LMFDB - Embedded newform 2009.1.i.b.901.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2009,1,Mod(901,2009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2009, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5, 3]))

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

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

chi := DirichletCharacter("2009.901");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2009 = 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2009.i (of order $6$, degree $2$, not 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:	$1.00262161038$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	6.0.64827.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 $ x^6 - x^5 + 3x^4 + 5x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 287)
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.23639903.1
Artin image:	$C_3\times D_7$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{21} - \cdots)$

Embedding invariants

Embedding label			901.2
Root			$0.222521 - 0.385418i$ of defining polynomial
Character	$\chi$	$=$	2009.901
Dual form			2009.1.i.b.1844.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.222521 - 0.385418i) q^{2} +(0.900969 + 1.56052i) q^{3} +(0.400969 + 0.694498i) q^{4} +0.801938 q^{6} +0.801938 q^{8} +(-1.12349 + 1.94594i) q^{9} +O(q^{10})$	\(q+(0.222521 - 0.385418i) q^{2} +(0.900969 + 1.56052i) q^{3} +(0.400969 + 0.694498i) q^{4} +0.801938 q^{6} +0.801938 q^{8} +(-1.12349 + 1.94594i) q^{9} +(-0.722521 + 1.25144i) q^{12} -0.445042 q^{13} +(-0.222521 + 0.385418i) q^{16} +(-0.623490 - 1.07992i) q^{17} +(0.500000 + 0.866025i) q^{18} +(-0.623490 + 1.07992i) q^{19} +(0.900969 - 1.56052i) q^{23} +(0.722521 + 1.25144i) q^{24} +(-0.500000 - 0.866025i) q^{25} +(-0.0990311 + 0.171527i) q^{26} -2.24698 q^{27} +(0.500000 + 0.866025i) q^{32} -0.554958 q^{34} -1.80194 q^{36} +(0.222521 - 0.385418i) q^{37} +(0.277479 + 0.480608i) q^{38} +(-0.400969 - 0.694498i) q^{39} +1.00000 q^{41} +1.24698 q^{43} +(-0.400969 - 0.694498i) q^{46} +(0.222521 - 0.385418i) q^{47} -0.801938 q^{48} -0.445042 q^{50} +(1.12349 - 1.94594i) q^{51} +(-0.178448 - 0.309081i) q^{52} +(-0.500000 + 0.866025i) q^{54} -2.24698 q^{57} +(0.500000 - 0.866025i) q^{68} +3.24698 q^{69} +(-0.900969 + 1.56052i) q^{72} +(-0.0990311 - 0.171527i) q^{74} +(0.900969 - 1.56052i) q^{75} -1.00000 q^{76} -0.356896 q^{78} +(-0.900969 - 1.56052i) q^{81} +(0.222521 - 0.385418i) q^{82} +(0.277479 - 0.480608i) q^{86} +(0.222521 - 0.385418i) q^{89} +1.44504 q^{92} +(-0.0990311 - 0.171527i) q^{94} +(-0.900969 + 1.56052i) q^{96} -1.80194 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} + q^{3} - 2 q^{4} - 4 q^{6} - 4 q^{8} - 2 q^{9}+O(q^{10}) $ 6 * q + q^2 + q^3 - 2 * q^4 - 4 * q^6 - 4 * q^8 - 2 * q^9	$ 6 q + q^{2} + q^{3} - 2 q^{4} - 4 q^{6} - 4 q^{8} - 2 q^{9} - 4 q^{12} - 2 q^{13} - q^{16} + q^{17} + 3 q^{18} + q^{19} + q^{23} + 4 q^{24} - 3 q^{25} - 5 q^{26} - 4 q^{27} + 3 q^{32} - 4 q^{34} - 2 q^{36} + q^{37} + 2 q^{38} + 2 q^{39} + 6 q^{41} - 2 q^{43} + 2 q^{46} + q^{47} + 4 q^{48} - 2 q^{50} + 2 q^{51} + 3 q^{52} - 3 q^{54} - 4 q^{57} + 3 q^{68} + 10 q^{69} - q^{72} - 5 q^{74} + q^{75} - 6 q^{76} + 6 q^{78} - q^{81} + q^{82} + 2 q^{86} + q^{89} + 8 q^{92} - 5 q^{94} - q^{96} - 2 q^{97}+O(q^{100}) $ 6 * q + q^2 + q^3 - 2 * q^4 - 4 * q^6 - 4 * q^8 - 2 * q^9 - 4 * q^12 - 2 * q^13 - q^16 + q^17 + 3 * q^18 + q^19 + q^23 + 4 * q^24 - 3 * q^25 - 5 * q^26 - 4 * q^27 + 3 * q^32 - 4 * q^34 - 2 * q^36 + q^37 + 2 * q^38 + 2 * q^39 + 6 * q^41 - 2 * q^43 + 2 * q^46 + q^47 + 4 * q^48 - 2 * q^50 + 2 * q^51 + 3 * q^52 - 3 * q^54 - 4 * q^57 + 3 * q^68 + 10 * q^69 - q^72 - 5 * q^74 + q^75 - 6 * q^76 + 6 * q^78 - q^81 + q^82 + 2 * q^86 + q^89 + 8 * q^92 - 5 * q^94 - q^96 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2009\mathbb{Z}\right)^\times$.

