LMFDB - Embedded newform 3751.1.cf.b.118.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3751,1,Mod(118,3751)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3751, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([9, 10])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3751.118"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 3751 = 11^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3751.cf (of order $30$, degree $8$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.87199286239$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{30})$
Coefficient field:	16.0.26873856000000000000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 2x^{14} - 8x^{10} - 16x^{8} - 32x^{6} + 128x^{2} + 256 $ x^16 + 2x^14 - 8x^10 - 16x^8 - 32x^6 + 128*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.1279091.2

Embedding invariants

Embedding label			118.2
Root			$-0.575212 + 1.29195i$ of defining polynomial
Character	$\chi$	$=$	3751.118
Dual form			3751.1.cf.b.2702.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.309017 - 0.951057i) q^{4} +(0.913545 - 0.406737i) q^{5} +(0.294032 + 1.38331i) q^{7} +(0.104528 + 0.994522i) q^{9} +(-0.809017 - 0.587785i) q^{16} +(-1.40647 - 0.147826i) q^{17} +(-0.104528 - 0.994522i) q^{20} +1.00000 q^{23} +(1.40647 + 0.147826i) q^{28} +(1.34500 + 0.437016i) q^{29} +(0.913545 + 0.406737i) q^{31} +(0.831254 + 1.14412i) q^{35} +(0.978148 + 0.207912i) q^{36} +(0.978148 - 0.207912i) q^{37} +(0.500000 + 0.866025i) q^{45} +(-0.309017 - 0.951057i) q^{47} +(-0.913545 + 0.406737i) q^{49} +(-0.104528 - 0.994522i) q^{53} +(0.978148 - 0.207912i) q^{59} +(0.831254 - 1.14412i) q^{61} +(-1.34500 + 0.437016i) q^{63} +(-0.809017 + 0.587785i) q^{64} +(-0.500000 + 0.866025i) q^{67} +(-0.575212 + 1.29195i) q^{68} +(-0.913545 + 0.406737i) q^{71} +(1.05097 - 0.946294i) q^{73} +(-0.575212 + 1.29195i) q^{79} +(-0.978148 - 0.207912i) q^{80} +(-0.978148 + 0.207912i) q^{81} +(0.575212 + 1.29195i) q^{83} +(-1.34500 + 0.437016i) q^{85} -1.00000 q^{89} +(0.309017 - 0.951057i) q^{92} +(-0.809017 + 0.587785i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{4} + 2 q^{5} - 2 q^{9} - 4 q^{16} + 2 q^{20} + 16 q^{23} + 2 q^{31} - 2 q^{36} - 2 q^{37} + 8 q^{45} + 4 q^{47} - 2 q^{49} + 2 q^{53} - 2 q^{59} - 4 q^{64} - 8 q^{67} - 2 q^{71} + 2 q^{80} + 2 q^{81}+ \cdots - 4 q^{97}+O(q^{100}) $ 16 * q - 4 * q^4 + 2 * q^5 - 2 * q^9 - 4 * q^16 + 2 * q^20 + 16 * q^23 + 2 * q^31 - 2 * q^36 - 2 * q^37 + 8 * q^45 + 4 * q^47 - 2 * q^49 + 2 * q^53 - 2 * q^59 - 4 * q^64 - 8 * q^67 - 2 * q^71 + 2 * q^80 + 2 * q^81 - 16 * q^89 - 4 * q^92 - 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3751\mathbb{Z}\right)^\times$.

$n$	$2421$	$2543$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{3}{10}\right)$

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			0		0.809017	−	0.587785i	$-0.200000\pi$
$2$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$3$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$3$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$4$	0.309017	−	0.951057i	0.309017	−	0.951057i
$5$	0.913545	−	0.406737i	0.913545	−	0.406737i	0.104528	−	0.994522i	$-0.466667\pi$
$5$	0.913545	−	0.406737i	0.913545	−	0.406737i	0.809017	+	0.587785i	$0.200000\pi$
$6$		0			0
$7$	0.294032	+	1.38331i	0.294032	+	1.38331i	0.838671	+	0.544639i	$0.183333\pi$
$7$	0.294032	+	1.38331i	0.294032	+	1.38331i	−0.544639	+	0.838671i	$0.683333\pi$
$8$		0			0
$9$	0.104528	+	0.994522i	0.104528	+	0.994522i
$10$		0			0
$11$		0			0
$11$		0			0
$12$		0			0
$13$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$13$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$14$		0			0
$15$		0			0
$16$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$17$	−1.40647	−	0.147826i	−1.40647	−	0.147826i	−0.629320	−	0.777146i	$-0.716667\pi$
$17$	−1.40647	−	0.147826i	−1.40647	−	0.147826i	−0.777146	+	0.629320i	$0.783333\pi$
$18$		0			0
$19$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$19$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$20$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$21$		0			0
$22$		0			0
$23$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$23$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$	1.40647	+	0.147826i	1.40647	+	0.147826i
$29$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$29$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$30$		0			0
$31$	0.913545	+	0.406737i	0.913545	+	0.406737i
$31$	0.913545	+	0.406737i	0.913545	+	0.406737i
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.831254	+	1.14412i	0.831254	+	1.14412i
$36$	0.978148	+	0.207912i	0.978148	+	0.207912i
$37$	0.978148	−	0.207912i	0.978148	−	0.207912i	0.309017	−	0.951057i	$-0.400000\pi$
$37$	0.978148	−	0.207912i	0.978148	−	0.207912i	0.669131	+	0.743145i	$0.266667\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$41$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$42$		0			0
