	0.907216	+	0.420664i	$0.138203\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	29.6985i	0.294044i	0.989133	+	0.147022i	$0.0469689\pi$
$101$	29.6985i	0.294044i	−0.989133	+	0.147022i	$0.953031\pi$
$102$	0	0
$103$	28.0000	0.271845	0.135922	−	0.990719i	$-0.456600\pi$
$103$	28.0000	0.271845	0.135922	+	0.990719i	$0.456600\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	−56.0000	−0.513761	−0.256881	−	0.966443i	$-0.582695\pi$
$109$	−56.0000	−0.513761	−0.256881	+	0.966443i	$0.582695\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	156.978i	1.38918i	0.719404	+	0.694592i	$0.244415\pi$
$113$	156.978i	1.38918i	−0.719404	+	0.694592i	$0.755585\pi$
$114$	0	0
$115$	−72.0000	−0.626087
$116$	0	0
$117$	0	0
$118$	0	0
$119$	50.9117i	0.427829i
$120$	0	0
$121$	−167.000	−1.38017
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 135.765i	− 1.08612i
$126$	0	0
$127$	−92.0000	−0.724409	−0.362205	−	0.932099i	$-0.617976\pi$
$127$	−92.0000	−0.724409	−0.362205	+	0.932099i	$0.617976\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	169.706i	1.29546i	0.761869	+	0.647731i	$0.224282\pi$
$131$	169.706i	1.29546i	−0.761869	+	0.647731i	$0.775718\pi$
$132$	0	0
$133$	−64.0000	−0.481203
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 156.978i	− 1.14582i	−0.819617	−	0.572911i	$-0.805814\pi$
$137$	− 156.978i	− 1.14582i	0.819617	−	0.572911i	$-0.194186\pi$
$138$	0	0
$139$	152.000	1.09353	0.546763	−	0.837288i	$-0.315860\pi$
$139$	152.000	1.09353	0.546763	+	0.837288i	$0.315860\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	135.765i	0.949402i
$144$	0	0
$145$	18.0000	0.124138
$146$	0	0
$147$	0	0
$148$	0	0
$149$	275.772i	1.85082i	0.378972	+	0.925408i	$0.376278\pi$
$149$	275.772i	1.85082i	−0.378972	+	0.925408i	$0.623722\pi$
$150$	0	0
$151$	148.000	0.980132	0.490066	−	0.871685i	$-0.336973\pi$
$151$	148.000	0.980132	0.490066	+	0.871685i	$0.336973\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	186.676i	1.20436i
$156$	0	0
$157$	82.0000	0.522293	0.261146	−	0.965299i	$-0.415899\pi$
$157$	82.0000	0.522293	0.261146	+	0.965299i	$0.415899\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 67.8823i	− 0.421629i
$162$	0	0
$163$	56.0000	0.343558	0.171779	−	0.985135i	$-0.445048\pi$
$163$	56.0000	0.343558	0.171779	+	0.985135i	$0.445048\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	33.9411i	0.203240i	0.994823	+	0.101620i	$0.0324026\pi$
$167$	33.9411i	0.203240i	−0.994823	+	0.101620i	$0.967597\pi$
$168$	0	0
$169$	−105.000	−0.621302
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 173.948i	− 1.00548i	−0.864437	−	0.502741i	$-0.832325\pi$
$173$	− 173.948i	− 1.00548i	0.864437	−	0.502741i	$-0.167675\pi$
$174$	0	0
$175$	28.0000	0.160000
$176$	0	0
$177$	0	0
$178$	0	0
$179$	203.647i	1.13769i	0.822444	+	0.568846i	$0.192610\pi$
$179$	203.647i	1.13769i	−0.822444	+	0.568846i	$0.807390\pi$
$180$	0	0
$181$	232.000	1.28177	0.640884	−	0.767638i	$-0.278568\pi$
$181$	232.000	1.28177	0.640884	+	0.767638i	$0.278568\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 144.250i	− 0.779729i
$186$	0	0
$187$	216.000	1.15508
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 33.9411i	− 0.177702i	−0.996045	−	0.0888511i	$-0.971680\pi$
$191$	− 33.9411i	− 0.177702i	0.996045	−	0.0888511i	$-0.0283195\pi$
$192$	0	0
$193$	206.000	1.06736	0.533679	−	0.845687i	$-0.320809\pi$
$193$	206.000	1.06736	0.533679	+	0.845687i	$0.320809\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	165.463i	0.839914i	0.907544	+	0.419957i	$0.137955\pi$
$197$	165.463i	0.839914i	−0.907544	+	0.419957i	$0.862045\pi$
$198$	0	0
$199$	−20.0000	−0.100503	−0.0502513	−	0.998737i	$-0.516002\pi$
$199$	−20.0000	−0.100503	−0.0502513	+	0.998737i	$0.516002\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	16.9706i	0.0835988i
$204$	0	0
$205$	−198.000	−0.965854
$206$	0	0
$207$	0	0
$208$	0	0
$209$	271.529i	1.29918i
$210$	0	0
$211$	296.000	1.40284	0.701422	−	0.712746i	$-0.252549\pi$
$211$	296.000	1.40284	0.701422	+	0.712746i	$0.252549\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	169.706i	0.789328i
$216$	0	0
$217$	−176.000	−0.811060
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 101.823i	− 0.460739i
$222$	0	0
$223$	436.000	1.95516	0.977578	−	0.210571i	$-0.0675325\pi$
