LMFDB - Embedded newform 1911.1.w.e.1598.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1911,1,Mod(116,1911)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1911, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 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("1911.116");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1911 = 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1911.w (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:	$0.953713239142$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.339738624.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 4x^{6} + 14x^{4} + 8x^{2} + 4 $ x^8 + 4x^6 + 14x^4 + 8*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{8}$
Projective field:	Galois closure of 8.2.90724673403.2

Embedding invariants

Embedding label			1598.2
Root			$-0.923880 + 1.60021i$ of defining polynomial
Character	$\chi$	$=$	1911.1598
Dual form			1911.1.w.e.116.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.382683 + 0.662827i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(0.207107 + 0.358719i) q^{4} +(-0.923880 + 1.60021i) q^{5} +0.765367 q^{6} -1.08239 q^{8} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})$	\(q+(-0.382683 + 0.662827i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(0.207107 + 0.358719i) q^{4} +(-0.923880 + 1.60021i) q^{5} +0.765367 q^{6} -1.08239 q^{8} +(-0.500000 + 0.866025i) q^{9} +(-0.707107 - 1.22474i) q^{10} +(0.382683 + 0.662827i) q^{11} +(0.207107 - 0.358719i) q^{12} -1.00000 q^{13} +1.84776 q^{15} +(0.207107 - 0.358719i) q^{16} +(-0.382683 - 0.662827i) q^{18} -0.765367 q^{20} -0.585786 q^{22} +(0.541196 + 0.937379i) q^{24} +(-1.20711 - 2.09077i) q^{25} +(0.382683 - 0.662827i) q^{26} +1.00000 q^{27} +(-0.707107 + 1.22474i) q^{30} +(-0.382683 - 0.662827i) q^{32} +(0.382683 - 0.662827i) q^{33} -0.414214 q^{36} +(0.500000 + 0.866025i) q^{39} +(1.00000 - 1.73205i) q^{40} -1.84776 q^{41} +(-0.158513 + 0.274552i) q^{44} +(-0.923880 - 1.60021i) q^{45} +(0.923880 - 1.60021i) q^{47} -0.414214 q^{48} +1.84776 q^{50} +(-0.207107 - 0.358719i) q^{52} +(-0.382683 + 0.662827i) q^{54} -1.41421 q^{55} +(0.382683 + 0.662827i) q^{59} +(0.382683 + 0.662827i) q^{60} +1.00000 q^{64} +(0.923880 - 1.60021i) q^{65} +(0.292893 + 0.507306i) q^{66} -1.84776 q^{71} +(0.541196 - 0.937379i) q^{72} +(-1.20711 + 2.09077i) q^{75} -0.765367 q^{78} +(-0.707107 + 1.22474i) q^{79} +(0.382683 + 0.662827i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(0.707107 - 1.22474i) q^{82} +0.765367 q^{83} +(-0.414214 - 0.717439i) q^{88} +(-0.382683 + 0.662827i) q^{89} +1.41421 q^{90} +(0.707107 + 1.22474i) q^{94} +(-0.382683 + 0.662827i) q^{96} -0.765367 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} - 4 q^{4} - 4 q^{9}+O(q^{10}) $ 8 * q - 4 * q^3 - 4 * q^4 - 4 * q^9	$ 8 q - 4 q^{3} - 4 q^{4} - 4 q^{9} - 4 q^{12} - 8 q^{13} - 4 q^{16} - 16 q^{22} - 4 q^{25} + 8 q^{27} + 8 q^{36} + 4 q^{39} + 8 q^{40} + 8 q^{48} + 4 q^{52} + 8 q^{64} + 8 q^{66} - 4 q^{75} - 4 q^{81} + 8 q^{88}+O(q^{100}) $ 8 * q - 4 * q^3 - 4 * q^4 - 4 * q^9 - 4 * q^12 - 8 * q^13 - 4 * q^16 - 16 * q^22 - 4 * q^25 + 8 * q^27 + 8 * q^36 + 4 * q^39 + 8 * q^40 + 8 * q^48 + 4 * q^52 + 8 * q^64 + 8 * q^66 - 4 * q^75 - 4 * q^81 + 8 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1911\mathbb{Z}\right)^\times$.