$n$	$493$	$785$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$2$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$3$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$3$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$4$	0.400969	+	0.694498i	0.400969	+	0.694498i
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$	0.801938			0.801938
$7$		0			0
$7$		0			0
$8$	0.801938			0.801938
$9$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	−0.722521	+	1.25144i	−0.722521	+	1.25144i
$13$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$13$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$14$		0			0
$15$		0			0
$16$	−0.222521	+	0.385418i	−0.222521	+	0.385418i
$17$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$17$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$18$	0.500000	+	0.866025i	0.500000	+	0.866025i
$19$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$19$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$23$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$24$	0.722521	+	1.25144i	0.722521	+	1.25144i
$25$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$26$	−0.0990311	+	0.171527i	−0.0990311	+	0.171527i
$27$	−2.24698			−2.24698
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		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
$33$		0			0
$34$	−0.554958			−0.554958
$35$		0			0
$36$	−1.80194			−1.80194
$37$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$37$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$38$	0.277479	+	0.480608i	0.277479	+	0.480608i
$39$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$40$		0			0
$41$	1.00000			1.00000
$41$	1.00000			1.00000
$42$		0			0
$43$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$43$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$44$		0			0
$45$		0			0
$46$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$47$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$47$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$48$	−0.801938			−0.801938
$49$		0			0
$50$	−0.445042			−0.445042
$51$	1.12349	−	1.94594i	1.12349	−	1.94594i
$52$	−0.178448	−	0.309081i	−0.178448	−	0.309081i
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$55$		0			0
$56$		0			0
$57$	−2.24698			−2.24698
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$	0.500000	−	0.866025i	0.500000	−	0.866025i
$69$	3.24698			3.24698
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−0.900969	+	1.56052i	−0.900969	+	1.56052i
$73$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$73$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$74$	−0.0990311	−	0.171527i	−0.0990311	−	0.171527i
$75$	0.900969	−	1.56052i	0.900969	−	1.56052i
$76$	−1.00000			−1.00000
$77$		0			0
$78$	−0.356896			−0.356896
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$		0			0
$81$	−0.900969	−	1.56052i	−0.900969	−	1.56052i
$82$	0.222521	−	0.385418i	0.222521	−	0.385418i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$	0.277479	−	0.480608i	0.277479	−	0.480608i
$87$		0			0
$88$		0			0
$89$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$89$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$90$		0			0
$91$		0			0
$92$	1.44504			1.44504
$93$		0			0
$94$	−0.0990311	−	0.171527i	−0.0990311	−	0.171527i
$95$		0			0
$96$	−0.900969	+	1.56052i	−0.900969	+	1.56052i
$97$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$97$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$98$		0			0
$99$		0			0
$100$	0.400969	−	0.694498i	0.400969	−	0.694498i
$101$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$101$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$102$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$	−0.356896			−0.356896
$105$		0			0
$106$		0			0
$107$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$107$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$108$	−0.900969	−	1.56052i	−0.900969	−	1.56052i
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$		0			0
$111$	0.801938			0.801938
$112$		0			0
$113$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$113$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$114$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$115$		0			0
$116$		0			0
$117$	0.500000	−	0.866025i	0.500000	−	0.866025i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$122$		0			0
