LMFDB - Embedded newform 1776.1.z.c.221.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1776,1,Mod(221,1776)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1776, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2, 2]))

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

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

chi := DirichletCharacter("1776.221");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1776 = 2^{4} \cdot 3 \cdot 37 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1776.z (of order $4$, degree $2$, 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:	$0.886339462436$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{32})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 1 $ x^16 + 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			221.1
Root			$-0.555570 - 0.831470i$ of defining polynomial
Character	$\chi$	$=$	1776.221
Dual form			1776.1.z.c.1109.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.980785 + 0.195090i) q^{2} +(-0.707107 - 0.707107i) q^{3} +(0.923880 - 0.382683i) q^{4} +(-0.275899 + 0.275899i) q^{5} +(0.831470 + 0.555570i) q^{6} -1.84776i q^{7} +(-0.831470 + 0.555570i) q^{8} +1.00000i q^{9} +O(q^{10})$	\(q+(-0.980785 + 0.195090i) q^{2} +(-0.707107 - 0.707107i) q^{3} +(0.923880 - 0.382683i) q^{4} +(-0.275899 + 0.275899i) q^{5} +(0.831470 + 0.555570i) q^{6} -1.84776i q^{7} +(-0.831470 + 0.555570i) q^{8} +1.00000i q^{9} +(0.216773 - 0.324423i) q^{10} +(-0.923880 - 0.382683i) q^{12} +(0.360480 + 1.81225i) q^{14} +0.390181 q^{15} +(0.707107 - 0.707107i) q^{16} +1.96157 q^{17} +(-0.195090 - 0.980785i) q^{18} +(-0.149316 + 0.360480i) q^{20} +(-1.30656 + 1.30656i) q^{21} -1.11114i q^{23} +(0.980785 + 0.195090i) q^{24} +0.847759i q^{25} +(0.707107 - 0.707107i) q^{27} +(-0.707107 - 1.70711i) q^{28} +(-1.17588 - 1.17588i) q^{29} +(-0.382683 + 0.0761205i) q^{30} +(-0.555570 + 0.831470i) q^{32} +(-1.92388 + 0.382683i) q^{34} +(0.509796 + 0.509796i) q^{35} +(0.382683 + 0.923880i) q^{36} +(0.707107 - 0.707107i) q^{37} +(0.0761205 - 0.382683i) q^{40} +(1.02656 - 1.53636i) q^{42} +(-0.275899 - 0.275899i) q^{45} +(0.216773 + 1.08979i) q^{46} -1.00000 q^{48} -2.41421 q^{49} +(-0.165390 - 0.831470i) q^{50} +(-1.38704 - 1.38704i) q^{51} +(-0.555570 + 0.831470i) q^{54} +(1.02656 + 1.53636i) q^{56} +(1.38268 + 0.923880i) q^{58} +(-1.17588 + 1.17588i) q^{59} +(0.360480 - 0.149316i) q^{60} +1.84776 q^{63} +(0.382683 - 0.923880i) q^{64} +(-0.541196 - 0.541196i) q^{67} +(1.81225 - 0.750661i) q^{68} +(-0.785695 + 0.785695i) q^{69} +(-0.599456 - 0.400544i) q^{70} +(-0.555570 - 0.831470i) q^{72} -0.765367i q^{73} +(-0.555570 + 0.831470i) q^{74} +(0.599456 - 0.599456i) q^{75} +0.390181i q^{80} -1.00000 q^{81} +(-0.707107 + 1.70711i) q^{84} +(-0.541196 + 0.541196i) q^{85} +1.66294i q^{87} -1.66294i q^{89} +(0.324423 + 0.216773i) q^{90} +(-0.425215 - 1.02656i) q^{92} +(0.980785 - 0.195090i) q^{96} +(2.36783 - 0.470990i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 16 q^{34} + 16 q^{40} - 16 q^{48} - 16 q^{49} + 16 q^{58} - 16 q^{81}+O(q^{100}) $ 16 * q - 16 * q^34 + 16 * q^40 - 16 * q^48 - 16 * q^49 + 16 * q^58 - 16 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1776\mathbb{Z}\right)^\times$.

$n$	$223$	$593$	$1297$	$1333$
$\chi(n)$	$1$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	−0.980785	+	0.195090i	−0.980785	+	0.195090i
$2$	−0.980785	+	0.195090i	−0.980785	+	0.195090i
$3$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$3$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$4$	0.923880	−	0.382683i	0.923880	−	0.382683i
$5$	−0.275899	+	0.275899i	−0.275899	+	0.275899i	−0.831470	−	0.555570i	$-0.812500\pi$
$5$	−0.275899	+	0.275899i	−0.275899	+	0.275899i	0.555570	+	0.831470i	$0.312500\pi$
$6$	0.831470	+	0.555570i	0.831470	+	0.555570i
$7$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$7$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$8$	−0.831470	+	0.555570i	−0.831470	+	0.555570i
$9$			1.00000i			1.00000i
$10$	0.216773	−	0.324423i	0.216773	−	0.324423i
$11$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$11$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$12$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$	0.360480	+	1.81225i	0.360480	+	1.81225i
$15$	0.390181			0.390181
$16$	0.707107	−	0.707107i	0.707107	−	0.707107i
$17$	1.96157			1.96157			0.980785	−	0.195090i	$-0.0625000\pi$
$17$	1.96157			1.96157			0.980785	+	0.195090i	$0.0625000\pi$
$18$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$	−0.149316	+	0.360480i	−0.149316	+	0.360480i
$21$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$22$		0			0
$23$		−	1.11114i		−	1.11114i	−0.831470	−	0.555570i	$-0.812500\pi$
$23$		−	1.11114i		−	1.11114i	0.831470	−	0.555570i	$-0.187500\pi$
$24$	0.980785	+	0.195090i	0.980785	+	0.195090i
$25$			0.847759i			0.847759i
$26$		0			0
$27$	0.707107	−	0.707107i	0.707107	−	0.707107i
$28$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$29$	−1.17588	−	1.17588i	−1.17588	−	1.17588i	−0.980785	−	0.195090i	$-0.937500\pi$
$29$	−1.17588	−	1.17588i	−1.17588	−	1.17588i	−0.195090	−	0.980785i	$-0.562500\pi$
$30$	−0.382683	+	0.0761205i	−0.382683	+	0.0761205i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$33$		0			0
$34$	−1.92388	+	0.382683i	−1.92388	+	0.382683i
$35$	0.509796	+	0.509796i	0.509796	+	0.509796i
$36$	0.382683	+	0.923880i	0.382683	+	0.923880i
$37$	0.707107	−	0.707107i	0.707107	−	0.707107i