$n$	$638$	$1471$	$1522$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$	−0.382683	+	0.662827i	−0.382683	+	0.662827i	−0.991445	−	0.130526i	$-0.958333\pi$
$2$	−0.382683	+	0.662827i	−0.382683	+	0.662827i	0.608761	+	0.793353i	$0.291667\pi$
$3$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$3$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$4$	0.207107	+	0.358719i	0.207107	+	0.358719i
$5$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.130526	+	0.991445i	$0.541667\pi$
$5$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.793353	+	0.608761i	$0.791667\pi$
$6$	0.765367			0.765367
$7$		0			0
$7$		0			0
$8$	−1.08239			−1.08239
$9$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$10$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$11$	0.382683	+	0.662827i	0.382683	+	0.662827i	0.991445	−	0.130526i	$-0.0416667\pi$
$11$	0.382683	+	0.662827i	0.382683	+	0.662827i	−0.608761	+	0.793353i	$0.708333\pi$
$12$	0.207107	−	0.358719i	0.207107	−	0.358719i
$13$	−1.00000			−1.00000
$13$	−1.00000			−1.00000
$14$		0			0
$15$	1.84776			1.84776
$16$	0.207107	−	0.358719i	0.207107	−	0.358719i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$	−0.382683	−	0.662827i	−0.382683	−	0.662827i
$19$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$	−0.765367			−0.765367
$21$		0			0
$22$	−0.585786			−0.585786
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	0.541196	+	0.937379i	0.541196	+	0.937379i
$25$	−1.20711	−	2.09077i	−1.20711	−	2.09077i
$26$	0.382683	−	0.662827i	0.382683	−	0.662827i
$27$	1.00000			1.00000
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	−0.382683	−	0.662827i	−0.382683	−	0.662827i
$33$	0.382683	−	0.662827i	0.382683	−	0.662827i
$34$		0			0
$35$		0			0
$36$	−0.414214			−0.414214
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$	0.500000	+	0.866025i	0.500000	+	0.866025i
$40$	1.00000	−	1.73205i	1.00000	−	1.73205i
$41$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$41$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−0.158513	+	0.274552i	−0.158513	+	0.274552i
$45$	−0.923880	−	1.60021i	−0.923880	−	1.60021i
$46$		0			0
$47$	0.923880	−	1.60021i	0.923880	−	1.60021i	0.130526	−	0.991445i	$-0.458333\pi$
$47$	0.923880	−	1.60021i	0.923880	−	1.60021i	0.793353	−	0.608761i	$-0.208333\pi$
$48$	−0.414214			−0.414214
$49$		0			0
$50$	1.84776			1.84776
$51$		0			0
$52$	−0.207107	−	0.358719i	−0.207107	−	0.358719i
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$	−0.382683	+	0.662827i	−0.382683	+	0.662827i
$55$	−1.41421			−1.41421
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.382683	+	0.662827i	0.382683	+	0.662827i	0.991445	−	0.130526i	$-0.0416667\pi$
$59$	0.382683	+	0.662827i	0.382683	+	0.662827i	−0.608761	+	0.793353i	$0.708333\pi$
$60$	0.382683	+	0.662827i	0.382683	+	0.662827i
$61$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$61$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$62$		0			0
$63$		0			0
$64$	1.00000			1.00000
$65$	0.923880	−	1.60021i	0.923880	−	1.60021i
$66$	0.292893	+	0.507306i	0.292893	+	0.507306i
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$71$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$72$	0.541196	−	0.937379i	0.541196	−	0.937379i
$73$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$73$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$74$		0			0
$75$	−1.20711	+	2.09077i	−1.20711	+	2.09077i
$76$		0			0
$77$		0			0
$78$	−0.765367			−0.765367
$79$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$79$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$80$	0.382683	+	0.662827i	0.382683	+	0.662827i
$81$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$82$	0.707107	−	1.22474i	0.707107	−	1.22474i
$83$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$83$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−0.414214	−	0.717439i	−0.414214	−	0.717439i
$89$	−0.382683	+	0.662827i	−0.382683	+	0.662827i	−0.991445	−	0.130526i	$-0.958333\pi$
$89$	−0.382683	+	0.662827i	−0.382683	+	0.662827i	0.608761	+	0.793353i	$0.291667\pi$
$90$	1.41421			1.41421
$91$		0			0
$92$		0			0
$93$		0			0
$94$	0.707107	+	1.22474i	0.707107	+	1.22474i
$95$		0			0
$96$	−0.382683	+	0.662827i	−0.382683	+	0.662827i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$	−0.765367			−0.765367
$100$	0.500000	−	0.866025i	0.500000	−	0.866025i
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$103$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$104$	1.08239			1.08239
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$	0.207107	+	0.358719i	0.207107	+	0.358719i
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$	0.541196	−	0.937379i	0.541196	−	0.937379i
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.500000	−	0.866025i	0.500000	−	0.866025i
$118$	−0.585786			−0.585786
$119$		0			0
$120$	−2.00000			−2.00000
$121$	0.207107	−	0.358719i	0.207107	−	0.358719i
$122$		0			0
$123$	0.923880	+	1.60021i	0.923880	+	1.60021i
$124$		0			0
$125$	2.61313			2.61313
$126$		0			0
