LMFDB - Embedded newform 408.1.bg.a.299.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [408,1,Mod(11,408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(408, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([8, 8, 8, 7]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("408.11");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 408 = 2^{3} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	408.bg (of order $16$, degree $8$, 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:	$0.203618525154$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{16}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{16} - \cdots)$

Embedding invariants

Embedding label			299.1
Root			$-0.923880 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	408.299
Dual form			408.1.bg.a.131.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.923880 + 0.382683i) q^{2} +(-0.707107 + 0.707107i) q^{3} +(0.707107 + 0.707107i) q^{4} +(-0.923880 + 0.382683i) q^{6} +(0.382683 + 0.923880i) q^{8} -1.00000i q^{9} +O(q^{10})$	\(q+(0.923880 + 0.382683i) q^{2} +(-0.707107 + 0.707107i) q^{3} +(0.707107 + 0.707107i) q^{4} +(-0.923880 + 0.382683i) q^{6} +(0.382683 + 0.923880i) q^{8} -1.00000i q^{9} +(-0.923880 + 0.617317i) q^{11} -1.00000 q^{12} +1.00000i q^{16} +(0.382683 - 0.923880i) q^{17} +(0.382683 - 0.923880i) q^{18} +(0.541196 - 1.30656i) q^{19} +(-1.08979 + 0.216773i) q^{22} +(-0.923880 - 0.382683i) q^{24} +(-0.923880 + 0.382683i) q^{25} +(0.707107 + 0.707107i) q^{27} +(-0.382683 + 0.923880i) q^{32} +(0.216773 - 1.08979i) q^{33} +(0.707107 - 0.707107i) q^{34} +(0.707107 - 0.707107i) q^{36} +(1.00000 - 1.00000i) q^{38} +(0.324423 - 1.63099i) q^{41} +(-0.292893 - 0.707107i) q^{43} +(-1.08979 - 0.216773i) q^{44} +(-0.707107 - 0.707107i) q^{48} +(0.923880 + 0.382683i) q^{49} -1.00000 q^{50} +(0.382683 + 0.923880i) q^{51} +(0.382683 + 0.923880i) q^{54} +(0.541196 + 1.30656i) q^{57} +(-0.707107 - 1.70711i) q^{59} +(-0.707107 + 0.707107i) q^{64} +(0.617317 - 0.923880i) q^{66} +1.84776i q^{67} +(0.923880 - 0.382683i) q^{68} +(0.923880 - 0.382683i) q^{72} +(-1.63099 + 0.324423i) q^{73} +(0.382683 - 0.923880i) q^{75} +(1.30656 - 0.541196i) q^{76} -1.00000 q^{81} +(0.923880 - 1.38268i) q^{82} +(-0.292893 + 0.707107i) q^{83} -0.765367i q^{86} +(-0.923880 - 0.617317i) q^{88} +(-1.30656 + 1.30656i) q^{89} +(-0.382683 - 0.923880i) q^{96} +(0.382683 + 1.92388i) q^{97} +(0.707107 + 0.707107i) q^{98} +(0.617317 + 0.923880i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{12} + 8 q^{38} - 8 q^{43} - 8 q^{50} + 8 q^{66} - 8 q^{81} - 8 q^{83} + 8 q^{99}+O(q^{100}) $ 8 * q - 8 * q^12 + 8 * q^38 - 8 * q^43 - 8 * q^50 + 8 * q^66 - 8 * q^81 - 8 * q^83 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/408\mathbb{Z}\right)^\times$.

$n$	$103$	$137$	$205$	$241$
$\chi(n)$	$-1$	$-1$	$-1$	$e\left(\frac{3}{16}\right)$

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 $

$2$	0.923880	+	0.382683i	0.923880	+	0.382683i
$2$	0.923880	+	0.382683i	0.923880	+	0.382683i
$3$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$3$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$4$	0.707107	+	0.707107i	0.707107	+	0.707107i
$5$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$5$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$6$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$7$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$7$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$8$	0.382683	+	0.923880i	0.382683	+	0.923880i
$9$		−	1.00000i		−	1.00000i
$10$		0			0
$11$	−0.923880	+	0.617317i	−0.923880	+	0.617317i	−0.923880	−	0.382683i	$-0.875000\pi$
$11$	−0.923880	+	0.617317i	−0.923880	+	0.617317i			1.00000i	$0.5\pi$
$12$	−1.00000			−1.00000
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$			1.00000i			1.00000i
$17$	0.382683	−	0.923880i	0.382683	−	0.923880i
$17$	0.382683	−	0.923880i	0.382683	−	0.923880i
$18$	0.382683	−	0.923880i	0.382683	−	0.923880i
$19$	0.541196	−	1.30656i	0.541196	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$19$	0.541196	−	1.30656i	0.541196	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$20$		0			0
$21$		0			0
$22$	−1.08979	+	0.216773i	−1.08979	+	0.216773i
$23$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$23$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$24$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$25$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$26$		0			0
$27$	0.707107	+	0.707107i	0.707107	+	0.707107i
$28$		0			0
$29$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$29$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$30$		0			0
$31$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$31$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$32$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$33$	0.216773	−	1.08979i	0.216773	−	1.08979i
$34$	0.707107	−	0.707107i	0.707107	−	0.707107i
$35$		0			0
$36$	0.707107	−	0.707107i	0.707107	−	0.707107i
$37$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$37$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$38$	1.00000	−	1.00000i	1.00000	−	1.00000i
$39$		0			0
$40$		0			0
$41$	0.324423	−	1.63099i	0.324423	−	1.63099i	−0.382683	−	0.923880i	$-0.625000\pi$
$41$	0.324423	−	1.63099i	0.324423	−	1.63099i	0.707107	−	0.707107i	$-0.250000\pi$
$42$		0			0
$43$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$43$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	−1.00000			$\pi$