$223$	436.000	1.95516	0.977578	+	0.210571i	$0.0675325\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 16.9706i	− 0.0747602i	−0.999301	−	0.0373801i	$-0.988099\pi$
$227$	− 16.9706i	− 0.0747602i	0.999301	−	0.0373801i	$-0.0119012\pi$
$228$	0	0
$229$	−8.00000	−0.0349345	−0.0174672	−	0.999847i	$-0.505560\pi$
$229$	−8.00000	−0.0349345	−0.0174672	+	0.999847i	$0.505560\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.7279i	0.0546263i	0.999627	+	0.0273131i	$0.00869512\pi$
$233$	12.7279i	0.0546263i	−0.999627	+	0.0273131i	$0.991305\pi$
$234$	0	0
$235$	−360.000	−1.53191
$236$	0	0
$237$	0	0
$238$	0	0
$239$	135.765i	0.568052i	0.958817	+	0.284026i	$0.0916703\pi$
$239$	135.765i	0.568052i	−0.958817	+	0.284026i	$0.908330\pi$
$240$	0	0
$241$	32.0000	0.132780	0.0663900	−	0.997794i	$-0.478852\pi$
$241$	32.0000	0.132780	0.0663900	+	0.997794i	$0.478852\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	140.007i	0.571458i
$246$	0	0
$247$	128.000	0.518219
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 50.9117i	− 0.202835i	−0.994844	−	0.101418i	$-0.967662\pi$
$251$	− 50.9117i	− 0.202835i	0.994844	−	0.101418i	$-0.0323379\pi$
$252$	0	0
$253$	−288.000	−1.13834
$254$	0	0
$255$	0	0
$256$	0	0
$257$	182.434i	0.709858i	0.934893	+	0.354929i	$0.115495\pi$
$257$	182.434i	0.709858i	−0.934893	+	0.354929i	$0.884505\pi$
$258$	0	0
$259$	136.000	0.525097
$260$	0	0
$261$	0	0
$262$	0	0
$263$	373.352i	1.41959i	0.704408	+	0.709795i	$0.251213\pi$
$263$	373.352i	1.41959i	−0.704408	+	0.709795i	$0.748787\pi$
$264$	0	0
$265$	162.000	0.611321
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 343.654i	− 1.27752i	−0.769404	−	0.638762i	$-0.779447\pi$
$269$	− 343.654i	− 1.27752i	0.769404	−	0.638762i	$-0.220553\pi$
$270$	0	0
$271$	−380.000	−1.40221	−0.701107	−	0.713056i	$-0.747310\pi$
$271$	−380.000	−1.40221	−0.701107	+	0.713056i	$0.747310\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 118.794i	− 0.431978i
$276$	0	0
$277$	328.000	1.18412	0.592058	−	0.805896i	$-0.298316\pi$
$277$	328.000	1.18412	0.592058	+	0.805896i	$0.298316\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 284.257i	− 1.01159i	−0.862654	−	0.505795i	$-0.831199\pi$
$281$	− 284.257i	− 1.01159i	0.862654	−	0.505795i	$-0.168801\pi$
$282$	0	0
$283$	−208.000	−0.734982	−0.367491	−	0.930027i	$-0.619783\pi$
$283$	−208.000	−0.734982	−0.367491	+	0.930027i	$0.619783\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 186.676i	− 0.650440i
$288$	0	0
$289$	127.000	0.439446
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 436.992i	− 1.49144i	−0.666259	−	0.745720i	$-0.732106\pi$
$293$	− 436.992i	− 1.49144i	0.666259	−	0.745720i	$-0.267894\pi$
$294$	0	0
$295$	−144.000	−0.488136
$296$	0	0
$297$	0	0
$298$	0	0
$299$	135.765i	0.454062i
$300$	0	0
$301$	−160.000	−0.531561
$302$	0	0
$303$	0	0
$304$	0	0
$305$	212.132i	0.695515i
$306$	0	0
$307$	−520.000	−1.69381	−0.846906	−	0.531743i	$-0.821537\pi$
$307$	−520.000	−1.69381	−0.846906	+	0.531743i	$0.821537\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 373.352i	− 1.20049i	−0.799816	−	0.600245i	$-0.795070\pi$
$311$	− 373.352i	− 1.20049i	0.799816	−	0.600245i	$-0.204930\pi$
$312$	0	0
$313$	−94.0000	−0.300319	−0.150160	−	0.988662i	$-0.547979\pi$
$313$	−94.0000	−0.300319	−0.150160	+	0.988662i	$0.547979\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	335.169i	1.05731i	0.848835	+	0.528657i	$0.177304\pi$
$317$	335.169i	1.05731i	−0.848835	+	0.528657i	$0.822696\pi$
$318$	0	0
$319$	72.0000	0.225705
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 203.647i	− 0.630485i
$324$	0	0
$325$	−56.0000	−0.172308
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 339.411i	− 1.03165i
$330$	0	0
$331$	536.000	1.61934	0.809668	−	0.586889i	$-0.199647\pi$
$331$	536.000	1.61934	0.809668	+	0.586889i	$0.199647\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 33.9411i	− 0.101317i
$336$	0	0
$337$	−208.000	−0.617211	−0.308605	−	0.951190i	$-0.599862\pi$
$337$	−208.000	−0.617211	−0.308605	+	0.951190i	$0.599862\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	746.705i	2.18975i
$342$	0	0
$343$	−328.000	−0.956268
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 288.500i	− 0.831411i	−0.909499	−	0.415705i	$-0.863535\pi$
$347$	− 288.500i	− 0.831411i	0.909499	−	0.415705i	$-0.136465\pi$
$348$	0	0
$349$	238.000	0.681948	0.340974	−	0.940073i	$-0.389243\pi$