$37$	0.707107	−	0.707107i	0.707107	−	0.707107i
$38$		0			0
$39$		0			0
$40$	0.0761205	−	0.382683i	0.0761205	−	0.382683i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$	1.02656	−	1.53636i	1.02656	−	1.53636i
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	−0.275899	−	0.275899i	−0.275899	−	0.275899i
$46$	0.216773	+	1.08979i	0.216773	+	1.08979i
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	−1.00000			−1.00000
$49$	−2.41421			−2.41421
$50$	−0.165390	−	0.831470i	−0.165390	−	0.831470i
$51$	−1.38704	−	1.38704i	−1.38704	−	1.38704i
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$55$		0			0
$56$	1.02656	+	1.53636i	1.02656	+	1.53636i
$57$		0			0
$58$	1.38268	+	0.923880i	1.38268	+	0.923880i
$59$	−1.17588	+	1.17588i	−1.17588	+	1.17588i	−0.195090	+	0.980785i	$0.562500\pi$
$59$	−1.17588	+	1.17588i	−1.17588	+	1.17588i	−0.980785	+	0.195090i	$0.937500\pi$
$60$	0.360480	−	0.149316i	0.360480	−	0.149316i
$61$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$61$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$62$		0			0
$63$	1.84776			1.84776
$64$	0.382683	−	0.923880i	0.382683	−	0.923880i
$65$		0			0
$66$		0			0
$67$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$67$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$68$	1.81225	−	0.750661i	1.81225	−	0.750661i
$69$	−0.785695	+	0.785695i	−0.785695	+	0.785695i
$70$	−0.599456	−	0.400544i	−0.599456	−	0.400544i
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−0.555570	−	0.831470i	−0.555570	−	0.831470i
$73$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$73$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$74$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$75$	0.599456	−	0.599456i	0.599456	−	0.599456i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$			0.390181i			0.390181i
$81$	−1.00000			−1.00000
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$85$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$86$		0			0
$87$			1.66294i			1.66294i
$88$		0			0
$89$		−	1.66294i		−	1.66294i	−0.555570	−	0.831470i	$-0.687500\pi$
$89$		−	1.66294i		−	1.66294i	0.555570	−	0.831470i	$-0.312500\pi$
$90$	0.324423	+	0.216773i	0.324423	+	0.216773i
$91$		0			0
$92$	−0.425215	−	1.02656i	−0.425215	−	1.02656i
$93$		0			0
$94$		0			0
$95$		0			0
$96$	0.980785	−	0.195090i	0.980785	−	0.195090i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	2.36783	−	0.470990i	2.36783	−	0.470990i
$99$		0			0
$100$	0.324423	+	0.783227i	0.324423	+	0.783227i
$101$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$101$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$102$	1.63099	+	1.08979i	1.63099	+	1.08979i
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		−	0.720960i		−	0.720960i
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$	0.382683	−	0.923880i	0.382683	−	0.923880i
$109$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$109$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$110$		0			0
$111$	−1.00000			−1.00000
$112$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$113$	−1.11114			−1.11114			−0.555570	−	0.831470i	$-0.687500\pi$
$113$	−1.11114			−1.11114			−0.555570	+	0.831470i	$0.687500\pi$
$114$		0			0
$115$	0.306563	+	0.306563i	0.306563	+	0.306563i
$116$	−1.53636	−	0.636379i	−1.53636	−	0.636379i
$117$		0			0
$118$	0.923880	−	1.38268i	0.923880	−	1.38268i
$119$		−	3.62451i		−	3.62451i
$120$	−0.324423	+	0.216773i	−0.324423	+	0.216773i
$121$		−	1.00000i		−	1.00000i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.509796	−	0.509796i	−0.509796	−	0.509796i
$126$	−1.81225	+	0.360480i	−1.81225	+	0.360480i
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	−0.195090	+	0.980785i	−0.195090	+	0.980785i
$129$		0			0
$130$		0			0
$131$	−1.17588	−	1.17588i	−1.17588	−	1.17588i	−0.980785	−	0.195090i	$-0.937500\pi$
$131$	−1.17588	−	1.17588i	−1.17588	−	1.17588i	−0.195090	−	0.980785i	$-0.562500\pi$
$132$		0			0
$133$		0			0
$134$	0.636379	+	0.425215i	0.636379	+	0.425215i
$135$			0.390181i			0.390181i
$136$	−1.63099	+	1.08979i	−1.63099	+	1.08979i
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$	0.617317	−	0.923880i	0.617317	−	0.923880i
$139$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$139$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$140$	0.666080	+	0.275899i	0.666080	+	0.275899i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.707107	+	0.707107i	0.707107	+	0.707107i
$145$	0.648847			0.648847
$146$	0.149316	+	0.750661i	0.149316	+	0.750661i
$147$	1.70711	+	1.70711i	1.70711	+	1.70711i
$148$	0.382683	−	0.923880i	0.382683	−	0.923880i
$149$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$149$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$150$	−0.470990	+	0.704886i	−0.470990	+	0.704886i
$151$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$151$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$152$		0			0
$153$			1.96157i			1.96157i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.541196	+	0.541196i	0.541196	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$157$	0.541196	+	0.541196i	0.541196	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$158$		0			0
$159$		0			0
$160$	−0.0761205	−	0.382683i	−0.0761205	−	0.382683i
$161$	−2.05312			−2.05312
$162$	0.980785	−	0.195090i	0.980785	−	0.195090i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	1.96157i		−	1.96157i	−0.195090	−	0.980785i	$-0.562500\pi$
$167$		−	1.96157i		−	1.96157i	0.195090	−	0.980785i	$-0.437500\pi$