$127$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$127$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$128$		0			0
$129$		0			0
$130$	0.707107	+	1.22474i	0.707107	+	1.22474i
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$	0.317025			0.317025
$133$		0			0
$134$		0			0
$135$	−0.923880	+	1.60021i	−0.923880	+	1.60021i
$136$		0			0
$137$	−0.923880	−	1.60021i	−0.923880	−	1.60021i	−0.793353	−	0.608761i	$-0.791667\pi$
$137$	−0.923880	−	1.60021i	−0.923880	−	1.60021i	−0.130526	−	0.991445i	$-0.541667\pi$
$138$		0			0
$139$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$139$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$140$		0			0
$141$	−1.84776			−1.84776
$142$	0.707107	−	1.22474i	0.707107	−	1.22474i
$143$	−0.382683	−	0.662827i	−0.382683	−	0.662827i
$144$	0.207107	+	0.358719i	0.207107	+	0.358719i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.130526	+	0.991445i	$0.541667\pi$
$149$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.793353	+	0.608761i	$0.791667\pi$
$150$	−0.923880	−	1.60021i	−0.923880	−	1.60021i
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−0.207107	+	0.358719i	−0.207107	+	0.358719i
$157$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$157$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$158$	−0.541196	−	0.937379i	−0.541196	−	0.937379i
$159$		0			0
$160$	1.41421			1.41421
$161$		0			0
$162$	0.765367			0.765367
$163$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$163$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$164$	−0.382683	−	0.662827i	−0.382683	−	0.662827i
$165$	0.707107	+	1.22474i	0.707107	+	1.22474i
$166$	−0.292893	+	0.507306i	−0.292893	+	0.507306i
$167$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$167$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$168$		0			0
$169$	1.00000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		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
$176$	0.317025			0.317025
$177$	0.382683	−	0.662827i	0.382683	−	0.662827i
$178$	−0.292893	−	0.507306i	−0.292893	−	0.507306i
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$	0.382683	−	0.662827i	0.382683	−	0.662827i
$181$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$181$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	0.765367			0.765367
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$	−1.84776			−1.84776
$196$		0			0
$197$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$197$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$198$	0.292893	−	0.507306i	0.292893	−	0.507306i
$199$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$199$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$200$	1.30656	+	2.26303i	1.30656	+	2.26303i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	1.70711	−	2.95680i	1.70711	−	2.95680i
$206$	−0.541196	−	0.937379i	−0.541196	−	0.937379i
$207$		0			0
$208$	−0.207107	+	0.358719i	−0.207107	+	0.358719i
$209$		0			0
$210$		0			0
$211$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$211$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$212$		0			0
$213$	0.923880	+	1.60021i	0.923880	+	1.60021i
$214$		0			0
$215$		0			0
$216$	−1.08239			−1.08239
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.292893	−	0.507306i	−0.292893	−	0.507306i
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	2.41421			2.41421
$226$		0			0
$227$	0.923880	+	1.60021i	0.923880	+	1.60021i	0.793353	+	0.608761i	$0.208333\pi$
$227$	0.923880	+	1.60021i	0.923880	+	1.60021i	0.130526	+	0.991445i	$0.458333\pi$
$228$		0			0
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$		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	$0.666667\pi$
$234$	0.382683	+	0.662827i	0.382683	+	0.662827i
$235$	1.70711	+	2.95680i	1.70711	+	2.95680i
$236$	−0.158513	+	0.274552i	−0.158513	+	0.274552i
$237$	1.41421			1.41421
$238$		0			0
$239$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$239$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$240$	0.382683	−	0.662827i	0.382683	−	0.662827i
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	0.158513	+	0.274552i	0.158513	+	0.274552i
$243$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$244$		0			0
$245$		0			0
$246$	−1.41421			−1.41421
$247$		0			0
$248$		0			0
$249$	−0.382683	−	0.662827i	−0.382683	−	0.662827i
$250$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$	0.541196	−	0.937379i	0.541196	−	0.937379i
$255$		0			0
$256$	0.500000	+	0.866025i	0.500000	+	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$		0			0
$260$	0.765367			0.765367
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$	−0.414214	+	0.717439i	−0.414214	+	0.717439i
$265$		0			0
$266$		0			0
$267$	0.765367			0.765367
$268$		0			0
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$		0			0
$273$		0			0
$274$	1.41421			1.41421
$275$	0.923880	−	1.60021i	0.923880	−	1.60021i
$276$		0			0
$277$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$277$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$278$	0.541196	−	0.937379i	0.541196	−	0.937379i
