LMFDB - Embedded newform 3879.1.g.c.1723.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3879,1,Mod(430,3879)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3879, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 3]))

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

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

chi := DirichletCharacter("3879.430");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3879 = 3^{2} \cdot 431 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3879.g (of order $6$, 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:	$1.93587318400$
Analytic rank:	$0$
Dimension:	$36$
Relative dimension:	$18$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{63})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{36} - x^{33} + x^{27} - x^{24} + x^{18} - x^{12} + x^{9} - x^{3} + 1 $ x^36 - x^33 + x^27 - x^24 + x^18 - x^12 + x^9 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{63}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{63} - \cdots)$

Embedding invariants

Embedding label			1723.15
Root			$-0.411287 + 0.911506i$ of defining polynomial
Character	$\chi$	$=$	3879.1723
Dual form			3879.1.g.c.430.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.900969 + 1.56052i) q^{2} +(-0.124344 + 0.992239i) q^{3} +(-1.12349 + 1.94594i) q^{4} +(0.583744 - 1.01107i) q^{5} +(-1.66044 + 0.699935i) q^{6} -2.24698 q^{8} +(-0.969077 - 0.246757i) q^{9} +O(q^{10})$	\(q+(0.900969 + 1.56052i) q^{2} +(-0.124344 + 0.992239i) q^{3} +(-1.12349 + 1.94594i) q^{4} +(0.583744 - 1.01107i) q^{5} +(-1.66044 + 0.699935i) q^{6} -2.24698 q^{8} +(-0.969077 - 0.246757i) q^{9} +2.10374 q^{10} +(0.939693 + 1.62760i) q^{11} +(-1.79114 - 1.35674i) q^{12} +(0.930642 + 0.704934i) q^{15} +(-0.900969 - 1.56052i) q^{16} +(-0.488038 - 1.73459i) q^{18} +0.912421 q^{19} +(1.31166 + 2.27186i) q^{20} +(-1.69327 + 2.93283i) q^{22} +(-0.980172 + 1.69771i) q^{23} +(0.279398 - 2.22954i) q^{24} +(-0.181513 - 0.314390i) q^{25} +(0.365341 - 0.930874i) q^{27} +(-0.878222 - 1.52112i) q^{29} +(-0.261587 + 2.08741i) q^{30} +(0.500000 - 0.866025i) q^{32} +(-1.73181 + 0.730019i) q^{33} +(1.56892 - 1.60854i) q^{36} +(0.822063 + 1.42386i) q^{38} +(-1.31166 + 2.27186i) q^{40} +(-0.365341 + 0.632789i) q^{41} -4.22294 q^{44} +(-0.815183 + 0.835765i) q^{45} -3.53242 q^{46} +(1.66044 - 0.699935i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(0.327076 - 0.566512i) q^{50} +1.08509 q^{53} +(1.78181 - 0.268565i) q^{54} +2.19416 q^{55} +(-0.113454 + 0.905340i) q^{57} +(1.58250 - 2.74097i) q^{58} +(0.969077 - 1.67849i) q^{59} +(-2.41733 + 1.01899i) q^{60} +(-0.955573 - 1.65510i) q^{61} +(-2.69952 - 2.04480i) q^{66} +(-1.56265 - 1.18366i) q^{69} +(2.17750 + 0.554459i) q^{72} +(0.334520 - 0.141012i) q^{75} +(-1.02510 + 1.77552i) q^{76} -2.10374 q^{80} +(0.878222 + 0.478254i) q^{81} -1.31664 q^{82} +(1.61852 - 0.682264i) q^{87} +(-2.11147 - 3.65717i) q^{88} +(-2.03869 - 0.519113i) q^{90} +(-2.20243 - 3.81472i) q^{92} +(0.532620 - 0.922525i) q^{95} +(0.797133 + 0.603804i) q^{96} +(0.0249307 + 0.0431812i) q^{97} -1.80194 q^{98} +(-0.509014 - 1.80914i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 6 q^{2} - 12 q^{4} - 24 q^{8}+O(q^{10}) $ 36 * q + 6 * q^2 - 12 * q^4 - 24 * q^8	$ 36 q + 6 q^{2} - 12 q^{4} - 24 q^{8} + 3 q^{15} - 6 q^{16} - 18 q^{25} + 3 q^{27} + 6 q^{30} + 18 q^{32} + 3 q^{33} - 3 q^{41} + 3 q^{45} - 18 q^{49} + 6 q^{50} + 6 q^{54} + 6 q^{55} + 3 q^{57} - 18 q^{60} - 3 q^{61} + 6 q^{66} - 6 q^{69} + 3 q^{75} + 12 q^{82} - 6 q^{87} + 6 q^{90} - 3 q^{95} - 12 q^{98} + 3 q^{99}+O(q^{100}) $ 36 * q + 6 * q^2 - 12 * q^4 - 24 * q^8 + 3 * q^15 - 6 * q^16 - 18 * q^25 + 3 * q^27 + 6 * q^30 + 18 * q^32 + 3 * q^33 - 3 * q^41 + 3 * q^45 - 18 * q^49 + 6 * q^50 + 6 * q^54 + 6 * q^55 + 3 * q^57 - 18 * q^60 - 3 * q^61 + 6 * q^66 - 6 * q^69 + 3 * q^75 + 12 * q^82 - 6 * q^87 + 6 * q^90 - 3 * q^95 - 12 * q^98 + 3 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3879\mathbb{Z}\right)^\times$.