$123$	0.900969	+	1.56052i	0.900969	+	1.56052i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$127$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$128$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$129$	1.12349	+	1.94594i	1.12349	+	1.94594i
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$	0.722521	−	1.25144i	0.722521	−	1.25144i
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	0.801938			0.801938
$142$		0			0
$143$		0			0
$144$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	0.356896			0.356896
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$153$	2.80194			2.80194
$154$		0			0
$155$		0			0
$156$	0.321552	−	0.556945i	0.321552	−	0.556945i
$157$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$157$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−0.801938			−0.801938
$163$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$163$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$164$	0.400969	+	0.694498i	0.400969	+	0.694498i
$165$		0			0
$166$		0			0
$167$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$167$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$168$		0			0
$169$	−0.801938			−0.801938
$170$		0			0
$171$	−1.40097	−	2.42655i	−1.40097	−	2.42655i
$172$	0.500000	+	0.866025i	0.500000	+	0.866025i
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−0.0990311	−	0.171527i	−0.0990311	−	0.171527i
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$181$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$182$		0			0
$183$		0			0
$184$	0.722521	−	1.25144i	0.722521	−	1.25144i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	0.356896			0.356896
$189$		0			0
$190$		0			0
$191$		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	$0.333333\pi$
$194$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$195$		0			0
$196$		0			0
$197$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$197$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$198$		0			0
$199$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$199$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$200$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$201$		0			0
$202$	0.801938			0.801938
$203$		0			0
$204$	1.80194			1.80194
$205$		0			0
$206$		0			0
$207$	2.02446	+	3.50647i	2.02446	+	3.50647i
$208$	0.0990311	−	0.171527i	0.0990311	−	0.171527i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$	0.277479	+	0.480608i	0.277479	+	0.480608i
$215$		0			0
$216$	−1.80194			−1.80194
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.277479	+	0.480608i	0.277479	+	0.480608i
$222$	0.178448	−	0.309081i	0.178448	−	0.309081i
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	2.24698			2.24698
$226$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$227$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$227$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$228$	−0.900969	−	1.56052i	−0.900969	−	1.56052i
$229$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$229$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$	−0.222521	−	0.385418i	−0.222521	−	0.385418i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	0.222521	+	0.385418i	0.222521	+	0.385418i
$243$	0.500000	−	0.866025i	0.500000	−	0.866025i
$244$		0			0
$245$		0			0
$246$	0.801938			0.801938
$247$	0.277479	−	0.480608i	0.277479	−	0.480608i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$	−0.0990311	+	0.171527i	−0.0990311	+	0.171527i
$255$		0			0
$256$	0.222521	+	0.385418i	0.222521	+	0.385418i
$257$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$257$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$258$	1.00000			1.00000
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	0.801938			0.801938
$268$		0			0
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$	0.554958			0.554958
$273$		0			0
$274$		0			0
$275$		0			0
$276$	1.30194	+	2.25502i	1.30194	+	2.25502i
$277$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$277$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	0.178448	−	0.309081i	0.178448	−	0.309081i
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−2.24698			−2.24698
$289$	−0.277479	+	0.480608i	−0.277479	+	0.480608i
$290$		0			0
$291$	−1.62349	−	2.81197i	−1.62349	−	2.81197i
$292$		0			0
$293$	2.00000			2.00000			1.00000			$0$
$293$	2.00000			2.00000			1.00000			$0$
$294$		0			0
$295$		0			0
$296$	0.178448	−	0.309081i	0.178448	−	0.309081i
$297$		0			0
$298$		0			0