$279$		0			0
$280$		0			0
$281$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$281$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$282$	0.707107	−	1.22474i	0.707107	−	1.22474i
$283$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$283$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$284$	−0.382683	−	0.662827i	−0.382683	−	0.662827i
$285$		0			0
$286$	0.585786			0.585786
$287$		0			0
$288$	0.765367			0.765367
$289$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$293$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$294$		0			0
$295$	−1.41421			−1.41421
$296$		0			0
$297$	0.382683	+	0.662827i	0.382683	+	0.662827i
$298$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$299$		0			0
$300$	−1.00000			−1.00000
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	1.41421			1.41421
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$	−0.541196	−	0.937379i	−0.541196	−	0.937379i
$313$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$313$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$314$	−1.08239			−1.08239
$315$		0			0
$316$	−0.585786			−0.585786
$317$	−0.382683	+	0.662827i	−0.382683	+	0.662827i	−0.991445	−	0.130526i	$-0.958333\pi$
$317$	−0.382683	+	0.662827i	−0.382683	+	0.662827i	0.608761	+	0.793353i	$0.291667\pi$
$318$		0			0
$319$		0			0
$320$	−0.923880	+	1.60021i	−0.923880	+	1.60021i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.207107	−	0.358719i	0.207107	−	0.358719i
$325$	1.20711	+	2.09077i	1.20711	+	2.09077i
$326$		0			0
$327$		0			0
$328$	2.00000			2.00000
$329$		0			0
$330$	−1.08239			−1.08239
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$	0.158513	+	0.274552i	0.158513	+	0.274552i
$333$		0			0
$334$	0.292893	−	0.507306i	0.292893	−	0.507306i
$335$		0			0
$336$		0			0
$337$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$337$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$338$	−0.382683	+	0.662827i	−0.382683	+	0.662827i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		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
$352$	0.292893	−	0.507306i	0.292893	−	0.507306i
$353$	−0.382683	−	0.662827i	−0.382683	−	0.662827i	0.608761	−	0.793353i	$-0.291667\pi$
$353$	−0.382683	−	0.662827i	−0.382683	−	0.662827i	−0.991445	+	0.130526i	$0.958333\pi$
$354$	0.292893	+	0.507306i	0.292893	+	0.507306i
$355$	1.70711	−	2.95680i	1.70711	−	2.95680i
$356$	−0.317025			−0.317025
$357$		0			0
$358$		0			0
$359$	−0.382683	+	0.662827i	−0.382683	+	0.662827i	−0.991445	−	0.130526i	$-0.958333\pi$
$359$	−0.382683	+	0.662827i	−0.382683	+	0.662827i	0.608761	+	0.793353i	$0.291667\pi$
$360$	1.00000	+	1.73205i	1.00000	+	1.73205i
$361$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$362$	−0.541196	+	0.937379i	−0.541196	+	0.937379i
$363$	−0.414214			−0.414214
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$367$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$368$		0			0
$369$	0.923880	−	1.60021i	0.923880	−	1.60021i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$373$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$374$		0			0
$375$	−1.30656	−	2.26303i	−1.30656	−	2.26303i
$376$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	0.707107	+	1.22474i	0.707107	+	1.22474i
$382$		0			0
$383$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.130526	+	0.991445i	$0.541667\pi$
$383$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.793353	+	0.608761i	$0.791667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$	0.707107	−	1.22474i	0.707107	−	1.22474i
$391$		0			0
$392$		0			0
$393$		0			0
$394$	0.292893	−	0.507306i	0.292893	−	0.507306i
$395$	−1.30656	−	2.26303i	−1.30656	−	2.26303i
$396$	−0.158513	−	0.274552i	−0.158513	−	0.274552i
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$	−1.53073			−1.53073
$399$		0			0
$400$	−1.00000			−1.00000
$401$	0.382683	−	0.662827i	0.382683	−	0.662827i	−0.608761	−	0.793353i	$-0.708333\pi$
$401$	0.382683	−	0.662827i	0.382683	−	0.662827i	0.991445	+	0.130526i	$0.0416667\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	1.84776			1.84776
$406$		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	$0.333333\pi$
$410$	1.30656	+	2.26303i	1.30656	+	2.26303i
$411$	−0.923880	+	1.60021i	−0.923880	+	1.60021i
$412$	−0.585786			−0.585786
$413$		0			0
$414$		0			0
$415$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$416$	0.382683	+	0.662827i	0.382683	+	0.662827i
$417$	0.707107	+	1.22474i	0.707107	+	1.22474i
$418$		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			$\pi$
$422$	−0.541196	+	0.937379i	−0.541196	+	0.937379i
$423$	0.923880	+	1.60021i	0.923880	+	1.60021i
$424$		0			0
$425$		0			0
$426$	−1.41421			−1.41421
$427$		0			0
$428$		0			0
$429$	−0.382683	+	0.662827i	−0.382683	+	0.662827i
$430$		0			0
$431$	0.923880	+	1.60021i	0.923880	+	1.60021i	0.793353	+	0.608761i	$0.208333\pi$
$431$	0.923880	+	1.60021i	0.923880	+	1.60021i	0.130526	+	0.991445i	$0.458333\pi$