$44$	−1.08979	−	0.216773i	−1.08979	−	0.216773i
$45$		0			0
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$49$	0.923880	+	0.382683i	0.923880	+	0.382683i
$50$	−1.00000			−1.00000
$51$	0.382683	+	0.923880i	0.382683	+	0.923880i
$52$		0			0
$53$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$53$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$54$	0.382683	+	0.923880i	0.382683	+	0.923880i
$55$		0			0
$56$		0			0
$57$	0.541196	+	1.30656i	0.541196	+	1.30656i
$58$		0			0
$59$	−0.707107	−	1.70711i	−0.707107	−	1.70711i	−0.707107	−	0.707107i	$-0.750000\pi$
$59$	−0.707107	−	1.70711i	−0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$60$		0			0
$61$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$61$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$62$		0			0
$63$		0			0
$64$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$65$		0			0
$66$	0.617317	−	0.923880i	0.617317	−	0.923880i
$67$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$67$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$68$	0.923880	−	0.382683i	0.923880	−	0.382683i
$69$		0			0
$70$		0			0
$71$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$71$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$72$	0.923880	−	0.382683i	0.923880	−	0.382683i
$73$	−1.63099	+	0.324423i	−1.63099	+	0.324423i	−0.923880	−	0.382683i	$-0.875000\pi$
$73$	−1.63099	+	0.324423i	−1.63099	+	0.324423i	−0.707107	+	0.707107i	$0.750000\pi$
$74$		0			0
$75$	0.382683	−	0.923880i	0.382683	−	0.923880i
$76$	1.30656	−	0.541196i	1.30656	−	0.541196i
$77$		0			0
$78$		0			0
$79$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$79$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$80$		0			0
$81$	−1.00000			−1.00000
$82$	0.923880	−	1.38268i	0.923880	−	1.38268i
$83$	−0.292893	+	0.707107i	−0.292893	+	0.707107i	0.707107	+	0.707107i	$0.250000\pi$
$83$	−0.292893	+	0.707107i	−0.292893	+	0.707107i	−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		−	0.765367i		−	0.765367i
$87$		0			0
$88$	−0.923880	−	0.617317i	−0.923880	−	0.617317i
$89$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$89$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$97$	0.382683	+	1.92388i	0.382683	+	1.92388i	0.382683	+	0.923880i	$0.375000\pi$
$97$	0.382683	+	1.92388i	0.382683	+	1.92388i			1.00000i	$0.5\pi$
$98$	0.707107	+	0.707107i	0.707107	+	0.707107i
$99$	0.617317	+	0.923880i	0.617317	+	0.923880i
$100$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$			1.00000i			1.00000i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.0761205	+	0.382683i	0.0761205	+	0.382683i	1.00000			$0$
$107$	0.0761205	+	0.382683i	0.0761205	+	0.382683i	−0.923880	+	0.382683i	$0.875000\pi$
$108$			1.00000i			1.00000i
$109$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$109$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.324423	+	0.216773i	−0.324423	+	0.216773i	−0.707107	−	0.707107i	$-0.750000\pi$
$113$	−0.324423	+	0.216773i	−0.324423	+	0.216773i	0.382683	+	0.923880i	$0.375000\pi$
$114$			1.41421i			1.41421i
$115$		0			0
$116$		0			0
$117$		0			0
$118$		−	1.84776i		−	1.84776i
$119$		0			0
$120$		0			0
$121$	0.0897902	−	0.216773i	0.0897902	−	0.216773i
$122$		0			0
$123$	0.923880	+	1.38268i	0.923880	+	1.38268i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$127$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$128$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$129$	0.707107	+	0.292893i	0.707107	+	0.292893i
$130$		0			0
$131$	1.63099	−	0.324423i	1.63099	−	0.324423i	0.707107	−	0.707107i	$-0.250000\pi$
$131$	1.63099	−	0.324423i	1.63099	−	0.324423i	0.923880	+	0.382683i	$0.125000\pi$
$132$	0.923880	−	0.617317i	0.923880	−	0.617317i
$133$		0			0
$134$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$135$		0			0
$136$	1.00000			1.00000
$137$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$137$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$138$		0			0
$139$	−1.63099	−	1.08979i	−1.63099	−	1.08979i	−0.923880	−	0.382683i	$-0.875000\pi$
$139$	−1.63099	−	1.08979i	−1.63099	−	1.08979i	−0.707107	−	0.707107i	$-0.750000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	1.00000			1.00000
$145$		0			0
$146$	−1.63099	−	0.324423i	−1.63099	−	0.324423i
$147$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$148$		0			0
$149$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$149$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$150$	0.707107	−	0.707107i	0.707107	−	0.707107i
$151$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$151$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$152$	1.41421			1.41421
$153$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$163$	0.382683	−	1.92388i	0.382683	−	1.92388i		−	1.00000i	$-0.5\pi$
$163$	0.382683	−	1.92388i	0.382683	−	1.92388i	0.382683	−	0.923880i	$-0.375000\pi$
$164$	1.38268	−	0.923880i	1.38268	−	0.923880i
$165$		0			0
$166$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$167$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$167$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$168$		0			0
$169$		−	1.00000i		−	1.00000i
$170$		0			0
$171$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$172$	0.292893	−	0.707107i	0.292893	−	0.707107i
$173$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$173$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$174$		0			0
$175$		0			0
$176$	−0.617317	−	0.923880i	−0.617317	−	0.923880i
$177$	1.70711	+	0.707107i	1.70711	+	0.707107i
