LMFDB - Embedded newform 209.1.m.a.75.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [209,1,Mod(37,209)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(209, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([2, 5]))

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

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

chi := DirichletCharacter("209.37");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 209 = 11 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	209.m (of order $10$, degree $4$, 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:	$0.104304587640$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + x^{2} - x + 1 $ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.5285401.1

Embedding invariants

Embedding label			75.1
Root			$0.809017 - 0.587785i$ of defining polynomial
Character	$\chi$	$=$	209.75
Dual form			209.1.m.a.170.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.809017 - 0.587785i) q^{4} +(-0.500000 - 1.53884i) q^{5} +(1.30902 + 0.951057i) q^{7} +(0.309017 - 0.951057i) q^{9} +(-0.809017 + 0.587785i) q^{11} +(0.309017 + 0.951057i) q^{16} +(0.190983 + 0.587785i) q^{17} +(-0.809017 + 0.587785i) q^{19} +(-0.500000 + 1.53884i) q^{20} +0.618034 q^{23} +(-1.30902 + 0.951057i) q^{25} +(-0.500000 - 1.53884i) q^{28} +(0.809017 - 2.48990i) q^{35} +(-0.809017 + 0.587785i) q^{36} +0.618034 q^{43} +1.00000 q^{44} -1.61803 q^{45} +(-0.500000 + 0.363271i) q^{47} +(0.500000 + 1.53884i) q^{49} +(1.30902 + 0.951057i) q^{55} +(0.190983 + 0.587785i) q^{61} +(1.30902 - 0.951057i) q^{63} +(0.309017 - 0.951057i) q^{64} +(0.190983 - 0.587785i) q^{68} +(-1.61803 - 1.17557i) q^{73} +1.00000 q^{76} -1.61803 q^{77} +(1.30902 - 0.951057i) q^{80} +(-0.809017 - 0.587785i) q^{81} +(-0.500000 - 1.53884i) q^{83} +(0.809017 - 0.587785i) q^{85} +(-0.500000 - 0.363271i) q^{92} +(1.30902 + 0.951057i) q^{95} +(0.309017 + 0.951057i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{4} - 2 q^{5} + 3 q^{7} - q^{9} - q^{11} - q^{16} + 3 q^{17} - q^{19} - 2 q^{20} - 2 q^{23} - 3 q^{25} - 2 q^{28} + q^{35} - q^{36} - 2 q^{43} + 4 q^{44} - 2 q^{45} - 2 q^{47} + 2 q^{49}+ \cdots - q^{99}+O(q^{100}) $ 4 * q - q^4 - 2 * q^5 + 3 * q^7 - q^9 - q^11 - q^16 + 3 * q^17 - q^19 - 2 * q^20 - 2 * q^23 - 3 * q^25 - 2 * q^28 + q^35 - q^36 - 2 * q^43 + 4 * q^44 - 2 * q^45 - 2 * q^47 + 2 * q^49 + 3 * q^55 + 3 * q^61 + 3 * q^63 - q^64 + 3 * q^68 - 2 * q^73 + 4 * q^76 - 2 * q^77 + 3 * q^80 - q^81 - 2 * q^83 + q^85 - 2 * q^92 + 3 * q^95 - q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/209\mathbb{Z}\right)^\times$.