$n$	$2593$	$3449$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$2$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$3$	−0.124344	+	0.992239i	−0.124344	+	0.992239i
$3$	−0.124344	+	0.992239i	−0.124344	+	0.992239i
$4$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$5$	0.583744	−	1.01107i	0.583744	−	1.01107i	−0.411287	−	0.911506i	$-0.634921\pi$
$5$	0.583744	−	1.01107i	0.583744	−	1.01107i	0.995031	−	0.0995678i	$-0.0317460\pi$
$6$	−1.66044	+	0.699935i	−1.66044	+	0.699935i
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$	−2.24698			−2.24698
$9$	−0.969077	−	0.246757i	−0.969077	−	0.246757i
$10$	2.10374			2.10374
$11$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$11$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$12$	−1.79114	−	1.35674i	−1.79114	−	1.35674i
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$	0.930642	+	0.704934i	0.930642	+	0.704934i
$16$	−0.900969	−	1.56052i	−0.900969	−	1.56052i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	−0.488038	−	1.73459i	−0.488038	−	1.73459i
$19$	0.912421			0.912421			0.456211	−	0.889872i	$-0.349206\pi$
$19$	0.912421			0.912421			0.456211	+	0.889872i	$0.349206\pi$
$20$	1.31166	+	2.27186i	1.31166	+	2.27186i
$21$		0			0
$22$	−1.69327	+	2.93283i	−1.69327	+	2.93283i
$23$	−0.980172	+	1.69771i	−0.980172	+	1.69771i	−0.318487	+	0.947927i	$0.603175\pi$
$23$	−0.980172	+	1.69771i	−0.980172	+	1.69771i	−0.661686	+	0.749781i	$0.730159\pi$
$24$	0.279398	−	2.22954i	0.279398	−	2.22954i
$25$	−0.181513	−	0.314390i	−0.181513	−	0.314390i
$26$		0			0
$27$	0.365341	−	0.930874i	0.365341	−	0.930874i
$28$		0			0
$29$	−0.878222	−	1.52112i	−0.878222	−	1.52112i	−0.853291	−	0.521435i	$-0.825397\pi$
$29$	−0.878222	−	1.52112i	−0.878222	−	1.52112i	−0.0249307	−	0.999689i	$-0.507937\pi$
$30$	−0.261587	+	2.08741i	−0.261587	+	2.08741i
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	0.500000	−	0.866025i	0.500000	−	0.866025i
$33$	−1.73181	+	0.730019i	−1.73181	+	0.730019i
$34$		0			0
$35$		0			0
$36$	1.56892	−	1.60854i	1.56892	−	1.60854i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	0.822063	+	1.42386i	0.822063	+	1.42386i
$39$		0			0
$40$	−1.31166	+	2.27186i	−1.31166	+	2.27186i
$41$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	−0.988831	−	0.149042i	$-0.952381\pi$
$41$	−0.365341	+	0.632789i	−0.365341	+	0.632789i	0.623490	+	0.781831i	$0.285714\pi$
$42$		0			0
$43$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$43$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$44$	−4.22294			−4.22294
$45$	−0.815183	+	0.835765i	−0.815183	+	0.835765i
$46$	−3.53242			−3.53242
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$	1.66044	−	0.699935i	1.66044	−	0.699935i
$49$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$50$	0.327076	−	0.566512i	0.327076	−	0.566512i
$51$		0			0
$52$		0			0
$53$	1.08509			1.08509			0.542546	−	0.840026i	$-0.317460\pi$
$53$	1.08509			1.08509			0.542546	+	0.840026i	$0.317460\pi$
$54$	1.78181	−	0.268565i	1.78181	−	0.268565i
$55$	2.19416			2.19416
$56$		0			0
$57$	−0.113454	+	0.905340i	−0.113454	+	0.905340i
$58$	1.58250	−	2.74097i	1.58250	−	2.74097i
$59$	0.969077	−	1.67849i	0.969077	−	1.67849i	0.270840	−	0.962624i	$-0.412698\pi$
$59$	0.969077	−	1.67849i	0.969077	−	1.67849i	0.698237	−	0.715867i	$-0.253968\pi$
$60$	−2.41733	+	1.01899i	−2.41733	+	1.01899i
$61$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.733052	−	0.680173i	$-0.761905\pi$
$61$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.222521	−	0.974928i	$-0.571429\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$	−2.69952	−	2.04480i	−2.69952	−	2.04480i
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$	−1.56265	−	1.18366i	−1.56265	−	1.18366i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	2.17750	+	0.554459i	2.17750	+	0.554459i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	0.334520	−	0.141012i	0.334520	−	0.141012i
$76$	−1.02510	+	1.77552i	−1.02510	+	1.77552i
$77$		0			0
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$	−2.10374			−2.10374
$81$	0.878222	+	0.478254i	0.878222	+	0.478254i
$82$	−1.31664			−1.31664
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	1.61852	−	0.682264i	1.61852	−	0.682264i
$88$	−2.11147	−	3.65717i	−2.11147	−	3.65717i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−2.03869	−	0.519113i	−2.03869	−	0.519113i
$91$		0			0
$92$	−2.20243	−	3.81472i	−2.20243	−	3.81472i
$93$		0			0
$94$		0			0
$95$	0.532620	−	0.922525i	0.532620	−	0.922525i
$96$	0.797133	+	0.603804i	0.797133	+	0.603804i
$97$	0.0249307	+	0.0431812i	0.0249307	+	0.0431812i	0.878222	−	0.478254i	$-0.158730\pi$
$97$	0.0249307	+	0.0431812i	0.0249307	+	0.0431812i	−0.853291	+	0.521435i	$0.825397\pi$
$98$	−1.80194			−1.80194
$99$	−0.509014	−	1.80914i	−0.509014	−	1.80914i
$100$	0.815714			0.815714
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$	0.977635	+	1.69331i	0.977635	+	1.69331i
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	1.40097	+	1.75676i	1.40097	+	1.75676i
$109$	0.149460			0.149460			0.0747301	−	0.997204i	$-0.476190\pi$
$109$	0.149460			0.149460			0.0747301	+	0.997204i	$0.476190\pi$
$110$	1.97687	+	3.42404i	1.97687	+	3.42404i
$111$		0			0
$112$		0			0
$113$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$	−1.51502	+	0.638636i	−1.51502	+	0.638636i
$115$	1.14434	+	1.98205i	1.14434	+	1.98205i
$116$	3.94669			3.94669
$117$		0			0
$118$	3.49243			3.49243
$119$		0			0
$120$	−2.09113	−	1.58397i	−2.09113	−	1.58397i
$121$	−1.26604	+	2.19285i	−1.26604	+	2.19285i
$122$	1.72188	−	2.98239i	1.72188	−	2.98239i
$123$	−0.582450	−	0.441189i	−0.582450	−	0.441189i
$124$		0			0
$125$	0.743658			0.743658
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$	0.525096	−	4.19017i	0.525096	−	4.19017i
$133$		0			0