$178$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$179$	1.30656	−	0.541196i	1.30656	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$179$	1.30656	−	0.541196i	1.30656	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$180$		0			0
$181$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$181$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	0.216773	+	1.08979i	0.216773	+	1.08979i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$191$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$192$		−	1.00000i		−	1.00000i
$193$	0.324423	−	0.216773i	0.324423	−	0.216773i	−0.382683	−	0.923880i	$-0.625000\pi$
$193$	0.324423	−	0.216773i	0.324423	−	0.216773i	0.707107	+	0.707107i	$0.250000\pi$
$194$	−0.382683	+	1.92388i	−0.382683	+	1.92388i
$195$		0			0
$196$	0.382683	+	0.923880i	0.382683	+	0.923880i
$197$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$197$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$198$	0.216773	+	1.08979i	0.216773	+	1.08979i
$199$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$199$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$200$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$201$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$202$		0			0
$203$		0			0
$204$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.306563	+	1.54120i	0.306563	+	1.54120i
$210$		0			0
$211$	−1.63099	−	0.324423i	−1.63099	−	0.324423i	−0.707107	−	0.707107i	$-0.750000\pi$
$211$	−1.63099	−	0.324423i	−1.63099	−	0.324423i	−0.923880	+	0.382683i	$0.875000\pi$
$212$		0			0
$213$		0			0
$214$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i
$215$		0			0
$216$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$217$		0			0
$218$		0			0
$219$	0.923880	−	1.38268i	0.923880	−	1.38268i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$223$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$224$		0			0
$225$	0.382683	+	0.923880i	0.382683	+	0.923880i
$226$	−0.382683	+	0.0761205i	−0.382683	+	0.0761205i
$227$	0.216773	+	0.324423i	0.216773	+	0.324423i	0.923880	−	0.382683i	$-0.125000\pi$
$227$	0.216773	+	0.324423i	0.216773	+	0.324423i	−0.707107	+	0.707107i	$0.750000\pi$
$228$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$229$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$229$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.92388	−	0.382683i	1.92388	−	0.382683i	0.923880	−	0.382683i	$-0.125000\pi$
$233$	1.92388	−	0.382683i	1.92388	−	0.382683i	1.00000			$0$
$234$		0			0
$235$		0			0
$236$	0.707107	−	1.70711i	0.707107	−	1.70711i
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	0.923880	+	0.617317i	0.923880	+	0.617317i	0.923880	−	0.382683i	$-0.125000\pi$
$241$	0.923880	+	0.617317i	0.923880	+	0.617317i			1.00000i	$0.5\pi$
$242$	0.165911	−	0.165911i	0.165911	−	0.165911i
$243$	0.707107	−	0.707107i	0.707107	−	0.707107i
$244$		0			0
$245$		0			0
$246$	0.324423	+	1.63099i	0.324423	+	1.63099i
$247$		0			0
$248$		0			0
$249$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$250$		0			0
$251$	0.541196	+	0.541196i	0.541196	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$251$	0.541196	+	0.541196i	0.541196	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−1.00000			−1.00000
$257$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$257$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$258$	0.541196	+	0.541196i	0.541196	+	0.541196i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	1.63099	+	0.324423i	1.63099	+	0.324423i
$263$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$263$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$264$	1.08979	−	0.216773i	1.08979	−	0.216773i
$265$		0			0
$266$		0			0
$267$		−	1.84776i		−	1.84776i
$268$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$269$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$269$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	0.923880	+	0.382683i	0.923880	+	0.382683i
$273$		0			0
$274$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$275$	0.617317	−	0.923880i	0.617317	−	0.923880i
$276$		0			0
$277$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$277$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$278$	−1.08979	−	1.63099i	−1.08979	−	1.63099i
$279$		0			0
$280$		0			0
$281$	−1.30656	+	0.541196i	−1.30656	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$281$	−1.30656	+	0.541196i	−1.30656	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$282$		0			0
$283$	0.617317	+	0.923880i	0.617317	+	0.923880i	1.00000			$0$
$283$	0.617317	+	0.923880i	0.617317	+	0.923880i	−0.382683	+	0.923880i	$0.625000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.923880	+	0.382683i	0.923880	+	0.382683i
$289$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$290$		0			0
$291$	−1.63099	−	1.08979i	−1.63099	−	1.08979i
$292$	−1.38268	−	0.923880i	−1.38268	−	0.923880i
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$	−1.00000			−1.00000
$295$		0			0
$296$		0			0
$297$	−1.08979	−	0.216773i	−1.08979	−	0.216773i
$298$		0			0
$299$		0			0
$300$	0.923880	−	0.382683i	0.923880	−	0.382683i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	1.30656	+	0.541196i	1.30656	+	0.541196i
$305$		0			0
$306$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$311$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$312$		0			0
$313$	−0.382683	−	0.0761205i	−0.382683	−	0.0761205i		−	1.00000i	$-0.5\pi$