$349$	238.000	0.681948	0.340974	+	0.940073i	$0.389243\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	224.860i	0.636997i	0.947923	+	0.318499i	$0.103179\pi$
$353$	224.860i	0.636997i	−0.947923	+	0.318499i	$0.896821\pi$
$354$	0	0
$355$	−216.000	−0.608451
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 560.029i	− 1.55997i	−0.625799	−	0.779984i	$-0.715227\pi$
$359$	− 560.029i	− 1.55997i	0.625799	−	0.779984i	$-0.284773\pi$
$360$	0	0
$361$	−105.000	−0.290859
$362$	0	0
$363$	0	0
$364$	0	0
$365$	67.8823i	0.185979i
$366$	0	0
$367$	−284.000	−0.773842	−0.386921	−	0.922113i	$-0.626461\pi$
$367$	−284.000	−0.773842	−0.386921	+	0.922113i	$0.626461\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	152.735i	0.411685i
$372$	0	0
$373$	190.000	0.509383	0.254692	−	0.967022i	$-0.418026\pi$
$373$	190.000	0.509383	0.254692	+	0.967022i	$0.418026\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 33.9411i	− 0.0900295i
$378$	0	0
$379$	−160.000	−0.422164	−0.211082	−	0.977468i	$-0.567699\pi$
$379$	−160.000	−0.422164	−0.211082	+	0.977468i	$0.567699\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 271.529i	− 0.708953i	−0.935065	−	0.354477i	$-0.884659\pi$
$383$	− 271.529i	− 0.708953i	0.935065	−	0.354477i	$-0.115341\pi$
$384$	0	0
$385$	−288.000	−0.748052
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 403.051i	− 1.03612i	−0.855344	−	0.518060i	$-0.826654\pi$
$389$	− 403.051i	− 1.03612i	0.855344	−	0.518060i	$-0.173346\pi$
$390$	0	0
$391$	216.000	0.552430
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 322.441i	− 0.816306i
$396$	0	0
$397$	−146.000	−0.367758	−0.183879	−	0.982949i	$-0.558865\pi$
$397$	−146.000	−0.367758	−0.183879	+	0.982949i	$0.558865\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 326.683i	− 0.814672i	−0.913278	−	0.407336i	$-0.866458\pi$
$401$	− 326.683i	− 0.814672i	0.913278	−	0.407336i	$-0.133542\pi$
$402$	0	0
$403$	352.000	0.873449
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 576.999i	− 1.41769i
$408$	0	0
$409$	368.000	0.899756	0.449878	−	0.893090i	$-0.351468\pi$
$409$	368.000	0.899756	0.449878	+	0.893090i	$0.351468\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 135.765i	− 0.328728i
$414$	0	0
$415$	−504.000	−1.21446
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 390.323i	− 0.931558i	−0.884901	−	0.465779i	$-0.845774\pi$
$419$	− 390.323i	− 0.931558i	0.884901	−	0.465779i	$-0.154226\pi$
$420$	0	0
$421$	40.0000	0.0950119	0.0475059	−	0.998871i	$-0.484873\pi$
$421$	40.0000	0.0950119	0.0475059	+	0.998871i	$0.484873\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	89.0955i	0.209636i
$426$	0	0
$427$	−200.000	−0.468384
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 152.735i	− 0.354374i	−0.984177	−	0.177187i	$-0.943300\pi$
$431$	− 152.735i	− 0.354374i	0.984177	−	0.177187i	$-0.0566997\pi$
$432$	0	0
$433$	542.000	1.25173	0.625866	−	0.779931i	$-0.284746\pi$
$433$	542.000	1.25173	0.625866	+	0.779931i	$0.284746\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	271.529i	0.621348i
$438$	0	0
$439$	4.00000	0.00911162	0.00455581	−	0.999990i	$-0.498550\pi$
$439$	4.00000	0.00911162	0.00455581	+	0.999990i	$0.498550\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 322.441i	− 0.727857i	−0.931427	−	0.363929i	$-0.881435\pi$
$443$	− 322.441i	− 0.727857i	0.931427	−	0.363929i	$-0.118565\pi$
$444$	0	0
$445$	−54.0000	−0.121348
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 216.375i	− 0.481904i	−0.970537	−	0.240952i	$-0.922540\pi$
$449$	− 216.375i	− 0.481904i	0.970537	−	0.240952i	$-0.0774596\pi$
$450$	0	0
$451$	−792.000	−1.75610
$452$	0	0
$453$	0	0
$454$	0	0
$455$	135.765i	0.298384i
$456$	0	0
$457$	−400.000	−0.875274	−0.437637	−	0.899152i	$-0.644184\pi$
$457$	−400.000	−0.875274	−0.437637	+	0.899152i	$0.644184\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 301.227i	− 0.653422i	−0.945124	−	0.326711i	$-0.894060\pi$
$461$	− 301.227i	− 0.653422i	0.945124	−	0.326711i	$-0.105940\pi$
$462$	0	0
$463$	604.000	1.30454	0.652268	−	0.757989i	$-0.273818\pi$
$463$	604.000	1.30454	0.652268	+	0.757989i	$0.273818\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 356.382i	− 0.763130i	−0.924342	−	0.381565i	$-0.875385\pi$
$467$	− 356.382i	− 0.763130i	0.924342	−	0.381565i	$-0.124615\pi$
$468$	0	0
$469$	32.0000	0.0682303
$470$	0	0
$471$	0	0
$472$	0	0
$473$	678.823i	1.43514i