$168$	0.360480	−	1.81225i	0.360480	−	1.81225i
$169$			1.00000i			1.00000i
$170$	0.425215	−	0.636379i	0.425215	−	0.636379i
$171$		0			0
$172$		0			0
$173$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$173$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$174$	−0.324423	−	1.63099i	−0.324423	−	1.63099i
$175$	1.56645			1.56645
$176$		0			0
$177$	1.66294			1.66294
$178$	0.324423	+	1.63099i	0.324423	+	1.63099i
$179$	1.38704	+	1.38704i	1.38704	+	1.38704i	0.831470	+	0.555570i	$0.187500\pi$
$179$	1.38704	+	1.38704i	1.38704	+	1.38704i	0.555570	+	0.831470i	$0.312500\pi$
$180$	−0.360480	−	0.149316i	−0.360480	−	0.149316i
$181$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$181$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$182$		0			0
$183$		0			0
$184$	0.617317	+	0.923880i	0.617317	+	0.923880i
$185$			0.390181i			0.390181i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$190$		0			0
$191$	1.96157			1.96157			0.980785	−	0.195090i	$-0.0625000\pi$
$191$	1.96157			1.96157			0.980785	+	0.195090i	$0.0625000\pi$
$192$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	−2.23044	+	0.923880i	−2.23044	+	0.923880i
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−0.470990	−	0.704886i	−0.470990	−	0.704886i
$201$			0.765367i			0.765367i
$202$		0			0
$203$	−2.17273	+	2.17273i	−2.17273	+	2.17273i
$204$	−1.81225	−	0.750661i	−1.81225	−	0.750661i
$205$		0			0
$206$		0			0
$207$	1.11114			1.11114
$208$		0			0
$209$		0			0
$210$	0.140652	+	0.707107i	0.140652	+	0.707107i
$211$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$211$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$	−0.195090	+	0.980785i	−0.195090	+	0.980785i
$217$		0			0
$218$		0			0
$219$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$220$		0			0
$221$		0			0
$222$	0.980785	−	0.195090i	0.980785	−	0.195090i
$223$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$223$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$224$	1.53636	+	1.02656i	1.53636	+	1.02656i
$225$	−0.847759			−0.847759
$226$	1.08979	−	0.216773i	1.08979	−	0.216773i
$227$	0.275899	+	0.275899i	0.275899	+	0.275899i	0.831470	−	0.555570i	$-0.187500\pi$
$227$	0.275899	+	0.275899i	0.275899	+	0.275899i	−0.555570	+	0.831470i	$0.687500\pi$
$228$		0			0
$229$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$229$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$230$	−0.360480	−	0.240865i	−0.360480	−	0.240865i
$231$		0			0
$232$	1.63099	+	0.324423i	1.63099	+	0.324423i
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$	−0.636379	+	1.53636i	−0.636379	+	1.53636i
$237$		0			0
$238$	0.707107	+	3.55487i	0.707107	+	3.55487i
$239$	1.66294			1.66294			0.831470	−	0.555570i	$-0.187500\pi$
$239$	1.66294			1.66294			0.831470	+	0.555570i	$0.187500\pi$
$240$	0.275899	−	0.275899i	0.275899	−	0.275899i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	0.195090	+	0.980785i	0.195090	+	0.980785i
$243$	0.707107	+	0.707107i	0.707107	+	0.707107i
$244$		0			0
$245$	0.666080	−	0.666080i	0.666080	−	0.666080i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.599456	+	0.400544i	0.599456	+	0.400544i
$251$	−0.275899	+	0.275899i	−0.275899	+	0.275899i	−0.831470	−	0.555570i	$-0.812500\pi$
$251$	−0.275899	+	0.275899i	−0.275899	+	0.275899i	0.555570	+	0.831470i	$0.312500\pi$
$252$	1.70711	−	0.707107i	1.70711	−	0.707107i
$253$		0			0
$254$		0			0
$255$	0.765367			0.765367
$256$		−	1.00000i		−	1.00000i
$257$	1.11114			1.11114			0.555570	−	0.831470i	$-0.312500\pi$
$257$	1.11114			1.11114			0.555570	+	0.831470i	$0.312500\pi$
$258$		0			0
$259$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$260$		0			0
$261$	1.17588	−	1.17588i	1.17588	−	1.17588i
$262$	1.38268	+	0.923880i	1.38268	+	0.923880i
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−1.17588	+	1.17588i	−1.17588	+	1.17588i
$268$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$269$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$269$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$270$	−0.0761205	−	0.382683i	−0.0761205	−	0.382683i
$271$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$271$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$272$	1.38704	−	1.38704i	1.38704	−	1.38704i
$273$		0			0
$274$		0			0
$275$		0			0
$276$	−0.425215	+	1.02656i	−0.425215	+	1.02656i
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$	−0.425215	+	0.636379i	−0.425215	+	0.636379i
$279$		0			0
$280$	−0.707107	−	0.140652i	−0.707107	−	0.140652i
$281$			1.11114i			1.11114i	0.831470	+	0.555570i	$0.187500\pi$
$281$			1.11114i			1.11114i	−0.831470	+	0.555570i	$0.812500\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.831470	−	0.555570i	−0.831470	−	0.555570i
$289$	2.84776			2.84776
$290$	−0.636379	+	0.126584i	−0.636379	+	0.126584i
$291$		0			0
$292$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$	−2.00735	−	1.34127i	−2.00735	−	1.34127i
$295$		−	0.648847i		−	0.648847i
$296$	−0.195090	+	0.980785i	−0.195090	+	0.980785i
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.324423	−	0.783227i	0.324423	−	0.783227i
$301$		0			0
$302$	−0.275899	−	1.38704i	−0.275899	−	1.38704i
$303$		0			0
$304$		0			0
$305$		0			0
$306$	−0.382683	−	1.92388i	−0.382683	−	1.92388i
$307$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$307$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			1.11114i			1.11114i	0.831470	+	0.555570i	$0.187500\pi$
$311$			1.11114i			1.11114i	−0.831470	+	0.555570i	$0.812500\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$	−0.636379	−	0.425215i	−0.636379	−	0.425215i