$313$	−0.382683	−	0.0761205i	−0.382683	−	0.0761205i	−0.382683	+	0.923880i	$0.625000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$317$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−0.324423	−	0.216773i	−0.324423	−	0.216773i
$322$		0			0
$323$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$324$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$325$		0			0
$326$	1.08979	−	1.63099i	1.08979	−	1.63099i
$327$		0			0
$328$	1.63099	−	0.324423i	1.63099	−	0.324423i
$329$		0			0
$330$		0			0
$331$	1.70711	−	0.707107i	1.70711	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$331$	1.70711	−	0.707107i	1.70711	−	0.707107i	1.00000			$0$
$332$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.08979	−	1.63099i	1.08979	−	1.63099i	0.382683	−	0.923880i	$-0.375000\pi$
$337$	1.08979	−	1.63099i	1.08979	−	1.63099i	0.707107	−	0.707107i	$-0.250000\pi$
$338$	0.382683	−	0.923880i	0.382683	−	0.923880i
$339$	0.0761205	−	0.382683i	0.0761205	−	0.382683i
$340$		0			0
$341$		0			0
$342$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$343$		0			0
$344$	0.541196	−	0.541196i	0.541196	−	0.541196i
$345$		0			0
$346$		0			0
$347$	−0.324423	+	1.63099i	−0.324423	+	1.63099i	0.382683	+	0.923880i	$0.375000\pi$
$347$	−0.324423	+	1.63099i	−0.324423	+	1.63099i	−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$349$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$350$		0			0
$351$		0			0
$352$	−0.216773	−	1.08979i	−0.216773	−	1.08979i
$353$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$353$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$354$	1.30656	+	1.30656i	1.30656	+	1.30656i
$355$		0			0
$356$	−1.84776			−1.84776
$357$		0			0
$358$	1.41421			1.41421
$359$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$359$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$360$		0			0
$361$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$362$		0			0
$363$	0.0897902	+	0.216773i	0.0897902	+	0.216773i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$367$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$368$		0			0
$369$	−1.63099	−	0.324423i	−1.63099	−	0.324423i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	−0.216773	+	1.08979i	−0.216773	+	1.08979i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.08979	−	0.216773i	1.08979	−	0.216773i	0.382683	−	0.923880i	$-0.375000\pi$
$379$	1.08979	−	0.216773i	1.08979	−	0.216773i	0.707107	+	0.707107i	$0.250000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$383$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$384$	0.382683	−	0.923880i	0.382683	−	0.923880i
$385$		0			0
$386$	0.382683	−	0.0761205i	0.382683	−	0.0761205i
$387$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$388$	−1.08979	+	1.63099i	−1.08979	+	1.63099i
$389$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$389$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$390$		0			0
$391$		0			0
$392$			1.00000i			1.00000i
$393$	−0.923880	+	1.38268i	−0.923880	+	1.38268i
$394$		0			0
$395$		0			0
$396$	−0.216773	+	1.08979i	−0.216773	+	1.08979i
$397$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$397$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$398$		0			0
$399$		0			0
$400$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$401$	1.92388	+	0.382683i	1.92388	+	0.382683i	1.00000			$0$
$401$	1.92388	+	0.382683i	1.92388	+	0.382683i	0.923880	+	0.382683i	$0.125000\pi$
$402$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$409$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$409$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$410$		0			0
$411$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.92388	−	0.382683i	1.92388	−	0.382683i
$418$	−0.306563	+	1.54120i	−0.306563	+	1.54120i
$419$	0.324423	−	0.216773i	0.324423	−	0.216773i	−0.382683	−	0.923880i	$-0.625000\pi$
$419$	0.324423	−	0.216773i	0.324423	−	0.216773i	0.707107	+	0.707107i	$0.250000\pi$
$420$		0			0
$421$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$421$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$422$	−1.38268	−	0.923880i	−1.38268	−	0.923880i
$423$		0			0
$424$		0			0
$425$			1.00000i			1.00000i
$426$		0			0
$427$		0			0
$428$	−0.216773	+	0.324423i	−0.216773	+	0.324423i
$429$		0			0
$430$		0			0
$431$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$431$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$432$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$433$	−1.30656	+	0.541196i	−1.30656	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$433$	−1.30656	+	0.541196i	−1.30656	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	1.38268	−	0.923880i	1.38268	−	0.923880i
$439$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$439$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$440$		0			0
$441$	0.382683	−	0.923880i	0.382683	−	0.923880i
$442$		0			0
$443$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$443$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i	0.923880	+	0.382683i	$0.125000\pi$
$449$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i	−1.00000			$\pi$
$450$			1.00000i			1.00000i
$451$	0.707107	+	1.70711i	0.707107	+	1.70711i
$452$	−0.382683	−	0.0761205i	−0.382683	−	0.0761205i
$453$		0			0
$454$	0.0761205	+	0.382683i	0.0761205	+	0.382683i
$455$		0			0
$456$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$457$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$457$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$458$		0			0
$459$	0.923880	−	0.382683i	0.923880	−	0.382683i
$460$		0			0