$134$		0			0
$135$	−0.727916	−	0.912778i	−0.727916	−	0.912778i
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$	0.439234	−	3.50501i	0.439234	−	3.50501i
$139$	0.853291	−	1.47794i	0.853291	−	1.47794i	−0.0249307	−	0.999689i	$-0.507937\pi$
$139$	0.853291	−	1.47794i	0.853291	−	1.47794i	0.878222	−	0.478254i	$-0.158730\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.488038	+	1.73459i	0.488038	+	1.73459i
$145$	−2.05063			−2.05063
$146$		0			0
$147$	−0.797133	−	0.603804i	−0.797133	−	0.603804i
$148$		0			0
$149$	−0.921476	+	1.59604i	−0.921476	+	1.59604i	−0.124344	+	0.992239i	$0.539683\pi$
$149$	−0.921476	+	1.59604i	−0.921476	+	1.59604i	−0.797133	+	0.603804i	$0.793651\pi$
$150$	0.521445	+	0.394980i	0.521445	+	0.394980i
$151$	0.583744	+	1.01107i	0.583744	+	1.01107i	0.995031	+	0.0995678i	$0.0317460\pi$
$151$	0.583744	+	1.01107i	0.583744	+	1.01107i	−0.411287	+	0.911506i	$0.634921\pi$
$152$	−2.05019			−2.05019
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$157$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$158$		0			0
$159$	−0.134924	+	1.07667i	−0.134924	+	1.07667i
$160$	−0.583744	−	1.01107i	−0.583744	−	1.01107i
$161$		0			0
$162$	0.0449236	+	1.80138i	0.0449236	+	1.80138i
$163$	1.96034			1.96034			0.980172	−	0.198146i	$-0.0634921\pi$
$163$	1.96034			1.96034			0.980172	+	0.198146i	$0.0634921\pi$
$164$	−0.820914	−	1.42186i	−0.820914	−	1.42186i
$165$	−0.272830	+	2.17713i	−0.272830	+	2.17713i
$166$		0			0
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$170$		0			0
$171$	−0.884207	−	0.225147i	−0.884207	−	0.225147i
$172$		0			0
$173$	0.661686	+	1.14607i	0.661686	+	1.14607i	0.980172	+	0.198146i	$0.0634921\pi$
$173$	0.661686	+	1.14607i	0.661686	+	1.14607i	−0.318487	+	0.947927i	$0.603175\pi$
$174$	2.52293	+	1.91104i	2.52293	+	1.91104i
$175$		0			0
$176$	1.69327	−	2.93283i	1.69327	−	2.93283i
$177$	1.54497	+	1.17027i	1.54497	+	1.17027i
$178$		0			0
$179$	−1.46610			−1.46610			−0.733052	−	0.680173i	$-0.761905\pi$
$179$	−1.46610			−1.46610			−0.733052	+	0.680173i	$0.761905\pi$
$180$	−0.710501	−	2.52527i	−0.710501	−	2.52527i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	1.76108	−	0.742355i	1.76108	−	0.742355i
$184$	2.20243	−	3.81472i	2.20243	−	3.81472i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	1.91950			1.91950
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$	−0.0449236	+	0.0778099i	−0.0449236	+	0.0778099i
$195$		0			0
$196$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$197$	1.39647			1.39647			0.698237	−	0.715867i	$-0.253968\pi$
$197$	1.39647			1.39647			0.698237	+	0.715867i	$0.253968\pi$
$198$	2.36460	−	2.42431i	2.36460	−	2.42431i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	0.407857	+	0.706429i	0.407857	+	0.706429i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	0.426531	+	0.738773i	0.426531	+	0.738773i
$206$		0			0
$207$	1.36879	−	1.40335i	1.36879	−	1.40335i
$208$		0			0
$209$	0.857396	+	1.48505i	0.857396	+	1.48505i
$210$		0			0
$211$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$211$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$212$	−1.21909	+	2.11153i	−1.21909	+	2.11153i
$213$		0			0
$214$		0			0
$215$		0			0
$216$	−0.820914	+	2.09165i	−0.820914	+	2.09165i
$217$		0			0
$218$	0.134659	+	0.233236i	0.134659	+	0.233236i
$219$		0			0
$220$	−2.46511	+	4.26970i	−2.46511	+	4.26970i
$221$		0			0
$222$		0			0
$223$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$223$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$224$		0			0
$225$	0.0983223	+	0.349458i	0.0983223	+	0.349458i
$226$		0			0
$227$	0.853291	+	1.47794i	0.853291	+	1.47794i	0.878222	+	0.478254i	$0.158730\pi$
$227$	0.853291	+	1.47794i	0.853291	+	1.47794i	−0.0249307	+	0.999689i	$0.507937\pi$
$228$	−1.63427	−	1.23792i	−1.63427	−	1.23792i
$229$	0.797133	−	1.38067i	0.797133	−	1.38067i	−0.124344	−	0.992239i	$-0.539683\pi$
$229$	0.797133	−	1.38067i	0.797133	−	1.38067i	0.921476	−	0.388435i	$-0.126984\pi$
$230$	−2.06203	+	3.57154i	−2.06203	+	3.57154i
$231$		0			0
$232$	1.97335	+	3.41794i	1.97335	+	3.41794i
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$	2.17750	+	3.77154i	2.17750	+	3.77154i
$237$		0			0
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$	0.261587	−	2.08741i	0.261587	−	2.08741i
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	−4.56267			−4.56267
$243$	−0.583744	+	0.811938i	−0.583744	+	0.811938i
$244$	4.29431			4.29431
$245$	0.583744	+	1.01107i	0.583744	+	1.01107i
$246$	0.163716	−	1.30643i	0.163716	−	1.30643i
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.670013	+	1.16050i	0.670013	+	1.16050i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$	−3.68424			−3.68424
$254$		0			0
$255$		0			0
$256$	0.900969	−	1.56052i	0.900969	−	1.56052i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.475716	+	1.69079i	0.475716	+	1.69079i
$262$		0			0
$263$	−0.698237	−	1.20938i	−0.698237	−	1.20938i	−0.969077	−	0.246757i	$-0.920635\pi$
$263$	−0.698237	−	1.20938i	−0.698237	−	1.20938i	0.270840	−	0.962624i	$-0.412698\pi$
$264$	3.89134	−	1.64034i	3.89134	−	1.64034i
$265$	0.633416	−	1.09711i	0.633416	−	1.09711i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$	0.768582	−	1.95832i	0.768582	−	1.95832i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.341134	−	0.590861i	0.341134	−	0.590861i
$276$	4.05897	−	1.71100i	4.05897	−	1.71100i
$277$	−0.921476	−	1.59604i	−0.921476	−	1.59604i	−0.797133	−	0.603804i	$-0.793651\pi$
$277$	−0.921476	−	1.59604i	−0.921476	−	1.59604i	−0.124344	−	0.992239i	$-0.539683\pi$
$278$	3.07515			3.07515
$279$		0			0
$280$		0			0
$281$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$281$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$282$		0			0
$283$	0.969077	−	1.67849i	0.969077	−	1.67849i	0.270840	−	0.962624i	$-0.412698\pi$