$315$	−0.509796	+	0.509796i	−0.509796	+	0.509796i
$316$		0			0
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$	0.149316	+	0.360480i	0.149316	+	0.360480i
$321$		0			0
$322$	2.01367	−	0.400544i	2.01367	−	0.400544i
$323$		0			0
$324$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$331$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$332$		0			0
$333$	0.707107	+	0.707107i	0.707107	+	0.707107i
$334$	0.382683	+	1.92388i	0.382683	+	1.92388i
$335$	0.298631			0.298631
$336$			1.84776i			1.84776i
$337$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$337$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$338$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$339$	0.785695	+	0.785695i	0.785695	+	0.785695i
$340$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$341$		0			0
$342$		0			0
$343$			2.61313i			2.61313i
$344$		0			0
$345$		−	0.433546i		−	0.433546i
$346$		0			0
$347$	0.785695	−	0.785695i	0.785695	−	0.785695i	−0.195090	−	0.980785i	$-0.562500\pi$
$347$	0.785695	−	0.785695i	0.785695	−	0.785695i	0.980785	+	0.195090i	$0.0625000\pi$
$348$	0.636379	+	1.53636i	0.636379	+	1.53636i
$349$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$349$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$350$	−1.53636	+	0.305600i	−1.53636	+	0.305600i
$351$		0			0
$352$		0			0
$353$	0.390181			0.390181			0.195090	−	0.980785i	$-0.437500\pi$
$353$	0.390181			0.390181			0.195090	+	0.980785i	$0.437500\pi$
$354$	−1.63099	+	0.324423i	−1.63099	+	0.324423i
$355$		0			0
$356$	−0.636379	−	1.53636i	−0.636379	−	1.53636i
$357$	−2.56292	+	2.56292i	−2.56292	+	2.56292i
$358$	−1.63099	−	1.08979i	−1.63099	−	1.08979i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	0.382683	+	0.0761205i	0.382683	+	0.0761205i
$361$			1.00000i			1.00000i
$362$	−0.425215	+	0.636379i	−0.425215	+	0.636379i
$363$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$364$		0			0
$365$	0.211164	+	0.211164i	0.211164	+	0.211164i
$366$		0			0
$367$	−2.00000			−2.00000			−1.00000			$\pi$
$367$	−2.00000			−2.00000			−1.00000			$\pi$
$368$	−0.785695	−	0.785695i	−0.785695	−	0.785695i
$369$		0			0
$370$	−0.0761205	−	0.382683i	−0.0761205	−	0.382683i
$371$		0			0
$372$		0			0
$373$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$373$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$374$		0			0
$375$			0.720960i			0.720960i
$376$		0			0
$377$		0			0
$378$	1.53636	+	1.02656i	1.53636	+	1.02656i
$379$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$379$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$380$		0			0
$381$		0			0
$382$	−1.92388	+	0.382683i	−1.92388	+	0.382683i
$383$	−1.96157			−1.96157			−0.980785	−	0.195090i	$-0.937500\pi$
$383$	−1.96157			−1.96157			−0.980785	+	0.195090i	$0.937500\pi$
$384$	0.831470	−	0.555570i	0.831470	−	0.555570i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.38704	−	1.38704i	1.38704	−	1.38704i	0.555570	−	0.831470i	$-0.312500\pi$
$389$	1.38704	−	1.38704i	1.38704	−	1.38704i	0.831470	−	0.555570i	$-0.187500\pi$
$390$		0			0
$391$		−	2.17958i		−	2.17958i
$392$	2.00735	−	1.34127i	2.00735	−	1.34127i
$393$			1.66294i			1.66294i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.541196	+	0.541196i	0.541196	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$397$	0.541196	+	0.541196i	0.541196	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$398$		0			0
$399$		0			0
$400$	0.599456	+	0.599456i	0.599456	+	0.599456i
$401$	−0.390181			−0.390181			−0.195090	−	0.980785i	$-0.562500\pi$
$401$	−0.390181			−0.390181			−0.195090	+	0.980785i	$0.562500\pi$
$402$	−0.149316	−	0.750661i	−0.149316	−	0.750661i
$403$		0			0
$404$		0			0
$405$	0.275899	−	0.275899i	0.275899	−	0.275899i
$406$	1.70711	−	2.55487i	1.70711	−	2.55487i
$407$		0			0
$408$	1.92388	+	0.382683i	1.92388	+	0.382683i
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.17273	+	2.17273i	2.17273	+	2.17273i
$414$	−1.08979	+	0.216773i	−1.08979	+	0.216773i
$415$		0			0
$416$		0			0
$417$	−0.765367			−0.765367
$418$		0			0
$419$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$419$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$420$	−0.275899	−	0.666080i	−0.275899	−	0.666080i
$421$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$421$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$422$	−1.17588	−	0.785695i	−1.17588	−	0.785695i
$423$		0			0
$424$		0			0
$425$			1.66294i			1.66294i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1.11114			−1.11114			−0.555570	−	0.831470i	$-0.687500\pi$
$431$	−1.11114			−1.11114			−0.555570	+	0.831470i	$0.687500\pi$
$432$		−	1.00000i		−	1.00000i
$433$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$433$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$434$		0			0
$435$	−0.458804	−	0.458804i	−0.458804	−	0.458804i
$436$		0			0
$437$		0			0
$438$	0.425215	−	0.636379i	0.425215	−	0.636379i
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		−	2.41421i		−	2.41421i
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$445$	0.458804	+	0.458804i	0.458804	+	0.458804i
$446$	1.38704	−	0.275899i	1.38704	−	0.275899i
$447$		0			0
$448$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$449$	1.96157			1.96157			0.980785	−	0.195090i	$-0.0625000\pi$
$449$	1.96157			1.96157			0.980785	+	0.195090i	$0.0625000\pi$
$450$	0.831470	−	0.165390i	0.831470	−	0.165390i
$451$		0			0
$452$	−1.02656	+	0.425215i	−1.02656	+	0.425215i
