LMFDB - Embedded newform 164.5.d.b.163.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [164,5,Mod(163,164)] 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("164.163"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(164, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 164 = 2^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	164.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,-8,0,32,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$16.9526739458$
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:	$ 2^{4}\cdot 3\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			163.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	164.163
Dual form			164.5.d.b.163.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{2} +16.0000 q^{4} +14.0000 q^{5} -64.0000 q^{8} -81.0000 q^{9} -56.0000 q^{10} -240.000i q^{13} +256.000 q^{16} +480.000i q^{17} +324.000 q^{18} +224.000 q^{20} -429.000 q^{25} +960.000i q^{26} -1680.00i q^{29} -1024.00 q^{32} -1920.00i q^{34} -1296.00 q^{36} -2162.00 q^{37} -896.000 q^{40} +(-1519.00 + 720.000i) q^{41} -1134.00 q^{45} -2401.00 q^{49} +1716.00 q^{50} -3840.00i q^{52} -5040.00i q^{53} +6720.00i q^{58} -6958.00 q^{61} +4096.00 q^{64} -3360.00i q^{65} +7680.00i q^{68} +5184.00 q^{72} -1442.00 q^{73} +8648.00 q^{74} +3584.00 q^{80} +6561.00 q^{81} +(6076.00 - 2880.00i) q^{82} +6720.00i q^{85} +12480.0i q^{89} +4536.00 q^{90} -18720.0i q^{97} +9604.00 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 32 q^{4} + 28 q^{5} - 128 q^{8} - 162 q^{9} - 112 q^{10} + 512 q^{16} + 648 q^{18} + 448 q^{20} - 858 q^{25} - 2048 q^{32} - 2592 q^{36} - 4324 q^{37} - 1792 q^{40} - 3038 q^{41} - 2268 q^{45}+ \cdots + 19208 q^{98}+O(q^{100}) $ 2 * q - 8 * q^2 + 32 * q^4 + 28 * q^5 - 128 * q^8 - 162 * q^9 - 112 * q^10 + 512 * q^16 + 648 * q^18 + 448 * q^20 - 858 * q^25 - 2048 * q^32 - 2592 * q^36 - 4324 * q^37 - 1792 * q^40 - 3038 * q^41 - 2268 * q^45 - 4802 * q^49 + 3432 * q^50 - 13916 * q^61 + 8192 * q^64 + 10368 * q^72 - 2884 * q^73 + 17296 * q^74 + 7168 * q^80 + 13122 * q^81 + 12152 * q^82 + 9072 * q^90 + 19208 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/164\mathbb{Z}\right)^\times$.

$n$	$83$	$129$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−4.00000			−1.00000
$2$	−4.00000			−1.00000
$3$		0			0			−	1.00000i	$-0.5\pi$
$3$		0			0				1.00000i	$0.5\pi$
$4$	16.0000			1.00000
$5$	14.0000			0.560000			0.280000	−	0.960000i	$-0.409666\pi$
$5$	14.0000			0.560000			0.280000	+	0.960000i	$0.409666\pi$
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	−64.0000			−1.00000
$9$	−81.0000			−1.00000
$10$	−56.0000			−0.560000
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$		−	240.000i		−	1.42012i	−0.704142	−	0.710059i	$-0.748668\pi$
$13$		−	240.000i		−	1.42012i	0.704142	−	0.710059i	$-0.251332\pi$
$14$		0			0
$15$		0			0
$16$	256.000			1.00000
$17$			480.000i			1.66090i	0.557093	+	0.830450i	$0.311917\pi$
$17$			480.000i			1.66090i	−0.557093	+	0.830450i	$0.688083\pi$
$18$	324.000			1.00000
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	224.000			0.560000
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−429.000			−0.686400
$26$			960.000i			1.42012i
$27$		0			0
$28$		0			0
$29$		−	1680.00i		−	1.99762i	−0.0487515	−	0.998811i	$-0.515524\pi$
$29$		−	1680.00i		−	1.99762i	0.0487515	−	0.998811i	$-0.484476\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−1024.00			−1.00000
$33$		0			0
$34$		−	1920.00i		−	1.66090i
$35$		0			0
$36$	−1296.00			−1.00000
$37$	−2162.00			−1.57925			−0.789627	−	0.613587i	$-0.789726\pi$
$37$	−2162.00			−1.57925			−0.789627	+	0.613587i	$0.789726\pi$
$38$		0			0
$39$		0			0
$40$	−896.000			−0.560000
$41$	−1519.00	+	720.000i	−0.903629	+	0.428316i
$41$	−1519.00	+	720.000i	−0.903629	+	0.428316i