$43$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$43$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$44$		0			0
$45$	0.500000	+	0.866025i	0.500000	+	0.866025i
$46$		0			0
$47$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	−0.978148	−	0.207912i	$-0.933333\pi$
$47$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	0.669131	−	0.743145i	$-0.266667\pi$
$48$		0			0
$49$	−0.913545	+	0.406737i	−0.913545	+	0.406737i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.104528	−	0.994522i	−0.104528	−	0.994522i	−0.913545	−	0.406737i	$-0.866667\pi$
$53$	−0.104528	−	0.994522i	−0.104528	−	0.994522i	0.809017	−	0.587785i	$-0.200000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.978148	−	0.207912i	0.978148	−	0.207912i	0.309017	−	0.951057i	$-0.400000\pi$
$59$	0.978148	−	0.207912i	0.978148	−	0.207912i	0.669131	+	0.743145i	$0.266667\pi$
$60$		0			0
$61$	0.831254	−	1.14412i	0.831254	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$61$	0.831254	−	1.14412i	0.831254	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$62$		0			0
$63$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$64$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$65$		0			0
$66$		0			0
$67$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$67$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$68$	−0.575212	+	1.29195i	−0.575212	+	1.29195i
$69$		0			0
$70$		0			0
$71$	−0.913545	+	0.406737i	−0.913545	+	0.406737i	−0.809017	−	0.587785i	$-0.800000\pi$
$71$	−0.913545	+	0.406737i	−0.913545	+	0.406737i	−0.104528	+	0.994522i	$0.533333\pi$
$72$		0			0
$73$	1.05097	−	0.946294i	1.05097	−	0.946294i	0.0523360	−	0.998630i	$-0.483333\pi$
$73$	1.05097	−	0.946294i	1.05097	−	0.946294i	0.998630	+	0.0523360i	$0.0166667\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.575212	+	1.29195i	−0.575212	+	1.29195i	0.358368	+	0.933580i	$0.383333\pi$
$79$	−0.575212	+	1.29195i	−0.575212	+	1.29195i	−0.933580	+	0.358368i	$0.883333\pi$
$80$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$81$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$82$		0			0
$83$	0.575212	+	1.29195i	0.575212	+	1.29195i	0.933580	+	0.358368i	$0.116667\pi$
$83$	0.575212	+	1.29195i	0.575212	+	1.29195i	−0.358368	+	0.933580i	$0.616667\pi$
$84$		0			0
$85$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$89$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$		0			0
$92$	0.309017	−	0.951057i	0.309017	−	0.951057i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$97$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$101$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$102$		0			0
$103$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$103$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.05097	−	0.946294i	−1.05097	−	0.946294i	−0.0523360	−	0.998630i	$-0.516667\pi$
$107$	−1.05097	−	0.946294i	−1.05097	−	0.946294i	−0.998630	+	0.0523360i	$0.983333\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	0.575212	−	1.29195i	0.575212	−	1.29195i
$113$	−0.978148	−	0.207912i	−0.978148	−	0.207912i	−0.309017	−	0.951057i	$-0.600000\pi$
$113$	−0.978148	−	0.207912i	−0.978148	−	0.207912i	−0.669131	+	0.743145i	$0.733333\pi$
$114$		0			0
$115$	0.913545	−	0.406737i	0.913545	−	0.406737i
$116$	0.831254	−	1.14412i	0.831254	−	1.14412i
$117$		0			0
$118$		0			0
$119$	−0.209057	−	1.98904i	−0.209057	−	1.98904i
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	0.669131	−	0.743145i	0.669131	−	0.743145i
$125$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$126$		0			0
$127$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$127$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$131$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$137$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$138$		0			0
$139$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$139$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$140$	1.34500	−	0.437016i	1.34500	−	0.437016i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.500000	−	0.866025i	0.500000	−	0.866025i
$145$	1.40647	−	0.147826i	1.40647	−	0.147826i
$146$		0			0
$147$		0			0
$148$	0.104528	−	0.994522i	0.104528	−	0.994522i
$149$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$149$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$150$		0			0
$151$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$151$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$152$		0			0
$153$		−	1.41421i		−	1.41421i
$154$		0			0
$155$	1.00000			1.00000
$156$		0			0
$157$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$157$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.294032	+	1.38331i	0.294032	+	1.38331i
$162$		0			0
$163$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$163$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$167$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$168$		0			0
