LMFDB - Embedded newform 3879.1.g.c.430.6

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			430.6
Root			$-0.0249307 + 0.999689i$ of defining polynomial
Character	$\chi$	$=$	3879.430
Dual form			3879.1.g.c.1723.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.623490 + 1.07992i) q^{2} +(0.995031 + 0.0995678i) q^{3} +(-0.277479 - 0.480608i) q^{4} +(-0.878222 - 1.52112i) q^{5} +(-0.727916 + 1.01247i) q^{6} -0.554958 q^{8} +(0.980172 + 0.198146i) q^{9} +O(q^{10})$	\(q+(-0.623490 + 1.07992i) q^{2} +(0.995031 + 0.0995678i) q^{3} +(-0.277479 - 0.480608i) q^{4} +(-0.878222 - 1.52112i) q^{5} +(-0.727916 + 1.01247i) q^{6} -0.554958 q^{8} +(0.980172 + 0.198146i) q^{9} +2.19025 q^{10} +(0.939693 - 1.62760i) q^{11} +(-0.228247 - 0.505848i) q^{12} +(-0.722402 - 1.60101i) q^{15} +(0.623490 - 1.07992i) q^{16} +(-0.825109 + 0.934962i) q^{18} -1.93815 q^{19} +(-0.487376 + 0.844160i) q^{20} +(1.17178 + 2.02958i) q^{22} +(-0.456211 - 0.790180i) q^{23} +(-0.552200 - 0.0552560i) q^{24} +(-1.04255 + 1.80574i) q^{25} +(0.955573 + 0.294755i) q^{27} +(-0.921476 + 1.59604i) q^{29} +(2.17936 + 0.218078i) q^{30} +(0.500000 + 0.866025i) q^{32} +(1.09708 - 1.52594i) q^{33} +(-0.176747 - 0.526060i) q^{36} +(1.20842 - 2.09304i) q^{38} +(0.487376 + 0.844160i) q^{40} +(-0.955573 - 1.65510i) q^{41} -1.04298 q^{44} +(-0.559404 - 1.66498i) q^{45} +1.13777 q^{46} +(0.727916 - 1.01247i) q^{48} +(-0.500000 - 0.866025i) q^{49} +(-1.30003 - 2.25172i) q^{50} +1.39647 q^{53} +(-0.914101 + 0.848162i) q^{54} -3.30103 q^{55} +(-1.92852 - 0.192978i) q^{57} +(-1.14906 - 1.99023i) q^{58} +(-0.980172 - 1.69771i) q^{59} +(-0.569006 + 0.791439i) q^{60} +(-0.0747301 + 0.129436i) q^{61} +(0.963874 + 2.13616i) q^{66} +(-0.375267 - 0.831677i) q^{69} +(-0.543955 - 0.109963i) q^{72} +(-1.21716 + 1.69297i) q^{75} +(0.537797 + 0.931492i) q^{76} -2.19025 q^{80} +(0.921476 + 0.388435i) q^{81} +2.38316 q^{82} +(-1.07581 + 1.49636i) q^{87} +(-0.521490 + 0.903247i) q^{88} +(2.14682 + 0.433989i) q^{90} +(-0.253178 + 0.438517i) q^{92} +(1.70213 + 2.94817i) q^{95} +(0.411287 + 0.911506i) q^{96} +(0.797133 - 1.38067i) q^{97} +1.24698 q^{98} +(1.24356 - 1.40913i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/3879\mathbb{Z}\right)^\times$.

$n$	$2593$	$3449$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$2$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$3$	0.995031	+	0.0995678i	0.995031	+	0.0995678i
$3$	0.995031	+	0.0995678i	0.995031	+	0.0995678i
$4$	−0.277479	−	0.480608i	−0.277479	−	0.480608i
$5$	−0.878222	−	1.52112i	−0.878222	−	1.52112i	−0.853291	−	0.521435i	$-0.825397\pi$
$5$	−0.878222	−	1.52112i	−0.878222	−	1.52112i	−0.0249307	−	0.999689i	$-0.507937\pi$
$6$	−0.727916	+	1.01247i	−0.727916	+	1.01247i
$7$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$7$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$8$	−0.554958			−0.554958
$9$	0.980172	+	0.198146i	0.980172	+	0.198146i
$10$	2.19025			2.19025
$11$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$11$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$12$	−0.228247	−	0.505848i	−0.228247	−	0.505848i
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$		0			0
$15$	−0.722402	−	1.60101i	−0.722402	−	1.60101i
$16$	0.623490	−	1.07992i	0.623490	−	1.07992i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	−0.825109	+	0.934962i	−0.825109	+	0.934962i
$19$	−1.93815			−1.93815			−0.969077	−	0.246757i	$-0.920635\pi$
$19$	−1.93815			−1.93815			−0.969077	+	0.246757i	$0.920635\pi$
$20$	−0.487376	+	0.844160i	−0.487376	+	0.844160i
$21$		0			0
$22$	1.17178	+	2.02958i	1.17178	+	2.02958i
$23$	−0.456211	−	0.790180i	−0.456211	−	0.790180i	0.542546	−	0.840026i	$-0.317460\pi$
$23$	−0.456211	−	0.790180i	−0.456211	−	0.790180i	−0.998757	+	0.0498459i	$0.984127\pi$
$24$	−0.552200	−	0.0552560i	−0.552200	−	0.0552560i
$25$	−1.04255	+	1.80574i	−1.04255	+	1.80574i
$26$		0			0
$27$	0.955573	+	0.294755i	0.955573	+	0.294755i
$28$		0			0
$29$	−0.921476	+	1.59604i	−0.921476	+	1.59604i	−0.124344	+	0.992239i	$0.539683\pi$
$29$	−0.921476	+	1.59604i	−0.921476	+	1.59604i	−0.797133	+	0.603804i	$0.793651\pi$
$30$	2.17936	+	0.218078i	2.17936	+	0.218078i
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	0.500000	+	0.866025i	0.500000	+	0.866025i
$33$	1.09708	−	1.52594i	1.09708	−	1.52594i
$34$		0			0
$35$		0			0
$36$	−0.176747	−	0.526060i	−0.176747	−	0.526060i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	1.20842	−	2.09304i	1.20842	−	2.09304i
$39$		0			0
$40$	0.487376	+	0.844160i	0.487376	+	0.844160i
$41$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.733052	−	0.680173i	$-0.761905\pi$
$41$	−0.955573	−	1.65510i	−0.955573	−	1.65510i	−0.222521	−	0.974928i	$-0.571429\pi$
$42$		0			0
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$	−1.04298			−1.04298
$45$	−0.559404	−	1.66498i	−0.559404	−	1.66498i
$46$	1.13777			1.13777
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$	0.727916	−	1.01247i	0.727916	−	1.01247i
$49$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$50$	−1.30003	−	2.25172i	−1.30003	−	2.25172i
$51$		0			0
$52$		0			0
$53$	1.39647			1.39647			0.698237	−	0.715867i	$-0.253968\pi$
$53$	1.39647			1.39647			0.698237	+	0.715867i	$0.253968\pi$
$54$	−0.914101	+	0.848162i	−0.914101	+	0.848162i