$283$	0.969077	−	1.67849i	0.969077	−	1.67849i	0.698237	−	0.715867i	$-0.253968\pi$
$284$		0			0
$285$	0.849138	+	0.643197i	0.849138	+	0.643197i
$286$		0			0
$287$		0			0
$288$	−0.698237	+	0.715867i	−0.698237	+	0.715867i
$289$	1.00000			1.00000
$290$	−1.84755	−	3.20005i	−1.84755	−	3.20005i
$291$	−0.0459461	+	0.0193679i	−0.0459461	+	0.0193679i
$292$		0			0
$293$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$293$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$294$	0.224060	−	1.78795i	0.224060	−	1.78795i
$295$	−1.13139	−	1.95962i	−1.13139	−	1.95962i
$296$		0			0
$297$	1.85839	−	0.280108i	1.85839	−	0.280108i
$298$	−3.32089			−3.32089
$299$		0			0
$300$	−0.101429	+	0.809383i	−0.101429	+	0.809383i
$301$		0			0
$302$	−1.05187	+	1.82189i	−1.05187	+	1.82189i
$303$		0			0
$304$	−0.822063	−	1.42386i	−0.822063	−	1.42386i
$305$	−2.23124			−2.23124
$306$		0			0
$307$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$307$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$	3.38654			3.38654
$315$		0			0
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$	−1.80173	+	0.759495i	−1.80173	+	0.759495i
$319$	1.65052	−	2.85878i	1.65052	−	2.85878i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−1.91733	+	1.17165i	−1.91733	+	1.17165i
$325$		0			0
$326$	1.76621	+	3.05917i	1.76621	+	3.05917i
$327$	−0.0185844	+	0.148300i	−0.0185844	+	0.148300i
$328$	0.820914	−	1.42186i	0.820914	−	1.42186i
$329$		0			0
$330$	−3.64327	+	1.53577i	−3.64327	+	1.53577i
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.733052	−	1.26968i	0.733052	−	1.26968i	−0.222521	−	0.974928i	$-0.571429\pi$
$337$	0.733052	−	1.26968i	0.733052	−	1.26968i	0.955573	−	0.294755i	$-0.0952381\pi$
$338$	0.900969	−	1.56052i	0.900969	−	1.56052i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	−0.445296	−	1.58268i	−0.445296	−	1.58268i
$343$		0			0
$344$		0			0
$345$	−2.10896	+	0.889002i	−2.10896	+	0.889002i
$346$	−1.19232	+	2.06515i	−1.19232	+	2.06515i
$347$	−0.955573	+	1.65510i	−0.955573	+	1.65510i	−0.222521	+	0.974928i	$0.571429\pi$
$347$	−0.955573	+	1.65510i	−0.955573	+	1.65510i	−0.733052	+	0.680173i	$0.761905\pi$
$348$	−0.490746	+	3.91606i	−0.490746	+	3.91606i
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$		0			0
$351$		0			0
$352$	1.87939			1.87939
$353$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$353$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$354$	−0.434262	+	3.46533i	−0.434262	+	3.46533i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−1.32091	−	2.28789i	−1.32091	−	2.28789i
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$	1.83170	−	1.87795i	1.83170	−	1.87795i
$361$	−0.167487			−0.167487
$362$		0			0
$363$	−2.01841	−	1.52889i	−2.01841	−	1.52889i
$364$		0			0
$365$		0			0
$366$	2.74514	+	2.07936i	2.74514	+	2.07936i
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$	3.53242			3.53242
$369$	0.510189	−	0.523071i	0.510189	−	0.523071i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$	−0.0924692	+	0.737887i	−0.0924692	+	0.737887i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.149460			0.149460			0.0747301	−	0.997204i	$-0.476190\pi$
$379$	0.149460			0.149460			0.0747301	+	0.997204i	$0.476190\pi$
$380$	1.19679	+	2.07290i	1.19679	+	2.07290i
$381$		0			0
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$	0.921476	−	0.388435i	0.921476	−	0.388435i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.112038			−0.112038
$389$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$389$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$390$		0			0
$391$		0			0
$392$	1.12349	−	1.94594i	1.12349	−	1.94594i
$393$		0			0
$394$	1.25818	+	2.17923i	1.25818	+	2.17923i
$395$		0			0
$396$	4.09236	+	1.04204i	4.09236	+	1.04204i
$397$	1.39647			1.39647			0.698237	−	0.715867i	$-0.253968\pi$
$397$	1.39647			1.39647			0.698237	+	0.715867i	$0.253968\pi$
$398$		0			0
$399$		0			0
$400$	−0.327076	+	0.566512i	−0.327076	+	0.566512i
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	0.996206	−	0.608769i	0.996206	−	0.608769i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$	−0.768582	+	1.33122i	−0.768582	+	1.33122i
$411$		0			0
$412$		0			0
$413$		0			0
$414$	3.42319	+	0.871651i	3.42319	+	0.871651i
$415$		0			0
$416$		0			0
$417$	1.36037	+	1.03044i	1.36037	+	1.03044i
$418$	−1.54497	+	2.67597i	−1.54497	+	2.67597i
$419$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$421$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$422$		0			0
$423$		0			0
$424$	−2.43818			−2.43818
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.00000			1.00000
$431$	1.00000			1.00000
$432$	−1.78181	+	0.268565i	−1.78181	+	0.268565i
$433$	−0.822574			−0.822574			−0.411287	−	0.911506i	$-0.634921\pi$
$433$	−0.822574			−0.822574			−0.411287	+	0.911506i	$0.634921\pi$
$434$		0			0
$435$	0.254982	−	2.03471i	0.254982	−	2.03471i
$436$	−0.167917	+	0.290841i	−0.167917	+	0.290841i
$437$	−0.894330	+	1.54903i	−0.894330	+	1.54903i
$438$		0			0
$439$	−0.995031	−	1.72344i	−0.995031	−	1.72344i	−0.583744	−	0.811938i	$-0.698413\pi$
$439$	−0.995031	−	1.72344i	−0.995031	−	1.72344i	−0.411287	−	0.911506i	$-0.634921\pi$
$440$	−4.93023			−4.93023
$441$	0.698237	−	0.715867i	0.698237	−	0.715867i
$442$		0			0
$443$	0.998757	+	1.72990i	0.998757	+	1.72990i	0.542546	+	0.840026i	$0.317460\pi$
$443$	0.998757	+	1.72990i	0.998757	+	1.72990i	0.456211	+	0.889872i	$0.349206\pi$
$444$		0			0
$445$		0			0
$446$	−1.69327	+	2.93283i	−1.69327	+	2.93283i
$447$	−1.46908	−	1.11278i	−1.46908	−	1.11278i
$448$		0			0
$449$	0.912421			0.912421			0.456211	−	0.889872i	$-0.349206\pi$
$449$	0.912421			0.912421			0.456211	+	0.889872i	$0.349206\pi$
$450$	−0.456753	+	0.468285i	−0.456753	+	0.468285i
$451$	−1.37323			−1.37323