$474$	0	0
$475$	−112.000	−0.235789
$476$	0	0
$477$	0	0
$478$	0	0
$479$	526.087i	1.09830i	0.835723	+	0.549152i	$0.185049\pi$
$479$	526.087i	1.09830i	−0.835723	+	0.549152i	$0.814951\pi$
$480$	0	0
$481$	−272.000	−0.565489
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 746.705i	− 1.53960i
$486$	0	0
$487$	−596.000	−1.22382	−0.611910	−	0.790928i	$-0.709598\pi$
$487$	−596.000	−1.22382	−0.611910	+	0.790928i	$0.709598\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	271.529i	0.553012i	0.961012	+	0.276506i	$0.0891766\pi$
$491$	271.529i	0.553012i	−0.961012	+	0.276506i	$0.910823\pi$
$492$	0	0
$493$	−54.0000	−0.109533
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 203.647i	− 0.409752i
$498$	0	0
$499$	224.000	0.448898	0.224449	−	0.974486i	$-0.427942\pi$
$499$	224.000	0.448898	0.224449	+	0.974486i	$0.427942\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	865.499i	1.72067i	0.509726	+	0.860337i	$0.329747\pi$
$503$	865.499i	1.72067i	−0.509726	+	0.860337i	$0.670253\pi$
$504$	0	0
$505$	126.000	0.249505
$506$	0	0
$507$	0	0
$508$	0	0
$509$	479.418i	0.941883i	0.882164	+	0.470941i	$0.156086\pi$
$509$	479.418i	0.941883i	−0.882164	+	0.470941i	$0.843914\pi$
$510$	0	0
$511$	−64.0000	−0.125245
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 118.794i	− 0.230668i
$516$	0	0
$517$	−1440.00	−2.78530
$518$	0	0
$519$	0	0
$520$	0	0
$521$	521.845i	1.00162i	0.865557	+	0.500811i	$0.166965\pi$
$521$	521.845i	1.00162i	−0.865557	+	0.500811i	$0.833035\pi$
$522$	0	0
$523$	−736.000	−1.40727	−0.703633	−	0.710564i	$-0.748440\pi$
$523$	−736.000	−1.40727	−0.703633	+	0.710564i	$0.748440\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 560.029i	− 1.06267i
$528$	0	0
$529$	241.000	0.455577
$530$	0	0
$531$	0	0
$532$	0	0
$533$	373.352i	0.700474i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	560.029i	1.03901i
$540$	0	0
$541$	808.000	1.49353	0.746765	−	0.665088i	$-0.231606\pi$
$541$	808.000	1.49353	0.746765	+	0.665088i	$0.231606\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	237.588i	0.435941i
$546$	0	0
$547$	536.000	0.979890	0.489945	−	0.871753i	$-0.337017\pi$
$547$	536.000	0.979890	0.489945	+	0.871753i	$0.337017\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 67.8823i	− 0.123198i
$552$	0	0
$553$	304.000	0.549729
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 165.463i	− 0.297061i	−0.988908	−	0.148531i	$-0.952546\pi$
$557$	− 165.463i	− 0.297061i	0.988908	−	0.148531i	$-0.0474543\pi$
$558$	0	0
$559$	320.000	0.572451
$560$	0	0
$561$	0	0
$562$	0	0
$563$	322.441i	0.572719i	0.958122	+	0.286359i	$0.0924451\pi$
$563$	322.441i	0.572719i	−0.958122	+	0.286359i	$0.907555\pi$
$564$	0	0
$565$	666.000	1.17876
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 156.978i	− 0.275883i	−0.990440	−	0.137942i	$-0.955951\pi$
$569$	− 156.978i	− 0.275883i	0.990440	−	0.137942i	$-0.0440487\pi$
$570$	0	0
$571$	368.000	0.644483	0.322242	−	0.946657i	$-0.395564\pi$
$571$	368.000	0.644483	0.322242	+	0.946657i	$0.395564\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 118.794i	− 0.206598i
$576$	0	0
$577$	−142.000	−0.246101	−0.123050	−	0.992400i	$-0.539268\pi$
$577$	−142.000	−0.246101	−0.123050	+	0.992400i	$0.539268\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 475.176i	− 0.817858i
$582$	0	0
$583$	648.000	1.11149
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 373.352i	− 0.636035i	−0.948085	−	0.318017i	$-0.896983\pi$
$587$	− 373.352i	− 0.636035i	0.948085	−	0.318017i	$-0.103017\pi$
$588$	0	0
$589$	704.000	1.19525
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 1107.33i	− 1.86733i	−0.358142	−	0.933667i	$-0.616590\pi$
$593$	− 1107.33i	− 1.86733i	0.358142	−	0.933667i	$-0.383410\pi$
$594$	0	0
$595$	216.000	0.363025
$596$	0	0
$597$	0	0
$598$	0	0
$599$	797.616i	1.33158i	0.746139	+	0.665790i	$0.231905\pi$
$599$	797.616i	1.33158i	−0.746139	+	0.665790i	$0.768095\pi$
$600$	0	0
$601$	158.000	0.262895	0.131448	−	0.991323i	$-0.458037\pi$
$601$	158.000	0.262895	0.131448	+	0.991323i	$0.458037\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	708.521i	1.17111i
$606$	0	0
$607$	−332.000	−0.546952	−0.273476	−	0.961879i	$-0.588173\pi$
$607$	−332.000	−0.546952	−0.273476	+	0.961879i	$0.588173\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	678.823i	1.11100i
$612$	0	0