$299$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$300$	1.44504			1.44504
$301$		0			0
$302$		0			0
$303$	−1.62349	+	2.81197i	−1.62349	+	2.81197i
$304$	−0.277479	−	0.480608i	−0.277479	−	0.480608i
$305$		0			0
$306$	0.623490	−	1.07992i	0.623490	−	1.07992i
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$311$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$312$	−0.321552	−	0.556945i	−0.321552	−	0.556945i
$313$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$313$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$314$	−0.554958			−0.554958
$315$		0			0
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−2.24698			−2.24698
$322$		0			0
$323$	1.55496			1.55496
$324$	0.722521	−	1.25144i	0.722521	−	1.25144i
$325$	0.222521	+	0.385418i	0.222521	+	0.385418i
$326$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$327$		0			0
$328$	0.801938			0.801938
$329$		0			0
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$	0.500000	+	0.866025i	0.500000	+	0.866025i
$334$	0.277479	−	0.480608i	0.277479	−	0.480608i
$335$		0			0
$336$		0			0
$337$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$337$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$338$	−0.178448	+	0.309081i	−0.178448	+	0.309081i
$339$	−1.62349	−	2.81197i	−1.62349	−	2.81197i
$340$		0			0
$341$		0			0
$342$	−1.24698			−1.24698
$343$		0			0
$344$	1.00000			1.00000
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$	1.00000			1.00000
$352$		0			0
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$	0.356896			0.356896
$357$		0			0
$358$		0			0
$359$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$359$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$360$		0			0
$361$	−0.277479	−	0.480608i	−0.277479	−	0.480608i
$362$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$363$	−1.80194			−1.80194
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$	0.400969	+	0.694498i	0.400969	+	0.694498i
$369$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$373$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$374$		0			0
$375$		0			0
$376$	0.178448	−	0.309081i	0.178448	−	0.309081i
$377$		0			0
$378$		0			0
$379$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$379$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$380$		0			0
$381$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$382$		0			0
$383$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$383$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$384$	−1.80194			−1.80194
$385$		0			0
$386$		0			0
$387$	−1.40097	+	2.42655i	−1.40097	+	2.42655i
$388$	−0.722521	−	1.25144i	−0.722521	−	1.25144i
$389$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$389$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$390$		0			0
$391$	−2.24698			−2.24698
$392$		0			0
$393$		0			0
$394$	0.277479	−	0.480608i	0.277479	−	0.480608i
$395$		0			0
$396$		0			0
$397$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$397$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$398$	0.198062			0.198062
$399$		0			0
$400$	0.445042			0.445042
$401$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$401$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$402$		0			0
$403$		0			0
$404$	−0.722521	+	1.25144i	−0.722521	+	1.25144i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	0.900969	−	1.56052i	0.900969	−	1.56052i
$409$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$409$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	1.80194			1.80194
$415$		0			0
$416$	−0.222521	−	0.385418i	−0.222521	−	0.385418i
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$	0.500000	+	0.866025i	0.500000	+	0.866025i
$424$		0			0
$425$	−0.623490	+	1.07992i	−0.623490	+	1.07992i
$426$		0			0
$427$		0			0
$428$	−1.00000			−1.00000
$429$		0			0
$430$		0			0
$431$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$431$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$432$	0.500000	−	0.866025i	0.500000	−	0.866025i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.12349	+	1.94594i	1.12349	+	1.94594i
$438$		0			0
$439$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$439$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$440$		0			0
$441$		0			0
$442$	0.246980			0.246980
$443$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$443$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$444$	0.321552	+	0.556945i	0.321552	+	0.556945i