$n$	$78$	$134$
$\chi(n)$	$-1$	$e\left(\frac{3}{5}\right)$

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		0.309017	−	0.951057i	$-0.400000\pi$
$2$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$3$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$3$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$4$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$5$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$5$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$6$		0			0
$7$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$7$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$8$		0			0
$9$	0.309017	−	0.951057i	0.309017	−	0.951057i
$10$		0			0
$11$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$11$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$12$		0			0
$13$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$13$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$14$		0			0
$15$		0			0
$16$	0.309017	+	0.951057i	0.309017	+	0.951057i
$17$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$17$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$18$		0			0
$19$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$19$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$20$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$21$		0			0
$22$		0			0
$23$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$23$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$24$		0			0
$25$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$26$		0			0
$27$		0			0
$28$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$29$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$29$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$30$		0			0
$31$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$31$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.809017	−	2.48990i	0.809017	−	2.48990i
$36$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$37$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$37$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$41$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$42$		0			0
$43$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$43$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$44$	1.00000			1.00000
$45$	−1.61803			−1.61803
$46$		0			0
$47$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$47$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$48$		0			0
$49$	0.500000	+	1.53884i	0.500000	+	1.53884i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$53$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$54$		0			0
$55$	1.30902	+	0.951057i	1.30902	+	0.951057i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$59$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$60$		0			0
$61$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$61$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$62$		0			0
$63$	1.30902	−	0.951057i	1.30902	−	0.951057i
$64$	0.309017	−	0.951057i	0.309017	−	0.951057i
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	0.190983	−	0.587785i	0.190983	−	0.587785i
$69$		0			0
$70$		0			0
$71$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$71$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$72$		0			0
$73$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$73$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$74$		0			0
$75$		0			0
$76$	1.00000			1.00000
$77$	−1.61803			−1.61803
$78$		0			0
$79$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$79$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$80$	1.30902	−	0.951057i	1.30902	−	0.951057i
$81$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$82$		0			0
$83$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$83$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$84$		0			0
$85$	0.809017	−	0.587785i	0.809017	−	0.587785i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$93$		0			0
$94$		0			0
$95$	1.30902	+	0.951057i	1.30902	+	0.951057i
$96$		0			0
$97$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$97$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$98$		0			0
$99$	0.309017	+	0.951057i	0.309017	+	0.951057i
$100$	1.61803			1.61803
$101$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$101$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$102$		0			0
$103$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$103$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$107$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$113$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$113$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$114$		0			0
$115$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$120$		0			0
$121$	0.309017	−	0.951057i	0.309017	−	0.951057i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.809017	+	0.587785i	0.809017	+	0.587785i
$126$		0			0
$127$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$127$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$131$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$132$		0			0
$133$	−1.61803			−1.61803
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$137$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$138$		0			0
$139$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$139$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$140$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	1.00000			1.00000
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$149$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$150$		0			0
$151$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$151$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$152$		0			0
$153$	0.618034			0.618034
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$157$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.809017	+	0.587785i	0.809017	+	0.587785i
$162$		0			0
$163$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$163$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$167$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$168$		0			0
$169$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$170$		0			0
$171$	0.309017	+	0.951057i	0.309017	+	0.951057i
$172$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$173$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$173$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$174$		0			0
$175$	−2.61803			−2.61803
$176$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$177$		0			0
$178$		0			0
$179$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$179$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$180$	1.30902	+	0.951057i	1.30902	+	0.951057i
$181$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$181$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$188$	0.618034			0.618034
$189$		0			0
$190$		0			0
$191$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$191$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$192$		0			0
$193$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$193$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$194$		0			0
$195$		0			0
$196$	0.500000	−	1.53884i	0.500000	−	1.53884i
$197$	2.00000			2.00000			1.00000			$0$
$197$	2.00000			2.00000			1.00000			$0$
$198$		0			0
$199$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$199$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.190983	−	0.587785i	0.190983	−	0.587785i
$208$		0			0
$209$	0.309017	−	0.951057i	0.309017	−	0.951057i
$210$		0			0
$211$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$211$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$221$		0			0
$222$		0			0
$223$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$223$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$224$		0			0
$225$	0.500000	+	1.53884i	0.500000	+	1.53884i
$226$		0			0
$227$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$227$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$228$		0			0
$229$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$229$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$233$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$234$		0			0
$235$	0.809017	+	0.587785i	0.809017	+	0.587785i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$239$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$	0.190983	−	0.587785i	0.190983	−	0.587785i
$245$	2.11803	−	1.53884i	2.11803	−	1.53884i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$251$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$252$	−1.61803			−1.61803
$253$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$254$		0			0
$255$		0			0