$613$	−578.000	−0.942904	−0.471452	−	0.881892i	$-0.656270\pi$
$613$	−578.000	−0.942904	−0.471452	+	0.881892i	$0.656270\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	55.1543i	0.0893911i	0.999001	+	0.0446956i	$0.0142318\pi$
$617$	55.1543i	0.0893911i	−0.999001	+	0.0446956i	$0.985768\pi$
$618$	0	0
$619$	896.000	1.44750	0.723748	−	0.690064i	$-0.242418\pi$
$619$	896.000	1.44750	0.723748	+	0.690064i	$0.242418\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 50.9117i	− 0.0817202i
$624$	0	0
$625$	−401.000	−0.641600
$626$	0	0
$627$	0	0
$628$	0	0
$629$	432.749i	0.687996i
$630$	0	0
$631$	−20.0000	−0.0316957	−0.0158479	−	0.999874i	$-0.505045\pi$
$631$	−20.0000	−0.0316957	−0.0158479	+	0.999874i	$0.505045\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	390.323i	0.614682i
$636$	0	0
$637$	264.000	0.414443
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 258.801i	− 0.403746i	−0.979412	−	0.201873i	$-0.935297\pi$
$641$	− 258.801i	− 0.403746i	0.979412	−	0.201873i	$-0.0647028\pi$
$642$	0	0
$643$	728.000	1.13219	0.566096	−	0.824339i	$-0.308453\pi$
$643$	728.000	1.13219	0.566096	+	0.824339i	$0.308453\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 458.205i	− 0.708200i	−0.935208	−	0.354100i	$-0.884787\pi$
$647$	− 458.205i	− 0.708200i	0.935208	−	0.354100i	$-0.115213\pi$
$648$	0	0
$649$	−576.000	−0.887519
$650$	0	0
$651$	0	0
$652$	0	0
$653$	301.227i	0.461298i	0.973037	+	0.230649i	$0.0740849\pi$
$653$	301.227i	0.461298i	−0.973037	+	0.230649i	$0.925915\pi$
$654$	0	0
$655$	720.000	1.09924
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1052.17i	1.59662i	0.602244	+	0.798312i	$0.294273\pi$
$659$	1052.17i	1.59662i	−0.602244	+	0.798312i	$0.705727\pi$
$660$	0	0
$661$	−62.0000	−0.0937973	−0.0468986	−	0.998900i	$-0.514934\pi$
$661$	−62.0000	−0.0937973	−0.0468986	+	0.998900i	$0.514934\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	271.529i	0.408314i
$666$	0	0
$667$	72.0000	0.107946
$668$	0	0
$669$	0	0
$670$	0	0
$671$	848.528i	1.26457i
$672$	0	0
$673$	−670.000	−0.995542	−0.497771	−	0.867308i	$-0.665848\pi$
$673$	−670.000	−0.995542	−0.497771	+	0.867308i	$0.665848\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1294.01i	− 1.91138i	−0.294372	−	0.955691i	$-0.595111\pi$
$677$	− 1294.01i	− 1.91138i	0.294372	−	0.955691i	$-0.404889\pi$
$678$	0	0
$679$	704.000	1.03682
$680$	0	0
$681$	0	0
$682$	0	0
$683$	560.029i	0.819954i	0.912096	+	0.409977i	$0.134463\pi$
$683$	560.029i	0.819954i	−0.912096	+	0.409977i	$0.865537\pi$
$684$	0	0
$685$	−666.000	−0.972263
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 305.470i	− 0.443353i
$690$	0	0
$691$	−40.0000	−0.0578871	−0.0289436	−	0.999581i	$-0.509214\pi$
$691$	−40.0000	−0.0578871	−0.0289436	+	0.999581i	$0.509214\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 644.881i	− 0.927887i
$696$	0	0
$697$	594.000	0.852224
$698$	0	0
$699$	0	0
$700$	0	0
$701$	954.594i	1.36176i	0.732395	+	0.680880i	$0.238403\pi$
$701$	954.594i	1.36176i	−0.732395	+	0.680880i	$0.761597\pi$
$702$	0	0
$703$	−544.000	−0.773826
$704$	0	0
$705$	0	0
$706$	0	0
$707$	118.794i	0.168025i
$708$	0	0
$709$	−968.000	−1.36530	−0.682652	−	0.730744i	$-0.739173\pi$
$709$	−968.000	−1.36530	−0.682652	+	0.730744i	$0.739173\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	746.705i	1.04727i
$714$	0	0
$715$	576.000	0.805594
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1170.97i	1.62861i	0.580439	+	0.814304i	$0.302881\pi$
$719$	1170.97i	1.62861i	−0.580439	+	0.814304i	$0.697119\pi$
$720$	0	0
$721$	112.000	0.155340
$722$	0	0
$723$	0	0
$724$	0	0
$725$	29.6985i	0.0409634i
$726$	0	0
$727$	508.000	0.698762	0.349381	−	0.936981i	$-0.386392\pi$
$727$	508.000	0.698762	0.349381	+	0.936981i	$0.386392\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 509.117i	− 0.696466i
$732$	0	0
$733$	1144.00	1.56071	0.780355	−	0.625337i	$-0.215039\pi$
$733$	1144.00	1.56071	0.780355	+	0.625337i	$0.215039\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 135.765i	− 0.184212i
$738$	0	0
$739$	−304.000	−0.411367	−0.205683	−	0.978619i	$-0.565942\pi$
$739$	−304.000	−0.411367	−0.205683	+	0.978619i	$0.565942\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	848.528i	1.14203i	0.820940	+	0.571015i	$0.193450\pi$
$743$	848.528i	1.14203i	−0.820940	+	0.571015i	$0.806550\pi$
$744$	0	0
$745$	1170.00	1.57047