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$449$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$450$	0.500000	−	0.866025i	0.500000	−	0.866025i
$451$		0			0
$452$	−0.722521	−	1.25144i	−0.722521	−	1.25144i
$453$		0			0
$454$	−0.890084			−0.890084
$455$		0			0
$456$	−1.80194			−1.80194
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$459$	1.40097	+	2.42655i	1.40097	+	2.42655i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$	0.801938			0.801938
$469$		0			0
$470$		0			0
$471$	1.12349	−	1.94594i	1.12349	−	1.94594i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	1.24698			1.24698
$476$		0			0
$477$		0			0
$478$		0			0
$479$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$479$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$480$		0			0
$481$	−0.0990311	+	0.171527i	−0.0990311	+	0.171527i
$482$		0			0
$483$		0			0
$484$	−0.801938			−0.801938
$485$		0			0
$486$	−0.222521	−	0.385418i	−0.222521	−	0.385418i
$487$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$487$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$488$		0			0
$489$	3.24698			3.24698
$490$		0			0
$491$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$491$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$492$	−0.722521	+	1.25144i	−0.722521	+	1.25144i
$493$		0			0
$494$	−0.123490	−	0.213891i	−0.123490	−	0.213891i
$495$		0			0
$496$		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			0
$501$	1.12349	+	1.94594i	1.12349	+	1.94594i
$502$		0			0
$503$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$503$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.722521	−	1.25144i	−0.722521	−	1.25144i
$508$	−0.178448	−	0.309081i	−0.178448	−	0.309081i
$509$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$509$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$510$		0			0
$511$		0			0
$512$	−0.801938			−0.801938
$513$	1.40097	−	2.42655i	1.40097	−	2.42655i
$514$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$515$		0			0
$516$	−0.900969	+	1.56052i	−0.900969	+	1.56052i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$521$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$522$		0			0
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.445042			−0.445042
$534$	0.178448	−	0.309081i	0.178448	−	0.309081i
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$541$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$542$		0			0
$543$	−1.62349	−	2.81197i	−1.62349	−	2.81197i
$544$	0.623490	−	1.07992i	0.623490	−	1.07992i
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	2.60388			2.60388
$553$		0			0
$554$	0.801938			0.801938
$555$		0			0
$556$		0			0
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$	−0.554958			−0.554958
$560$		0			0
$561$		0			0
$562$		0			0
$563$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$563$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$564$	0.321552	+	0.556945i	0.321552	+	0.556945i
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$569$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$570$		0			0
$571$		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
$575$	−1.80194			−1.80194
$576$		0			0
$577$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$577$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$578$	0.123490	+	0.213891i	0.123490	+	0.213891i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	−1.44504			−1.44504
$583$		0			0
$584$		0			0
$585$		0			0
$586$	0.445042	−	0.770835i	0.445042	−	0.770835i
$587$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$587$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	1.12349	+	1.94594i	1.12349	+	1.94594i
$592$	0.0990311	+	0.171527i	0.0990311	+	0.171527i
$593$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$593$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$598$	0.178448	+	0.309081i	0.178448	+	0.309081i
$599$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$599$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$600$	0.722521	−	1.25144i	0.722521	−	1.25144i
$601$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$601$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	0.722521	+	1.25144i	0.722521	+	1.25144i
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$	−1.24698			−1.24698
$609$		0			0
$610$		0			0
$611$	−0.0990311	+	0.171527i	−0.0990311	+	0.171527i
$612$	1.12349	+	1.94594i	1.12349	+	1.94594i