$169$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.294032	−	1.38331i	0.294032	−	1.38331i	−0.544639	−	0.838671i	$-0.683333\pi$
$173$	0.294032	−	1.38331i	0.294032	−	1.38331i	0.838671	−	0.544639i	$-0.183333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	0.669131	−	0.743145i	0.669131	−	0.743145i	−0.309017	−	0.951057i	$-0.600000\pi$
$179$	0.669131	−	0.743145i	0.669131	−	0.743145i	0.978148	+	0.207912i	$0.0666667\pi$
$180$	0.978148	−	0.207912i	0.978148	−	0.207912i
$181$	0.913545	−	0.406737i	0.913545	−	0.406737i	0.104528	−	0.994522i	$-0.466667\pi$
$181$	0.913545	−	0.406737i	0.913545	−	0.406737i	0.809017	+	0.587785i	$0.200000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.809017	−	0.587785i	0.809017	−	0.587785i
$186$		0			0
$187$		0			0
$188$	−1.00000			−1.00000
$189$		0			0
$190$		0			0
$191$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$191$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$192$		0			0
$193$	−1.40647	−	0.147826i	−1.40647	−	0.147826i	−0.629320	−	0.777146i	$-0.716667\pi$
$193$	−1.40647	−	0.147826i	−1.40647	−	0.147826i	−0.777146	+	0.629320i	$0.783333\pi$
$194$		0			0
$195$		0			0
$196$	0.104528	+	0.994522i	0.104528	+	0.994522i
$197$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$197$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$198$		0			0
$199$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$199$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.209057	+	1.98904i	−0.209057	+	1.98904i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.104528	+	0.994522i	0.104528	+	0.994522i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.575212	+	1.29195i	−0.575212	+	1.29195i	0.358368	+	0.933580i	$0.383333\pi$
$211$	−0.575212	+	1.29195i	−0.575212	+	1.29195i	−0.933580	+	0.358368i	$0.883333\pi$
$212$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.294032	+	1.38331i	−0.294032	+	1.38331i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.669131	−	0.743145i	0.669131	−	0.743145i	−0.309017	−	0.951057i	$-0.600000\pi$
$223$	0.669131	−	0.743145i	0.669131	−	0.743145i	0.978148	+	0.207912i	$0.0666667\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$227$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$228$		0			0
$229$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$229$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$233$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$234$		0			0
$235$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$236$	0.104528	−	0.994522i	0.104528	−	0.994522i
$237$		0			0
$238$		0			0
$239$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$239$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$		0			0
$243$		0			0
$244$	−0.831254	−	1.14412i	−0.831254	−	1.14412i
$245$	−0.669131	+	0.743145i	−0.669131	+	0.743145i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	0.913545	+	0.406737i	0.913545	+	0.406737i	0.809017	−	0.587785i	$-0.200000\pi$
$251$	0.913545	+	0.406737i	0.913545	+	0.406737i	0.104528	+	0.994522i	$0.466667\pi$
$252$			1.41421i			1.41421i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.309017	+	0.951057i	0.309017	+	0.951057i
$257$	−0.669131	−	0.743145i	−0.669131	−	0.743145i	0.309017	−	0.951057i	$-0.400000\pi$
$257$	−0.669131	−	0.743145i	−0.669131	−	0.743145i	−0.978148	+	0.207912i	$0.933333\pi$
$258$		0			0
$259$	0.575212	+	1.29195i	0.575212	+	1.29195i
$260$		0			0
$261$	−0.294032	+	1.38331i	−0.294032	+	1.38331i
$262$		0			0
$263$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$263$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$264$		0			0
$265$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$266$		0			0
$267$		0			0
$268$	0.669131	+	0.743145i	0.669131	+	0.743145i
$269$	0.913545	−	0.406737i	0.913545	−	0.406737i	0.104528	−	0.994522i	$-0.466667\pi$
$269$	0.913545	−	0.406737i	0.913545	−	0.406737i	0.809017	+	0.587785i	$0.200000\pi$
$270$		0			0
$271$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$271$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$272$	1.05097	+	0.946294i	1.05097	+	0.946294i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$277$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$278$		0			0
$279$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$280$		0			0
$281$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$281$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$282$		0			0
$283$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$283$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$284$	0.104528	+	0.994522i	0.104528	+	0.994522i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.978148	+	0.207912i	0.978148	+	0.207912i
$290$		0			0
$291$		0			0
$292$	−0.575212	−	1.29195i	−0.575212	−	1.29195i
$293$	−0.294032	−	1.38331i	−0.294032	−	1.38331i	−0.838671	−	0.544639i	$-0.816667\pi$
$293$	−0.294032	−	1.38331i	−0.294032	−	1.38331i	0.544639	−	0.838671i	$-0.316667\pi$
$294$		0			0