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−188.000	−0.250333	−0.125166	−	0.992136i	$-0.539946\pi$
$751$	−188.000	−0.250333	−0.125166	+	0.992136i	$0.539946\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 627.911i	− 0.831670i
$756$	0	0
$757$	1240.00	1.63804	0.819022	−	0.573761i	$-0.194516\pi$
$757$	1240.00	1.63804	0.819022	+	0.573761i	$0.194516\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	156.978i	0.206278i	0.994667	+	0.103139i	$0.0328887\pi$
$761$	156.978i	0.206278i	−0.994667	+	0.103139i	$0.967111\pi$
$762$	0	0
$763$	−224.000	−0.293578
$764$	0	0
$765$	0	0
$766$	0	0
$767$	271.529i	0.354014i
$768$	0	0
$769$	−910.000	−1.18336	−0.591678	−	0.806175i	$-0.701534\pi$
$769$	−910.000	−1.18336	−0.591678	+	0.806175i	$0.701534\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1387.34i	− 1.79475i	−0.441266	−	0.897376i	$-0.645471\pi$
$773$	− 1387.34i	− 1.79475i	0.441266	−	0.897376i	$-0.354529\pi$
$774$	0	0
$775$	−308.000	−0.397419
$776$	0	0
$777$	0	0
$778$	0	0
$779$	746.705i	0.958543i
$780$	0	0
$781$	−864.000	−1.10627
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 347.897i	− 0.443180i
$786$	0	0
$787$	−1360.00	−1.72808	−0.864041	−	0.503422i	$-0.832074\pi$
$787$	−1360.00	−1.72808	−0.864041	+	0.503422i	$0.832074\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	627.911i	0.793819i
$792$	0	0
$793$	400.000	0.504414
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 106.066i	− 0.133082i	−0.997784	−	0.0665408i	$-0.978804\pi$
$797$	− 106.066i	− 0.133082i	0.997784	−	0.0665408i	$-0.0211963\pi$
$798$	0	0
$799$	1080.00	1.35169
$800$	0	0
$801$	0	0
$802$	0	0
$803$	271.529i	0.338143i
$804$	0	0
$805$	−288.000	−0.357764
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1107.33i	1.36876i	0.729124	+	0.684381i	$0.239928\pi$
$809$	1107.33i	1.36876i	−0.729124	+	0.684381i	$0.760072\pi$
$810$	0	0
$811$	−160.000	−0.197287	−0.0986436	−	0.995123i	$-0.531450\pi$
$811$	−160.000	−0.197287	−0.0986436	+	0.995123i	$0.531450\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 237.588i	− 0.291519i
$816$	0	0
$817$	640.000	0.783354
$818$	0	0
$819$	0	0
$820$	0	0
$821$	436.992i	0.532268i	0.963936	+	0.266134i	$0.0857464\pi$
$821$	436.992i	0.532268i	−0.963936	+	0.266134i	$0.914254\pi$
$822$	0	0
$823$	−332.000	−0.403402	−0.201701	−	0.979447i	$-0.564647\pi$
$823$	−332.000	−0.403402	−0.201701	+	0.979447i	$0.564647\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 101.823i	− 0.123124i	−0.998103	−	0.0615619i	$-0.980392\pi$
$827$	− 101.823i	− 0.123124i	0.998103	−	0.0615619i	$-0.0196082\pi$
$828$	0	0
$829$	−632.000	−0.762364	−0.381182	−	0.924500i	$-0.624483\pi$
$829$	−632.000	−0.762364	−0.381182	+	0.924500i	$0.624483\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 420.021i	− 0.504227i
$834$	0	0
$835$	144.000	0.172455
$836$	0	0
$837$	0	0
$838$	0	0
$839$	729.734i	0.869767i	0.900487	+	0.434883i	$0.143210\pi$
$839$	729.734i	0.869767i	−0.900487	+	0.434883i	$0.856790\pi$
$840$	0	0
$841$	823.000	0.978597
$842$	0	0
$843$	0	0
$844$	0	0
$845$	445.477i	0.527192i
$846$	0	0
$847$	−668.000	−0.788666
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 576.999i	− 0.678025i
$852$	0	0
$853$	−446.000	−0.522860	−0.261430	−	0.965222i	$-0.584194\pi$
$853$	−446.000	−0.522860	−0.261430	+	0.965222i	$0.584194\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	428.507i	0.500008i	0.968245	+	0.250004i	$0.0804319\pi$
$857$	428.507i	0.500008i	−0.968245	+	0.250004i	$0.919568\pi$
$858$	0	0
$859$	728.000	0.847497	0.423749	−	0.905780i	$-0.360714\pi$
$859$	728.000	0.847497	0.423749	+	0.905780i	$0.360714\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 916.410i	− 1.06189i	−0.847407	−	0.530945i	$-0.821837\pi$
$863$	− 916.410i	− 1.06189i	0.847407	−	0.530945i	$-0.178163\pi$
$864$	0	0
$865$	−738.000	−0.853179
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1289.76i	− 1.48419i
$870$	0	0
$871$	−64.0000	−0.0734788
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 543.058i	− 0.620638i
$876$	0	0
$877$	910.000	1.03763	0.518814	−	0.854887i	$-0.326374\pi$
$877$	910.000	1.03763	0.518814	+	0.854887i	$0.326374\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	929.138i	1.05464i	0.849667	+	0.527320i	$0.176803\pi$
$881$	929.138i	1.05464i	−0.849667	+	0.527320i	$0.823197\pi$
$882$	0	0
$883$	1064.00	1.20498	0.602492	−	0.798125i	$-0.294175\pi$