$453$	1.00000	−	1.00000i	1.00000	−	1.00000i
$454$	−0.324423	−	0.216773i	−0.324423	−	0.216773i
$455$		0			0
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$	1.11114	−	1.66294i	1.11114	−	1.66294i
$459$	1.38704	−	1.38704i	1.38704	−	1.38704i
$460$	0.400544	+	0.165911i	0.400544	+	0.165911i
$461$	−1.38704	−	1.38704i	−1.38704	−	1.38704i	−0.831470	−	0.555570i	$-0.812500\pi$
$461$	−1.38704	−	1.38704i	−1.38704	−	1.38704i	−0.555570	−	0.831470i	$-0.687500\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$	−1.66294			−1.66294
$465$		0			0
$466$		0			0
$467$	1.17588	+	1.17588i	1.17588	+	1.17588i	0.980785	+	0.195090i	$0.0625000\pi$
$467$	1.17588	+	1.17588i	1.17588	+	1.17588i	0.195090	+	0.980785i	$0.437500\pi$
$468$		0			0
$469$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$470$		0			0
$471$		−	0.765367i		−	0.765367i
$472$	0.324423	−	1.63099i	0.324423	−	1.63099i
$473$		0			0
$474$		0			0
$475$		0			0
$476$	−1.38704	−	3.34861i	−1.38704	−	3.34861i
$477$		0			0
$478$	−1.63099	+	0.324423i	−1.63099	+	0.324423i
$479$	0.390181			0.390181			0.195090	−	0.980785i	$-0.437500\pi$
$479$	0.390181			0.390181			0.195090	+	0.980785i	$0.437500\pi$
$480$	−0.216773	+	0.324423i	−0.216773	+	0.324423i
$481$		0			0
$482$		0			0
$483$	1.45177	+	1.45177i	1.45177	+	1.45177i
$484$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$485$		0			0
$486$	−0.831470	−	0.555570i	−0.831470	−	0.555570i
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$	−0.523336	+	0.783227i	−0.523336	+	0.783227i
$491$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$491$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$492$		0			0
$493$	−2.30656	−	2.30656i	−2.30656	−	2.30656i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$499$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$500$	−0.666080	−	0.275899i	−0.666080	−	0.275899i
$501$	−1.38704	+	1.38704i	−1.38704	+	1.38704i
$502$	0.216773	−	0.324423i	0.216773	−	0.324423i
$503$			0.390181i			0.390181i	0.980785	+	0.195090i	$0.0625000\pi$
$503$			0.390181i			0.390181i	−0.980785	+	0.195090i	$0.937500\pi$
$504$	−1.53636	+	1.02656i	−1.53636	+	1.02656i
$505$		0			0
$506$		0			0
$507$	0.707107	−	0.707107i	0.707107	−	0.707107i
$508$		0			0
$509$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$509$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$510$	−0.750661	+	0.149316i	−0.750661	+	0.149316i
$511$	−1.41421			−1.41421
$512$	0.195090	+	0.980785i	0.195090	+	0.980785i
$513$		0			0
$514$	−1.08979	+	0.216773i	−1.08979	+	0.216773i
$515$		0			0
$516$		0			0
$517$		0			0
$518$	1.53636	+	1.02656i	1.53636	+	1.02656i
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$	−0.923880	+	1.38268i	−0.923880	+	1.38268i
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$	−1.53636	−	0.636379i	−1.53636	−	0.636379i
$525$	−1.10765	−	1.10765i	−1.10765	−	1.10765i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.234633			−0.234633
$530$		0			0
$531$	−1.17588	−	1.17588i	−1.17588	−	1.17588i
$532$		0			0
$533$		0			0
$534$	0.923880	−	1.38268i	0.923880	−	1.38268i
$535$		0			0
$536$	0.750661	+	0.149316i	0.750661	+	0.149316i
$537$		−	1.96157i		−	1.96157i
$538$		0			0
$539$		0			0
$540$	0.149316	+	0.360480i	0.149316	+	0.360480i
$541$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$541$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$542$	−0.750661	+	0.149316i	−0.750661	+	0.149316i
$543$	−0.765367			−0.765367
$544$	−1.08979	+	1.63099i	−1.08979	+	1.63099i
$545$		0			0
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	0.216773	−	1.08979i	0.216773	−	1.08979i
$553$		0			0
$554$		0			0
$555$	0.275899	−	0.275899i	0.275899	−	0.275899i
$556$	0.292893	−	0.707107i	0.292893	−	0.707107i
$557$	0.785695	+	0.785695i	0.785695	+	0.785695i	0.980785	−	0.195090i	$-0.0625000\pi$
$557$	0.785695	+	0.785695i	0.785695	+	0.785695i	−0.195090	+	0.980785i	$0.562500\pi$
$558$		0			0
$559$		0			0
$560$	0.720960			0.720960
$561$		0			0
$562$	−0.216773	−	1.08979i	−0.216773	−	1.08979i
$563$	1.38704	+	1.38704i	1.38704	+	1.38704i	0.831470	+	0.555570i	$0.187500\pi$
$563$	1.38704	+	1.38704i	1.38704	+	1.38704i	0.555570	+	0.831470i	$0.312500\pi$
$564$		0			0
$565$	0.306563	−	0.306563i	0.306563	−	0.306563i
$566$		0			0
$567$			1.84776i			1.84776i
$568$		0			0
$569$			1.11114i			1.11114i	0.831470	+	0.555570i	$0.187500\pi$
$569$			1.11114i			1.11114i	−0.831470	+	0.555570i	$0.812500\pi$
$570$		0			0
$571$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$571$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$572$		0			0
$573$	−1.38704	−	1.38704i	−1.38704	−	1.38704i
$574$		0			0
$575$	0.941979			0.941979
$576$	0.923880	+	0.382683i	0.923880	+	0.382683i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−2.79304	+	0.555570i	−2.79304	+	0.555570i
$579$		0			0
$580$	0.599456	−	0.248303i	0.599456	−	0.248303i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.425215	+	0.636379i	0.425215	+	0.636379i
$585$		0			0
$586$		0			0
$587$	1.38704	−	1.38704i	1.38704	−	1.38704i	0.555570	−	0.831470i	$-0.312500\pi$
$587$	1.38704	−	1.38704i	1.38704	−	1.38704i	0.831470	−	0.555570i	$-0.187500\pi$
$588$	2.23044	+	0.923880i	2.23044	+	0.923880i
$589$		0			0
$590$	0.126584	+	0.636379i	0.126584	+	0.636379i
$591$		0			0
$592$		−	1.00000i		−	1.00000i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$	1.00000	+	1.00000i	1.00000	+	1.00000i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	−0.165390	+	0.831470i	−0.165390	+	0.831470i