$452$		0			0
$453$	−1.07581	+	0.453493i	−1.07581	+	0.453493i
$454$	−1.53758	+	2.66316i	−1.53758	+	2.66316i
$455$		0			0
$456$	0.254928	−	2.03428i	0.254928	−	2.03428i
$457$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$457$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$458$	2.87277			2.87277
$459$		0			0
$460$	−5.14261			−5.14261
$461$	0.318487	+	0.551635i	0.318487	+	0.551635i	0.980172	−	0.198146i	$-0.0634921\pi$
$461$	0.318487	+	0.551635i	0.318487	+	0.551635i	−0.661686	+	0.749781i	$0.730159\pi$
$462$		0			0
$463$	0.0249307	−	0.0431812i	0.0249307	−	0.0431812i	−0.853291	−	0.521435i	$-0.825397\pi$
$463$	0.0249307	−	0.0431812i	0.0249307	−	0.0431812i	0.878222	+	0.478254i	$0.158730\pi$
$464$	−1.58250	+	2.74097i	−1.58250	+	2.74097i
$465$		0			0
$466$		0			0
$467$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$467$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	1.49812	+	1.13478i	1.49812	+	1.13478i
$472$	−2.17750	+	3.77154i	−2.17750	+	3.77154i
$473$		0			0
$474$		0			0
$475$	−0.165617	−	0.286856i	−0.165617	−	0.286856i
$476$		0			0
$477$	−1.05154	−	0.267755i	−1.05154	−	0.267755i
$478$		0			0
$479$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.733052	−	0.680173i	$-0.761905\pi$
$479$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.222521	−	0.974928i	$-0.571429\pi$
$480$	1.07581	−	0.453493i	1.07581	−	0.453493i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−2.84478	−	4.92730i	−2.84478	−	4.92730i
$485$	0.0582125			0.0582125
$486$	−1.79298	−	0.179415i	−1.79298	−	0.179415i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	2.14715	+	3.71898i	2.14715	+	3.71898i
$489$	−0.243757	+	1.94513i	−0.243757	+	1.94513i
$490$	−1.05187	+	1.82189i	−1.05187	+	1.82189i
$491$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	−0.900969	−	0.433884i	$-0.857143\pi$
$491$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	0.826239	+	0.563320i	$0.190476\pi$
$492$	1.51291	−	0.637743i	1.51291	−	0.637743i
$493$		0			0
$494$		0			0
$495$	−2.12631	−	0.541425i	−2.12631	−	0.541425i
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$	−0.835492	+	1.44712i	−0.835492	+	1.44712i
$501$		0			0
$502$		0			0
$503$	0.541681			0.541681			0.270840	−	0.962624i	$-0.412698\pi$
$503$	0.541681			0.541681			0.270840	+	0.962624i	$0.412698\pi$
$504$		0			0
$505$		0			0
$506$	−3.31939	−	5.74935i	−3.31939	−	5.74935i
$507$	0.921476	−	0.388435i	0.921476	−	0.388435i
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$		0			0
$512$	2.24698			2.24698
$513$	0.333345	−	0.849349i	0.333345	−	0.849349i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−1.21946	+	0.514044i	−1.21946	+	0.514044i
$520$		0			0
$521$	0.149460			0.149460			0.0747301	−	0.997204i	$-0.476190\pi$
$521$	0.149460			0.149460			0.0747301	+	0.997204i	$0.476190\pi$
$522$	−2.20992	+	2.26572i	−2.20992	+	2.26572i
$523$	0.730682			0.730682			0.365341	−	0.930874i	$-0.380952\pi$
$523$	0.730682			0.730682			0.365341	+	0.930874i	$0.380952\pi$
$524$		0			0
$525$		0			0
$526$	1.25818	−	2.17923i	1.25818	−	2.17923i
$527$		0			0
$528$	2.69952	+	2.04480i	2.69952	+	2.04480i
$529$	−1.42148	−	2.46207i	−1.42148	−	2.46207i
$530$	2.28275			2.28275
$531$	−1.35329	+	1.38746i	−1.35329	+	1.38746i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.182301	−	1.45473i	0.182301	−	1.45473i
$538$		0			0
$539$	−1.87939			−1.87939
$540$	2.59402	−	0.390986i	2.59402	−	0.390986i
$541$	−1.99751			−1.99751			−0.998757	−	0.0498459i	$-0.984127\pi$
$541$	−1.99751			−1.99751			−0.998757	+	0.0498459i	$0.984127\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.0872464	−	0.151115i	0.0872464	−	0.151115i
$546$		0			0
$547$	−0.995031	−	1.72344i	−0.995031	−	1.72344i	−0.583744	−	0.811938i	$-0.698413\pi$
$547$	−0.995031	−	1.72344i	−0.995031	−	1.72344i	−0.411287	−	0.911506i	$-0.634921\pi$
$548$		0			0
$549$	0.517616	+	1.83972i	0.517616	+	1.83972i
$550$	1.22940			1.22940
$551$	−0.801308	−	1.38791i	−0.801308	−	1.38791i
$552$	3.51125	+	2.65967i	3.51125	+	2.65967i
$553$		0			0
$554$	1.66044	−	2.87597i	1.66044	−	2.87597i
$555$		0			0
$556$	1.91733	+	3.32091i	1.91733	+	3.32091i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	0.0747301	+	0.997204i	$0.476190\pi$
$563$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	−0.900969	+	0.433884i	$0.857143\pi$
$564$		0			0
$565$		0			0
$566$	3.49243			3.49243
$567$		0			0
$568$		0			0
$569$	0.998757	+	1.72990i	0.998757	+	1.72990i	0.542546	+	0.840026i	$0.317460\pi$
$569$	0.998757	+	1.72990i	0.998757	+	1.72990i	0.456211	+	0.889872i	$0.349206\pi$
$570$	−0.238677	+	1.90460i	−0.238677	+	1.90460i
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.711658			0.711658
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	0.900969	+	1.56052i	0.900969	+	1.56052i
$579$		0			0
$580$	2.30386	−	3.99040i	2.30386	−	3.99040i
$581$		0			0
$582$	−0.0716201	−	0.0542501i	−0.0716201	−	0.0542501i
$583$	1.01965	+	1.76609i	1.01965	+	1.76609i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$587$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$588$	2.07054	−	0.872805i	2.07054	−	0.872805i
$589$		0			0
$590$	2.03869	−	3.53111i	2.03869	−	3.53111i
$591$	−0.173643	+	1.38564i	−0.173643	+	1.38564i
$592$		0			0
$593$	−0.0498614			−0.0498614			−0.0249307	−	0.999689i	$-0.507937\pi$
$593$	−0.0498614			−0.0498614			−0.0249307	+	0.999689i	$0.507937\pi$
$594$	2.11147	+	2.64770i	2.11147	+	2.64770i
$595$		0			0
$596$	−2.07054	−	3.58628i	−2.07054	−	3.58628i
$597$		0			0
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$	−0.751661	+	0.316852i	−0.751661	+	0.316852i
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$		0			0
$604$	−2.62332			−2.62332
$605$	1.47809	+	2.56013i	1.47809	+	2.56013i
$606$		0			0
$607$	−0.955573	+	1.65510i	−0.955573	+	1.65510i	−0.222521	+	0.974928i	$0.571429\pi$