$883$	1064.00	1.20498	0.602492	+	0.798125i	$0.294175\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1391.59i	− 1.56887i	−0.620212	−	0.784434i	$-0.712953\pi$
$887$	− 1391.59i	− 1.56887i	0.620212	−	0.784434i	$-0.287047\pi$
$888$	0	0
$889$	−368.000	−0.413948
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1357.65i	1.52032i
$894$	0	0
$895$	864.000	0.965363
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 186.676i	− 0.207649i
$900$	0	0
$901$	−486.000	−0.539401
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 984.293i	− 1.08762i
$906$	0	0
$907$	−1768.00	−1.94928	−0.974642	−	0.223771i	$-0.928163\pi$
$907$	−1768.00	−1.94928	−0.974642	+	0.223771i	$0.928163\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 237.588i	− 0.260799i	−0.991462	−	0.130399i	$-0.958374\pi$
$911$	− 237.588i	− 0.260799i	0.991462	−	0.130399i	$-0.0416260\pi$
$912$	0	0
$913$	−2016.00	−2.20811
$914$	0	0
$915$	0	0
$916$	0	0
$917$	678.823i	0.740264i
$918$	0	0
$919$	−380.000	−0.413493	−0.206746	−	0.978395i	$-0.566288\pi$
$919$	−380.000	−0.413493	−0.206746	+	0.978395i	$0.566288\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	407.294i	0.441271i
$924$	0	0
$925$	238.000	0.257297
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 666.095i	− 0.717002i	−0.933529	−	0.358501i	$-0.883288\pi$
$929$	− 666.095i	− 0.717002i	0.933529	−	0.358501i	$-0.116712\pi$
$930$	0	0
$931$	528.000	0.567132
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 916.410i	− 0.980118i
$936$	0	0
$937$	−178.000	−0.189968	−0.0949840	−	0.995479i	$-0.530280\pi$
$937$	−178.000	−0.189968	−0.0949840	+	0.995479i	$0.530280\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 436.992i	− 0.464391i	−0.972669	−	0.232196i	$-0.925409\pi$
$941$	− 436.992i	− 0.464391i	0.972669	−	0.232196i	$-0.0745909\pi$
$942$	0	0
$943$	−792.000	−0.839873
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1798.88i	− 1.89956i	−0.312924	−	0.949778i	$-0.601309\pi$
$947$	− 1798.88i	− 1.89956i	0.312924	−	0.949778i	$-0.398691\pi$
$948$	0	0
$949$	128.000	0.134879
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1310.98i	− 1.37563i	−0.725886	−	0.687815i	$-0.758570\pi$
$953$	− 1310.98i	− 1.37563i	0.725886	−	0.687815i	$-0.241430\pi$
$954$	0	0
$955$	−144.000	−0.150785
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 627.911i	− 0.654756i
$960$	0	0
$961$	975.000	1.01457
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 873.984i	− 0.905683i
$966$	0	0
$967$	−1700.00	−1.75801	−0.879007	−	0.476808i	$-0.841794\pi$
$967$	−1700.00	−1.75801	−0.879007	+	0.476808i	$0.841794\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 458.205i	− 0.471890i	−0.971766	−	0.235945i	$-0.924181\pi$
$971$	− 458.205i	− 0.471890i	0.971766	−	0.235945i	$-0.0758185\pi$
$972$	0	0
$973$	608.000	0.624872
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 759.433i	− 0.777311i	−0.921383	−	0.388655i	$-0.872940\pi$
$977$	− 759.433i	− 0.777311i	0.921383	−	0.388655i	$-0.127060\pi$
$978$	0	0
$979$	−216.000	−0.220633
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1052.17i	− 1.07037i	−0.844734	−	0.535186i	$-0.820242\pi$
$983$	− 1052.17i	− 1.07037i	0.844734	−	0.535186i	$-0.179758\pi$
$984$	0	0
$985$	702.000	0.712690
$986$	0	0
$987$	0	0
$988$	0	0
$989$	678.823i	0.686373i
$990$	0	0
$991$	772.000	0.779011	0.389506	−	0.921024i	$-0.372646\pi$
$991$	772.000	0.779011	0.389506	+	0.921024i	$0.372646\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	84.8528i	0.0852792i
$996$	0	0
$997$	−194.000	−0.194584	−0.0972919	−	0.995256i	$-0.531018\pi$
$997$	−194.000	−0.194584	−0.0972919	+	0.995256i	$0.531018\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.3.e.f.449.1		2
3.2	odd	2	inner	576.3.e.f.449.2		2
4.3	odd	2		576.3.e.c.449.1		2
8.3	odd	2		18.3.b.a.17.1	✓	2
8.5	even	2		144.3.e.b.17.2		2
12.11	even	2		576.3.e.c.449.2		2
16.3	odd	4		2304.3.h.f.2177.1		4
16.5	even	4		2304.3.h.c.2177.4		4
16.11	odd	4		2304.3.h.f.2177.4		4
16.13	even	4		2304.3.h.c.2177.1		4
24.5	odd	2		144.3.e.b.17.1		2
24.11	even	2		18.3.b.a.17.2	yes	2
40.3	even	4		450.3.b.b.449.2		4
40.13	odd	4		3600.3.c.b.449.2		4
40.19	odd	2		450.3.d.f.251.2		2
40.27	even	4		450.3.b.b.449.3		4
40.29	even	2		3600.3.l.d.1601.2		2
40.37	odd	4		3600.3.c.b.449.4		4
48.5	odd	4		2304.3.h.c.2177.2		4
48.11	even	4		2304.3.h.f.2177.2		4