$55$	−3.30103			−3.30103
$56$		0			0
$57$	−1.92852	−	0.192978i	−1.92852	−	0.192978i
$58$	−1.14906	−	1.99023i	−1.14906	−	1.99023i
$59$	−0.980172	−	1.69771i	−0.980172	−	1.69771i	−0.661686	−	0.749781i	$-0.730159\pi$
$59$	−0.980172	−	1.69771i	−0.980172	−	1.69771i	−0.318487	−	0.947927i	$-0.603175\pi$
$60$	−0.569006	+	0.791439i	−0.569006	+	0.791439i
$61$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	−0.900969	−	0.433884i	$-0.857143\pi$
$61$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	0.826239	+	0.563320i	$0.190476\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$	0.963874	+	2.13616i	0.963874	+	2.13616i
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$	−0.375267	−	0.831677i	−0.375267	−	0.831677i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−0.543955	−	0.109963i	−0.543955	−	0.109963i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−1.21716	+	1.69297i	−1.21716	+	1.69297i
$76$	0.537797	+	0.931492i	0.537797	+	0.931492i
$77$		0			0
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$	−2.19025			−2.19025
$81$	0.921476	+	0.388435i	0.921476	+	0.388435i
$82$	2.38316			2.38316
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1.07581	+	1.49636i	−1.07581	+	1.49636i
$88$	−0.521490	+	0.903247i	−0.521490	+	0.903247i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	2.14682	+	0.433989i	2.14682	+	0.433989i
$91$		0			0
$92$	−0.253178	+	0.438517i	−0.253178	+	0.438517i
$93$		0			0
$94$		0			0
$95$	1.70213	+	2.94817i	1.70213	+	2.94817i
$96$	0.411287	+	0.911506i	0.411287	+	0.911506i
$97$	0.797133	−	1.38067i	0.797133	−	1.38067i	−0.124344	−	0.992239i	$-0.539683\pi$
$97$	0.797133	−	1.38067i	0.797133	−	1.38067i	0.921476	−	0.388435i	$-0.126984\pi$
$98$	1.24698			1.24698
$99$	1.24356	−	1.40913i	1.24356	−	1.40913i
$100$	1.15714			1.15714
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$	−0.870687	+	1.50807i	−0.870687	+	1.50807i
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	−0.123490	−	0.541044i	−0.123490	−	0.541044i
$109$	0.730682			0.730682			0.365341	−	0.930874i	$-0.380952\pi$
$109$	0.730682			0.730682			0.365341	+	0.930874i	$0.380952\pi$
$110$	2.05816	−	3.56484i	2.05816	−	3.56484i
$111$		0			0
$112$		0			0
$113$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$113$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$114$	1.41081	−	1.96232i	1.41081	−	1.96232i
$115$	−0.801308	+	1.38791i	−0.801308	+	1.38791i
$116$	1.02276			1.02276
$117$		0			0
$118$	2.44451			2.44451
$119$		0			0
$120$	0.400903	+	0.888493i	0.400903	+	0.888493i
$121$	−1.26604	−	2.19285i	−1.26604	−	2.19285i
$122$	−0.0931869	−	0.161404i	−0.0931869	−	0.161404i
$123$	−0.786030	−	1.74202i	−0.786030	−	1.74202i
$124$		0			0
$125$	1.90590			1.90590
$126$		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	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$	−1.03780	−	0.103847i	−1.03780	−	0.103847i
$133$		0			0
$134$		0			0
$135$	−0.390845	−	1.71241i	−0.390845	−	1.71241i
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$	1.13212	+	0.113285i	1.13212	+	0.113285i
$139$	0.124344	+	0.215370i	0.124344	+	0.215370i	0.921476	−	0.388435i	$-0.126984\pi$
$139$	0.124344	+	0.215370i	0.124344	+	0.215370i	−0.797133	+	0.603804i	$0.793651\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.825109	−	0.934962i	0.825109	−	0.934962i
$145$	3.23704			3.23704
$146$		0			0
$147$	−0.411287	−	0.911506i	−0.411287	−	0.911506i
$148$		0			0
$149$	0.583744	+	1.01107i	0.583744	+	1.01107i	0.995031	+	0.0995678i	$0.0317460\pi$
$149$	0.583744	+	1.01107i	0.583744	+	1.01107i	−0.411287	+	0.911506i	$0.634921\pi$
$150$	−1.06937	−	2.36998i	−1.06937	−	2.36998i
$151$	−0.878222	+	1.52112i	−0.878222	+	1.52112i	−0.0249307	+	0.999689i	$0.507937\pi$
$151$	−0.878222	+	1.52112i	−0.878222	+	1.52112i	−0.853291	+	0.521435i	$0.825397\pi$
$152$	1.07559			1.07559
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$157$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$158$		0			0
$159$	1.38953	+	0.139044i	1.38953	+	0.139044i
$160$	0.878222	−	1.52112i	0.878222	−	1.52112i
$161$		0			0
$162$	−0.994008	+	0.752932i	−0.994008	+	0.752932i
$163$	0.912421			0.912421			0.456211	−	0.889872i	$-0.349206\pi$
$163$	0.912421			0.912421			0.456211	+	0.889872i	$0.349206\pi$
$164$	−0.530303	+	0.918512i	−0.530303	+	0.918512i
$165$	−3.28463	−	0.328677i	−3.28463	−	0.328677i
$166$		0			0
$167$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$167$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$168$		0			0
$169$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$170$		0			0
$171$	−1.89973	−	0.384038i	−1.89973	−	0.384038i
$172$		0			0
$173$	0.998757	−	1.72990i	0.998757	−	1.72990i	0.456211	−	0.889872i	$-0.349206\pi$
$173$	0.998757	−	1.72990i	0.998757	−	1.72990i	0.542546	−	0.840026i	$-0.317460\pi$
$174$	−0.945189	−	2.09475i	−0.945189	−	2.09475i
$175$		0			0
$176$	−1.17178	−	2.02958i	−1.17178	−	2.02958i
$177$	−0.806265	−	1.78687i	−0.806265	−	1.78687i
$178$		0			0
$179$	1.65248			1.65248			0.826239	−	0.563320i	$-0.190476\pi$
$179$	1.65248			1.65248			0.826239	+	0.563320i	$0.190476\pi$
$180$	−0.644980	+	0.730851i	−0.644980	+	0.730851i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	−0.0872464	+	0.121352i	−0.0872464	+	0.121352i