$461$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$461$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$		0			0
$466$	1.92388	+	0.382683i	1.92388	+	0.382683i
$467$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$467$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	1.30656	−	1.30656i	1.30656	−	1.30656i
$473$	0.707107	+	0.472474i	0.707107	+	0.472474i
$474$		0			0
$475$			1.41421i			1.41421i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$479$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$480$		0			0
$481$		0			0
$482$	0.617317	+	0.923880i	0.617317	+	0.923880i
$483$		0			0
$484$	0.216773	−	0.0897902i	0.216773	−	0.0897902i
$485$		0			0
$486$	0.923880	−	0.382683i	0.923880	−	0.382683i
$487$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$487$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$488$		0			0
$489$	1.08979	+	1.63099i	1.08979	+	1.63099i
$490$		0			0
$491$	−0.541196	+	1.30656i	−0.541196	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$491$	−0.541196	+	1.30656i	−0.541196	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$492$	−0.324423	+	1.63099i	−0.324423	+	1.63099i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		−	0.765367i		−	0.765367i
$499$	−0.923880	+	0.617317i	−0.923880	+	0.617317i	−0.923880	−	0.382683i	$-0.875000\pi$
$499$	−0.923880	+	0.617317i	−0.923880	+	0.617317i			1.00000i	$0.5\pi$
$500$		0			0
$501$		0			0
$502$	0.292893	+	0.707107i	0.292893	+	0.707107i
$503$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$503$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.707107	+	0.707107i	0.707107	+	0.707107i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$513$	1.30656	−	0.541196i	1.30656	−	0.541196i
$514$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$515$		0			0
$516$	0.292893	+	0.707107i	0.292893	+	0.707107i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.923880	−	0.617317i	0.923880	−	0.617317i		−	1.00000i	$-0.5\pi$
$521$	0.923880	−	0.617317i	0.923880	−	0.617317i	0.923880	+	0.382683i	$0.125000\pi$
$522$		0			0
$523$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$523$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$524$	1.38268	+	0.923880i	1.38268	+	0.923880i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	1.08979	+	0.216773i	1.08979	+	0.216773i
$529$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$530$		0			0
$531$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$532$		0			0
$533$		0			0
$534$	0.707107	−	1.70711i	0.707107	−	1.70711i
$535$		0			0
$536$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$537$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$538$		0			0
$539$	−1.08979	+	0.216773i	−1.08979	+	0.216773i
$540$		0			0
$541$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$541$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$542$		0			0
$543$		0			0
$544$	0.707107	+	0.707107i	0.707107	+	0.707107i
$545$		0			0
$546$		0			0
$547$	1.63099	+	1.08979i	1.63099	+	1.08979i	0.923880	+	0.382683i	$0.125000\pi$
$547$	1.63099	+	1.08979i	1.63099	+	1.08979i	0.707107	+	0.707107i	$0.250000\pi$
$548$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$549$		0			0
$550$	0.923880	−	0.617317i	0.923880	−	0.617317i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−0.382683	−	1.92388i	−0.382683	−	1.92388i
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−0.923880	−	0.617317i	−0.923880	−	0.617317i
$562$	−1.41421			−1.41421
$563$	0.707107	+	0.292893i	0.707107	+	0.292893i	0.707107	−	0.707107i	$-0.250000\pi$
$563$	0.707107	+	0.292893i	0.707107	+	0.292893i			1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$	0.216773	+	1.08979i	0.216773	+	1.08979i
$567$		0			0
$568$		0			0
$569$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$569$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	−1.00000			$\pi$
$570$		0			0
$571$	−0.324423	+	1.63099i	−0.324423	+	1.63099i	0.382683	+	0.923880i	$0.375000\pi$
$571$	−0.324423	+	1.63099i	−0.324423	+	1.63099i	−0.707107	+	0.707107i	$0.750000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.707107	+	0.707107i	0.707107	+	0.707107i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$579$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i
$580$		0			0
$581$		0			0
$582$	−1.08979	−	1.63099i	−1.08979	−	1.63099i
$583$		0			0
$584$	−0.923880	−	1.38268i	−0.923880	−	1.38268i
$585$		0			0
$586$		0			0
$587$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$587$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$588$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.292893	−	0.707107i	0.292893	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$593$	0.292893	−	0.707107i	0.292893	−	0.707107i	1.00000			$0$
$594$	−0.923880	−	0.617317i	−0.923880	−	0.617317i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$599$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$600$	1.00000			1.00000
$601$	−0.324423	+	0.216773i	−0.324423	+	0.216773i	−0.707107	−	0.707107i	$-0.750000\pi$
$601$	−0.324423	+	0.216773i	−0.324423	+	0.216773i	0.382683	+	0.923880i	$0.375000\pi$
$602$		0			0
$603$	1.84776			1.84776
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$607$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$608$	1.00000	+	1.00000i	1.00000	+	1.00000i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.324423	−	1.63099i	−0.324423	−	1.63099i	−0.707107	−	0.707107i	$-0.750000\pi$
$617$	−0.324423	−	1.63099i	−0.324423	−	1.63099i	0.382683	−	0.923880i	$-0.375000\pi$