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−1134.00			−0.560000
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−2401.00			−1.00000
$50$	1716.00			0.686400
$51$		0			0
$52$		−	3840.00i		−	1.42012i
$53$		−	5040.00i		−	1.79423i	−0.441794	−	0.897116i	$-0.645658\pi$
$53$		−	5040.00i		−	1.79423i	0.441794	−	0.897116i	$-0.354342\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$			6720.00i			1.99762i
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	−6958.00			−1.86993			−0.934964	−	0.354743i	$-0.884568\pi$
$61$	−6958.00			−1.86993			−0.934964	+	0.354743i	$0.884568\pi$
$62$		0			0
$63$		0			0
$64$	4096.00			1.00000
$65$		−	3360.00i		−	0.795266i
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$			7680.00i			1.66090i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	5184.00			1.00000
$73$	−1442.00			−0.270595			−0.135297	−	0.990805i	$-0.543199\pi$
$73$	−1442.00			−0.270595			−0.135297	+	0.990805i	$0.543199\pi$
$74$	8648.00			1.57925
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	3584.00			0.560000
$81$	6561.00			1.00000
$82$	6076.00	−	2880.00i	0.903629	−	0.428316i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$			6720.00i			0.930104i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			12480.0i			1.57556i	0.615958	+	0.787779i	$0.288769\pi$
$89$			12480.0i			1.57556i	−0.615958	+	0.787779i	$0.711231\pi$
$90$	4536.00			0.560000
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	18720.0i		−	1.98958i	−0.101924	−	0.994792i	$-0.532500\pi$
$97$		−	18720.0i		−	1.98958i	0.101924	−	0.994792i	$-0.467500\pi$
$98$	9604.00			1.00000
$99$		0			0
$100$	−6864.00			−0.686400
$101$			7920.00i			0.776394i	0.921576	+	0.388197i	$0.126902\pi$
$101$			7920.00i			0.776394i	−0.921576	+	0.388197i	$0.873098\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$			15360.0i			1.42012i
$105$		0			0
$106$			20160.0i			1.79423i
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$		−	21840.0i		−	1.83823i	−0.393990	−	0.919115i	$-0.628906\pi$
$109$		−	21840.0i		−	1.83823i	0.393990	−	0.919115i	$-0.371094\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	24638.0			1.92952			0.964758	−	0.263137i	$-0.0847572\pi$
$113$	24638.0			1.92952			0.964758	+	0.263137i	$0.0847572\pi$
$114$		0			0
$115$		0			0
$116$		−	26880.0i		−	1.99762i
$117$			19440.0i			1.42012i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−14641.0			−1.00000
$122$	27832.0			1.86993
$123$		0			0
$124$		0			0
$125$	−14756.0			−0.944384
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−16384.0			−1.00000
$129$		0			0
$130$			13440.0i			0.795266i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		−	30720.0i		−	1.66090i
$137$			36960.0i			1.96920i	0.174810	+	0.984602i	$0.444069\pi$
$137$			36960.0i			1.96920i	−0.174810	+	0.984602i	$0.555931\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−20736.0			−1.00000
$145$		−	23520.0i		−	1.11867i
$146$	5768.00			0.270595
$147$		0			0
$148$	−34592.0			−1.57925
$149$		−	28560.0i		−	1.28643i	−0.765686	−	0.643214i	$-0.777600\pi$
$149$		−	28560.0i		−	1.28643i	0.765686	−	0.643214i	$-0.222400\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		−	38880.0i		−	1.66090i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			44880.0i			1.82076i	0.413769	+	0.910382i	$0.364212\pi$
$157$			44880.0i			1.82076i	−0.413769	+	0.910382i	$0.635788\pi$
$158$		0			0
$159$		0			0
$160$	−14336.0			−0.560000
$161$		0			0
$162$	−26244.0			−1.00000
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$	−24304.0	+	11520.0i	−0.903629	+	0.428316i
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−29039.0			−1.01674
$170$		−	26880.0i		−	0.930104i
$171$		0			0
$172$		0			0
$173$	49042.0			1.63861			0.819306	−	0.573357i	$-0.194359\pi$