$184$	0.253178	+	0.438517i	0.253178	+	0.438517i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	−4.24504			−4.24504
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$	0.994008	+	1.72167i	0.994008	+	1.72167i
$195$		0			0
$196$	−0.277479	+	0.480608i	−0.277479	+	0.480608i
$197$	−0.636973			−0.636973			−0.318487	−	0.947927i	$-0.603175\pi$
$197$	−0.636973			−0.636973			−0.318487	+	0.947927i	$0.603175\pi$
$198$	0.746391	+	2.22152i	0.746391	+	2.22152i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	0.578570	−	1.00211i	0.578570	−	1.00211i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.67841	+	2.90709i	−1.67841	+	2.90709i
$206$		0			0
$207$	−0.290594	−	0.864909i	−0.290594	−	0.864909i
$208$		0			0
$209$	−1.82127	+	3.15453i	−1.82127	+	3.15453i
$210$		0			0
$211$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$211$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$212$	−0.387492	−	0.671156i	−0.387492	−	0.671156i
$213$		0			0
$214$		0			0
$215$		0			0
$216$	−0.530303	−	0.163577i	−0.530303	−	0.163577i
$217$		0			0
$218$	−0.455573	+	0.789075i	−0.455573	+	0.789075i
$219$		0			0
$220$	0.915968	+	1.58650i	0.915968	+	1.58650i
$221$		0			0
$222$		0			0
$223$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$223$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$224$		0			0
$225$	−1.37968	+	1.56336i	−1.37968	+	1.56336i
$226$		0			0
$227$	0.124344	−	0.215370i	0.124344	−	0.215370i	−0.797133	−	0.603804i	$-0.793651\pi$
$227$	0.124344	−	0.215370i	0.124344	−	0.215370i	0.921476	+	0.388435i	$0.126984\pi$
$228$	0.442378	+	0.980411i	0.442378	+	0.980411i
$229$	0.411287	+	0.712370i	0.411287	+	0.712370i	0.995031	−	0.0995678i	$-0.0317460\pi$
$229$	0.411287	+	0.712370i	0.411287	+	0.712370i	−0.583744	+	0.811938i	$0.698413\pi$
$230$	−0.999215	−	1.73069i	−0.999215	−	1.73069i
$231$		0			0
$232$	0.511381	−	0.885737i	0.511381	−	0.885737i
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$	−0.543955	+	0.942157i	−0.543955	+	0.942157i
$237$		0			0
$238$		0			0
$239$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$239$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$240$	−2.17936	−	0.218078i	−2.17936	−	0.218078i
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	3.15746			3.15746
$243$	0.878222	+	0.478254i	0.878222	+	0.478254i
$244$	0.0829441			0.0829441
$245$	−0.878222	+	1.52112i	−0.878222	+	1.52112i
$246$	2.37132	+	0.237286i	2.37132	+	0.237286i
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−1.18831	+	2.05822i	−1.18831	+	2.05822i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$	−1.71479			−1.71479
$254$		0			0
$255$		0			0
$256$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−1.21946	+	1.38181i	−1.21946	+	1.38181i
$262$		0			0
$263$	0.318487	−	0.551635i	0.318487	−	0.551635i	−0.661686	−	0.749781i	$-0.730159\pi$
$263$	0.318487	−	0.551635i	0.318487	−	0.551635i	0.980172	+	0.198146i	$0.0634921\pi$
$264$	−0.608833	+	0.846835i	−0.608833	+	0.846835i
$265$	−1.22641	−	2.12421i	−1.22641	−	2.12421i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$	2.09294	+	0.645587i	2.09294	+	0.645587i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.95935	+	3.39369i	1.95935	+	3.39369i
$276$	−0.295582	+	0.411129i	−0.295582	+	0.411129i
$277$	0.583744	−	1.01107i	0.583744	−	1.01107i	−0.411287	−	0.911506i	$-0.634921\pi$
$277$	0.583744	−	1.01107i	0.583744	−	1.01107i	0.995031	−	0.0995678i	$-0.0317460\pi$
$278$	−0.310108			−0.310108
$279$		0			0
$280$		0			0
$281$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$281$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$282$		0			0
$283$	−0.980172	−	1.69771i	−0.980172	−	1.69771i	−0.661686	−	0.749781i	$-0.730159\pi$
$283$	−0.980172	−	1.69771i	−0.980172	−	1.69771i	−0.318487	−	0.947927i	$-0.603175\pi$
$284$		0			0
$285$	1.40013	+	3.10300i	1.40013	+	3.10300i
$286$		0			0
$287$		0			0
$288$	0.318487	+	0.947927i	0.318487	+	0.947927i
$289$	1.00000			1.00000
$290$	−2.01826	+	3.49573i	−2.01826	+	3.49573i
$291$	0.930642	−	1.29444i	0.930642	−	1.29444i
$292$		0			0
$293$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$293$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$294$	1.24078	+	0.124159i	1.24078	+	0.124159i
$295$	−1.72162	+	2.98193i	−1.72162	+	2.98193i
$296$		0			0
$297$	1.37769	−	1.27831i	1.37769	−	1.27831i
$298$	−1.45583			−1.45583
$299$		0			0
$300$	1.15139	+	0.115214i	1.15139	+	0.115214i
$301$		0			0
$302$	−1.09512	−	1.89681i	−1.09512	−	1.89681i
$303$		0			0
$304$	−1.20842	+	2.09304i	−1.20842	+	2.09304i
$305$	0.262518			0.262518
$306$		0			0
$307$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$307$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$	−2.34356			−2.34356
$315$		0			0
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$	−1.01652	+	1.41389i	−1.01652	+	1.41389i
$319$	1.73181	+	2.99958i	1.73181	+	2.99958i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.0690056	−	0.550651i	−0.0690056	−	0.550651i
$325$		0			0
$326$	−0.568885	+	0.985338i	−0.568885	+	0.985338i
$327$	0.727051	+	0.0727524i	0.727051	+	0.0727524i
$328$	0.530303	+	0.918512i	0.530303	+	0.918512i
$329$		0			0
$330$	2.40288	−	3.34220i	2.40288	−	3.34220i
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	−0.900969	−	0.433884i	$-0.857143\pi$
$337$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	0.0747301	−	0.997204i	$-0.476190\pi$