$295$	0.809017	−	0.587785i	0.809017	−	0.587785i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.294032	−	1.38331i	0.294032	−	1.38331i
$306$		0			0
$307$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$307$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$311$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$312$		0			0
$313$	−0.104528	+	0.994522i	−0.104528	+	0.994522i	0.809017	+	0.587785i	$0.200000\pi$
$313$	−0.104528	+	0.994522i	−0.104528	+	0.994522i	−0.913545	+	0.406737i	$0.866667\pi$
$314$		0			0
$315$	−1.05097	+	0.946294i	−1.05097	+	0.946294i
$316$	1.05097	+	0.946294i	1.05097	+	0.946294i
$317$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$317$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$318$		0			0
$319$		0			0
$320$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.22474	−	0.707107i	1.22474	−	0.707107i
$330$		0			0
$331$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$331$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$332$	1.40647	−	0.147826i	1.40647	−	0.147826i
$333$	0.309017	+	0.951057i	0.309017	+	0.951057i
$334$		0			0
$335$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$336$		0			0
$337$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$337$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$338$		0			0
$339$		0			0
$340$			1.41421i			1.41421i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.40647	−	0.147826i	−1.40647	−	0.147826i	−0.629320	−	0.777146i	$-0.716667\pi$
$347$	−1.40647	−	0.147826i	−1.40647	−	0.147826i	−0.777146	+	0.629320i	$0.783333\pi$
$348$		0			0
$349$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$349$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.453990	+	0.891007i	$0.650000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$353$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$354$		0			0
$355$	−0.669131	+	0.743145i	−0.669131	+	0.743145i
$356$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$357$		0			0
$358$		0			0
$359$	1.05097	−	0.946294i	1.05097	−	0.946294i	0.0523360	−	0.998630i	$-0.483333\pi$
$359$	1.05097	−	0.946294i	1.05097	−	0.946294i	0.998630	+	0.0523360i	$0.0166667\pi$
$360$		0			0
$361$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.575212	−	1.29195i	0.575212	−	1.29195i
$366$		0			0
$367$	0.669131	+	0.743145i	0.669131	+	0.743145i	0.978148	−	0.207912i	$-0.0666667\pi$
$367$	0.669131	+	0.743145i	0.669131	+	0.743145i	−0.309017	+	0.951057i	$0.600000\pi$
$368$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$369$		0			0
$370$		0			0
$371$	1.34500	−	0.437016i	1.34500	−	0.437016i
$372$		0			0
$373$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$373$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$379$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.104528	−	0.994522i	−0.104528	−	0.994522i	−0.913545	−	0.406737i	$-0.866667\pi$
$383$	−0.104528	−	0.994522i	−0.104528	−	0.994522i	0.809017	−	0.587785i	$-0.200000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.309017	+	0.951057i	0.309017	+	0.951057i
$389$	−0.978148	+	0.207912i	−0.978148	+	0.207912i	−0.669131	−	0.743145i	$-0.733333\pi$
$389$	−0.978148	+	0.207912i	−0.978148	+	0.207912i	−0.309017	+	0.951057i	$0.600000\pi$
$390$		0			0
$391$	−1.40647	−	0.147826i	−1.40647	−	0.147826i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			1.41421i			1.41421i
$396$		0			0
$397$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$397$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$401$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$409$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	0.575212	+	1.29195i	0.575212	+	1.29195i
$414$		0			0
$415$	1.05097	+	0.946294i	1.05097	+	0.946294i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$419$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	0.978148	+	0.207912i	0.978148	+	0.207912i	0.669131	−	0.743145i	$-0.266667\pi$
$421$	0.978148	+	0.207912i	0.978148	+	0.207912i	0.309017	+	0.951057i	$0.400000\pi$
$422$		0			0
$423$	0.913545	−	0.406737i	0.913545	−	0.406737i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.82709	+	0.813473i	1.82709	+	0.813473i
$428$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$429$		0			0
$430$		0			0
$431$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$431$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$432$		0			0
$433$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$433$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$439$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$440$		0			0
$441$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$442$		0			0
$443$	0.978148	+	0.207912i	0.978148	+	0.207912i	0.669131	−	0.743145i	$-0.266667\pi$
$443$	0.978148	+	0.207912i	0.978148	+	0.207912i	0.309017	+	0.951057i	$0.400000\pi$
$444$		0			0
$445$	−0.913545	+	0.406737i	−0.913545	+	0.406737i
$446$		0			0
$447$		0			0
$448$	−1.05097	−	0.946294i	−1.05097	−	0.946294i