$173$	49042.0			1.63861			0.819306	+	0.573357i	$0.194359\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		−	49920.0i		−	1.57556i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−18144.0			−0.560000
$181$		−	13680.0i		−	0.417570i	−0.977962	−	0.208785i	$-0.933049\pi$
$181$		−	13680.0i		−	0.417570i	0.977962	−	0.208785i	$-0.0669508\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−30268.0			−0.884383
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$			63840.0i			1.71387i	0.515423	+	0.856936i	$0.327635\pi$
$193$			63840.0i			1.71387i	−0.515423	+	0.856936i	$0.672365\pi$
$194$			74880.0i			1.98958i
$195$		0			0
$196$	−38416.0			−1.00000
$197$	74482.0			1.91919			0.959597	−	0.281378i	$-0.0907915\pi$
$197$	74482.0			1.91919			0.959597	+	0.281378i	$0.0907915\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	27456.0			0.686400
$201$		0			0
$202$		−	31680.0i		−	0.776394i
$203$		0			0
$204$		0			0
$205$	−21266.0	+	10080.0i	−0.506032	+	0.239857i
$206$		0			0
$207$		0			0
$208$		−	61440.0i		−	1.42012i
$209$		0			0
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		−	80640.0i		−	1.79423i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$			87360.0i			1.83823i
$219$		0			0
$220$		0			0
$221$	115200.			2.35867
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	34749.0			0.686400
$226$	−98552.0			−1.92952
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		−	53040.0i		−	1.01142i	−0.862703	−	0.505711i	$-0.831230\pi$
$229$		−	53040.0i		−	1.01142i	0.862703	−	0.505711i	$-0.168770\pi$
$230$		0			0
$231$		0			0
$232$			107520.i			1.99762i
$233$			87360.0i			1.60917i	0.593840	+	0.804583i	$0.297611\pi$
$233$			87360.0i			1.60917i	−0.593840	+	0.804583i	$0.702389\pi$
$234$		−	77760.0i		−	1.42012i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	58562.0			1.00828			0.504141	−	0.863621i	$-0.331809\pi$
$241$	58562.0			1.00828			0.504141	+	0.863621i	$0.331809\pi$
$242$	58564.0			1.00000
$243$		0			0
$244$	−111328.			−1.86993
$245$	−33614.0			−0.560000
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	59024.0			0.944384
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	65536.0			1.00000
$257$		−	32640.0i		−	0.494179i	−0.968993	−	0.247089i	$-0.920526\pi$
$257$		−	32640.0i		−	0.494179i	0.968993	−	0.247089i	$-0.0794741\pi$
$258$		0			0
$259$		0			0
$260$		−	53760.0i		−	0.795266i
$261$			136080.i			1.99762i
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		−	70560.0i		−	1.00477i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−125678.			−1.73682			−0.868410	−	0.495847i	$-0.834858\pi$
$269$	−125678.			−1.73682			−0.868410	+	0.495847i	$0.834858\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$			122880.i			1.66090i
$273$		0			0
$274$		−	147840.i		−	1.96920i
$275$		0			0
$276$		0			0
$277$	100558.			1.31056			0.655280	−	0.755386i	$-0.272551\pi$
$277$	100558.			1.31056			0.655280	+	0.755386i	$0.272551\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	147840.i		−	1.87232i	−0.351579	−	0.936158i	$-0.614355\pi$
$281$		−	147840.i		−	1.87232i	0.351579	−	0.936158i	$-0.385645\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	82944.0			1.00000
$289$	−146879.			−1.75859
$290$			94080.0i			1.11867i
$291$		0			0
$292$	−23072.0			−0.270595
$293$			77520.0i			0.902981i	0.892276	+	0.451490i	$0.149107\pi$
$293$			77520.0i			0.902981i	−0.892276	+	0.451490i	$0.850893\pi$
$294$		0			0
$295$		0			0
$296$	138368.			1.57925
$297$		0			0
$298$			114240.i			1.28643i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−97412.0			−1.04716
$306$			155520.i			1.66090i
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	31200.0i		−	0.318468i	−0.987241	−	0.159234i	$-0.949098\pi$
$313$		−	31200.0i		−	0.318468i	0.987241	−	0.159234i	$-0.0509024\pi$