$338$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	1.59919	−	1.81210i	1.59919	−	1.81210i
$343$		0			0
$344$		0			0
$345$	−0.935517	+	1.30122i	−0.935517	+	1.30122i
$346$	1.24543	+	2.15715i	1.24543	+	2.15715i
$347$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	0.826239	−	0.563320i	$-0.190476\pi$
$347$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	−0.900969	+	0.433884i	$0.857143\pi$
$348$	1.01768	+	0.101834i	1.01768	+	0.101834i
$349$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$		0			0
$352$	1.87939			1.87939
$353$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$353$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$354$	2.43236	+	0.243395i	2.43236	+	0.243395i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−1.03030	+	1.78454i	−1.03030	+	1.78454i
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$	0.310446	+	0.923994i	0.310446	+	0.923994i
$361$	2.75644			2.75644
$362$		0			0
$363$	−1.04142	−	2.30801i	−1.04142	−	2.30801i
$364$		0			0
$365$		0			0
$366$	−0.0766531	−	0.169881i	−0.0766531	−	0.169881i
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$	−1.13777			−1.13777
$369$	−0.608674	−	1.81163i	−0.608674	−	1.81163i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$	1.89643	+	0.189767i	1.89643	+	0.189767i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.730682			0.730682			0.365341	−	0.930874i	$-0.380952\pi$
$379$	0.730682			0.730682			0.365341	+	0.930874i	$0.380952\pi$
$380$	0.944610	−	1.63611i	0.944610	−	1.63611i
$381$		0			0
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$	−0.583744	+	0.811938i	−0.583744	+	0.811938i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.884750			−0.884750
$389$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$389$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$390$		0			0
$391$		0			0
$392$	0.277479	+	0.480608i	0.277479	+	0.480608i
$393$		0			0
$394$	0.397146	−	0.687878i	0.397146	−	0.687878i
$395$		0			0
$396$	−1.02230	−	0.206662i	−1.02230	−	0.206662i
$397$	−0.636973			−0.636973			−0.318487	−	0.947927i	$-0.603175\pi$
$397$	−0.636973			−0.636973			−0.318487	+	0.947927i	$0.603175\pi$
$398$		0			0
$399$		0			0
$400$	1.30003	+	2.25172i	1.30003	+	2.25172i
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.218403	−	1.74281i	−0.218403	−	1.74281i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$409$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$410$	−2.09294	−	3.62508i	−2.09294	−	3.62508i
$411$		0			0
$412$		0			0
$413$		0			0
$414$	1.11521	+	0.225445i	1.11521	+	0.225445i
$415$		0			0
$416$		0			0
$417$	0.102282	+	0.226680i	0.102282	+	0.226680i
$418$	−2.27109	−	3.93364i	−2.27109	−	3.93364i
$419$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$419$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$420$		0			0
$421$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$421$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$	−0.774984			−0.774984
$425$		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
$432$	0.914101	−	0.848162i	0.914101	−	0.848162i
$433$	−0.0498614			−0.0498614			−0.0249307	−	0.999689i	$-0.507937\pi$
$433$	−0.0498614			−0.0498614			−0.0249307	+	0.999689i	$0.507937\pi$
$434$		0			0
$435$	3.22096	+	0.322305i	3.22096	+	0.322305i
$436$	−0.202749	−	0.351172i	−0.202749	−	0.351172i
$437$	0.884207	+	1.53149i	0.884207	+	1.53149i
$438$		0			0
$439$	0.853291	−	1.47794i	0.853291	−	1.47794i	−0.0249307	−	0.999689i	$-0.507937\pi$
$439$	0.853291	−	1.47794i	0.853291	−	1.47794i	0.878222	−	0.478254i	$-0.158730\pi$
$440$	1.83194			1.83194
$441$	−0.318487	−	0.947927i	−0.318487	−	0.947927i
$442$		0			0
$443$	−0.270840	+	0.469109i	−0.270840	+	0.469109i	−0.969077	−	0.246757i	$-0.920635\pi$
$443$	−0.270840	+	0.469109i	−0.270840	+	0.469109i	0.698237	+	0.715867i	$0.253968\pi$
$444$		0			0
$445$		0			0
$446$	1.17178	+	2.02958i	1.17178	+	2.02958i
$447$	0.480172	+	1.06417i	0.480172	+	1.06417i
$448$		0			0
$449$	−1.93815			−1.93815			−0.969077	−	0.246757i	$-0.920635\pi$
$449$	−1.93815			−1.93815			−0.969077	+	0.246757i	$0.920635\pi$
$450$	−0.828087	−	2.46468i	−0.828087	−	2.46468i
$451$	−3.59178			−3.59178
$452$		0			0
$453$	−1.02531	+	1.42612i	−1.02531	+	1.42612i
$454$	0.155054	+	0.268562i	0.155054	+	0.268562i
$455$		0			0
$456$	1.07025	+	0.107095i	1.07025	+	0.107095i
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$	−1.02573			−1.02573
$459$		0			0
$460$	0.889385			0.889385
$461$	−0.542546	+	0.939718i	−0.542546	+	0.939718i	0.456211	+	0.889872i	$0.349206\pi$
$461$	−0.542546	+	0.939718i	−0.542546	+	0.939718i	−0.998757	+	0.0498459i	$0.984127\pi$
$462$		0			0
$463$	0.797133	+	1.38067i	0.797133	+	1.38067i	0.921476	+	0.388435i	$0.126984\pi$
$463$	0.797133	+	1.38067i	0.797133	+	1.38067i	−0.124344	+	0.992239i	$0.539683\pi$
$464$	1.14906	+	1.99023i	1.14906	+	1.99023i
$465$		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
$471$	0.772967	+	1.71307i	0.772967	+	1.71307i
$472$	0.543955	+	0.942157i	0.543955	+	0.942157i
$473$		0			0
$474$		0			0
$475$	2.02062	−	3.49981i	2.02062	−	3.49981i
$476$		0			0
$477$	1.36879	+	0.276706i	1.36879	+	0.276706i
$478$		0			0
$479$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	−0.900969	−	0.433884i	$-0.857143\pi$