$432$	0.207107	−	0.358719i	0.207107	−	0.358719i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$439$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$440$	1.53073			1.53073
$441$		0			0
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$446$		0			0
$447$	1.84776			1.84776
$448$		0			0
$449$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$449$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$450$	−0.923880	+	1.60021i	−0.923880	+	1.60021i
$451$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$452$		0			0
$453$		0			0
$454$	−1.41421			−1.41421
$455$		0			0
$456$		0			0
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$461$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$462$		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	$0.666667\pi$
$468$	0.414214			0.414214
$469$		0			0
$470$	−2.61313			−2.61313
$471$	0.707107	−	1.22474i	0.707107	−	1.22474i
$472$	−0.414214	−	0.717439i	−0.414214	−	0.717439i
$473$		0			0
$474$	−0.541196	+	0.937379i	−0.541196	+	0.937379i
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−0.292893	+	0.507306i	−0.292893	+	0.507306i
$479$	−0.923880	−	1.60021i	−0.923880	−	1.60021i	−0.793353	−	0.608761i	$-0.791667\pi$
$479$	−0.923880	−	1.60021i	−0.923880	−	1.60021i	−0.130526	−	0.991445i	$-0.541667\pi$
$480$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.171573			0.171573
$485$		0			0
$486$	−0.382683	−	0.662827i	−0.382683	−	0.662827i
$487$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$487$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$	−0.382683	+	0.662827i	−0.382683	+	0.662827i
$493$		0			0
$494$		0			0
$495$	0.707107	−	1.22474i	0.707107	−	1.22474i
$496$		0			0
$497$		0			0
$498$	0.585786			0.585786
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$	0.541196	+	0.937379i	0.541196	+	0.937379i
$501$	0.382683	+	0.662827i	0.382683	+	0.662827i
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$508$	−0.292893	−	0.507306i	−0.292893	−	0.507306i
$509$	0.923880	−	1.60021i	0.923880	−	1.60021i	0.130526	−	0.991445i	$-0.458333\pi$
$509$	0.923880	−	1.60021i	0.923880	−	1.60021i	0.793353	−	0.608761i	$-0.208333\pi$
$510$		0			0
$511$		0			0
$512$	−0.765367			−0.765367
$513$		0			0
$514$		0			0
$515$	−1.30656	−	2.26303i	−1.30656	−	2.26303i
$516$		0			0
$517$	1.41421			1.41421
$518$		0			0
$519$		0			0
$520$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$523$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−0.158513	−	0.274552i	−0.158513	−	0.274552i
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$		0			0
$531$	−0.765367			−0.765367
$532$		0			0
$533$	1.84776			1.84776
$534$	−0.292893	+	0.507306i	−0.292893	+	0.507306i
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	−0.765367			−0.765367
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−2.00000			−2.00000			−1.00000			$\pi$
$547$	−2.00000			−2.00000			−1.00000			$\pi$
$548$	0.382683	−	0.662827i	0.382683	−	0.662827i
$549$		0			0
$550$	0.707107	+	1.22474i	0.707107	+	1.22474i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−0.292893	−	0.507306i	−0.292893	−	0.507306i
$557$	0.923880	+	1.60021i	0.923880	+	1.60021i	0.793353	+	0.608761i	$0.208333\pi$
$557$	0.923880	+	1.60021i	0.923880	+	1.60021i	0.130526	+	0.991445i	$0.458333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$	−0.382683	−	0.662827i	−0.382683	−	0.662827i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	2.00000			2.00000
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$571$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$572$	0.158513	−	0.274552i	0.158513	−	0.274552i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$577$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$577$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$578$	−0.382683	−	0.662827i	−0.382683	−	0.662827i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	0.923880	+	1.60021i	0.923880	+	1.60021i
$586$	0.292893	−	0.507306i	0.292893	−	0.507306i
$587$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$587$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$588$		0			0
$589$		0			0
$590$	0.541196	−	0.937379i	0.541196	−	0.937379i
$591$	0.382683	+	0.662827i	0.382683	+	0.662827i
$592$		0			0
$593$	−0.382683	+	0.662827i	−0.382683	+	0.662827i	−0.991445	−	0.130526i	$-0.958333\pi$
$593$	−0.382683	+	0.662827i	−0.382683	+	0.662827i	0.608761	+	0.793353i	$0.291667\pi$
$594$	−0.585786			−0.585786
$595$		0			0
$596$	−0.765367			−0.765367
$597$	1.00000	−	1.73205i	1.00000	−	1.73205i
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$	1.30656	−	2.26303i	1.30656	−	2.26303i
$601$	−2.00000			−2.00000			−1.00000			$\pi$
$601$	−2.00000			−2.00000			−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.382683	+	0.662827i	0.382683	+	0.662827i
$606$		0			0
$607$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$607$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−0.923880	+	1.60021i	−0.923880	+	1.60021i
$612$		0			0
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$		0			0
$615$	−3.41421			−3.41421
$616$		0			0