$314$		−	179520.i		−	1.82076i
$315$		0			0
$316$		0			0
$317$		−	92400.0i		−	0.919504i	−0.888047	−	0.459752i	$-0.847938\pi$
$317$		−	92400.0i		−	0.919504i	0.888047	−	0.459752i	$-0.152062\pi$
$318$		0			0
$319$		0			0
$320$	57344.0			0.560000
$321$		0			0
$322$		0			0
$323$		0			0
$324$	104976.			1.00000
$325$			102960.i			0.974769i
$326$		0			0
$327$		0			0
$328$	97216.0	−	46080.0i	0.903629	−	0.428316i
$329$		0			0
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$	175122.			1.57925
$334$		0			0
$335$		0			0
$336$		0			0
$337$	104638.			0.921361			0.460680	−	0.887566i	$-0.347605\pi$
$337$	104638.			0.921361			0.460680	+	0.887566i	$0.347605\pi$
$338$	116156.			1.01674
$339$		0			0
$340$			107520.i			0.930104i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−196168.			−1.63861
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$	−114002.			−0.935969			−0.467985	−	0.883737i	$-0.655020\pi$
$349$	−114002.			−0.935969			−0.467985	+	0.883737i	$0.655020\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	46718.0			0.374917			0.187458	−	0.982273i	$-0.439975\pi$
$353$	46718.0			0.374917			0.187458	+	0.982273i	$0.439975\pi$
$354$		0			0
$355$		0			0
$356$			199680.i			1.57556i
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$	72576.0			0.560000
$361$	−130321.			−1.00000
$362$			54720.0i			0.417570i
$363$		0			0
$364$		0			0
$365$	−20188.0			−0.151533
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	123039.	−	58320.0i	0.903629	−	0.428316i
$370$	121072.			0.884383
$371$		0			0
$372$		0			0
$373$	24242.0			0.174241			0.0871206	−	0.996198i	$-0.472233\pi$
$373$	24242.0			0.174241			0.0871206	+	0.996198i	$0.472233\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−403200.			−2.83686
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		−	255360.i		−	1.71387i
$387$		0			0
$388$		−	299520.i		−	1.98958i
$389$	−159758.			−1.05576			−0.527878	−	0.849320i	$-0.677012\pi$
$389$	−159758.			−1.05576			−0.527878	+	0.849320i	$0.677012\pi$
$390$		0			0
$391$		0			0
$392$	153664.			1.00000
$393$		0			0
$394$	−297928.			−1.91919
$395$		0			0
$396$		0			0
$397$		−	296400.i		−	1.88060i	−0.340342	−	0.940302i	$-0.610543\pi$
$397$		−	296400.i		−	1.88060i	0.340342	−	0.940302i	$-0.389457\pi$
$398$		0			0
$399$		0			0
$400$	−109824.			−0.686400
$401$	−315202.			−1.96020			−0.980100	−	0.198506i	$-0.936391\pi$
$401$	−315202.			−1.96020			−0.980100	+	0.198506i	$0.936391\pi$
$402$		0			0
$403$		0			0
$404$			126720.i			0.776394i
$405$	91854.0			0.560000
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−276962.			−1.65567			−0.827835	−	0.560972i	$-0.810427\pi$
$409$	−276962.			−1.65567			−0.827835	+	0.560972i	$0.810427\pi$
$410$	85064.0	−	40320.0i	0.506032	−	0.239857i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$			245760.i			1.42012i
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		−	48720.0i		−	0.274880i	−0.990510	−	0.137440i	$-0.956113\pi$
$421$		−	48720.0i		−	0.274880i	0.990510	−	0.137440i	$-0.0438874\pi$
$422$		0			0
$423$		0			0
$424$			322560.i			1.79423i
$425$		−	205920.i		−	1.14004i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	−290878.			−1.55144			−0.775720	−	0.631077i	$-0.782613\pi$
$433$	−290878.			−1.55144			−0.775720	+	0.631077i	$0.782613\pi$
$434$		0			0
$435$		0			0
$436$		−	349440.i		−	1.83823i
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	194481.			1.00000
$442$	−460800.			−2.35867
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$			174720.i			0.882313i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	89602.0			0.444452			0.222226	−	0.974995i	$-0.428668\pi$
$449$	89602.0			0.444452			0.222226	+	0.974995i	$0.428668\pi$
$450$	−138996.			−0.686400
$451$		0			0
$452$	394208.			1.92952