$479$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	0.826239	+	0.563320i	$0.190476\pi$
$480$	1.02531	−	1.42612i	1.02531	−	1.42612i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.702602	+	1.21694i	−0.702602	+	1.21694i
$485$	−2.80024			−2.80024
$486$	−1.06404	+	0.650219i	−1.06404	+	0.650219i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	0.0414721	−	0.0718317i	0.0414721	−	0.0718317i
$489$	0.907887	+	0.0908478i	0.907887	+	0.0908478i
$490$	−1.09512	−	1.89681i	−1.09512	−	1.89681i
$491$	−0.365341	−	0.632789i	−0.365341	−	0.632789i	0.623490	−	0.781831i	$-0.285714\pi$
$491$	−0.365341	−	0.632789i	−0.365341	−	0.632789i	−0.988831	+	0.149042i	$0.952381\pi$
$492$	−0.619122	+	0.861146i	−0.619122	+	0.861146i
$493$		0			0
$494$		0			0
$495$	−3.23558	−	0.654087i	−3.23558	−	0.654087i
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$	−0.528848	−	0.915992i	−0.528848	−	0.915992i
$501$		0			0
$502$		0			0
$503$	−1.32337			−1.32337			−0.661686	−	0.749781i	$-0.730159\pi$
$503$	−1.32337			−1.32337			−0.661686	+	0.749781i	$0.730159\pi$
$504$		0			0
$505$		0			0
$506$	1.06915	−	1.85183i	1.06915	−	1.85183i
$507$	−0.583744	+	0.811938i	−0.583744	+	0.811938i
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$	0.554958			0.554958
$513$	−1.85205	−	0.571281i	−1.85205	−	0.571281i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	1.16604	−	1.62186i	1.16604	−	1.62186i
$520$		0			0
$521$	0.730682			0.730682			0.365341	−	0.930874i	$-0.380952\pi$
$521$	0.730682			0.730682			0.365341	+	0.930874i	$0.380952\pi$
$522$	−0.731922	−	2.17845i	−0.731922	−	2.17845i
$523$	1.91115			1.91115			0.955573	−	0.294755i	$-0.0952381\pi$
$523$	1.91115			1.91115			0.955573	+	0.294755i	$0.0952381\pi$
$524$		0			0
$525$		0			0
$526$	0.397146	+	0.687878i	0.397146	+	0.687878i
$527$		0			0
$528$	−0.963874	−	2.13616i	−0.963874	−	2.13616i
$529$	0.0837437	−	0.145048i	0.0837437	−	0.145048i
$530$	3.05862			3.05862
$531$	−0.624344	−	1.85826i	−0.624344	−	1.85826i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	1.64427	+	0.164534i	1.64427	+	0.164534i
$538$		0			0
$539$	−1.87939			−1.87939
$540$	−0.714544	+	0.663000i	−0.714544	+	0.663000i
$541$	0.541681			0.541681			0.270840	−	0.962624i	$-0.412698\pi$
$541$	0.541681			0.541681			0.270840	+	0.962624i	$0.412698\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−0.641701	−	1.11146i	−0.641701	−	1.11146i
$546$		0			0
$547$	0.853291	−	1.47794i	0.853291	−	1.47794i	−0.0249307	−	0.999689i	$-0.507937\pi$
$547$	0.853291	−	1.47794i	0.853291	−	1.47794i	0.878222	−	0.478254i	$-0.158730\pi$
$548$		0			0
$549$	−0.0988957	+	0.112062i	−0.0988957	+	0.112062i
$550$	−4.88653			−4.88653
$551$	1.78596	−	3.09338i	1.78596	−	3.09338i
$552$	0.208258	+	0.461546i	0.208258	+	0.461546i
$553$		0			0
$554$	0.727916	+	1.26079i	0.727916	+	1.26079i
$555$		0			0
$556$	0.0690056	−	0.119521i	0.0690056	−	0.119521i
$557$		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
$563$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.623490	+	0.781831i	$0.285714\pi$
$563$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.365341	+	0.930874i	$0.380952\pi$
$564$		0			0
$565$		0			0
$566$	2.44451			2.44451
$567$		0			0
$568$		0			0
$569$	−0.270840	+	0.469109i	−0.270840	+	0.469109i	−0.969077	−	0.246757i	$-0.920635\pi$
$569$	−0.270840	+	0.469109i	−0.270840	+	0.469109i	0.698237	+	0.715867i	$0.253968\pi$
$570$	−4.22395	−	0.422670i	−4.22395	−	0.422670i
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.90248			1.90248
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.623490	+	1.07992i	−0.623490	+	1.07992i
$579$		0			0
$580$	−0.898211	−	1.55575i	−0.898211	−	1.55575i
$581$		0			0
$582$	0.817645	+	1.81209i	0.817645	+	1.81209i
$583$	1.31226	−	2.27289i	1.31226	−	2.27289i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$	−0.323953	+	0.450592i	−0.323953	+	0.450592i
$589$		0			0
$590$	−2.14682	−	3.71840i	−2.14682	−	3.71840i
$591$	−0.633808	−	0.0634221i	−0.633808	−	0.0634221i
$592$		0			0
$593$	−1.59427			−1.59427			−0.797133	−	0.603804i	$-0.793651\pi$
$593$	−1.59427			−1.59427			−0.797133	+	0.603804i	$0.793651\pi$
$594$	0.521490	+	2.28480i	0.521490	+	2.28480i
$595$		0			0
$596$	0.323953	−	0.561104i	0.323953	−	0.561104i
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$	0.675473	−	0.939525i	0.675473	−	0.939525i
$601$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$601$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$		0			0
$604$	0.974752			0.974752
$605$	−2.22374	+	3.85162i	−2.22374	+	3.85162i
$606$		0			0
$607$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	0.826239	−	0.563320i	$-0.190476\pi$
$607$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	−0.900969	+	0.433884i	$0.857143\pi$
$608$	−0.969077	−	1.67849i	−0.969077	−	1.67849i
$609$		0			0
$610$	−0.163677	+	0.283498i	−0.163677	+	0.283498i
$611$		0			0
$612$		0			0
$613$	1.91115			1.91115			0.955573	−	0.294755i	$-0.0952381\pi$
$613$	1.91115			1.91115			0.955573	+	0.294755i	$0.0952381\pi$
$614$	−0.777479	+	1.34663i	−0.777479	+	1.34663i
$615$	−1.95952	+	2.72553i	−1.95952	+	2.72553i
$616$		0			0
$617$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$617$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$	−0.203033	−	0.889545i	−0.203033	−	0.889545i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.631259	−	1.09337i	−0.631259	−	1.09337i