$449$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$449$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$450$		0			0
$451$		0			0
$452$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$457$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$458$		0			0
$459$		0			0
$460$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$	−0.831254	−	1.14412i	−0.831254	−	1.14412i
$465$		0			0
$466$		0			0
$467$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$467$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$468$		0			0
$469$	−1.34500	−	0.437016i	−1.34500	−	0.437016i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$	−1.95630	−	0.415823i	−1.95630	−	0.415823i
$477$	0.978148	−	0.207912i	0.978148	−	0.207912i
$478$		0			0
$479$	1.40647	+	0.147826i	1.40647	+	0.147826i	0.777146	−	0.629320i	$-0.216667\pi$
$479$	1.40647	+	0.147826i	1.40647	+	0.147826i	0.629320	+	0.777146i	$0.283333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$486$		0			0
$487$	−1.95630	−	0.415823i	−1.95630	−	0.415823i	−0.978148	−	0.207912i	$-0.933333\pi$
$487$	−1.95630	−	0.415823i	−1.95630	−	0.415823i	−0.978148	−	0.207912i	$-0.933333\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.05097	−	0.946294i	1.05097	−	0.946294i	0.0523360	−	0.998630i	$-0.483333\pi$
$491$	1.05097	−	0.946294i	1.05097	−	0.946294i	0.998630	+	0.0523360i	$0.0166667\pi$
$492$		0			0
$493$	−1.82709	−	0.813473i	−1.82709	−	0.813473i
$494$		0			0
$495$		0			0
$496$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$497$	−0.831254	−	1.14412i	−0.831254	−	1.14412i
$498$		0			0
$499$	−0.978148	+	0.207912i	−0.978148	+	0.207912i	−0.669131	−	0.743145i	$-0.733333\pi$
$499$	−0.978148	+	0.207912i	−0.978148	+	0.207912i	−0.309017	+	0.951057i	$0.600000\pi$
$500$	0.809017	+	0.587785i	0.809017	+	0.587785i
$501$		0			0
$502$		0			0
$503$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$503$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.978148	+	0.207912i	0.978148	+	0.207912i	0.669131	−	0.743145i	$-0.266667\pi$
$509$	0.978148	+	0.207912i	0.978148	+	0.207912i	0.309017	+	0.951057i	$0.400000\pi$
$510$		0			0
$511$	1.61803	+	1.17557i	1.61803	+	1.17557i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.978148	+	0.207912i	−0.978148	+	0.207912i	−0.669131	−	0.743145i	$-0.733333\pi$
$521$	−0.978148	+	0.207912i	−0.978148	+	0.207912i	−0.309017	+	0.951057i	$0.600000\pi$
$522$		0			0
$523$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$523$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$524$	−0.294032	−	1.38331i	−0.294032	−	1.38331i
$525$		0			0
$526$		0			0
$527$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$528$		0			0
$529$		0			0
$530$		0			0
$531$	0.309017	+	0.951057i	0.309017	+	0.951057i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−1.34500	−	0.437016i	−1.34500	−	0.437016i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$541$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$547$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$548$		0			0
$549$	1.22474	+	0.707107i	1.22474	+	0.707107i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.95630	−	0.415823i	−1.95630	−	0.415823i
$554$		0			0
$555$		0			0
$556$	−0.831254	+	1.14412i	−0.831254	+	1.14412i
$557$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$557$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$558$		0			0
$559$		0			0
$560$		−	1.41421i		−	1.41421i
$561$		0			0
$562$		0			0
$563$	1.40647	−	0.147826i	1.40647	−	0.147826i	0.629320	−	0.777146i	$-0.283333\pi$
$563$	1.40647	−	0.147826i	1.40647	−	0.147826i	0.777146	+	0.629320i	$0.216667\pi$
$564$		0			0
$565$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$566$		0			0
$567$	−0.575212	−	1.29195i	−0.575212	−	1.29195i
$568$		0			0
$569$	0.294032	−	1.38331i	0.294032	−	1.38331i	−0.544639	−	0.838671i	$-0.683333\pi$
$569$	0.294032	−	1.38331i	0.294032	−	1.38331i	0.838671	−	0.544639i	$-0.183333\pi$
$570$		0			0
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$577$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$577$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$578$		0			0
$579$		0			0
$580$	0.294032	−	1.38331i	0.294032	−	1.38331i
$581$	−1.61803	+	1.17557i	−1.61803	+	1.17557i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$587$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.913545	−	0.406737i	−0.913545	−	0.406737i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	0.913545	−	0.406737i	0.913545	−	0.406737i	0.104528	−	0.994522i	$-0.466667\pi$
$599$	0.913545	−	0.406737i	0.913545	−	0.406737i	0.809017	+	0.587785i	$0.200000\pi$
$600$		0			0
$601$	0.294032	+	1.38331i	0.294032	+	1.38331i	0.838671	+	0.544639i	$0.183333\pi$
$601$	0.294032	+	1.38331i	0.294032	+	1.38331i	−0.544639	+	0.838671i	$0.683333\pi$
$602$		0			0
$603$	−0.913545	−	0.406737i	−0.913545	−	0.406737i
$604$		−	1.41421i		−	1.41421i
$605$		0			0
$606$		0			0