$613$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$613$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$617$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$618$		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
$621$	−2.02446	+	3.50647i	−2.02446	+	3.50647i
$622$	−0.554958			−0.554958
$623$		0			0
$624$	0.356896			0.356896
$625$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$626$	0.277479	+	0.480608i	0.277479	+	0.480608i
$627$		0			0
$628$	0.500000	−	0.866025i	0.500000	−	0.866025i
$629$	−0.554958			−0.554958
$630$		0			0
$631$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$631$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$643$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$643$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$644$		0			0
$645$		0			0
$646$	0.346011	−	0.599308i	0.346011	−	0.599308i
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$	−0.722521	−	1.25144i	−0.722521	−	1.25144i
$649$		0			0
$650$	0.198062			0.198062
$651$		0			0
$652$	1.44504			1.44504
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$	−0.222521	+	0.385418i	−0.222521	+	0.385418i
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$664$		0			0
$665$		0			0
$666$	0.445042			0.445042
$667$		0			0
$668$	0.500000	+	0.866025i	0.500000	+	0.866025i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$675$	1.12349	+	1.94594i	1.12349	+	1.94594i
$676$	−0.321552	−	0.556945i	−0.321552	−	0.556945i
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$	−1.44504			−1.44504
$679$		0			0
$680$		0			0
$681$	1.80194	−	3.12105i	1.80194	−	3.12105i
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$	1.12349	−	1.94594i	1.12349	−	1.94594i
$685$		0			0
$686$		0			0
$687$	3.24698			3.24698
$688$	−0.277479	+	0.480608i	−0.277479	+	0.480608i
$689$		0			0
$690$		0			0
$691$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$691$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$701$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$702$	0.222521	−	0.385418i	0.222521	−	0.385418i
$703$	0.277479	+	0.480608i	0.277479	+	0.480608i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$	0.178448	−	0.309081i	0.178448	−	0.309081i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−0.0990311	−	0.171527i	−0.0990311	−	0.171527i
$719$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$719$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$720$		0			0
$721$		0			0
$722$	−0.246980			−0.246980
$723$		0			0
$724$	−0.722521	−	1.25144i	−0.722521	−	1.25144i
$725$		0			0
$726$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$727$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$727$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.777479	−	1.34663i	−0.777479	−	1.34663i
$732$		0			0
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$		0			0
$735$		0			0
$736$	1.80194			1.80194
$737$		0			0
$738$	0.500000	+	0.866025i	0.500000	+	0.866025i
$739$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$739$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$740$		0			0
$741$	1.00000			1.00000
$742$		0			0
$743$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$743$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$744$		0			0
$745$		0			0
$746$	0.277479	+	0.480608i	0.277479	+	0.480608i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$	0.0990311	+	0.171527i	0.0990311	+	0.171527i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$	−0.356896			−0.356896
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$767$		0			0
$768$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	3.24698			3.24698
$772$		0			0
$773$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$773$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$774$	0.623490	+	1.07992i	0.623490	+	1.07992i
$775$		0			0
$776$	−1.44504			−1.44504
$777$		0			0
$778$	0.198062			0.198062
$779$	−0.623490	+	1.07992i	−0.623490	+	1.07992i
$780$		0			0
$781$		0			0
$782$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	0.500000	+	0.866025i	0.500000	+	0.866025i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	0.445042	+	0.770835i	0.445042	+	0.770835i
$795$		0			0
$796$	−0.178448	+	0.309081i	−0.178448	+	0.309081i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$	−0.554958			−0.554958
$800$	0.500000	−	0.866025i	0.500000	−	0.866025i
$801$	0.500000	+	0.866025i	0.500000	+	0.866025i