$626$		0			0
$627$	−2.12631	+	2.95752i	−2.12631	+	2.95752i
$628$	0.521490	−	0.903247i	0.521490	−	0.903247i
$629$		0			0
$630$		0			0
$631$	−1.16749			−1.16749			−0.583744	−	0.811938i	$-0.698413\pi$
$631$	−1.16749			−1.16749			−0.583744	+	0.811938i	$0.698413\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−0.318741	−	0.706403i	−0.318741	−	0.706403i
$637$		0			0
$638$	−4.31906			−4.31906
$639$		0			0
$640$	1.75644			1.75644
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$643$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.248687			−0.248687			−0.124344	−	0.992239i	$-0.539683\pi$
$647$	−0.248687			−0.248687			−0.124344	+	0.992239i	$0.539683\pi$
$648$	−0.511381	−	0.215565i	−0.511381	−	0.215565i
$649$	−3.68424			−3.68424
$650$		0			0
$651$		0			0
$652$	−0.253178	−	0.438517i	−0.253178	−	0.438517i
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$	−0.531875	+	0.739794i	−0.531875	+	0.739794i
$655$		0			0
$656$	−2.38316			−2.38316
$657$		0			0
$658$		0			0
$659$	−0.698237	+	1.20938i	−0.698237	+	1.20938i	0.270840	+	0.962624i	$0.412698\pi$
$659$	−0.698237	+	1.20938i	−0.698237	+	1.20938i	−0.969077	+	0.246757i	$0.920635\pi$
$660$	0.753451	+	1.66982i	0.753451	+	1.66982i
$661$	0.661686	+	1.14607i	0.661686	+	1.14607i	0.980172	+	0.198146i	$0.0634921\pi$
$661$	0.661686	+	1.14607i	0.661686	+	1.14607i	−0.318487	+	0.947927i	$0.603175\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1.68155			1.68155
$668$		0			0
$669$	1.09708	−	1.52594i	1.09708	−	1.52594i
$670$		0			0
$671$	0.140447	+	0.243261i	0.140447	+	0.243261i
$672$		0			0
$673$	0.733052	−	1.26968i	0.733052	−	1.26968i	−0.222521	−	0.974928i	$-0.571429\pi$
$673$	0.733052	−	1.26968i	0.733052	−	1.26968i	0.955573	−	0.294755i	$-0.0952381\pi$
$674$	2.06061			2.06061
$675$	−1.52848	+	1.41822i	−1.52848	+	1.41822i
$676$	0.554958			0.554958
$677$	−0.995031	+	1.72344i	−0.995031	+	1.72344i	−0.411287	+	0.911506i	$0.634921\pi$
$677$	−0.995031	+	1.72344i	−0.995031	+	1.72344i	−0.583744	+	0.811938i	$0.698413\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	0.145170	−	0.201919i	0.145170	−	0.201919i
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	0.342563	+	1.01959i	0.342563	+	1.01959i
$685$		0			0
$686$		0			0
$687$	0.338314	+	0.749781i	0.338314	+	0.749781i
$688$		0			0
$689$		0			0
$690$	−0.821928	−	1.82158i	−0.821928	−	1.82158i
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$	−1.10854			−1.10854
$693$		0			0
$694$	0.186374			0.186374
$695$	0.218403	−	0.378284i	0.218403	−	0.378284i
$696$	0.597031	−	0.830419i	0.597031	−	0.830419i
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.46610			−1.46610			−0.733052	−	0.680173i	$-0.761905\pi$
$701$	−1.46610			−1.46610			−0.733052	+	0.680173i	$0.761905\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.216536	−	0.375051i	−0.216536	−	0.375051i
$707$		0			0
$708$	−0.635060	+	0.883315i	−0.635060	+	0.883315i
$709$	−0.456211	+	0.790180i	−0.456211	+	0.790180i	−0.998757	−	0.0498459i	$-0.984127\pi$
$709$	−0.456211	+	0.790180i	−0.456211	+	0.790180i	0.542546	+	0.840026i	$0.317460\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.458528	−	0.794194i	−0.458528	−	0.794194i
$717$		0			0
$718$		0			0
$719$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$719$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$720$	−2.14682	−	0.433989i	−2.14682	−	0.433989i
$721$		0			0
$722$	−1.71861	+	2.97673i	−1.71861	+	2.97673i
$723$		0			0
$724$		0			0
$725$	−1.92136	−	3.32790i	−1.92136	−	3.32790i
$726$	3.14177	+	0.314382i	3.14177	+	0.314382i
$727$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$727$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$	0.826239	+	0.563320i	0.826239	+	0.563320i
$730$		0			0
$731$		0			0
$732$	0.0825320	+	0.00825857i	0.0825320	+	0.00825857i
$733$	0.998757	+	1.72990i	0.998757	+	1.72990i	0.542546	+	0.840026i	$0.317460\pi$
$733$	0.998757	+	1.72990i	0.998757	+	1.72990i	0.456211	+	0.889872i	$0.349206\pi$
$734$		0			0
$735$	−1.02531	+	1.42612i	−1.02531	+	1.42612i
$736$	0.456211	−	0.790180i	0.456211	−	0.790180i
$737$		0			0
$738$	2.33591	+	0.472214i	2.33591	+	0.472214i
$739$		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	$-0.666667\pi$
$743$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$744$		0			0
$745$	1.02531	−	1.77589i	1.02531	−	1.77589i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$	−1.38734	+	1.92967i	−1.38734	+	1.92967i
$751$	−0.995031	−	1.72344i	−0.995031	−	1.72344i	−0.583744	−	0.811938i	$-0.698413\pi$
$751$	−0.995031	−	1.72344i	−0.995031	−	1.72344i	−0.411287	−	0.911506i	$-0.634921\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	3.08509			3.08509
$756$		0			0
$757$	−1.97766			−1.97766			−0.988831	−	0.149042i	$-0.952381\pi$
$757$	−1.97766			−1.97766			−0.988831	+	0.149042i	$0.952381\pi$
$758$	−0.455573	+	0.789075i	−0.455573	+	0.789075i
$759$	−1.70627	−	0.170738i	−1.70627	−	0.170738i
$760$	−0.944610	−	1.63611i	−0.944610	−	1.63611i
$761$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.623490	+	0.781831i	$0.285714\pi$
$761$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.365341	+	0.930874i	$0.380952\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.512867	−	1.13663i	−0.512867	−	1.13663i