$618$		0			0
$619$	−1.92388	−	0.382683i	−1.92388	−	0.382683i	−0.923880	−	0.382683i	$-0.875000\pi$
$619$	−1.92388	−	0.382683i	−1.92388	−	0.382683i	−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.707107	−	0.707107i	0.707107	−	0.707107i
$626$	−0.324423	−	0.216773i	−0.324423	−	0.216773i
$627$	−1.30656	−	0.873017i	−1.30656	−	0.873017i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$631$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$632$		0			0
$633$	1.38268	−	0.923880i	1.38268	−	0.923880i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.08979	−	0.216773i	1.08979	−	0.216773i	0.382683	−	0.923880i	$-0.375000\pi$
$641$	1.08979	−	0.216773i	1.08979	−	0.216773i	0.707107	+	0.707107i	$0.250000\pi$
$642$	−0.216773	−	0.324423i	−0.216773	−	0.324423i
$643$	−0.923880	+	1.38268i	−0.923880	+	1.38268i			1.00000i	$0.5\pi$
$643$	−0.923880	+	1.38268i	−0.923880	+	1.38268i	−0.923880	+	0.382683i	$0.875000\pi$
$644$		0			0
$645$		0			0
$646$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$649$	1.70711	+	1.14065i	1.70711	+	1.14065i
$650$		0			0
$651$		0			0
$652$	1.63099	−	1.08979i	1.63099	−	1.08979i
$653$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$653$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$654$		0			0
$655$		0			0
$656$	1.63099	+	0.324423i	1.63099	+	0.324423i
$657$	0.324423	+	1.63099i	0.324423	+	1.63099i
$658$		0			0
$659$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$659$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$660$		0			0
$661$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$661$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$662$	1.84776			1.84776
$663$		0			0
$664$	−0.765367			−0.765367
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.216773	−	1.08979i	0.216773	−	1.08979i	−0.707107	−	0.707107i	$-0.750000\pi$
$673$	0.216773	−	1.08979i	0.216773	−	1.08979i	0.923880	−	0.382683i	$-0.125000\pi$
$674$	1.63099	−	1.08979i	1.63099	−	1.08979i
$675$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$676$	0.707107	−	0.707107i	0.707107	−	0.707107i
$677$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$677$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$678$	0.216773	−	0.324423i	0.216773	−	0.324423i
$679$		0			0
$680$		0			0
$681$	−0.382683	−	0.0761205i	−0.382683	−	0.0761205i
$682$		0			0
$683$	−0.923880	+	1.38268i	−0.923880	+	1.38268i			1.00000i	$0.5\pi$
$683$	−0.923880	+	1.38268i	−0.923880	+	1.38268i	−0.923880	+	0.382683i	$0.875000\pi$
$684$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	0.707107	−	0.292893i	0.707107	−	0.292893i
$689$		0			0
$690$		0			0
$691$	−0.617317	−	0.923880i	−0.617317	−	0.923880i	0.382683	−	0.923880i	$-0.375000\pi$
$691$	−0.617317	−	0.923880i	−0.617317	−	0.923880i	−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$	−0.923880	+	1.38268i	−0.923880	+	1.38268i
$695$		0			0
$696$		0			0
$697$	−1.38268	−	0.923880i	−1.38268	−	0.923880i
$698$		0			0
$699$	−1.08979	+	1.63099i	−1.08979	+	1.63099i
$700$		0			0
$701$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$701$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$702$		0			0
$703$		0			0
$704$	0.216773	−	1.08979i	0.216773	−	1.08979i
$705$		0			0
$706$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$707$		0			0
$708$	0.707107	+	1.70711i	0.707107	+	1.70711i
$709$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$709$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$710$		0			0
$711$		0			0
$712$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	1.30656	+	0.541196i	1.30656	+	0.541196i
$717$		0			0
$718$		0			0
$719$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$719$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$720$		0			0
$721$		0			0
$722$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$723$	−1.08979	+	0.216773i	−1.08979	+	0.216773i
$724$		0			0
$725$		0			0
$726$			0.234633i			0.234633i
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$			1.00000i			1.00000i
$730$		0			0
$731$	−0.765367			−0.765367
$732$		0			0
$733$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$733$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.14065	−	1.70711i	−1.14065	−	1.70711i
$738$	−1.38268	−	0.923880i	−1.38268	−	0.923880i
$739$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$739$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$743$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.707107	+	0.292893i	0.707107	+	0.292893i
$748$	−0.617317	+	0.923880i	−0.617317	+	0.923880i
$749$		0			0
$750$		0			0
$751$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$751$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$752$		0			0
$753$	−0.765367			−0.765367
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$757$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$758$	1.08979	+	0.216773i	1.08979	+	0.216773i
$759$		0			0
$760$		0			0
$761$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$761$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.707107	−	0.707107i	0.707107	−	0.707107i
$769$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$769$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$770$		0			0
$771$	0.707107	−	0.292893i	0.707107	−	0.292893i
$772$	0.382683	+	0.0761205i	0.382683	+	0.0761205i
$773$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$773$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$774$	−0.765367			−0.765367
$775$		0			0
$776$	−1.63099	+	1.08979i	−1.63099	+	1.08979i