$601$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$601$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$602$		0			0
$603$	0.541196	−	0.541196i	0.541196	−	0.541196i
$604$	0.541196	+	1.30656i	0.541196	+	1.30656i
$605$	0.275899	+	0.275899i	0.275899	+	0.275899i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$	3.07271			3.07271
$610$		0			0
$611$		0			0
$612$	0.750661	+	1.81225i	0.750661	+	1.81225i
$613$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$613$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$614$	1.17588	+	0.785695i	1.17588	+	0.785695i
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$619$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$620$		0			0
$621$	−0.785695	−	0.785695i	−0.785695	−	0.785695i
$622$	−0.216773	−	1.08979i	−0.216773	−	1.08979i
$623$	−3.07271			−3.07271
$624$		0			0
$625$	−0.566454			−0.566454
$626$		0			0
$627$		0			0
$628$	0.707107	+	0.292893i	0.707107	+	0.292893i
$629$	1.38704	−	1.38704i	1.38704	−	1.38704i
$630$	0.400544	−	0.599456i	0.400544	−	0.599456i
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$		−	1.41421i		−	1.41421i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−0.216773	−	0.324423i	−0.216773	−	0.324423i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$	−1.89684	+	0.785695i	−1.89684	+	0.785695i
$645$		0			0
$646$		0			0
$647$		−	0.390181i		−	0.390181i	−0.980785	−	0.195090i	$-0.937500\pi$
$647$		−	0.390181i		−	0.390181i	0.980785	−	0.195090i	$-0.0625000\pi$
$648$	0.831470	−	0.555570i	0.831470	−	0.555570i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.785695	−	0.785695i	−0.785695	−	0.785695i	0.195090	−	0.980785i	$-0.437500\pi$
$653$	−0.785695	−	0.785695i	−0.785695	−	0.785695i	−0.980785	+	0.195090i	$0.937500\pi$
$654$		0			0
$655$	0.648847			0.648847
$656$		0			0
$657$	0.765367			0.765367
$658$		0			0
$659$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$659$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$660$		0			0
$661$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$661$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.831470	−	0.555570i	−0.831470	−	0.555570i
$667$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$668$	−0.750661	−	1.81225i	−0.750661	−	1.81225i
$669$	1.00000	+	1.00000i	1.00000	+	1.00000i
$670$	−0.292893	+	0.0582601i	−0.292893	+	0.0582601i
$671$		0			0
$672$	−0.360480	−	1.81225i	−0.360480	−	1.81225i
$673$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$673$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$674$	−1.38704	+	0.275899i	−1.38704	+	0.275899i
$675$	0.599456	+	0.599456i	0.599456	+	0.599456i
$676$	0.382683	+	0.923880i	0.382683	+	0.923880i
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$	−0.923880	−	0.617317i	−0.923880	−	0.617317i
$679$		0			0
$680$	0.149316	−	0.750661i	0.149316	−	0.750661i
$681$		−	0.390181i		−	0.390181i
$682$		0			0
$683$	1.17588	−	1.17588i	1.17588	−	1.17588i	0.195090	−	0.980785i	$-0.437500\pi$
$683$	1.17588	−	1.17588i	1.17588	−	1.17588i	0.980785	−	0.195090i	$-0.0625000\pi$
$684$		0			0
$685$		0			0
$686$	−0.509796	−	2.56292i	−0.509796	−	2.56292i
$687$	2.00000			2.00000
$688$		0			0
$689$		0			0
$690$	0.0845805	+	0.425215i	0.0845805	+	0.425215i
$691$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$691$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$692$		0			0
$693$		0			0
$694$	−0.617317	+	0.923880i	−0.617317	+	0.923880i
$695$			0.298631i			0.298631i
$696$	−0.923880	−	1.38268i	−0.923880	−	1.38268i
$697$		0			0
$698$	−1.53636	−	1.02656i	−1.53636	−	1.02656i
$699$		0			0
$700$	1.44722	−	0.599456i	1.44722	−	0.599456i
$701$	1.17588	+	1.17588i	1.17588	+	1.17588i	0.980785	+	0.195090i	$0.0625000\pi$
$701$	1.17588	+	1.17588i	1.17588	+	1.17588i	0.195090	+	0.980785i	$0.437500\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.382683	+	0.0761205i	−0.382683	+	0.0761205i
$707$		0			0
$708$	1.53636	−	0.636379i	1.53636	−	0.636379i
$709$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$709$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$710$		0			0
$711$		0			0
$712$	0.923880	+	1.38268i	0.923880	+	1.38268i
$713$		0			0
$714$	2.01367	−	3.01367i	2.01367	−	3.01367i
$715$		0			0
$716$	1.81225	+	0.750661i	1.81225	+	0.750661i
$717$	−1.17588	−	1.17588i	−1.17588	−	1.17588i
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$	−0.390181			−0.390181
$721$		0			0
$722$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$723$		0			0
$724$	0.292893	−	0.707107i	0.292893	−	0.707107i
$725$	0.996859	−	0.996859i	0.996859	−	0.996859i
$726$	0.555570	−	0.831470i	0.555570	−	0.831470i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$		−	1.00000i		−	1.00000i
$730$	−0.248303	−	0.165911i	−0.248303	−	0.165911i
$731$		0			0
$732$		0			0
$733$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$733$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$734$	1.96157	−	0.390181i	1.96157	−	0.390181i
$735$	−0.941979			−0.941979
$736$	0.923880	+	0.617317i	0.923880	+	0.617317i
$737$		0			0
$738$		0			0
$739$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$739$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$740$	0.149316	+	0.360480i	0.149316	+	0.360480i
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$	−1.02656	+	1.53636i	−1.02656	+	1.53636i
$747$		0			0
$748$		0			0
$749$		0			0
$750$	−0.140652	−	0.707107i	−0.140652	−	0.707107i
$751$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$751$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$752$		0			0