$769$	0.318487	+	0.551635i	0.318487	+	0.551635i	0.980172	−	0.198146i	$-0.0634921\pi$
$769$	0.318487	+	0.551635i	0.318487	+	0.551635i	−0.661686	+	0.749781i	$0.730159\pi$
$770$		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
$776$	−0.442375	+	0.766216i	−0.442375	+	0.766216i
$777$		0			0
$778$	−0.955242	−	1.65453i	−0.955242	−	1.65453i
$779$	1.85205	+	3.20784i	1.85205	+	3.20784i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−1.35098	+	1.25353i	−1.35098	+	1.25353i
$784$	−1.24698			−1.24698
$785$	1.65052	−	2.85878i	1.65052	−	2.85878i
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	0.176747	+	0.306134i	0.176747	+	0.306134i
$789$	0.371829	−	0.517183i	0.371829	−	0.517183i
$790$		0			0
$791$		0			0
$792$	−0.690125	+	0.782007i	−0.690125	+	0.782007i
$793$		0			0
$794$	0.397146	−	0.687878i	0.397146	−	0.687878i
$795$	−1.00882	−	2.23577i	−1.00882	−	2.23577i
$796$		0			0
$797$	0.733052	+	1.26968i	0.733052	+	1.26968i	0.955573	+	0.294755i	$0.0952381\pi$
$797$	0.733052	+	1.26968i	0.733052	+	1.26968i	−0.222521	+	0.974928i	$0.571429\pi$
$798$		0			0
$799$		0			0
$800$	−2.08509			−2.08509
$801$		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			$\pi$
$810$	2.01826	+	0.850769i	2.01826	+	0.850769i
$811$	−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
$815$	−0.801308	−	1.38791i	−0.801308	−	1.38791i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	1.86289			1.86289
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	−0.270840	−	0.469109i	−0.270840	−	0.469109i	0.698237	−	0.715867i	$-0.253968\pi$
$823$	−0.270840	−	0.469109i	−0.270840	−	0.469109i	−0.969077	+	0.246757i	$0.920635\pi$
$824$		0			0
$825$	1.61171	+	3.57191i	1.61171	+	3.57191i
$826$		0			0
$827$	−0.0498614			−0.0498614			−0.0249307	−	0.999689i	$-0.507937\pi$
$827$	−0.0498614			−0.0498614			−0.0249307	+	0.999689i	$0.507937\pi$
$828$	−0.335048	+	0.379656i	−0.335048	+	0.379656i
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$	0.681513	−	0.947927i	0.681513	−	0.947927i
$832$		0			0
$833$		0			0
$834$	−0.308567	−	0.0308768i	−0.308567	−	0.0308768i
$835$		0			0
$836$	2.02146			2.02146
$837$		0			0
$838$		0			0
$839$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$		0			0
$841$	−1.19824	−	2.07541i	−1.19824	−	2.07541i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.75644			1.75644
$846$		0			0
$847$		0			0
$848$	0.870687	−	1.50807i	0.870687	−	1.50807i
$849$	−0.806265	−	1.78687i	−0.806265	−	1.78687i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$853$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$	1.08421	+	3.22699i	1.08421	+	3.22699i
$856$		0			0
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$859$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$860$		0			0
$861$		0			0
$862$	−0.623490	+	1.07992i	−0.623490	+	1.07992i
$863$	0.730682			0.730682			0.365341	−	0.930874i	$-0.380952\pi$
$863$	0.730682			0.730682			0.365341	+	0.930874i	$0.380952\pi$
$864$	0.222521	+	0.974928i	0.222521	+	0.974928i
$865$	−3.50852			−3.50852
$866$	0.0310881	−	0.0538461i	0.0310881	−	0.0538461i
$867$	0.995031	+	0.0995678i	0.995031	+	0.0995678i
$868$		0			0
$869$		0			0
$870$	−2.35630	+	3.27741i	−2.35630	+	3.27741i
$871$		0			0
$872$	−0.405498			−0.405498
$873$	1.05490	−	1.19535i	1.05490	−	1.19535i
$874$	−2.20518			−2.20518
$875$		0			0
$876$		0			0
$877$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$877$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$878$	1.06404	+	1.84296i	1.06404	+	1.84296i
$879$		0			0
$880$	−2.05816	+	3.56484i	−2.05816	+	3.56484i
$881$	−1.99751			−1.99751			−0.998757	−	0.0498459i	$-0.984127\pi$
$881$	−1.99751			−1.99751			−0.998757	+	0.0498459i	$0.984127\pi$
$882$	1.22226	+	0.247084i	1.22226	+	0.247084i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$	−2.00997	+	2.79569i	−2.00997	+	2.79569i
$886$	−0.337733	−	0.584970i	−0.337733	−	0.584970i
$887$	−0.698237	−	1.20938i	−0.698237	−	1.20938i	−0.969077	−	0.246757i	$-0.920635\pi$
$887$	−0.698237	−	1.20938i	−0.698237	−	1.20938i	0.270840	−	0.962624i	$-0.412698\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1.49812	−	1.13478i	1.49812	−	1.13478i
$892$	−1.04298			−1.04298
$893$		0			0
$894$	−1.44860	−	0.144954i	−1.44860	−	0.144954i
$895$	−1.45124	−	2.51362i	−1.45124	−	2.51362i
$896$		0			0
$897$		0			0
$898$	1.20842	−	2.09304i	1.20842	−	2.09304i
$899$		0			0
$900$	1.13420	+	0.229283i	1.13420	+	0.229283i
$901$		0			0
$902$	2.23944	−	3.87882i	2.23944	−	3.87882i
$903$		0			0
$904$		0			0
$905$		0			0
$906$	−0.900821	−	1.99642i	−0.900821	−	1.99642i
$907$	−0.921476	+	1.59604i	−0.921476	+	1.59604i	−0.124344	+	0.992239i	$0.539683\pi$
$907$	−0.921476	+	1.59604i	−0.921476	+	1.59604i	−0.797133	+	0.603804i	$0.793651\pi$
$908$	−0.138011			−0.138011
$909$		0			0
$910$		0			0
$911$	0.853291	−	1.47794i	0.853291	−	1.47794i	−0.0249307	−	0.999689i	$-0.507937\pi$
$911$	0.853291	−	1.47794i	0.853291	−	1.47794i	0.878222	−	0.478254i	$-0.158730\pi$
$912$	−1.41081	+	1.96232i	−1.41081	+	1.96232i
$913$		0			0
$914$		0			0
$915$	0.261214	+	0.0261384i	0.261214	+	0.0261384i
$916$	0.228247	−	0.395336i	0.228247	−	0.395336i
$917$		0			0
$918$		0			0
$919$	1.84295			1.84295			0.921476	−	0.388435i	$-0.126984\pi$
$919$	1.84295			1.84295			0.921476	+	0.388435i	$0.126984\pi$
$920$	0.444692	−	0.770230i	0.444692	−	0.770230i