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	285600.i		−	1.36750i	−0.729719	−	0.683748i	$-0.760349\pi$
$457$		−	285600.i		−	1.36750i	0.729719	−	0.683748i	$-0.239651\pi$
$458$			212160.i			1.01142i
$459$		0			0
$460$		0			0
$461$	152558.			0.717849			0.358925	−	0.933367i	$-0.383144\pi$
$461$	152558.			0.717849			0.358925	+	0.933367i	$0.383144\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		−	430080.i		−	1.99762i
$465$		0			0
$466$		−	349440.i		−	1.60917i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$			311040.i			1.42012i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$			408240.i			1.79423i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$			518880.i			2.24273i
$482$	−234248.			−1.00828
$483$		0			0
$484$	−234256.			−1.00000
$485$		−	262080.i		−	1.11417i
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	445312.			1.86993
$489$		0			0
$490$	134456.			0.560000
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	806400.			3.31785
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−236096.			−0.944384
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$			110880.i			0.434781i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			403920.i			1.55905i	0.626372	+	0.779525i	$0.284539\pi$
$509$			403920.i			1.55905i	−0.626372	+	0.779525i	$0.715461\pi$
$510$		0			0
$511$		0			0
$512$	−262144.			−1.00000
$513$		0			0
$514$			130560.i			0.494179i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$			215040.i			0.795266i
$521$		−	491040.i		−	1.80901i	−0.426461	−	0.904506i	$-0.640240\pi$
$521$		−	491040.i		−	1.80901i	0.426461	−	0.904506i	$-0.359760\pi$
$522$		−	544320.i		−	1.99762i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	279841.			1.00000
$530$			282240.i			1.00477i
$531$		0			0
$532$		0			0
$533$	172800.	+	364560.i	0.608260	+	1.28326i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	502712.			1.73682
$539$		0			0
$540$		0			0
$541$	120238.			0.410816			0.205408	−	0.978676i	$-0.434148\pi$
$541$	120238.			0.410816			0.205408	+	0.978676i	$0.434148\pi$
$542$		0			0
$543$		0			0
$544$		−	491520.i		−	1.66090i
$545$		−	305760.i		−	1.02941i
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$			591360.i			1.96920i
$549$	563598.			1.86993
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−402232.			−1.31056
$555$		0			0
$556$		0			0
$557$			351120.i			1.13174i	0.824496	+	0.565868i	$0.191459\pi$
$557$			351120.i			1.13174i	−0.824496	+	0.565868i	$0.808541\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$			591360.i			1.87232i
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$	344932.			1.08053
$566$		0			0
$567$		0			0
$568$		0			0
$569$	434078.			1.34074			0.670368	−	0.742029i	$-0.266136\pi$
$569$	434078.			1.34074			0.670368	+	0.742029i	$0.266136\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−331776.			−1.00000
$577$		−	110400.i		−	0.331602i	−0.986159	−	0.165801i	$-0.946979\pi$
$577$		−	110400.i		−	0.331602i	0.986159	−	0.165801i	$-0.0530210\pi$
$578$	587516.			1.75859
$579$		0			0
$580$		−	376320.i		−	1.11867i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	92288.0			0.270595
$585$			272160.i			0.795266i
$586$		−	310080.i		−	0.902981i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−553472.			−1.57925
$593$			684480.i			1.94649i	0.229777	+	0.973243i	$0.426200\pi$
$593$			684480.i			1.94649i	−0.229777	+	0.973243i	$0.573800\pi$
$594$		0			0
$595$		0			0
$596$		−	456960.i		−	1.28643i
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$			528960.i			1.46445i	0.681064	+	0.732224i	$0.261518\pi$
$601$			528960.i			1.46445i	−0.681064	+	0.732224i	$0.738482\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−204974.			−0.560000
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$	389648.			1.04716
$611$		0			0
$612$		−	622080.i		−	1.66090i