$753$	0.390181			0.390181
$754$		0			0
$755$	−0.390181	−	0.390181i	−0.390181	−	0.390181i
$756$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$	1.02656	−	1.53636i	1.02656	−	1.53636i
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$		0			0
$764$	1.81225	−	0.750661i	1.81225	−	0.750661i
$765$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$766$	1.92388	−	0.382683i	1.92388	−	0.382683i
$767$		0			0
$768$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−0.785695	−	0.785695i	−0.785695	−	0.785695i
$772$		0			0
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$			1.84776i			1.84776i
$778$	−1.08979	+	1.63099i	−1.08979	+	1.63099i
$779$		0			0
$780$		0			0
$781$		0			0
$782$	0.425215	+	2.13770i	0.425215	+	2.13770i
$783$	−1.66294			−1.66294
$784$	−1.70711	+	1.70711i	−1.70711	+	1.70711i
$785$	−0.298631			−0.298631
$786$	−0.324423	−	1.63099i	−0.324423	−	1.63099i
$787$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$787$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			2.05312i			2.05312i
$792$		0			0
$793$		0			0
$794$	−0.636379	−	0.425215i	−0.636379	−	0.425215i
$795$		0			0
$796$		0			0
$797$	−1.38704	−	1.38704i	−1.38704	−	1.38704i	−0.831470	−	0.555570i	$-0.812500\pi$
$797$	−1.38704	−	1.38704i	−1.38704	−	1.38704i	−0.555570	−	0.831470i	$-0.687500\pi$
$798$		0			0
$799$		0			0
$800$	−0.704886	−	0.470990i	−0.704886	−	0.470990i
$801$	1.66294			1.66294
$802$	0.382683	−	0.0761205i	0.382683	−	0.0761205i
$803$		0			0
$804$	0.292893	+	0.707107i	0.292893	+	0.707107i
$805$	0.566454	−	0.566454i	0.566454	−	0.566454i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1.11114i			1.11114i	0.831470	+	0.555570i	$0.187500\pi$
$809$			1.11114i			1.11114i	−0.831470	+	0.555570i	$0.812500\pi$
$810$	−0.216773	+	0.324423i	−0.216773	+	0.324423i
$811$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$811$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$812$	−1.17588	+	2.83881i	−1.17588	+	2.83881i
$813$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$814$		0			0
$815$		0			0
$816$	−1.96157			−1.96157
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$821$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$822$		0			0
$823$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$823$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$	−2.55487	−	1.70711i	−2.55487	−	1.70711i
$827$	0.275899	−	0.275899i	0.275899	−	0.275899i	−0.555570	−	0.831470i	$-0.687500\pi$
$827$	0.275899	−	0.275899i	0.275899	−	0.275899i	0.831470	+	0.555570i	$0.187500\pi$
$828$	1.02656	−	0.425215i	1.02656	−	0.425215i
$829$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$829$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.73565			−4.73565
$834$	0.750661	−	0.149316i	0.750661	−	0.149316i
$835$	0.541196	+	0.541196i	0.541196	+	0.541196i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$	0.400544	+	0.599456i	0.400544	+	0.599456i
$841$			1.76537i			1.76537i
$842$		0			0
$843$	0.785695	−	0.785695i	0.785695	−	0.785695i
$844$	1.30656	+	0.541196i	1.30656	+	0.541196i
$845$	−0.275899	−	0.275899i	−0.275899	−	0.275899i
$846$		0			0
$847$	−1.84776			−1.84776
$848$		0			0
$849$		0			0
$850$	−0.324423	−	1.63099i	−0.324423	−	1.63099i
$851$	−0.785695	−	0.785695i	−0.785695	−	0.785695i
$852$		0			0
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1.66294i			1.66294i	0.555570	+	0.831470i	$0.312500\pi$
$857$			1.66294i			1.66294i	−0.555570	+	0.831470i	$0.687500\pi$
$858$		0			0
$859$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$859$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$860$		0			0
$861$		0			0
$862$	1.08979	−	0.216773i	1.08979	−	0.216773i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	0.195090	+	0.980785i	0.195090	+	0.980785i
$865$		0			0
$866$	1.81225	−	0.360480i	1.81225	−	0.360480i
$867$	−2.01367	−	2.01367i	−2.01367	−	2.01367i
$868$		0			0
$869$		0			0
$870$	0.539496	+	0.360480i	0.539496	+	0.360480i
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.941979	+	0.941979i	−0.941979	+	0.941979i
$876$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$877$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$877$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	0.470990	+	2.36783i	0.470990	+	2.36783i
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$	−0.458804	+	0.458804i	−0.458804	+	0.458804i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$	0.831470	−	0.555570i	0.831470	−	0.555570i
$889$		0			0
$890$	−0.539496	−	0.360480i	−0.539496	−	0.360480i
$891$		0			0
$892$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$893$		0			0
$894$		0			0
$895$	−0.765367			−0.765367
$896$	1.81225	+	0.360480i	1.81225	+	0.360480i
$897$		0			0
$898$	−1.92388	+	0.382683i	−1.92388	+	0.382683i
$899$		0			0
$900$	−0.783227	+	0.324423i	−0.783227	+	0.324423i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	0.923880	−	0.617317i	0.923880	−	0.617317i
$905$			0.298631i			0.298631i
$906$	−0.785695	+	1.17588i	−0.785695	+	1.17588i
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$	0.360480	+	0.149316i	0.360480	+	0.149316i
$909$		0			0
$910$		0			0
$911$	−0.390181			−0.390181			−0.195090	−	0.980785i	$-0.562500\pi$
$911$	−0.390181			−0.390181			−0.195090	+	0.980785i	$0.562500\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.765367	+	1.84776i	−0.765367	+	1.84776i
$917$	−2.17273	+	2.17273i	−2.17273	+	2.17273i
$918$	−1.08979	+	1.63099i	−1.08979	+	1.63099i