$921$	1.24078	+	0.124159i	1.24078	+	0.124159i
$922$	−0.676544	−	1.17181i	−0.676544	−	1.17181i
$923$		0			0
$924$		0			0
$925$		0			0
$926$	−1.98802			−1.98802
$927$		0			0
$928$	−1.84295			−1.84295
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$	0.969077	+	1.67849i	0.969077	+	1.67849i
$932$		0			0
$933$		0			0
$934$	0.623490	−	1.07992i	0.623490	−	1.07992i
$935$		0			0
$936$		0			0
$937$	1.39647			1.39647			0.698237	−	0.715867i	$-0.253968\pi$
$937$	1.39647			1.39647			0.698237	+	0.715867i	$0.253968\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$	−2.33191	−	0.233343i	−2.33191	−	0.233343i
$943$	−0.871885	+	1.51015i	−0.871885	+	1.51015i
$944$	−2.44451			−2.44451
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$		0			0
$950$	2.51967	+	4.36419i	2.51967	+	4.36419i
$951$		0			0
$952$		0			0
$953$	1.96034			1.96034			0.980172	−	0.198146i	$-0.0634921\pi$
$953$	1.96034			1.96034			0.980172	+	0.198146i	$0.0634921\pi$
$954$	−1.15224	+	1.30565i	−1.15224	+	1.30565i
$955$		0			0
$956$		0			0
$957$	1.42454	+	3.15711i	1.42454	+	3.15711i
$958$	−0.0931869	−	0.161404i	−0.0931869	−	0.161404i
$959$		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	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$	0.702602	+	1.21694i	0.702602	+	1.21694i
$969$		0			0
$970$	1.74592	−	3.02402i	1.74592	−	3.02402i
$971$	1.91115			1.91115			0.955573	−	0.294755i	$-0.0952381\pi$
$971$	1.91115			1.91115			0.955573	+	0.294755i	$0.0952381\pi$
$972$	−0.0138355	−	0.554786i	−0.0138355	−	0.554786i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.0931869	+	0.161404i	0.0931869	+	0.161404i
$977$	0.583744	+	1.01107i	0.583744	+	1.01107i	0.995031	+	0.0995678i	$0.0317460\pi$
$977$	0.583744	+	1.01107i	0.583744	+	1.01107i	−0.411287	+	0.911506i	$0.634921\pi$
$978$	−0.664166	+	0.923799i	−0.664166	+	0.923799i
$979$		0			0
$980$	0.974752			0.974752
$981$	0.716194	+	0.144782i	0.716194	+	0.144782i
$982$	0.911146			0.911146
$983$	0.583744	−	1.01107i	0.583744	−	1.01107i	−0.411287	−	0.911506i	$-0.634921\pi$
$983$	0.583744	−	1.01107i	0.583744	−	1.01107i	0.995031	−	0.0995678i	$-0.0317460\pi$
$984$	0.436213	+	0.966748i	0.436213	+	0.966748i
$985$	0.559404	+	0.968916i	0.559404	+	0.968916i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	2.72371	−	3.08634i	2.72371	−	3.08634i
$991$		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
$997$	0.998757	−	1.72990i	0.998757	−	1.72990i	0.456211	−	0.889872i	$-0.349206\pi$
$997$	0.998757	−	1.72990i	0.998757	−	1.72990i	0.542546	−	0.840026i	$-0.317460\pi$
$998$		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.430.6	✓	36
9.4	even	3	inner	3879.1.g.c.1723.6	yes	36
431.430	odd	2	CM	3879.1.g.c.430.6	✓	36
3879.1723	odd	6	inner	3879.1.g.c.1723.6	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3879.1.g.c.430.6	✓	36	1.1	even	1	trivial
3879.1.g.c.430.6	✓	36	431.430	odd	2	CM
3879.1.g.c.1723.6	yes	36	9.4	even	3	inner
3879.1.g.c.1723.6	yes	36	3879.1723	odd	6	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:	\( N \)	\(=\)	\( 3879 = 3^{2} \cdot 431 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3879.g (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.623490 + 1.07992i) q^{2} +(0.995031 + 0.0995678i) q^{3} +(-0.277479 - 0.480608i) q^{4} +(-0.878222 - 1.52112i) q^{5} +(-0.727916 + 1.01247i) q^{6} -0.554958 q^{8} +(0.980172 + 0.198146i) q^{9} +O(q^{10})\)	\(q+(-0.623490 + 1.07992i) q^{2} +(0.995031 + 0.0995678i) q^{3} +(-0.277479 - 0.480608i) q^{4} +(-0.878222 - 1.52112i) q^{5} +(-0.727916 + 1.01247i) q^{6} -0.554958 q^{8} +(0.980172 + 0.198146i) q^{9} +2.19025 q^{10} +(0.939693 - 1.62760i) q^{11} +(-0.228247 - 0.505848i) q^{12} +(-0.722402 - 1.60101i) q^{15} +(0.623490 - 1.07992i) q^{16} +(-0.825109 + 0.934962i) q^{18} -1.93815 q^{19} +(-0.487376 + 0.844160i) q^{20} +(1.17178 + 2.02958i) q^{22} +(-0.456211 - 0.790180i) q^{23} +(-0.552200 - 0.0552560i) q^{24} +(-1.04255 + 1.80574i) q^{25} +(0.955573 + 0.294755i) q^{27} +(-0.921476 + 1.59604i) q^{29} +(2.17936 + 0.218078i) q^{30} +(0.500000 + 0.866025i) q^{32} +(1.09708 - 1.52594i) q^{33} +(-0.176747 - 0.526060i) q^{36} +(1.20842 - 2.09304i) q^{38} +(0.487376 + 0.844160i) q^{40} +(-0.955573 - 1.65510i) q^{41} -1.04298 q^{44} +(-0.559404 - 1.66498i) q^{45} +1.13777 q^{46} +(0.727916 - 1.01247i) q^{48} +(-0.500000 - 0.866025i) q^{49} +(-1.30003 - 2.25172i) q^{50} +1.39647 q^{53} +(-0.914101 + 0.848162i) q^{54} -3.30103 q^{55} +(-1.92852 - 0.192978i) q^{57} +(-1.14906 - 1.99023i) q^{58} +(-0.980172 - 1.69771i) q^{59} +(-0.569006 + 0.791439i) q^{60} +(-0.0747301 + 0.129436i) q^{61} +(0.963874 + 2.13616i) q^{66} +(-0.375267 - 0.831677i) q^{69} +(-0.543955 - 0.109963i) q^{72} +(-1.21716 + 1.69297i) q^{75} +(0.537797 + 0.931492i) q^{76} -2.19025 q^{80} +(0.921476 + 0.388435i) q^{81} +2.38316 q^{82} +(-1.07581 + 1.49636i) q^{87} +(-0.521490 + 0.903247i) q^{88} +(2.14682 + 0.433989i) q^{90} +(-0.253178 + 0.438517i) q^{92} +(1.70213 + 2.94817i) q^{95} +(0.411287 + 0.911506i) q^{96} +(0.797133 - 1.38067i) q^{97} +1.24698 q^{98} +(1.24356 - 1.40913i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 6 q^{2} - 12 q^{4} - 24 q^{8}+O(q^{10}) \) 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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