$617$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$617$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$618$	−0.541196	+	0.937379i	−0.541196	+	0.937379i
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	0.414214			0.414214
$625$	−1.20711	+	2.09077i	−1.20711	+	2.09077i
$626$		0			0
$627$		0			0
$628$	−0.292893	+	0.507306i	−0.292893	+	0.507306i
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	0.765367	−	1.32565i	0.765367	−	1.32565i
$633$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$634$	−0.292893	−	0.507306i	−0.292893	−	0.507306i
$635$	1.30656	−	2.26303i	1.30656	−	2.26303i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	0.923880	−	1.60021i	0.923880	−	1.60021i
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$	0.541196	+	0.937379i	0.541196	+	0.937379i
$649$	−0.292893	+	0.507306i	−0.292893	+	0.507306i
$650$	−1.84776			−1.84776
$651$		0			0
$652$		0			0
$653$		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
$656$	−0.382683	+	0.662827i	−0.382683	+	0.662827i
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$	−0.292893	+	0.507306i	−0.292893	+	0.507306i
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$		0			0
$664$	−0.828427			−0.828427
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−0.158513	−	0.274552i	−0.158513	−	0.274552i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−2.00000			−2.00000			−1.00000			$\pi$
$673$	−2.00000			−2.00000			−1.00000			$\pi$
$674$	0.541196	−	0.937379i	0.541196	−	0.937379i
$675$	−1.20711	−	2.09077i	−1.20711	−	2.09077i
$676$	0.207107	+	0.358719i	0.207107	+	0.358719i
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	0.923880	−	1.60021i	0.923880	−	1.60021i
$682$		0			0
$683$	−0.923880	−	1.60021i	−0.923880	−	1.60021i	−0.793353	−	0.608761i	$-0.791667\pi$
$683$	−0.923880	−	1.60021i	−0.923880	−	1.60021i	−0.130526	−	0.991445i	$-0.541667\pi$
$684$		0			0
$685$	3.41421			3.41421
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.30656	−	2.26303i	1.30656	−	2.26303i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	0.382683	−	0.662827i	0.382683	−	0.662827i
$703$		0			0
$704$	0.382683	+	0.662827i	0.382683	+	0.662827i
$705$	1.70711	−	2.95680i	1.70711	−	2.95680i
$706$	0.585786			0.585786
$707$		0			0
$708$	0.317025			0.317025
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$	1.30656	+	2.26303i	1.30656	+	2.26303i
$711$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$712$	0.414214	−	0.717439i	0.414214	−	0.717439i
$713$		0			0
$714$		0			0
$715$	1.41421			1.41421
$716$		0			0
$717$	−0.382683	−	0.662827i	−0.382683	−	0.662827i
$718$	−0.292893	−	0.507306i	−0.292893	−	0.507306i
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$	−0.765367			−0.765367
$721$		0			0
$722$	0.765367			0.765367
$723$		0			0
$724$	0.292893	+	0.507306i	0.292893	+	0.507306i
$725$		0			0
$726$	0.158513	−	0.274552i	0.158513	−	0.274552i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	1.00000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$	1.08239			1.08239
$735$		0			0
$736$		0			0
$737$		0			0
$738$	0.707107	+	1.22474i	0.707107	+	1.22474i
$739$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$739$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$743$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$744$		0			0
$745$	−1.70711	−	2.95680i	−1.70711	−	2.95680i
$746$	−0.541196	−	0.937379i	−0.541196	−	0.937379i
$747$	−0.382683	+	0.662827i	−0.382683	+	0.662827i
$748$		0			0
$749$		0			0
$750$	2.00000			2.00000
$751$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$751$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$752$	−0.382683	−	0.662827i	−0.382683	−	0.662827i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$757$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.923880	−	1.60021i	0.923880	−	1.60021i	0.130526	−	0.991445i	$-0.458333\pi$
$761$	0.923880	−	1.60021i	0.923880	−	1.60021i	0.793353	−	0.608761i	$-0.208333\pi$
$762$	−1.08239			−1.08239
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$767$	−0.382683	−	0.662827i	−0.382683	−	0.662827i
$768$	0.500000	−	0.866025i	0.500000	−	0.866025i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.382683	−	0.662827i	−0.382683	−	0.662827i	0.608761	−	0.793353i	$-0.291667\pi$
$773$	−0.382683	−	0.662827i	−0.382683	−	0.662827i	−0.991445	+	0.130526i	$0.958333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	−0.382683	−	0.662827i	−0.382683	−	0.662827i
$781$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.61313			−2.61313
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	−0.158513	−	0.274552i	−0.158513	−	0.274552i
$789$		0			0
$790$	2.00000			2.00000
$791$		0			0
$792$	0.828427			0.828427
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−0.414214	+	0.717439i	−0.414214	+	0.717439i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	−0.923880	+	1.60021i	−0.923880	+	1.60021i
$801$	−0.382683	−	0.662827i	−0.382683	−	0.662827i