$613$	746638.			1.98696			0.993480	−	0.114006i	$-0.0363684\pi$
$613$	746638.			1.98696			0.993480	+	0.114006i	$0.0363684\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−717278.			−1.88416			−0.942079	−	0.335392i	$-0.891131\pi$
$617$	−717278.			−1.88416			−0.942079	+	0.335392i	$0.891131\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	61541.0			0.157545
$626$			124800.i			0.318468i
$627$		0			0
$628$			718080.i			1.82076i
$629$		−	1.03776e6i		−	2.62298i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$			369600.i			0.919504i
$635$		0			0
$636$		0			0
$637$			576240.i			1.42012i
$638$		0			0
$639$		0			0
$640$	−229376.			−0.560000
$641$		−	487200.i		−	1.18574i	−0.805296	−	0.592872i	$-0.797994\pi$
$641$		−	487200.i		−	1.18574i	0.805296	−	0.592872i	$-0.202006\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−419904.			−1.00000
$649$		0			0
$650$		−	411840.i		−	0.974769i
$651$		0			0
$652$		0			0
$653$		−	720720.i		−	1.69021i	−0.534602	−	0.845104i	$-0.679538\pi$
$653$		−	720720.i		−	1.69021i	0.534602	−	0.845104i	$-0.320462\pi$
$654$		0			0
$655$		0			0
$656$	−388864.	+	184320.i	−0.903629	+	0.428316i
$657$	116802.			0.270595
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−513842.			−1.17605			−0.588026	−	0.808842i	$-0.700095\pi$
$661$	−513842.			−1.17605			−0.588026	+	0.808842i	$0.700095\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−700488.			−1.57925
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	850080.i		−	1.87685i	−0.345482	−	0.938425i	$-0.612285\pi$
$673$		−	850080.i		−	1.87685i	0.345482	−	0.938425i	$-0.387715\pi$
$674$	−418552.			−0.921361
$675$		0			0
$676$	−464624.			−1.01674
$677$	−905842.			−1.97640			−0.988201	−	0.153165i	$-0.951053\pi$
$677$	−905842.			−1.97640			−0.988201	+	0.153165i	$0.951053\pi$
$678$		0			0
$679$		0			0
$680$		−	430080.i		−	0.930104i
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$			517440.i			1.10275i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.20960e6			−2.54802
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	784672.			1.63861
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−345600.	−	729120.i	−0.711391	−	1.50084i
$698$	456008.			0.935969
$699$		0			0
$700$		0			0
$701$	−712402.			−1.44974			−0.724868	−	0.688887i	$-0.758099\pi$
$701$	−712402.			−1.44974			−0.724868	+	0.688887i	$0.758099\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−186872.			−0.374917
$707$		0			0
$708$		0			0
$709$			683760.i			1.36023i	0.733107	+	0.680113i	$0.238069\pi$
$709$			683760.i			1.36023i	−0.733107	+	0.680113i	$0.761931\pi$
$710$		0			0
$711$		0			0
$712$		−	798720.i		−	1.57556i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−290304.			−0.560000
$721$		0			0
$722$	521284.			1.00000
$723$		0			0
$724$		−	218880.i		−	0.417570i
$725$			720720.i			1.37117i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−531441.			−1.00000
$730$	80752.0			0.151533
$731$		0			0
$732$		0			0
$733$	1.02792e6			1.91316			0.956582	−	0.291463i	$-0.0941421\pi$
$733$	1.02792e6			1.91316			0.956582	+	0.291463i	$0.0941421\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−492156.	+	233280.i	−0.903629	+	0.428316i
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$	−484288.			−0.884383
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		−	399840.i		−	0.720400i
$746$	−96968.0			−0.174241
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$	1.61280e6			2.83686
$755$		0			0
$756$		0			0
$757$		−	1.11384e6i		−	1.94371i	−0.235584	−	0.971854i	$-0.575700\pi$
$757$		−	1.11384e6i		−	1.94371i	0.235584	−	0.971854i	$-0.424300\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.15216e6			1.98949			0.994747	−	0.102362i	$-0.0326400\pi$