$607$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$607$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−1.34500	−	0.437016i	−1.34500	−	0.437016i
$613$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$613$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$617$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$618$		0			0
$619$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	−0.978148	−	0.207912i	$-0.933333\pi$
$619$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	0.669131	−	0.743145i	$-0.266667\pi$
$620$	0.309017	−	0.951057i	0.309017	−	0.951057i
$621$		0			0
$622$		0			0
$623$	−0.294032	−	1.38331i	−0.294032	−	1.38331i
$624$		0			0
$625$	0.104528	+	0.994522i	0.104528	+	0.994522i
$626$		0			0
$627$		0			0
$628$	1.00000			1.00000
$629$	−1.40647	+	0.147826i	−1.40647	+	0.147826i
$630$		0			0
$631$	0.669131	+	0.743145i	0.669131	+	0.743145i	0.978148	−	0.207912i	$-0.0666667\pi$
$631$	0.669131	+	0.743145i	0.669131	+	0.743145i	−0.309017	+	0.951057i	$0.600000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$640$		0			0
$641$	0.978148	+	0.207912i	0.978148	+	0.207912i	0.669131	−	0.743145i	$-0.266667\pi$
$641$	0.978148	+	0.207912i	0.978148	+	0.207912i	0.309017	+	0.951057i	$0.400000\pi$
$642$		0			0
$643$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$643$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$644$	1.40647	+	0.147826i	1.40647	+	0.147826i
$645$		0			0
$646$		0			0
$647$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$647$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.309017	+	0.951057i	0.309017	+	0.951057i
$653$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$653$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$654$		0			0
$655$	0.831254	−	1.14412i	0.831254	−	1.14412i
$656$		0			0
$657$	1.05097	+	0.946294i	1.05097	+	0.946294i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$661$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1.34500	+	0.437016i	1.34500	+	0.437016i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$673$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$674$		0			0
$675$		0			0
$676$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$677$	0.575212	+	1.29195i	0.575212	+	1.29195i	0.933580	+	0.358368i	$0.116667\pi$
$677$	0.575212	+	1.29195i	0.575212	+	1.29195i	−0.358368	+	0.933580i	$0.616667\pi$
$678$		0			0
$679$	−1.05097	−	0.946294i	−1.05097	−	0.946294i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$683$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.104528	+	0.994522i	0.104528	+	0.994522i	0.913545	+	0.406737i	$0.133333\pi$
$691$	0.104528	+	0.994522i	0.104528	+	0.994522i	−0.809017	+	0.587785i	$0.800000\pi$
$692$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$693$		0			0
$694$		0			0
$695$	−1.40647	+	0.147826i	−1.40647	+	0.147826i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$701$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$709$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$710$		0			0
$711$	−1.34500	−	0.437016i	−1.34500	−	0.437016i
$712$		0			0
$713$	0.913545	+	0.406737i	0.913545	+	0.406737i
$714$		0			0
$715$		0			0
$716$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$717$		0			0
$718$		0			0
$719$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$719$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$720$	0.104528	−	0.994522i	0.104528	−	0.994522i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$725$		0			0
$726$		0			0
$727$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$727$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$733$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.40647	−	0.147826i	1.40647	−	0.147826i	0.629320	−	0.777146i	$-0.283333\pi$
$739$	1.40647	−	0.147826i	1.40647	−	0.147826i	0.777146	+	0.629320i	$0.216667\pi$
$740$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$741$		0			0
$742$		0			0
$743$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$743$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$748$		0			0
$749$	1.00000	−	1.73205i	1.00000	−	1.73205i
$750$		0			0
$751$	0.669131	−	0.743145i	0.669131	−	0.743145i	−0.309017	−	0.951057i	$-0.600000\pi$
$751$	0.669131	−	0.743145i	0.669131	−	0.743145i	0.978148	+	0.207912i	$0.0666667\pi$
$752$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$753$		0			0
$754$		0			0
$755$	1.05097	−	0.946294i	1.05097	−	0.946294i
$756$		0			0
$757$	0.104528	+	0.994522i	0.104528	+	0.994522i	0.913545	+	0.406737i	$0.133333\pi$
$757$	0.104528	+	0.994522i	0.104528	+	0.994522i	−0.809017	+	0.587785i	$0.800000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$761$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−0.575212	−	1.29195i	−0.575212	−	1.29195i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$769$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$770$		0			0
$771$		0			0
$772$	−0.575212	+	1.29195i	−0.575212	+	1.29195i
$773$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	−0.978148	−	0.207912i	$-0.933333\pi$