$607$	−0.955573	+	1.65510i	−0.955573	+	1.65510i	−0.733052	+	0.680173i	$0.761905\pi$
$608$	0.456211	−	0.790180i	0.456211	−	0.790180i
$609$		0			0
$610$	−2.01028	−	3.48190i	−2.01028	−	3.48190i
$611$		0			0
$612$		0			0
$613$	0.730682			0.730682			0.365341	−	0.930874i	$-0.380952\pi$
$613$	0.730682			0.730682			0.365341	+	0.930874i	$0.380952\pi$
$614$	−1.62349	−	2.81197i	−1.62349	−	2.81197i
$615$	−0.786076	+	0.331359i	−0.786076	+	0.331359i
$616$		0			0
$617$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$617$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$	1.22226	+	1.53266i	1.22226	+	1.53266i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.615619	−	1.06628i	0.615619	−	1.06628i
$626$		0			0
$627$	−1.58014	+	0.666085i	−1.58014	+	0.666085i
$628$	2.11147	+	3.65717i	2.11147	+	3.65717i
$629$		0			0
$630$		0			0
$631$	1.84295			1.84295			0.921476	−	0.388435i	$-0.126984\pi$
$631$	1.84295			1.84295			0.921476	+	0.388435i	$0.126984\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−1.94355	−	1.47218i	−1.94355	−	1.47218i
$637$		0			0
$638$	5.94826			5.94826
$639$		0			0
$640$	−1.16749			−1.16749
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$643$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.70658			−1.70658			−0.853291	−	0.521435i	$-0.825397\pi$
$647$	−1.70658			−1.70658			−0.853291	+	0.521435i	$0.825397\pi$
$648$	−1.97335	−	1.07463i	−1.97335	−	1.07463i
$649$	3.64254			3.64254
$650$		0			0
$651$		0			0
$652$	−2.20243	+	3.81472i	−2.20243	+	3.81472i
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$	−0.248170	+	0.104612i	−0.248170	+	0.104612i
$655$		0			0
$656$	1.31664			1.31664
$657$		0			0
$658$		0			0
$659$	−0.542546	−	0.939718i	−0.542546	−	0.939718i	−0.998757	−	0.0498459i	$-0.984127\pi$
$659$	−0.542546	−	0.939718i	−0.542546	−	0.939718i	0.456211	−	0.889872i	$-0.349206\pi$
$660$	−3.93005	−	2.97689i	−3.93005	−	2.97689i
$661$	−0.270840	+	0.469109i	−0.270840	+	0.469109i	−0.969077	−	0.246757i	$-0.920635\pi$
$661$	−0.270840	+	0.469109i	−0.270840	+	0.469109i	0.698237	+	0.715867i	$0.253968\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	3.44323			3.44323
$668$		0			0
$669$	−1.73181	+	0.730019i	−1.73181	+	0.730019i
$670$		0			0
$671$	1.79589	−	3.11057i	1.79589	−	3.11057i
$672$		0			0
$673$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.623490	+	0.781831i	$0.285714\pi$
$673$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.365341	+	0.930874i	$0.380952\pi$
$674$	2.64183			2.64183
$675$	−0.358972	+	0.0541063i	−0.358972	+	0.0541063i
$676$	2.24698			2.24698
$677$	0.124344	+	0.215370i	0.124344	+	0.215370i	0.921476	−	0.388435i	$-0.126984\pi$
$677$	0.124344	+	0.215370i	0.124344	+	0.215370i	−0.797133	+	0.603804i	$0.793651\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−1.57257	+	0.662896i	−1.57257	+	0.662896i
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	1.43152	−	1.46766i	1.43152	−	1.46766i
$685$		0			0
$686$		0			0
$687$	1.27084	+	0.962624i	1.27084	+	0.962624i
$688$		0			0
$689$		0			0
$690$	−3.28742	−	2.49012i	−3.28742	−	2.49012i
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$	−2.97359			−2.97359
$693$		0			0
$694$	−3.44377			−3.44377
$695$	−0.996206	−	1.72548i	−0.996206	−	1.72548i
$696$	−3.63678	+	1.53303i	−3.63678	+	1.53303i
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.97766			−1.97766			−0.988831	−	0.149042i	$-0.952381\pi$
$701$	−1.97766			−1.97766			−0.988831	+	0.149042i	$0.952381\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	0.312903	−	0.541964i	0.312903	−	0.541964i
$707$		0			0
$708$	−4.01302	+	1.69163i	−4.01302	+	1.69163i
$709$	−0.980172	−	1.69771i	−0.980172	−	1.69771i	−0.661686	−	0.749781i	$-0.730159\pi$
$709$	−0.980172	−	1.69771i	−0.980172	−	1.69771i	−0.318487	−	0.947927i	$-0.603175\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	1.64715	−	2.85295i	1.64715	−	2.85295i
$717$		0			0
$718$		0			0
$719$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$719$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$720$	2.03869	+	0.519113i	2.03869	+	0.519113i
$721$		0			0
$722$	−0.150901	−	0.261368i	−0.150901	−	0.261368i
$723$		0			0
$724$		0			0
$725$	−0.318818	+	0.552209i	−0.318818	+	0.552209i
$726$	0.567339	−	4.52726i	0.567339	−	4.52726i
$727$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$727$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$730$		0			0
$731$		0			0
$732$	−0.533970	+	4.26098i	−0.533970	+	4.26098i
$733$	0.661686	−	1.14607i	0.661686	−	1.14607i	−0.318487	−	0.947927i	$-0.603175\pi$
$733$	0.661686	−	1.14607i	0.661686	−	1.14607i	0.980172	−	0.198146i	$-0.0634921\pi$
$734$		0			0
$735$	−1.07581	+	0.453493i	−1.07581	+	0.453493i
$736$	0.980172	+	1.69771i	0.980172	+	1.69771i
$737$		0			0
$738$	1.27593	+	0.324892i	1.27593	+	0.324892i
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$743$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$744$		0			0
$745$	1.07581	+	1.86336i	1.07581	+	1.86336i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$	−1.23480	+	0.520513i	−1.23480	+	0.520513i
$751$	0.124344	−	0.215370i	0.124344	−	0.215370i	−0.797133	−	0.603804i	$-0.793651\pi$
$751$	0.124344	−	0.215370i	0.124344	−	0.215370i	0.921476	+	0.388435i	$0.126984\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.36303			1.36303
$756$		0			0
$757$	1.65248			1.65248			0.826239	−	0.563320i	$-0.190476\pi$
$757$	1.65248			1.65248			0.826239	+	0.563320i	$0.190476\pi$
$758$	0.134659	+	0.233236i	0.134659	+	0.233236i
$759$	0.458112	−	3.65565i	0.458112	−	3.65565i
$760$	−1.19679	+	2.07290i	−1.19679	+	2.07290i
$761$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	0.0747301	+	0.997204i	$0.476190\pi$