$802$	−0.0990311	−	0.171527i	−0.0990311	−	0.171527i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.722521	+	1.25144i	0.722521	+	1.25144i
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	0.500000	+	0.866025i	0.500000	+	0.866025i
$817$	−0.777479	+	1.34663i	−0.777479	+	1.34663i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$821$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$	−1.62349	+	2.81197i	−1.62349	+	2.81197i
$829$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$829$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$830$		0			0
$831$	−1.62349	+	2.81197i	−1.62349	+	2.81197i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$839$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$	0.445042			0.445042
$847$		0			0
$848$		0			0
$849$		0			0
$850$	0.277479	+	0.480608i	0.277479	+	0.480608i
$851$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$857$		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
$862$	−0.890084			−0.890084
$863$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$863$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$864$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$865$		0			0
$866$		0			0
$867$	−1.00000			−1.00000
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	2.02446	−	3.50647i	2.02446	−	3.50647i
$874$	1.00000			1.00000
$875$		0			0
$876$		0			0
$877$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$877$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$878$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$879$	1.80194	+	3.12105i	1.80194	+	3.12105i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	−0.222521	+	0.385418i	−0.222521	+	0.385418i
$885$		0			0
$886$	−0.0990311	−	0.171527i	−0.0990311	−	0.171527i
$887$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$887$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$888$	0.643104			0.643104
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	0.277479	+	0.480608i	0.277479	+	0.480608i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−1.44504			−1.44504
$898$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$899$		0			0
$900$	0.900969	+	1.56052i	0.900969	+	1.56052i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−1.44504			−1.44504
$905$		0			0
$906$		0			0
$907$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$907$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$908$	0.801938	−	1.38900i	0.801938	−	1.38900i
$909$	−4.04892			−4.04892
$910$		0			0
$911$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$911$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$912$	0.500000	−	0.866025i	0.500000	−	0.866025i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.44504			1.44504
$917$		0			0
$918$	1.24698			1.24698
$919$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$919$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.445042			−0.445042
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$929$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	1.12349	−	1.94594i	1.12349	−	1.94594i
$934$		0			0
$935$		0			0
$936$	0.400969	−	0.694498i	0.400969	−	0.694498i
$937$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$937$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$938$		0			0
$939$	−2.24698			−2.24698
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$943$	0.900969	−	1.56052i	0.900969	−	1.56052i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$947$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$948$		0			0
$949$		0			0
$950$	0.277479	−	0.480608i	0.277479	−	0.480608i
$951$		0			0
$952$		0			0
$953$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$953$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$	0.801938			0.801938
$959$		0			0
$960$		0			0
$961$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$962$	0.0440730	+	0.0763367i	0.0440730	+	0.0763367i
$963$	−1.40097	−	2.42655i	−1.40097	−	2.42655i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$969$	1.40097	+	2.42655i	1.40097	+	2.42655i
$970$		0			0
$971$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$971$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$972$	0.801938			0.801938
$973$		0			0
$974$	−0.554958			−0.554958
$975$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$976$		0			0
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$	0.722521	−	1.25144i	0.722521	−	1.25144i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$	0.722521	+	1.25144i	0.722521	+	1.25144i