$802$	0.292893	+	0.507306i	0.292893	+	0.507306i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$811$		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			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	1.41421			1.41421
$821$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.130526	+	0.991445i	$0.541667\pi$
$821$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.793353	+	0.608761i	$0.791667\pi$
$822$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$823$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$823$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$824$	0.765367	−	1.32565i	0.765367	−	1.32565i
$825$	−1.84776			−1.84776
$826$		0			0
$827$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$827$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$828$		0			0
$829$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$829$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$830$	−0.541196	−	0.937379i	−0.541196	−	0.937379i
$831$		0			0
$832$	−1.00000			−1.00000
$833$		0			0
$834$	−1.08239			−1.08239
$835$	0.707107	−	1.22474i	0.707107	−	1.22474i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$839$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$		0			0
$843$	−0.923880	−	1.60021i	−0.923880	−	1.60021i
$844$	0.292893	+	0.507306i	0.292893	+	0.507306i
$845$	−0.923880	+	1.60021i	−0.923880	+	1.60021i
$846$	−1.41421			−1.41421
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	−0.382683	+	0.662827i	−0.382683	+	0.662827i
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$	−0.292893	−	0.507306i	−0.292893	−	0.507306i
$859$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$859$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$860$		0			0
$861$		0			0
$862$	−1.41421			−1.41421
$863$	0.382683	−	0.662827i	0.382683	−	0.662827i	−0.608761	−	0.793353i	$-0.708333\pi$
$863$	0.382683	−	0.662827i	0.382683	−	0.662827i	0.991445	+	0.130526i	$0.0416667\pi$
$864$	−0.382683	−	0.662827i	−0.382683	−	0.662827i
$865$		0			0
$866$		0			0
$867$	1.00000			1.00000
$868$		0			0
$869$	−1.08239			−1.08239
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$	−0.541196	−	0.937379i	−0.541196	−	0.937379i
$879$	0.382683	+	0.662827i	0.382683	+	0.662827i
$880$	−0.292893	+	0.507306i	−0.292893	+	0.507306i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$883$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$	0.707107	+	1.22474i	0.707107	+	1.22474i
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$		0			0
$890$	1.08239			1.08239
$891$	0.382683	−	0.662827i	0.382683	−	0.662827i
$892$		0			0
$893$		0			0
$894$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.292893	+	0.507306i	−0.292893	+	0.507306i
$899$		0			0
$900$	0.500000	+	0.866025i	0.500000	+	0.866025i
$901$		0			0
$902$	1.08239			1.08239
$903$		0			0
$904$		0			0
$905$	−1.30656	+	2.26303i	−1.30656	+	2.26303i
$906$		0			0
$907$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$907$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$908$	−0.382683	+	0.662827i	−0.382683	+	0.662827i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	0.292893	+	0.507306i	0.292893	+	0.507306i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$919$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$920$		0			0
$921$		0			0
$922$	0.292893	−	0.507306i	0.292893	−	0.507306i
$923$	1.84776			1.84776
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$928$		0			0
$929$	0.382683	−	0.662827i	0.382683	−	0.662827i	−0.608761	−	0.793353i	$-0.708333\pi$
$929$	0.382683	−	0.662827i	0.382683	−	0.662827i	0.991445	+	0.130526i	$0.0416667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−0.541196	+	0.937379i	−0.541196	+	0.937379i
$937$	2.00000			2.00000			1.00000			$0$
$937$	2.00000			2.00000			1.00000			$0$
$938$		0			0
$939$		0			0
$940$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$941$	0.382683	+	0.662827i	0.382683	+	0.662827i	0.991445	−	0.130526i	$-0.0416667\pi$
$941$	0.382683	+	0.662827i	0.382683	+	0.662827i	−0.608761	+	0.793353i	$0.708333\pi$
$942$	0.541196	+	0.937379i	0.541196	+	0.937379i
$943$		0			0
$944$	0.317025			0.317025
$945$		0			0
$946$		0			0
$947$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.130526	+	0.991445i	$0.541667\pi$
$947$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.793353	+	0.608761i	$0.791667\pi$
$948$	0.292893	+	0.507306i	0.292893	+	0.507306i
$949$		0			0
$950$		0			0
$951$	0.765367			0.765367
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	0.158513	+	0.274552i	0.158513	+	0.274552i
$957$		0			0
$958$	1.41421			1.41421
$959$		0			0
$960$	1.84776			1.84776
$961$	−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		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	−0.224171	+	0.388275i	−0.224171	+	0.388275i
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$	−0.414214			−0.414214
$973$		0			0
$974$		0			0
$975$	1.20711	−	2.09077i	1.20711	−	2.09077i
$976$		0			0
$977$	0.382683	+	0.662827i	0.382683	+	0.662827i	0.991445	−	0.130526i	$-0.0416667\pi$
$977$	0.382683	+	0.662827i	0.382683	+	0.662827i	−0.608761	+	0.793353i	$0.708333\pi$