$773$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	0.669131	−	0.743145i	$-0.266667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.978148	+	0.207912i	0.978148	+	0.207912i
$785$	0.669131	+	0.743145i	0.669131	+	0.743145i
$786$		0			0
$787$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$787$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$788$	−1.05097	+	0.946294i	−1.05097	+	0.946294i
$789$		0			0
$790$		0			0
$791$		−	1.41421i		−	1.41421i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$797$	−0.913545	+	0.406737i	−0.913545	+	0.406737i	−0.809017	−	0.587785i	$-0.800000\pi$
$797$	−0.913545	+	0.406737i	−0.913545	+	0.406737i	−0.104528	+	0.994522i	$0.533333\pi$
$798$		0			0
$799$	0.294032	+	1.38331i	0.294032	+	1.38331i
$800$		0			0
$801$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	0.831254	+	1.14412i	0.831254	+	1.14412i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$809$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$810$		0			0
$811$	−0.294032	+	1.38331i	−0.294032	+	1.38331i	0.544639	+	0.838671i	$0.316667\pi$
$811$	−0.294032	+	1.38331i	−0.294032	+	1.38331i	−0.838671	+	0.544639i	$0.816667\pi$
$812$	1.82709	+	0.813473i	1.82709	+	0.813473i
$813$		0			0
$814$		0			0
$815$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$821$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$822$		0			0
$823$	0.913545	+	0.406737i	0.913545	+	0.406737i	0.809017	−	0.587785i	$-0.200000\pi$
$823$	0.913545	+	0.406737i	0.913545	+	0.406737i	0.104528	+	0.994522i	$0.466667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$827$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$828$	0.978148	+	0.207912i	0.978148	+	0.207912i
$829$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$829$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.34500	−	0.437016i	1.34500	−	0.437016i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$839$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$840$		0			0
$841$	0.809017	+	0.587785i	0.809017	+	0.587785i
$842$		0			0
$843$		0			0
$844$	1.05097	+	0.946294i	1.05097	+	0.946294i
$845$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$846$		0			0
$847$		0			0
$848$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$849$		0			0
$850$		0			0
$851$	0.978148	−	0.207912i	0.978148	−	0.207912i
$852$		0			0
$853$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$853$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$859$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.913545	+	0.406737i	−0.913545	+	0.406737i	−0.809017	−	0.587785i	$-0.800000\pi$
$863$	−0.913545	+	0.406737i	−0.913545	+	0.406737i	−0.104528	+	0.994522i	$0.533333\pi$
$864$		0			0
$865$	−0.294032	−	1.38331i	−0.294032	−	1.38331i
$866$		0			0
$867$		0			0
$868$	1.22474	+	0.707107i	1.22474	+	0.707107i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$874$		0			0
$875$	−1.40647	−	0.147826i	−1.40647	−	0.147826i
$876$		0			0
$877$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$877$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$881$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$882$		0			0
$883$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$883$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.294032	+	1.38331i	0.294032	+	1.38331i	0.838671	+	0.544639i	$0.183333\pi$
$887$	0.294032	+	1.38331i	0.294032	+	1.38331i	−0.544639	+	0.838671i	$0.683333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$893$		0			0
$894$		0			0
$895$	0.309017	−	0.951057i	0.309017	−	0.951057i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.05097	+	0.946294i	1.05097	+	0.946294i
$900$		0			0
$901$			1.41421i			1.41421i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.669131	−	0.743145i	0.669131	−	0.743145i
$906$		0			0
$907$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$907$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$911$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.33826	+	1.48629i	1.33826	+	1.48629i
$918$		0			0
$919$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$919$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$929$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$930$		0			0
$931$		0			0
$932$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.575212	−	1.29195i	0.575212	−	1.29195i	−0.358368	−	0.933580i	$-0.616667\pi$
$937$	0.575212	−	1.29195i	0.575212	−	1.29195i	0.933580	−	0.358368i	$-0.116667\pi$
$938$		0			0
$939$		0			0
$940$	−0.913545	+	0.406737i	−0.913545	+	0.406737i
$941$	0.575212	+	1.29195i	0.575212	+	1.29195i	0.933580	+	0.358368i	$0.116667\pi$
$941$	0.575212	+	1.29195i	0.575212	+	1.29195i	−0.358368	+	0.933580i	$0.616667\pi$
$942$		0			0
$943$		0			0
$944$	−0.913545	−	0.406737i	−0.913545	−	0.406737i
$945$		0			0
$946$		0			0
$947$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$947$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$953$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.669131	+	0.743145i	0.669131	+	0.743145i