$761$	1.15216e6			1.98949			0.994747	+	0.102362i	$0.0326400\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		−	544320.i		−	0.930104i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−257278.			−0.435061			−0.217530	−	0.976054i	$-0.569800\pi$
$769$	−257278.			−0.435061			−0.217530	+	0.976054i	$0.569800\pi$
$770$		0			0
$771$		0			0
$772$			1.02144e6i			1.71387i
$773$			583440.i			0.976421i	0.872726	+	0.488211i	$0.162350\pi$
$773$			583440.i			0.976421i	−0.872726	+	0.488211i	$0.837650\pi$
$774$		0			0
$775$		0			0
$776$			1.19808e6i			1.98958i
$777$		0			0
$778$	639032.			1.05576
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−614656.			−1.00000
$785$			628320.i			1.01963i
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	1.19171e6			1.91919
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$			1.66992e6i			2.65552i
$794$			1.18560e6i			1.88060i
$795$		0			0
$796$		0			0
$797$	−38318.0			−0.0603235			−0.0301617	−	0.999545i	$-0.509602\pi$
$797$	−38318.0			−0.0603235			−0.0301617	+	0.999545i	$0.509602\pi$
$798$		0			0
$799$		0			0
$800$	439296.			0.686400
$801$		−	1.01088e6i		−	1.57556i
$802$	1.26081e6			1.96020
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		−	506880.i		−	0.776394i
$809$		−	850080.i		−	1.29886i	−0.760421	−	0.649431i	$-0.775007\pi$
$809$		−	850080.i		−	1.29886i	0.760421	−	0.649431i	$-0.224993\pi$
$810$	−367416.			−0.560000
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.10785e6			1.65567
$819$		0			0
$820$	−340256.	+	161280.i	−0.506032	+	0.239857i
$821$	611918.			0.907835			0.453917	−	0.891044i	$-0.350026\pi$
$821$	611918.			0.907835			0.453917	+	0.891044i	$0.350026\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	−208082.			−0.302779			−0.151389	−	0.988474i	$-0.548375\pi$
$829$	−208082.			−0.302779			−0.151389	+	0.988474i	$0.548375\pi$
$830$		0			0
$831$		0			0
$832$		−	983040.i		−	1.42012i
$833$		−	1.15248e6i		−	1.66090i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−2.11512e6			−2.99049
$842$			194880.i			0.274880i
$843$		0			0
$844$		0			0
$845$	−406546.			−0.569372
$846$		0			0
$847$		0			0
$848$		−	1.29024e6i		−	1.79423i
$849$		0			0
$850$			823680.i			1.14004i
$851$		0			0
$852$		0			0
$853$	−1.28712e6			−1.76897			−0.884485	−	0.466569i	$-0.845490\pi$
$853$	−1.28712e6			−1.76897			−0.884485	+	0.466569i	$0.845490\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.25360e6			−1.70686			−0.853430	−	0.521207i	$-0.825482\pi$
$857$	−1.25360e6			−1.70686			−0.853430	+	0.521207i	$0.825482\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	686588.			0.917622
$866$	1.16351e6			1.55144
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$			1.39776e6i			1.83823i
$873$			1.51632e6i			1.98958i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.05384e6			1.37018			0.685088	−	0.728460i	$-0.259764\pi$
$877$	1.05384e6			1.37018			0.685088	+	0.728460i	$0.259764\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.00768e6			−1.29828			−0.649142	−	0.760667i	$-0.724872\pi$
$881$	−1.00768e6			−1.29828			−0.649142	+	0.760667i	$0.724872\pi$
$882$	−777924.			−1.00000
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$	1.84320e6			2.35867
$885$		0			0
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		−	698880.i		−	0.882313i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−358408.			−0.444452
$899$		0			0
$900$	555984.			0.686400
$901$	2.41920e6			2.98004
$902$		0			0
$903$		0			0
$904$	−1.57683e6			−1.92952
$905$		−	191520.i		−	0.233839i
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$		−	641520.i		−	0.776394i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$			1.14240e6i			1.36750i
$915$		0			0
$916$		−	848640.i		−	1.01142i
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$	−610232.			−0.717849
$923$		0			0
$924$		0			0
$925$	927498.			1.08400