$777$		0			0
$778$		0			0
$779$	−1.95541	−	1.30656i	−1.95541	−	1.30656i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$785$		0			0
$786$	−1.38268	+	0.923880i	−1.38268	+	0.923880i
$787$	0.382683	−	0.0761205i	0.382683	−	0.0761205i		−	1.00000i	$-0.5\pi$
$787$	0.382683	−	0.0761205i	0.382683	−	0.0761205i	0.382683	+	0.923880i	$0.375000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−0.617317	+	0.923880i	−0.617317	+	0.923880i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$797$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$798$		0			0
$799$		0			0
$800$		−	1.00000i		−	1.00000i
$801$	1.30656	+	1.30656i	1.30656	+	1.30656i
$802$	1.63099	+	1.08979i	1.63099	+	1.08979i
$803$	1.30656	−	1.30656i	1.30656	−	1.30656i
$804$		−	1.84776i		−	1.84776i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.92388	−	0.382683i	−1.92388	−	0.382683i	−0.923880	−	0.382683i	$-0.875000\pi$
$809$	−1.92388	−	0.382683i	−1.92388	−	0.382683i	−1.00000			$\pi$
$810$		0			0
$811$	−0.0761205	−	0.382683i	−0.0761205	−	0.382683i	0.923880	−	0.382683i	$-0.125000\pi$
$811$	−0.0761205	−	0.382683i	−0.0761205	−	0.382683i	−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$817$	−1.08239			−1.08239
$818$	1.70711	+	0.707107i	1.70711	+	0.707107i
$819$		0			0
$820$		0			0
$821$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$821$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$822$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$823$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$823$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$824$		0			0
$825$	0.216773	+	1.08979i	0.216773	+	1.08979i
$826$		0			0
$827$	1.63099	−	1.08979i	1.63099	−	1.08979i	0.707107	−	0.707107i	$-0.250000\pi$
$827$	1.63099	−	1.08979i	1.63099	−	1.08979i	0.923880	−	0.382683i	$-0.125000\pi$
$828$		0			0
$829$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$829$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.707107	−	0.707107i	0.707107	−	0.707107i
$834$	1.92388	+	0.382683i	1.92388	+	0.382683i
$835$		0			0
$836$	−0.873017	+	1.30656i	−0.873017	+	1.30656i
$837$		0			0
$838$	0.382683	−	0.0761205i	0.382683	−	0.0761205i
$839$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$839$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$840$		0			0
$841$	0.923880	−	0.382683i	0.923880	−	0.382683i
$842$		0			0
$843$	0.541196	−	1.30656i	0.541196	−	1.30656i
$844$	−0.923880	−	1.38268i	−0.923880	−	1.38268i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1.08979	−	0.216773i	−1.08979	−	0.216773i
$850$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$851$		0			0
$852$		0			0
$853$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$853$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$854$		0			0
$855$		0			0
$856$	−0.324423	+	0.216773i	−0.324423	+	0.216773i
$857$	0.216773	−	1.08979i	0.216773	−	1.08979i	−0.707107	−	0.707107i	$-0.750000\pi$
$857$	0.216773	−	1.08979i	0.216773	−	1.08979i	0.923880	−	0.382683i	$-0.125000\pi$
$858$		0			0
$859$	0.292893	+	0.707107i	0.292893	+	0.707107i	1.00000			$0$
$859$	0.292893	+	0.707107i	0.292893	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$865$		0			0
$866$	−1.41421			−1.41421
$867$	1.00000			1.00000
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1.92388	−	0.382683i	1.92388	−	0.382683i
$874$		0			0
$875$		0			0
$876$	1.63099	−	0.324423i	1.63099	−	0.324423i
$877$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$877$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.923880	+	0.617317i	0.923880	+	0.617317i	0.923880	−	0.382683i	$-0.125000\pi$
$881$	0.923880	+	0.617317i	0.923880	+	0.617317i			1.00000i	$0.5\pi$
$882$	0.707107	−	0.707107i	0.707107	−	0.707107i
$883$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$883$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$884$		0			0
$885$		0			0
$886$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$887$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$887$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.923880	−	0.617317i	0.923880	−	0.617317i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.216773	+	0.324423i	−0.216773	+	0.324423i
$899$		0			0
$900$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$901$		0			0
$902$			1.84776i			1.84776i
$903$		0			0
$904$	−0.324423	−	0.216773i	−0.324423	−	0.216773i
$905$		0			0
$906$		0			0
$907$	1.38268	−	0.923880i	1.38268	−	0.923880i	0.382683	−	0.923880i	$-0.375000\pi$
$907$	1.38268	−	0.923880i	1.38268	−	0.923880i	1.00000			$0$
$908$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i
$909$		0			0
$910$		0			0
$911$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$911$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$912$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$913$	−0.165911	−	0.834089i	−0.165911	−	0.834089i
$914$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	1.00000			1.00000
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.63099	+	1.08979i	−1.63099	+	1.08979i	−0.707107	+	0.707107i	$0.750000\pi$
$929$	−1.63099	+	1.08979i	−1.63099	+	1.08979i	−0.923880	+	0.382683i	$0.875000\pi$
$930$		0			0
$931$	1.00000	−	1.00000i	1.00000	−	1.00000i
$932$	1.63099	+	1.08979i	1.63099	+	1.08979i
$933$		0			0
$934$		−	1.41421i		−	1.41421i
$935$		0			0
$936$		0			0
$937$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$937$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$938$		0			0
$939$	0.324423	−	0.216773i	0.324423	−	0.216773i
$940$		0			0