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$	−0.425215	−	0.0845805i	−0.425215	−	0.0845805i
$921$			1.41421i			1.41421i
$922$	1.63099	+	1.08979i	1.63099	+	1.08979i
$923$		0			0
$924$		0			0
$925$	0.599456	+	0.599456i	0.599456	+	0.599456i
$926$		0			0
$927$		0			0
$928$	1.63099	−	0.324423i	1.63099	−	0.324423i
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	0.785695	−	0.785695i	0.785695	−	0.785695i
$934$	−1.38268	−	0.923880i	−1.38268	−	0.923880i
$935$		0			0
$936$		0			0
$937$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$937$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$938$	0.785695	−	1.17588i	0.785695	−	1.17588i
$939$		0			0
$940$		0			0
$941$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$941$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$942$	0.149316	+	0.750661i	0.149316	+	0.750661i
$943$		0			0
$944$			1.66294i			1.66294i
$945$	0.720960			0.720960
$946$		0			0
$947$	−1.38704	−	1.38704i	−1.38704	−	1.38704i	−0.831470	−	0.555570i	$-0.812500\pi$
$947$	−1.38704	−	1.38704i	−1.38704	−	1.38704i	−0.555570	−	0.831470i	$-0.687500\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$	2.01367	+	3.01367i	2.01367	+	3.01367i
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$956$	1.53636	−	0.636379i	1.53636	−	0.636379i
$957$		0			0
$958$	−0.382683	+	0.0761205i	−0.382683	+	0.0761205i
$959$		0			0
$960$	0.149316	−	0.360480i	0.149316	−	0.360480i
$961$	1.00000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$	−1.70711	−	1.14065i	−1.70711	−	1.14065i
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	0.555570	+	0.831470i	0.555570	+	0.831470i
$969$		0			0
$970$		0			0
$971$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$971$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$972$	0.923880	+	0.382683i	0.923880	+	0.382683i
$973$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.390181			0.390181			0.195090	−	0.980785i	$-0.437500\pi$
$977$	0.390181			0.390181			0.195090	+	0.980785i	$0.437500\pi$
$978$		0			0
$979$		0			0
$980$	0.360480	−	0.870275i	0.360480	−	0.870275i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$	2.71223	+	1.81225i	2.71223	+	1.81225i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$		0			0
$999$		−	1.00000i		−	1.00000i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1776.1.z.c.221.1	✓	16
3.2	odd	2	inner	1776.1.z.c.221.8	yes	16
16.5	even	4	inner	1776.1.z.c.1109.1	yes	16
37.36	even	2	inner	1776.1.z.c.221.8	yes	16
48.5	odd	4	inner	1776.1.z.c.1109.8	yes	16
111.110	odd	2	CM	1776.1.z.c.221.1	✓	16
592.517	even	4	inner	1776.1.z.c.1109.8	yes	16
1776.1109	odd	4	inner	1776.1.z.c.1109.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1776.1.z.c.221.1	✓	16	1.1	even	1	trivial
1776.1.z.c.221.1	✓	16	111.110	odd	2	CM
1776.1.z.c.221.8	yes	16	3.2	odd	2	inner
1776.1.z.c.221.8	yes	16	37.36	even	2	inner
1776.1.z.c.1109.1	yes	16	16.5	even	4	inner
1776.1.z.c.1109.1	yes	16	1776.1109	odd	4	inner
1776.1.z.c.1109.8	yes	16	48.5	odd	4	inner
1776.1.z.c.1109.8	yes	16	592.517	even	4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1776 = 2^{4} \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1776.z (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.980785 + 0.195090i) q^{2} +(-0.707107 - 0.707107i) q^{3} +(0.923880 - 0.382683i) q^{4} +(-0.275899 + 0.275899i) q^{5} +(0.831470 + 0.555570i) q^{6} -1.84776i q^{7} +(-0.831470 + 0.555570i) q^{8} +1.00000i q^{9} +O(q^{10})\)	\(q+(-0.980785 + 0.195090i) q^{2} +(-0.707107 - 0.707107i) q^{3} +(0.923880 - 0.382683i) q^{4} +(-0.275899 + 0.275899i) q^{5} +(0.831470 + 0.555570i) q^{6} -1.84776i q^{7} +(-0.831470 + 0.555570i) q^{8} +1.00000i q^{9} +(0.216773 - 0.324423i) q^{10} +(-0.923880 - 0.382683i) q^{12} +(0.360480 + 1.81225i) q^{14} +0.390181 q^{15} +(0.707107 - 0.707107i) q^{16} +1.96157 q^{17} +(-0.195090 - 0.980785i) q^{18} +(-0.149316 + 0.360480i) q^{20} +(-1.30656 + 1.30656i) q^{21} -1.11114i q^{23} +(0.980785 + 0.195090i) q^{24} +0.847759i q^{25} +(0.707107 - 0.707107i) q^{27} +(-0.707107 - 1.70711i) q^{28} +(-1.17588 - 1.17588i) q^{29} +(-0.382683 + 0.0761205i) q^{30} +(-0.555570 + 0.831470i) q^{32} +(-1.92388 + 0.382683i) q^{34} +(0.509796 + 0.509796i) q^{35} +(0.382683 + 0.923880i) q^{36} +(0.707107 - 0.707107i) q^{37} +(0.0761205 - 0.382683i) q^{40} +(1.02656 - 1.53636i) q^{42} +(-0.275899 - 0.275899i) q^{45} +(0.216773 + 1.08979i) q^{46} -1.00000 q^{48} -2.41421 q^{49} +(-0.165390 - 0.831470i) q^{50} +(-1.38704 - 1.38704i) q^{51} +(-0.555570 + 0.831470i) q^{54} +(1.02656 + 1.53636i) q^{56} +(1.38268 + 0.923880i) q^{58} +(-1.17588 + 1.17588i) q^{59} +(0.360480 - 0.149316i) q^{60} +1.84776 q^{63} +(0.382683 - 0.923880i) q^{64} +(-0.541196 - 0.541196i) q^{67} +(1.81225 - 0.750661i) q^{68} +(-0.785695 + 0.785695i) q^{69} +(-0.599456 - 0.400544i) q^{70} +(-0.555570 - 0.831470i) q^{72} -0.765367i q^{73} +(-0.555570 + 0.831470i) q^{74} +(0.599456 - 0.599456i) q^{75} +0.390181i q^{80} -1.00000 q^{81} +(-0.707107 + 1.70711i) q^{84} +(-0.541196 + 0.541196i) q^{85} +1.66294i q^{87} -1.66294i q^{89} +(0.324423 + 0.216773i) q^{90} +(-0.425215 - 1.02656i) q^{92} +(0.980785 - 0.195090i) q^{96} +(2.36783 - 0.470990i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 16 q^{34} + 16 q^{40} - 16 q^{48} - 16 q^{49} + 16 q^{58} - 16 q^{81}+O(q^{100}) \) 16 * q - 16 * q^34 + 16 * q^40 - 16 * q^48 - 16 * q^49 + 16 * q^58 - 16 * q^81

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