$761$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	−0.900969	+	0.433884i	$0.857143\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	1.43638	+	1.08802i	1.43638	+	1.08802i
$769$	−0.698237	+	1.20938i	−0.698237	+	1.20938i	0.270840	+	0.962624i	$0.412698\pi$
$769$	−0.698237	+	1.20938i	−0.698237	+	1.20938i	−0.969077	+	0.246757i	$0.920635\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$773$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$774$		0			0
$775$		0			0
$776$	−0.0560188	−	0.0970273i	−0.0560188	−	0.0970273i
$777$		0			0
$778$	1.38036	−	2.39086i	1.38036	−	2.39086i
$779$	−0.333345	+	0.577370i	−0.333345	+	0.577370i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−1.73683	+	0.261784i	−1.73683	+	0.261784i
$784$	1.80194			1.80194
$785$	−1.09708	−	1.90020i	−1.09708	−	1.90020i
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$	−1.56892	+	2.71746i	−1.56892	+	2.71746i
$789$	1.28682	−	0.542439i	1.28682	−	0.542439i
$790$		0			0
$791$		0			0
$792$	1.14374	+	4.06510i	1.14374	+	4.06510i
$793$		0			0
$794$	1.25818	+	2.17923i	1.25818	+	2.17923i
$795$	1.00983	+	0.764919i	1.00983	+	0.764919i
$796$		0			0
$797$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.365341	−	0.930874i	$-0.380952\pi$
$797$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.623490	−	0.781831i	$-0.285714\pi$
$798$		0			0
$799$		0			0
$800$	−0.363027			−0.363027
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	1.84755	+	1.00612i	1.84755	+	1.00612i
$811$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$811$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.14434	−	1.98205i	1.14434	−	1.98205i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	−1.91681			−1.91681
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	0.998757	−	1.72990i	0.998757	−	1.72990i	0.456211	−	0.889872i	$-0.349206\pi$
$823$	0.998757	−	1.72990i	0.998757	−	1.72990i	0.542546	−	0.840026i	$-0.317460\pi$
$824$		0			0
$825$	0.543857	+	0.411956i	0.543857	+	0.411956i
$826$		0			0
$827$	−0.822574			−0.822574			−0.411287	−	0.911506i	$-0.634921\pi$
$827$	−0.822574			−0.822574			−0.411287	+	0.911506i	$0.634921\pi$
$828$	1.19301	+	4.24022i	1.19301	+	4.24022i
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$	1.69824	−	0.715867i	1.69824	−	0.715867i
$832$		0			0
$833$		0			0
$834$	−0.382376	+	3.05129i	−0.382376	+	3.05129i
$835$		0			0
$836$	−3.85310			−3.85310
$837$		0			0
$838$		0			0
$839$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	−1.04255	+	1.80574i	−1.04255	+	1.80574i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.16749			−1.16749
$846$		0			0
$847$		0			0
$848$	−0.977635	−	1.69331i	−0.977635	−	1.69331i
$849$	1.54497	+	1.17027i	1.54497	+	1.17027i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$853$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$854$		0			0
$855$	−0.743790	+	0.762570i	−0.743790	+	0.762570i
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$859$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$	0.900969	+	1.56052i	0.900969	+	1.56052i
$863$	0.149460			0.149460			0.0747301	−	0.997204i	$-0.476190\pi$
$863$	0.149460			0.149460			0.0747301	+	0.997204i	$0.476190\pi$
$864$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$865$	1.54502			1.54502
$866$	−0.741114	−	1.28365i	−0.741114	−	1.28365i
$867$	−0.124344	+	0.992239i	−0.124344	+	0.992239i
$868$		0			0
$869$		0			0
$870$	3.40495	−	1.43530i	3.40495	−	1.43530i
$871$		0			0
$872$	−0.335834			−0.335834
$873$	−0.0135045	−	0.0479978i	−0.0135045	−	0.0479978i
$874$	−3.22305			−3.22305
$875$		0			0
$876$		0			0
$877$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$877$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$878$	1.79298	−	3.10554i	1.79298	−	3.10554i
$879$		0			0
$880$	−1.97687	−	3.42404i	−1.97687	−	3.42404i
$881$	−1.32337			−1.32337			−0.661686	−	0.749781i	$-0.730159\pi$
$881$	−1.32337			−1.32337			−0.661686	+	0.749781i	$0.730159\pi$
$882$	1.74622	+	0.444641i	1.74622	+	0.444641i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$	2.08509	−	0.878939i	2.08509	−	0.878939i
$886$	−1.79970	+	3.11717i	−1.79970	+	3.11717i
$887$	−0.542546	+	0.939718i	−0.542546	+	0.939718i	0.456211	+	0.889872i	$0.349206\pi$
$887$	−0.542546	+	0.939718i	−0.542546	+	0.939718i	−0.998757	+	0.0498459i	$0.984127\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.0468544	+	1.87880i	0.0468544	+	1.87880i
$892$	−4.22294			−4.22294
$893$		0			0
$894$	0.412931	−	3.29511i	0.412931	−	3.29511i
$895$	−0.855829	+	1.48234i	−0.855829	+	1.48234i
$896$		0			0
$897$		0			0
$898$	0.822063	+	1.42386i	0.822063	+	1.42386i
$899$		0			0
$900$	−0.790490	−	0.201283i	−0.790490	−	0.201283i
$901$		0			0
$902$	−1.23724	−	2.14296i	−1.23724	−	2.14296i
$903$		0			0
$904$		0			0
$905$		0			0
$906$	−1.67696	−	1.27025i	−1.67696	−	1.27025i
$907$	−0.878222	−	1.52112i	−0.878222	−	1.52112i	−0.853291	−	0.521435i	$-0.825397\pi$
$907$	−0.878222	−	1.52112i	−0.878222	−	1.52112i	−0.0249307	−	0.999689i	$-0.507937\pi$
$908$	−3.83465			−3.83465
$909$		0			0
$910$		0			0
$911$	−0.995031	−	1.72344i	−0.995031	−	1.72344i	−0.583744	−	0.811938i	$-0.698413\pi$
$911$	−0.995031	−	1.72344i	−0.995031	−	1.72344i	−0.411287	−	0.911506i	$-0.634921\pi$
$912$	1.51502	−	0.638636i	1.51502	−	0.638636i
$913$		0			0
$914$		0			0
$915$	0.277440	−	2.21392i	0.277440	−	2.21392i
$916$	1.79114	+	3.10235i	1.79114	+	3.10235i
$917$		0			0
$918$		0			0
$919$	1.75644			1.75644			0.878222	−	0.478254i	$-0.158730\pi$
$919$	1.75644			1.75644			0.878222	+	0.478254i	$0.158730\pi$
$920$	−2.57131	−	4.45363i	−2.57131	−	4.45363i
$921$	0.224060	−	1.78795i	0.224060	−	1.78795i
$922$	−0.573893	+	0.994012i	−0.573893	+	0.994012i
$923$		0			0
$924$		0			0
$925$		0			0
$926$	0.0898471			0.0898471
$927$		0			0
$928$	−1.75644			−1.75644