$941$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$941$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$942$		0			0
$943$		0			0
$944$	1.70711	−	0.707107i	1.70711	−	0.707107i
$945$		0			0
$946$	0.472474	+	0.707107i	0.472474	+	0.707107i
$947$	−1.92388	+	0.382683i	−1.92388	+	0.382683i	−0.923880	+	0.382683i	$0.875000\pi$
$947$	−1.92388	+	0.382683i	−1.92388	+	0.382683i	−1.00000			$\pi$
$948$		0			0
$949$		0			0
$950$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$951$		0			0
$952$		0			0
$953$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$953$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$962$		0			0
$963$	0.382683	−	0.0761205i	0.382683	−	0.0761205i
$964$	0.216773	+	1.08979i	0.216773	+	1.08979i
$965$		0			0
$966$		0			0
$967$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$967$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$968$	0.234633			0.234633
$969$	1.41421			1.41421
$970$		0			0
$971$	1.70711	+	0.707107i	1.70711	+	0.707107i	1.00000			$0$
$971$	1.70711	+	0.707107i	1.70711	+	0.707107i	0.707107	+	0.707107i	$0.250000\pi$
$972$	1.00000			1.00000
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.707107	+	1.70711i	0.707107	+	1.70711i	0.707107	+	0.707107i	$0.250000\pi$
$977$	0.707107	+	1.70711i	0.707107	+	1.70711i			1.00000i	$0.5\pi$
$978$	0.382683	+	1.92388i	0.382683	+	1.92388i
$979$	0.400544	−	2.01367i	0.400544	−	2.01367i
$980$		0			0
$981$		0			0
$982$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$983$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$983$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$984$	−0.923880	+	1.38268i	−0.923880	+	1.38268i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$991$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$992$		0			0
$993$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$994$		0			0
$995$		0			0
$996$	0.292893	−	0.707107i	0.292893	−	0.707107i
$997$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$997$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$998$	−1.08979	+	0.216773i	−1.08979	+	0.216773i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	408.1.bg.a.299.1	yes	8
3.2	odd	2		408.1.bg.b.299.1	yes	8
4.3	odd	2		1632.1.ds.b.911.1		8
8.3	odd	2	CM	408.1.bg.a.299.1	yes	8
8.5	even	2		1632.1.ds.b.911.1		8
12.11	even	2		1632.1.ds.a.911.1		8
17.12	odd	16		408.1.bg.b.131.1	yes	8
24.5	odd	2		1632.1.ds.a.911.1		8
24.11	even	2		408.1.bg.b.299.1	yes	8
51.29	even	16	inner	408.1.bg.a.131.1	✓	8
68.63	even	16		1632.1.ds.a.335.1		8
136.29	odd	16		1632.1.ds.a.335.1		8
136.131	even	16		408.1.bg.b.131.1	yes	8
204.131	odd	16		1632.1.ds.b.335.1		8
408.29	even	16		1632.1.ds.b.335.1		8
408.131	odd	16	inner	408.1.bg.a.131.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
408.1.bg.a.131.1	✓	8	51.29	even	16	inner
408.1.bg.a.131.1	✓	8	408.131	odd	16	inner
408.1.bg.a.299.1	yes	8	1.1	even	1	trivial
408.1.bg.a.299.1	yes	8	8.3	odd	2	CM
408.1.bg.b.131.1	yes	8	17.12	odd	16
408.1.bg.b.131.1	yes	8	136.131	even	16
408.1.bg.b.299.1	yes	8	3.2	odd	2
408.1.bg.b.299.1	yes	8	24.11	even	2
1632.1.ds.a.335.1		8	68.63	even	16
1632.1.ds.a.335.1		8	136.29	odd	16
1632.1.ds.a.911.1		8	12.11	even	2
1632.1.ds.a.911.1		8	24.5	odd	2
1632.1.ds.b.335.1		8	204.131	odd	16
1632.1.ds.b.335.1		8	408.29	even	16
1632.1.ds.b.911.1		8	4.3	odd	2
1632.1.ds.b.911.1		8	8.5	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:	\( N \)	\(=\)	\( 408 = 2^{3} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	408.bg (of order \(16\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(0.923880 + 0.382683i) q^{2} +(-0.707107 + 0.707107i) q^{3} +(0.707107 + 0.707107i) q^{4} +(-0.923880 + 0.382683i) q^{6} +(0.382683 + 0.923880i) q^{8} -1.00000i q^{9} +O(q^{10})\)	\(q+(0.923880 + 0.382683i) q^{2} +(-0.707107 + 0.707107i) q^{3} +(0.707107 + 0.707107i) q^{4} +(-0.923880 + 0.382683i) q^{6} +(0.382683 + 0.923880i) q^{8} -1.00000i q^{9} +(-0.923880 + 0.617317i) q^{11} -1.00000 q^{12} +1.00000i q^{16} +(0.382683 - 0.923880i) q^{17} +(0.382683 - 0.923880i) q^{18} +(0.541196 - 1.30656i) q^{19} +(-1.08979 + 0.216773i) q^{22} +(-0.923880 - 0.382683i) q^{24} +(-0.923880 + 0.382683i) q^{25} +(0.707107 + 0.707107i) q^{27} +(-0.382683 + 0.923880i) q^{32} +(0.216773 - 1.08979i) q^{33} +(0.707107 - 0.707107i) q^{34} +(0.707107 - 0.707107i) q^{36} +(1.00000 - 1.00000i) q^{38} +(0.324423 - 1.63099i) q^{41} +(-0.292893 - 0.707107i) q^{43} +(-1.08979 - 0.216773i) q^{44} +(-0.707107 - 0.707107i) q^{48} +(0.923880 + 0.382683i) q^{49} -1.00000 q^{50} +(0.382683 + 0.923880i) q^{51} +(0.382683 + 0.923880i) q^{54} +(0.541196 + 1.30656i) q^{57} +(-0.707107 - 1.70711i) q^{59} +(-0.707107 + 0.707107i) q^{64} +(0.617317 - 0.923880i) q^{66} +1.84776i q^{67} +(0.923880 - 0.382683i) q^{68} +(0.923880 - 0.382683i) q^{72} +(-1.63099 + 0.324423i) q^{73} +(0.382683 - 0.923880i) q^{75} +(1.30656 - 0.541196i) q^{76} -1.00000 q^{81} +(0.923880 - 1.38268i) q^{82} +(-0.292893 + 0.707107i) q^{83} -0.765367i q^{86} +(-0.923880 - 0.617317i) q^{88} +(-1.30656 + 1.30656i) q^{89} +(-0.382683 - 0.923880i) q^{96} +(0.382683 + 1.92388i) q^{97} +(0.707107 + 0.707107i) q^{98} +(0.617317 + 0.923880i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{12} + 8 q^{38} - 8 q^{43} - 8 q^{50} + 8 q^{66} - 8 q^{81} - 8 q^{83} + 8 q^{99}+O(q^{100}) \) 8 * q - 8 * q^12 + 8 * q^38 - 8 * q^43 - 8 * q^50 + 8 * q^66 - 8 * q^81 - 8 * q^83 + 8 * q^99

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