48.29	odd	4		2304.3.h.c.2177.3		4
48.35	even	4		2304.3.h.f.2177.3		4
56.3	even	6		882.3.s.d.863.2		4
56.11	odd	6		882.3.s.b.863.2		4
56.19	even	6		882.3.s.d.557.1		4
56.27	even	2		882.3.b.a.197.1		2
56.51	odd	6		882.3.s.b.557.1		4
72.5	odd	6		1296.3.q.f.593.1		4
72.11	even	6		162.3.d.b.53.2		4
72.13	even	6		1296.3.q.f.593.2		4
72.29	odd	6		1296.3.q.f.1025.2		4
72.43	odd	6		162.3.d.b.53.1		4
72.59	even	6		162.3.d.b.107.1		4
72.61	even	6		1296.3.q.f.1025.1		4
72.67	odd	6		162.3.d.b.107.2		4
88.43	even	2		2178.3.c.d.485.2		2
104.51	odd	2		3042.3.c.e.1691.2		2
104.83	even	4		3042.3.d.a.3041.1		4
104.99	even	4		3042.3.d.a.3041.4		4
120.29	odd	2		3600.3.l.d.1601.1		2
120.53	even	4		3600.3.c.b.449.1		4
120.59	even	2		450.3.d.f.251.1		2
120.77	even	4		3600.3.c.b.449.3		4
120.83	odd	4		450.3.b.b.449.4		4
120.107	odd	4		450.3.b.b.449.1		4
168.11	even	6		882.3.s.b.863.1		4
168.59	odd	6		882.3.s.d.863.1		4
168.83	odd	2		882.3.b.a.197.2		2
168.107	even	6		882.3.s.b.557.2		4
168.131	odd	6		882.3.s.d.557.2		4
264.131	odd	2		2178.3.c.d.485.1		2
312.83	odd	4		3042.3.d.a.3041.3		4
312.155	even	2		3042.3.c.e.1691.1		2
312.203	odd	4		3042.3.d.a.3041.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.3.b.a.17.1	✓	2	8.3	odd	2
18.3.b.a.17.2	yes	2	24.11	even	2
144.3.e.b.17.1		2	24.5	odd	2
144.3.e.b.17.2		2	8.5	even	2
162.3.d.b.53.1		4	72.43	odd	6
162.3.d.b.53.2		4	72.11	even	6
162.3.d.b.107.1		4	72.59	even	6
162.3.d.b.107.2		4	72.67	odd	6
450.3.b.b.449.1		4	120.107	odd	4
450.3.b.b.449.2		4	40.3	even	4
450.3.b.b.449.3		4	40.27	even	4
450.3.b.b.449.4		4	120.83	odd	4
450.3.d.f.251.1		2	120.59	even	2
450.3.d.f.251.2		2	40.19	odd	2
576.3.e.c.449.1		2	4.3	odd	2
576.3.e.c.449.2		2	12.11	even	2
576.3.e.f.449.1		2	1.1	even	1	trivial
576.3.e.f.449.2		2	3.2	odd	2	inner
882.3.b.a.197.1		2	56.27	even	2
882.3.b.a.197.2		2	168.83	odd	2
882.3.s.b.557.1		4	56.51	odd	6
882.3.s.b.557.2		4	168.107	even	6
882.3.s.b.863.1		4	168.11	even	6
882.3.s.b.863.2		4	56.11	odd	6
882.3.s.d.557.1		4	56.19	even	6
882.3.s.d.557.2		4	168.131	odd	6
882.3.s.d.863.1		4	168.59	odd	6
882.3.s.d.863.2		4	56.3	even	6
1296.3.q.f.593.1		4	72.5	odd	6
1296.3.q.f.593.2		4	72.13	even	6
1296.3.q.f.1025.1		4	72.61	even	6
1296.3.q.f.1025.2		4	72.29	odd	6
2178.3.c.d.485.1		2	264.131	odd	2
2178.3.c.d.485.2		2	88.43	even	2
2304.3.h.c.2177.1		4	16.13	even	4
2304.3.h.c.2177.2		4	48.5	odd	4
2304.3.h.c.2177.3		4	48.29	odd	4
2304.3.h.c.2177.4		4	16.5	even	4
2304.3.h.f.2177.1		4	16.3	odd	4
2304.3.h.f.2177.2		4	48.11	even	4
2304.3.h.f.2177.3		4	48.35	even	4
2304.3.h.f.2177.4		4	16.11	odd	4
3042.3.c.e.1691.1		2	312.155	even	2
3042.3.c.e.1691.2		2	104.51	odd	2
3042.3.d.a.3041.1		4	104.83	even	4
3042.3.d.a.3041.2		4	312.203	odd	4
3042.3.d.a.3041.3		4	312.83	odd	4
3042.3.d.a.3041.4		4	104.99	even	4
3600.3.c.b.449.1		4	120.53	even	4
3600.3.c.b.449.2		4	40.13	odd	4
3600.3.c.b.449.3		4	120.77	even	4
3600.3.c.b.449.4		4	40.37	odd	4
3600.3.l.d.1601.1		2	120.29	odd	2
3600.3.l.d.1601.2		2	40.29	even	2

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:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	576.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.24264i q^{5} +4.00000 q^{7} +O(q^{10})\)	\(q-4.24264i q^{5} +4.00000 q^{7} -16.9706i q^{11} -8.00000 q^{13} +12.7279i q^{17} -16.0000 q^{19} -16.9706i q^{23} +7.00000 q^{25} +4.24264i q^{29} -44.0000 q^{31} -16.9706i q^{35} +34.0000 q^{37} -46.6690i q^{41} -40.0000 q^{43} -84.8528i q^{47} -33.0000 q^{49} +38.1838i q^{53} -72.0000 q^{55} -33.9411i q^{59} -50.0000 q^{61} +33.9411i q^{65} +8.00000 q^{67} -50.9117i q^{71} -16.0000 q^{73} -67.8823i q^{77} +76.0000 q^{79} -118.794i q^{83} +54.0000 q^{85} -12.7279i q^{89} -32.0000 q^{91} +67.8823i q^{95} +176.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{7}+O(q^{10}) \) 2 * q + 8 * q^7	\( 2 q + 8 q^{7} - 16 q^{13} - 32 q^{19} + 14 q^{25} - 88 q^{31} + 68 q^{37} - 80 q^{43} - 66 q^{49} - 144 q^{55} - 100 q^{61} + 16 q^{67} - 32 q^{73} + 152 q^{79} + 108 q^{85} - 64 q^{91} + 352 q^{97}+O(q^{100}) \) 2 * q + 8 * q^7 - 16 * q^13 - 32 * q^19 + 14 * q^25 - 88 * q^31 + 68 * q^37 - 80 * q^43 - 66 * q^49 - 144 * q^55 - 100 * q^61 + 16 * q^67 - 32 * q^73 + 152 * q^79 + 108 * q^85 - 64 * q^91 + 352 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more