$962$		0			0
$963$	0.831254	−	1.14412i	0.831254	−	1.14412i
$964$		0			0
$965$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−0.978148	−	0.207912i	−0.978148	−	0.207912i	−0.309017	−	0.951057i	$-0.600000\pi$
$971$	−0.978148	−	0.207912i	−0.978148	−	0.207912i	−0.669131	+	0.743145i	$0.733333\pi$
$972$		0			0
$973$	0.209057	−	1.98904i	0.209057	−	1.98904i
$974$		0			0
$975$		0			0
$976$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$977$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$977$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$978$		0			0
$979$		0			0
$980$	0.500000	+	0.866025i	0.500000	+	0.866025i
$981$		0			0
$982$		0			0
$983$	−1.33826	−	1.48629i	−1.33826	−	1.48629i	−0.669131	−	0.743145i	$-0.733333\pi$
$983$	−1.33826	−	1.48629i	−1.33826	−	1.48629i	−0.669131	−	0.743145i	$-0.733333\pi$
$984$		0			0
$985$	−1.40647	−	0.147826i	−1.40647	−	0.147826i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$991$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.104528	−	0.994522i	0.104528	−	0.994522i
$996$		0			0
$997$	−1.05097	+	0.946294i	−1.05097	+	0.946294i	−0.998630	−	0.0523360i	$-0.983333\pi$
$997$	−1.05097	+	0.946294i	−1.05097	+	0.946294i	−0.0523360	+	0.998630i	$0.516667\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3751.1.cf.b.118.2		16
11.2	odd	10		3751.1.l.a.1451.2	yes	4
11.3	even	5	inner	3751.1.cf.b.2877.2		16
11.4	even	5	inner	3751.1.cf.b.645.1		16
11.5	even	5	inner	3751.1.cf.b.3621.2		16
11.6	odd	10	inner	3751.1.cf.b.3621.1		16
11.7	odd	10	inner	3751.1.cf.b.645.2		16
11.8	odd	10	inner	3751.1.cf.b.2877.1		16
11.9	even	5		3751.1.l.a.1451.1	✓	4
11.10	odd	2	inner	3751.1.cf.b.118.1		16
31.5	even	3	inner	3751.1.cf.b.2175.2		16
341.5	even	15	inner	3751.1.cf.b.1927.1		16
341.36	even	15	inner	3751.1.cf.b.1183.1		16
341.98	odd	6	inner	3751.1.cf.b.2175.1		16
341.129	odd	30	inner	3751.1.cf.b.1183.2		16
341.160	odd	30	inner	3751.1.cf.b.1927.2		16
341.191	even	15	inner	3751.1.cf.b.2702.1		16
341.222	odd	30		3751.1.l.a.3508.1	yes	4
341.284	even	15		3751.1.l.a.3508.2	yes	4
341.315	odd	30	inner	3751.1.cf.b.2702.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3751.1.l.a.1451.1	✓	4	11.9	even	5
3751.1.l.a.1451.2	yes	4	11.2	odd	10
3751.1.l.a.3508.1	yes	4	341.222	odd	30
3751.1.l.a.3508.2	yes	4	341.284	even	15
3751.1.cf.b.118.1		16	11.10	odd	2	inner
3751.1.cf.b.118.2		16	1.1	even	1	trivial
3751.1.cf.b.645.1		16	11.4	even	5	inner
3751.1.cf.b.645.2		16	11.7	odd	10	inner
3751.1.cf.b.1183.1		16	341.36	even	15	inner
3751.1.cf.b.1183.2		16	341.129	odd	30	inner
3751.1.cf.b.1927.1		16	341.5	even	15	inner
3751.1.cf.b.1927.2		16	341.160	odd	30	inner
3751.1.cf.b.2175.1		16	341.98	odd	6	inner
3751.1.cf.b.2175.2		16	31.5	even	3	inner
3751.1.cf.b.2702.1		16	341.191	even	15	inner
3751.1.cf.b.2702.2		16	341.315	odd	30	inner
3751.1.cf.b.2877.1		16	11.8	odd	10	inner
3751.1.cf.b.2877.2		16	11.3	even	5	inner
3751.1.cf.b.3621.1		16	11.6	odd	10	inner
3751.1.cf.b.3621.2		16	11.5	even	5	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3751 = 11^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3751.cf (of order \(30\), degree \(8\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.309017 - 0.951057i) q^{4} +(0.913545 - 0.406737i) q^{5} +(0.294032 + 1.38331i) q^{7} +(0.104528 + 0.994522i) q^{9} +(-0.809017 - 0.587785i) q^{16} +(-1.40647 - 0.147826i) q^{17} +(-0.104528 - 0.994522i) q^{20} +1.00000 q^{23} +(1.40647 + 0.147826i) q^{28} +(1.34500 + 0.437016i) q^{29} +(0.913545 + 0.406737i) q^{31} +(0.831254 + 1.14412i) q^{35} +(0.978148 + 0.207912i) q^{36} +(0.978148 - 0.207912i) q^{37} +(0.500000 + 0.866025i) q^{45} +(-0.309017 - 0.951057i) q^{47} +(-0.913545 + 0.406737i) q^{49} +(-0.104528 - 0.994522i) q^{53} +(0.978148 - 0.207912i) q^{59} +(0.831254 - 1.14412i) q^{61} +(-1.34500 + 0.437016i) q^{63} +(-0.809017 + 0.587785i) q^{64} +(-0.500000 + 0.866025i) q^{67} +(-0.575212 + 1.29195i) q^{68} +(-0.913545 + 0.406737i) q^{71} +(1.05097 - 0.946294i) q^{73} +(-0.575212 + 1.29195i) q^{79} +(-0.978148 - 0.207912i) q^{80} +(-0.978148 + 0.207912i) q^{81} +(0.575212 + 1.29195i) q^{83} +(-1.34500 + 0.437016i) q^{85} -1.00000 q^{89} +(0.309017 - 0.951057i) q^{92} +(-0.809017 + 0.587785i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{4} + 2 q^{5} - 2 q^{9} - 4 q^{16} + 2 q^{20} + 16 q^{23} + 2 q^{31} - 2 q^{36} - 2 q^{37} + 8 q^{45} + 4 q^{47} - 2 q^{49} + 2 q^{53} - 2 q^{59} - 4 q^{64} - 8 q^{67} - 2 q^{71} + 2 q^{80} + 2 q^{81}+ \cdots - 4 q^{97}+O(q^{100}) \) 16 * q - 4 * q^4 + 2 * q^5 - 2 * q^9 - 4 * q^16 + 2 * q^20 + 16 * q^23 + 2 * q^31 - 2 * q^36 - 2 * q^37 + 8 * q^45 + 4 * q^47 - 2 * q^49 + 2 * q^53 - 2 * q^59 - 4 * q^64 - 8 * q^67 - 2 * q^71 + 2 * q^80 + 2 * q^81 - 16 * q^89 - 4 * q^92 - 4 * q^97

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