$926$		0			0
$927$		0			0
$928$			1.72032e6i			1.99762i
$929$		−	474720.i		−	0.550055i	−0.961436	−	0.275027i	$-0.911313\pi$
$929$		−	474720.i		−	0.550055i	0.961436	−	0.275027i	$-0.0886870\pi$
$930$		0			0
$931$		0			0
$932$			1.39776e6i			1.60917i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		−	1.24416e6i		−	1.42012i
$937$		−	784320.i		−	0.893335i	−0.894700	−	0.446667i	$-0.852611\pi$
$937$		−	784320.i		−	0.893335i	0.894700	−	0.446667i	$-0.147389\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	425362.			0.480374			0.240187	−	0.970727i	$-0.422791\pi$
$941$	425362.			0.480374			0.240187	+	0.970727i	$0.422791\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$			346080.i			0.384277i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	303518.			0.334194			0.167097	−	0.985940i	$-0.446561\pi$
$953$	303518.			0.334194			0.167097	+	0.985940i	$0.446561\pi$
$954$		−	1.63296e6i		−	1.79423i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	923521.			1.00000
$962$		−	2.07552e6i		−	2.24273i
$963$		0			0
$964$	936992.			1.00828
$965$			893760.i			0.959768i
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	937024.			1.00000
$969$		0			0
$970$			1.04832e6i			1.11417i
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−1.78125e6			−1.86993
$977$			937440.i			0.982097i	0.871132	+	0.491048i	$0.163386\pi$
$977$			937440.i			0.982097i	−0.871132	+	0.491048i	$0.836614\pi$
$978$		0			0
$979$		0			0
$980$	−537824.			−0.560000
$981$			1.76904e6i			1.83823i
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$	1.04275e6			1.07475
$986$	−3.22560e6			−3.31785
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$			1.37640e6i			1.38470i	0.721564	+	0.692348i	$0.243424\pi$
$997$			1.37640e6i			1.38470i	−0.721564	+	0.692348i	$0.756576\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.5.d.b.163.1	✓	2
4.3	odd	2	CM	164.5.d.b.163.1	✓	2
41.40	even	2	inner	164.5.d.b.163.2	yes	2
164.163	odd	2	inner	164.5.d.b.163.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.5.d.b.163.1	✓	2	1.1	even	1	trivial
164.5.d.b.163.1	✓	2	4.3	odd	2	CM
164.5.d.b.163.2	yes	2	41.40	even	2	inner
164.5.d.b.163.2	yes	2	164.163	odd	2	inner

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:	\( N \)	\(=\)	\( 164 = 2^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	164.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} +14.0000 q^{5} -64.0000 q^{8} -81.0000 q^{9} -56.0000 q^{10} -240.000i q^{13} +256.000 q^{16} +480.000i q^{17} +324.000 q^{18} +224.000 q^{20} -429.000 q^{25} +960.000i q^{26} -1680.00i q^{29} -1024.00 q^{32} -1920.00i q^{34} -1296.00 q^{36} -2162.00 q^{37} -896.000 q^{40} +(-1519.00 + 720.000i) q^{41} -1134.00 q^{45} -2401.00 q^{49} +1716.00 q^{50} -3840.00i q^{52} -5040.00i q^{53} +6720.00i q^{58} -6958.00 q^{61} +4096.00 q^{64} -3360.00i q^{65} +7680.00i q^{68} +5184.00 q^{72} -1442.00 q^{73} +8648.00 q^{74} +3584.00 q^{80} +6561.00 q^{81} +(6076.00 - 2880.00i) q^{82} +6720.00i q^{85} +12480.0i q^{89} +4536.00 q^{90} -18720.0i q^{97} +9604.00 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 32 q^{4} + 28 q^{5} - 128 q^{8} - 162 q^{9} - 112 q^{10} + 512 q^{16} + 648 q^{18} + 448 q^{20} - 858 q^{25} - 2048 q^{32} - 2592 q^{36} - 4324 q^{37} - 1792 q^{40} - 3038 q^{41} - 2268 q^{45}+ \cdots + 19208 q^{98}+O(q^{100}) \) 2 * q - 8 * q^2 + 32 * q^4 + 28 * q^5 - 128 * q^8 - 162 * q^9 - 112 * q^10 + 512 * q^16 + 648 * q^18 + 448 * q^20 - 858 * q^25 - 2048 * q^32 - 2592 * q^36 - 4324 * q^37 - 1792 * q^40 - 3038 * q^41 - 2268 * q^45 - 4802 * q^49 + 3432 * q^50 - 13916 * q^61 + 8192 * q^64 + 10368 * q^72 - 2884 * q^73 + 17296 * q^74 + 7168 * q^80 + 13122 * q^81 + 12152 * q^82 + 9072 * q^90 + 19208 * q^98

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