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.445042			0.445042
$989$	1.12349	−	1.94594i	1.12349	−	1.94594i
$990$		0			0
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$997$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$998$		0			0
$999$	−0.500000	+	0.866025i	−0.500000	+	0.866025i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.1.i.b.901.2		6
7.2	even	3		287.1.d.a.286.2	✓	3
7.3	odd	6		2009.1.i.a.1844.2		6
7.4	even	3	inner	2009.1.i.b.1844.2		6
7.5	odd	6		287.1.d.b.286.2	yes	3
7.6	odd	2		2009.1.i.a.901.2		6
21.2	odd	6		2583.1.f.b.2008.2		3
21.5	even	6		2583.1.f.a.2008.2		3
41.40	even	2		2009.1.i.a.901.2		6
287.40	odd	6		287.1.d.a.286.2	✓	3
287.81	even	6		2009.1.i.a.1844.2		6
287.122	odd	6	inner	2009.1.i.b.1844.2		6
287.163	even	6		287.1.d.b.286.2	yes	3
287.286	odd	2	CM	2009.1.i.b.901.2		6
861.614	even	6		2583.1.f.b.2008.2		3
861.737	odd	6		2583.1.f.a.2008.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.1.d.a.286.2	✓	3	7.2	even	3
287.1.d.a.286.2	✓	3	287.40	odd	6
287.1.d.b.286.2	yes	3	7.5	odd	6
287.1.d.b.286.2	yes	3	287.163	even	6
2009.1.i.a.901.2		6	7.6	odd	2
2009.1.i.a.901.2		6	41.40	even	2
2009.1.i.a.1844.2		6	7.3	odd	6
2009.1.i.a.1844.2		6	287.81	even	6
2009.1.i.b.901.2		6	1.1	even	1	trivial
2009.1.i.b.901.2		6	287.286	odd	2	CM
2009.1.i.b.1844.2		6	7.4	even	3	inner
2009.1.i.b.1844.2		6	287.122	odd	6	inner
2583.1.f.a.2008.2		3	21.5	even	6
2583.1.f.a.2008.2		3	861.737	odd	6
2583.1.f.b.2008.2		3	21.2	odd	6
2583.1.f.b.2008.2		3	861.614	even	6

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:	\( N \)	\(=\)	\( 2009 = 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2009.i (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.222521 - 0.385418i) q^{2} +(0.900969 + 1.56052i) q^{3} +(0.400969 + 0.694498i) q^{4} +0.801938 q^{6} +0.801938 q^{8} +(-1.12349 + 1.94594i) q^{9} +O(q^{10})\)	\(q+(0.222521 - 0.385418i) q^{2} +(0.900969 + 1.56052i) q^{3} +(0.400969 + 0.694498i) q^{4} +0.801938 q^{6} +0.801938 q^{8} +(-1.12349 + 1.94594i) q^{9} +(-0.722521 + 1.25144i) q^{12} -0.445042 q^{13} +(-0.222521 + 0.385418i) q^{16} +(-0.623490 - 1.07992i) q^{17} +(0.500000 + 0.866025i) q^{18} +(-0.623490 + 1.07992i) q^{19} +(0.900969 - 1.56052i) q^{23} +(0.722521 + 1.25144i) q^{24} +(-0.500000 - 0.866025i) q^{25} +(-0.0990311 + 0.171527i) q^{26} -2.24698 q^{27} +(0.500000 + 0.866025i) q^{32} -0.554958 q^{34} -1.80194 q^{36} +(0.222521 - 0.385418i) q^{37} +(0.277479 + 0.480608i) q^{38} +(-0.400969 - 0.694498i) q^{39} +1.00000 q^{41} +1.24698 q^{43} +(-0.400969 - 0.694498i) q^{46} +(0.222521 - 0.385418i) q^{47} -0.801938 q^{48} -0.445042 q^{50} +(1.12349 - 1.94594i) q^{51} +(-0.178448 - 0.309081i) q^{52} +(-0.500000 + 0.866025i) q^{54} -2.24698 q^{57} +(0.500000 - 0.866025i) q^{68} +3.24698 q^{69} +(-0.900969 + 1.56052i) q^{72} +(-0.0990311 - 0.171527i) q^{74} +(0.900969 - 1.56052i) q^{75} -1.00000 q^{76} -0.356896 q^{78} +(-0.900969 - 1.56052i) q^{81} +(0.222521 - 0.385418i) q^{82} +(0.277479 - 0.480608i) q^{86} +(0.222521 - 0.385418i) q^{89} +1.44504 q^{92} +(-0.0990311 - 0.171527i) q^{94} +(-0.900969 + 1.56052i) q^{96} -1.80194 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} + q^{3} - 2 q^{4} - 4 q^{6} - 4 q^{8} - 2 q^{9}+O(q^{10}) \) 6 * q + q^2 + q^3 - 2 * q^4 - 4 * q^6 - 4 * q^8 - 2 * q^9	\( 6 q + q^{2} + q^{3} - 2 q^{4} - 4 q^{6} - 4 q^{8} - 2 q^{9} - 4 q^{12} - 2 q^{13} - q^{16} + q^{17} + 3 q^{18} + q^{19} + q^{23} + 4 q^{24} - 3 q^{25} - 5 q^{26} - 4 q^{27} + 3 q^{32} - 4 q^{34} - 2 q^{36} + q^{37} + 2 q^{38} + 2 q^{39} + 6 q^{41} - 2 q^{43} + 2 q^{46} + q^{47} + 4 q^{48} - 2 q^{50} + 2 q^{51} + 3 q^{52} - 3 q^{54} - 4 q^{57} + 3 q^{68} + 10 q^{69} - q^{72} - 5 q^{74} + q^{75} - 6 q^{76} + 6 q^{78} - q^{81} + q^{82} + 2 q^{86} + q^{89} + 8 q^{92} - 5 q^{94} - q^{96} - 2 q^{97}+O(q^{100}) \) 6 * q + q^2 + q^3 - 2 * q^4 - 4 * q^6 - 4 * q^8 - 2 * q^9 - 4 * q^12 - 2 * q^13 - q^16 + q^17 + 3 * q^18 + q^19 + q^23 + 4 * q^24 - 3 * q^25 - 5 * q^26 - 4 * q^27 + 3 * q^32 - 4 * q^34 - 2 * q^36 + q^37 + 2 * q^38 + 2 * q^39 + 6 * q^41 - 2 * q^43 + 2 * q^46 + q^47 + 4 * q^48 - 2 * q^50 + 2 * q^51 + 3 * q^52 - 3 * q^54 - 4 * q^57 + 3 * q^68 + 10 * q^69 - q^72 - 5 * q^74 + q^75 - 6 * q^76 + 6 * q^78 - q^81 + q^82 + 2 * q^86 + q^89 + 8 * q^92 - 5 * q^94 - q^96 - 2 * q^97

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