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$	−0.456211	+	0.790180i	−0.456211	+	0.790180i
$932$		0			0
$933$		0			0
$934$	−0.900969	−	1.56052i	−0.900969	−	1.56052i
$935$		0			0
$936$		0			0
$937$	1.08509			1.08509			0.542546	−	0.840026i	$-0.317460\pi$
$937$	1.08509			1.08509			0.542546	+	0.840026i	$0.317460\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$	−0.421094	+	3.36025i	−0.421094	+	3.36025i
$943$	−0.716194	−	1.24049i	−0.716194	−	1.24049i
$944$	−3.49243			−3.49243
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$		0			0
$950$	0.298431	−	0.516897i	0.298431	−	0.516897i
$951$		0			0
$952$		0			0
$953$	−1.93815			−1.93815			−0.969077	−	0.246757i	$-0.920635\pi$
$953$	−1.93815			−1.93815			−0.969077	+	0.246757i	$0.920635\pi$
$954$	−0.529566	−	1.88219i	−0.529566	−	1.88219i
$955$		0			0
$956$		0			0
$957$	2.63136	+	1.99318i	2.63136	+	1.99318i
$958$	1.72188	−	2.98239i	1.72188	−	2.98239i
$959$		0			0
$960$		0			0
$961$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$967$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$968$	2.84478	−	4.92730i	2.84478	−	4.92730i
$969$		0			0
$970$	0.0524477	+	0.0908420i	0.0524477	+	0.0908420i
$971$	0.730682			0.730682			0.365341	−	0.930874i	$-0.380952\pi$
$971$	0.730682			0.730682			0.365341	+	0.930874i	$0.380952\pi$
$972$	−0.924154	−	2.04814i	−0.924154	−	2.04814i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−1.72188	+	2.98239i	−1.72188	+	2.98239i
$977$	−0.921476	+	1.59604i	−0.921476	+	1.59604i	−0.124344	+	0.992239i	$0.539683\pi$
$977$	−0.921476	+	1.59604i	−0.921476	+	1.59604i	−0.797133	+	0.603804i	$0.793651\pi$
$978$	−3.25504	+	1.37211i	−3.25504	+	1.37211i
$979$		0			0
$980$	−2.62332			−2.62332
$981$	−0.144838	−	0.0368804i	−0.144838	−	0.0368804i
$982$	−0.269318			−0.269318
$983$	−0.921476	−	1.59604i	−0.921476	−	1.59604i	−0.797133	−	0.603804i	$-0.793651\pi$
$983$	−0.921476	−	1.59604i	−0.921476	−	1.59604i	−0.124344	−	0.992239i	$-0.539683\pi$
$984$	1.30875	+	0.991343i	1.30875	+	0.991343i
$985$	0.815183	−	1.41194i	0.815183	−	1.41194i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−1.07083	−	3.80596i	−1.07083	−	3.80596i
$991$		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
$997$	0.661686	+	1.14607i	0.661686	+	1.14607i	0.980172	+	0.198146i	$0.0634921\pi$
$997$	0.661686	+	1.14607i	0.661686	+	1.14607i	−0.318487	+	0.947927i	$0.603175\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3879.1.g.c.1723.15	yes	36
9.7	even	3	inner	3879.1.g.c.430.15	✓	36
431.430	odd	2	CM	3879.1.g.c.1723.15	yes	36
3879.430	odd	6	inner	3879.1.g.c.430.15	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3879.1.g.c.430.15	✓	36	9.7	even	3	inner
3879.1.g.c.430.15	✓	36	3879.430	odd	6	inner
3879.1.g.c.1723.15	yes	36	1.1	even	1	trivial
3879.1.g.c.1723.15	yes	36	431.430	odd	2	CM

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:	\( N \)	\(=\)	\( 3879 = 3^{2} \cdot 431 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3879.g (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.900969 + 1.56052i) q^{2} +(-0.124344 + 0.992239i) q^{3} +(-1.12349 + 1.94594i) q^{4} +(0.583744 - 1.01107i) q^{5} +(-1.66044 + 0.699935i) q^{6} -2.24698 q^{8} +(-0.969077 - 0.246757i) q^{9} +O(q^{10})\)	\(q+(0.900969 + 1.56052i) q^{2} +(-0.124344 + 0.992239i) q^{3} +(-1.12349 + 1.94594i) q^{4} +(0.583744 - 1.01107i) q^{5} +(-1.66044 + 0.699935i) q^{6} -2.24698 q^{8} +(-0.969077 - 0.246757i) q^{9} +2.10374 q^{10} +(0.939693 + 1.62760i) q^{11} +(-1.79114 - 1.35674i) q^{12} +(0.930642 + 0.704934i) q^{15} +(-0.900969 - 1.56052i) q^{16} +(-0.488038 - 1.73459i) q^{18} +0.912421 q^{19} +(1.31166 + 2.27186i) q^{20} +(-1.69327 + 2.93283i) q^{22} +(-0.980172 + 1.69771i) q^{23} +(0.279398 - 2.22954i) q^{24} +(-0.181513 - 0.314390i) q^{25} +(0.365341 - 0.930874i) q^{27} +(-0.878222 - 1.52112i) q^{29} +(-0.261587 + 2.08741i) q^{30} +(0.500000 - 0.866025i) q^{32} +(-1.73181 + 0.730019i) q^{33} +(1.56892 - 1.60854i) q^{36} +(0.822063 + 1.42386i) q^{38} +(-1.31166 + 2.27186i) q^{40} +(-0.365341 + 0.632789i) q^{41} -4.22294 q^{44} +(-0.815183 + 0.835765i) q^{45} -3.53242 q^{46} +(1.66044 - 0.699935i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(0.327076 - 0.566512i) q^{50} +1.08509 q^{53} +(1.78181 - 0.268565i) q^{54} +2.19416 q^{55} +(-0.113454 + 0.905340i) q^{57} +(1.58250 - 2.74097i) q^{58} +(0.969077 - 1.67849i) q^{59} +(-2.41733 + 1.01899i) q^{60} +(-0.955573 - 1.65510i) q^{61} +(-2.69952 - 2.04480i) q^{66} +(-1.56265 - 1.18366i) q^{69} +(2.17750 + 0.554459i) q^{72} +(0.334520 - 0.141012i) q^{75} +(-1.02510 + 1.77552i) q^{76} -2.10374 q^{80} +(0.878222 + 0.478254i) q^{81} -1.31664 q^{82} +(1.61852 - 0.682264i) q^{87} +(-2.11147 - 3.65717i) q^{88} +(-2.03869 - 0.519113i) q^{90} +(-2.20243 - 3.81472i) q^{92} +(0.532620 - 0.922525i) q^{95} +(0.797133 + 0.603804i) q^{96} +(0.0249307 + 0.0431812i) q^{97} -1.80194 q^{98} +(-0.509014 - 1.80914i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 6 q^{2} - 12 q^{4} - 24 q^{8}+O(q^{10}) \) 36 * q + 6 * q^2 - 12 * q^4 - 24 * q^8	\( 36 q + 6 q^{2} - 12 q^{4} - 24 q^{8} + 3 q^{15} - 6 q^{16} - 18 q^{25} + 3 q^{27} + 6 q^{30} + 18 q^{32} + 3 q^{33} - 3 q^{41} + 3 q^{45} - 18 q^{49} + 6 q^{50} + 6 q^{54} + 6 q^{55} + 3 q^{57} - 18 q^{60} - 3 q^{61} + 6 q^{66} - 6 q^{69} + 3 q^{75} + 12 q^{82} - 6 q^{87} + 6 q^{90} - 3 q^{95} - 12 q^{98} + 3 q^{99}+O(q^{100}) \) 36 * q + 6 * q^2 - 12 * q^4 - 24 * q^8 + 3 * q^15 - 6 * q^16 - 18 * q^25 + 3 * q^27 + 6 * q^30 + 18 * q^32 + 3 * q^33 - 3 * q^41 + 3 * q^45 - 18 * q^49 + 6 * q^50 + 6 * q^54 + 6 * q^55 + 3 * q^57 - 18 * q^60 - 3 * q^61 + 6 * q^66 - 6 * q^69 + 3 * q^75 + 12 * q^82 - 6 * q^87 + 6 * q^90 - 3 * q^95 - 12 * q^98 + 3 * q^99

Introduction

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