$978$		0			0
$979$	−0.585786			−0.585786
$980$		0			0
$981$		0			0
$982$		0			0
$983$	0.923880	+	1.60021i	0.923880	+	1.60021i	0.793353	+	0.608761i	$0.208333\pi$
$983$	0.923880	+	1.60021i	0.923880	+	1.60021i	0.130526	+	0.991445i	$0.458333\pi$
$984$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$985$	0.707107	−	1.22474i	0.707107	−	1.22474i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	0.541196	+	0.937379i	0.541196	+	0.937379i
$991$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$991$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3.69552			−3.69552
$996$	0.158513	−	0.274552i	0.158513	−	0.274552i
$997$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$997$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1911.1.w.e.1598.2		8
3.2	odd	2	inner	1911.1.w.e.1598.3		8
7.2	even	3		1911.1.h.e.1520.3	yes	4
7.3	odd	6		1911.1.w.f.116.2		8
7.4	even	3	inner	1911.1.w.e.116.2		8
7.5	odd	6		1911.1.h.d.1520.3	yes	4
7.6	odd	2		1911.1.w.f.1598.2		8
13.12	even	2	inner	1911.1.w.e.1598.3		8
21.2	odd	6		1911.1.h.e.1520.2	yes	4
21.5	even	6		1911.1.h.d.1520.2	✓	4
21.11	odd	6	inner	1911.1.w.e.116.3		8
21.17	even	6		1911.1.w.f.116.3		8
21.20	even	2		1911.1.w.f.1598.3		8
39.38	odd	2	CM	1911.1.w.e.1598.2		8
91.12	odd	6		1911.1.h.d.1520.2	✓	4
91.25	even	6	inner	1911.1.w.e.116.3		8
91.38	odd	6		1911.1.w.f.116.3		8
91.51	even	6		1911.1.h.e.1520.2	yes	4
91.90	odd	2		1911.1.w.f.1598.3		8
273.38	even	6		1911.1.w.f.116.2		8
273.116	odd	6	inner	1911.1.w.e.116.2		8
273.194	even	6		1911.1.h.d.1520.3	yes	4
273.233	odd	6		1911.1.h.e.1520.3	yes	4
273.272	even	2		1911.1.w.f.1598.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1911.1.h.d.1520.2	✓	4	21.5	even	6
1911.1.h.d.1520.2	✓	4	91.12	odd	6
1911.1.h.d.1520.3	yes	4	7.5	odd	6
1911.1.h.d.1520.3	yes	4	273.194	even	6
1911.1.h.e.1520.2	yes	4	21.2	odd	6
1911.1.h.e.1520.2	yes	4	91.51	even	6
1911.1.h.e.1520.3	yes	4	7.2	even	3
1911.1.h.e.1520.3	yes	4	273.233	odd	6
1911.1.w.e.116.2		8	7.4	even	3	inner
1911.1.w.e.116.2		8	273.116	odd	6	inner
1911.1.w.e.116.3		8	21.11	odd	6	inner
1911.1.w.e.116.3		8	91.25	even	6	inner
1911.1.w.e.1598.2		8	1.1	even	1	trivial
1911.1.w.e.1598.2		8	39.38	odd	2	CM
1911.1.w.e.1598.3		8	3.2	odd	2	inner
1911.1.w.e.1598.3		8	13.12	even	2	inner
1911.1.w.f.116.2		8	7.3	odd	6
1911.1.w.f.116.2		8	273.38	even	6
1911.1.w.f.116.3		8	21.17	even	6
1911.1.w.f.116.3		8	91.38	odd	6
1911.1.w.f.1598.2		8	7.6	odd	2
1911.1.w.f.1598.2		8	273.272	even	2
1911.1.w.f.1598.3		8	21.20	even	2
1911.1.w.f.1598.3		8	91.90	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1911.w (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.382683 + 0.662827i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(0.207107 + 0.358719i) q^{4} +(-0.923880 + 1.60021i) q^{5} +0.765367 q^{6} -1.08239 q^{8} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.382683 + 0.662827i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(0.207107 + 0.358719i) q^{4} +(-0.923880 + 1.60021i) q^{5} +0.765367 q^{6} -1.08239 q^{8} +(-0.500000 + 0.866025i) q^{9} +(-0.707107 - 1.22474i) q^{10} +(0.382683 + 0.662827i) q^{11} +(0.207107 - 0.358719i) q^{12} -1.00000 q^{13} +1.84776 q^{15} +(0.207107 - 0.358719i) q^{16} +(-0.382683 - 0.662827i) q^{18} -0.765367 q^{20} -0.585786 q^{22} +(0.541196 + 0.937379i) q^{24} +(-1.20711 - 2.09077i) q^{25} +(0.382683 - 0.662827i) q^{26} +1.00000 q^{27} +(-0.707107 + 1.22474i) q^{30} +(-0.382683 - 0.662827i) q^{32} +(0.382683 - 0.662827i) q^{33} -0.414214 q^{36} +(0.500000 + 0.866025i) q^{39} +(1.00000 - 1.73205i) q^{40} -1.84776 q^{41} +(-0.158513 + 0.274552i) q^{44} +(-0.923880 - 1.60021i) q^{45} +(0.923880 - 1.60021i) q^{47} -0.414214 q^{48} +1.84776 q^{50} +(-0.207107 - 0.358719i) q^{52} +(-0.382683 + 0.662827i) q^{54} -1.41421 q^{55} +(0.382683 + 0.662827i) q^{59} +(0.382683 + 0.662827i) q^{60} +1.00000 q^{64} +(0.923880 - 1.60021i) q^{65} +(0.292893 + 0.507306i) q^{66} -1.84776 q^{71} +(0.541196 - 0.937379i) q^{72} +(-1.20711 + 2.09077i) q^{75} -0.765367 q^{78} +(-0.707107 + 1.22474i) q^{79} +(0.382683 + 0.662827i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(0.707107 - 1.22474i) q^{82} +0.765367 q^{83} +(-0.414214 - 0.717439i) q^{88} +(-0.382683 + 0.662827i) q^{89} +1.41421 q^{90} +(0.707107 + 1.22474i) q^{94} +(-0.382683 + 0.662827i) q^{96} -0.765367 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} - 4 q^{4} - 4 q^{9}+O(q^{10}) \) 8 * q - 4 * q^3 - 4 * q^4 - 4 * q^9	\( 8 q - 4 q^{3} - 4 q^{4} - 4 q^{9} - 4 q^{12} - 8 q^{13} - 4 q^{16} - 16 q^{22} - 4 q^{25} + 8 q^{27} + 8 q^{36} + 4 q^{39} + 8 q^{40} + 8 q^{48} + 4 q^{52} + 8 q^{64} + 8 q^{66} - 4 q^{75} - 4 q^{81} + 8 q^{88}+O(q^{100}) \) 8 * q - 4 * q^3 - 4 * q^4 - 4 * q^9 - 4 * q^12 - 8 * q^13 - 4 * q^16 - 16 * q^22 - 4 * q^25 + 8 * q^27 + 8 * q^36 + 4 * q^39 + 8 * q^40 + 8 * q^48 + 4 * q^52 + 8 * q^64 + 8 * q^66 - 4 * q^75 - 4 * q^81 + 8 * q^88

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