$256$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$257$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$257$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.00000			2.00000			1.00000			$0$
$263$	2.00000			2.00000			1.00000			$0$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$269$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$270$		0			0
$271$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$271$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$272$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$273$		0			0
$274$		0			0
$275$	0.500000	−	1.53884i	0.500000	−	1.53884i
$276$		0			0
$277$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$277$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$281$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$282$		0			0
$283$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$283$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.500000	−	0.363271i	0.500000	−	0.363271i
$290$		0			0
$291$		0			0
$292$	0.618034	+	1.90211i	0.618034	+	1.90211i
$293$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$293$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	0.809017	+	0.587785i	0.809017	+	0.587785i
$302$		0			0
$303$		0			0
$304$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$305$	0.809017	−	0.587785i	0.809017	−	0.587785i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	1.30902	+	0.951057i	1.30902	+	0.951057i
$309$		0			0
$310$		0			0
$311$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$311$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$312$		0			0
$313$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$313$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$314$		0			0
$315$	−2.11803	−	1.53884i	−2.11803	−	1.53884i
$316$		0			0
$317$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$317$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$318$		0			0
$319$		0			0
$320$	−1.61803			−1.61803
$321$		0			0
$322$		0			0
$323$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$324$	0.309017	+	0.951057i	0.309017	+	0.951057i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.00000			−1.00000
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$337$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$338$		0			0
$339$		0			0
$340$	−1.00000			−1.00000
$341$		0			0
$342$		0			0
$343$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$347$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$348$		0			0
$349$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$349$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$353$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$359$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$360$		0			0
$361$	0.309017	−	0.951057i	0.309017	−	0.951057i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.00000	+	3.07768i	−1.00000	+	3.07768i
$366$		0			0
$367$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$367$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$368$	0.190983	+	0.587785i	0.190983	+	0.587785i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$379$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$380$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$381$		0			0
$382$		0			0
$383$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$383$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$384$		0			0
$385$	0.809017	+	2.48990i	0.809017	+	2.48990i
$386$		0			0
$387$	0.190983	−	0.587785i	0.190983	−	0.587785i
$388$		0			0
$389$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$389$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$390$		0			0
$391$	0.118034	+	0.363271i	0.118034	+	0.363271i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	0.309017	−	0.951057i	0.309017	−	0.951057i
$397$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$397$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$398$		0			0
$399$		0			0
$400$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$401$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$401$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$402$		0			0
$403$		0			0
$404$	1.30902	−	0.951057i	1.30902	−	0.951057i
$405$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$409$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$419$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$420$		0			0
$421$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$421$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$422$		0			0
$423$	0.190983	+	0.587785i	0.190983	+	0.587785i
$424$		0			0
$425$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$426$		0			0
$427$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$431$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$432$		0			0
$433$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$433$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	1.61803			1.61803
$442$		0			0
$443$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$443$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	1.30902	−	0.951057i	1.30902	−	0.951057i
$449$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$449$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$457$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$458$		0			0
$459$		0			0
$460$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$461$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$461$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$462$		0			0
$463$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$463$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$467$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$474$		0			0
$475$	0.500000	−	1.53884i	0.500000	−	1.53884i
$476$	0.809017	−	0.587785i	0.809017	−	0.587785i
$477$		0			0
$478$		0			0
$479$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$479$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$485$		0			0
$486$		0			0
$487$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$487$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$491$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	1.30902	−	0.951057i	1.30902	−	0.951057i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$499$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$500$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$501$		0			0
$502$		0			0
$503$	−1.61803	+	1.17557i	−1.61803	+	1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$503$	−1.61803	+	1.17557i	−1.61803	+	1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$504$		0			0
$505$	2.61803			2.61803
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$509$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$510$		0			0
$511$	−1.00000	−	3.07768i	−1.00000	−	3.07768i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	0.190983	−	0.587785i	0.190983	−	0.587785i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$521$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$522$		0			0
$523$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$523$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$524$	1.30902	+	0.951057i	1.30902	+	0.951057i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.618034			−0.618034
$530$		0			0
$531$		0			0
$532$	1.30902	+	0.951057i	1.30902	+	0.951057i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$540$		0			0
$541$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$541$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$547$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$548$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$549$	0.618034			0.618034
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	0.190983	+	0.587785i	0.190983	+	0.587785i
$557$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$557$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$558$		0			0
$559$		0			0
$560$	2.61803			2.61803
$561$		0			0
$562$		0			0
$563$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$563$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$568$		0			0
$569$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$569$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$570$		0			0
$571$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$571$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$576$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$577$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$577$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	0.809017	−	2.48990i	0.809017	−	2.48990i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$587$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$593$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$594$		0			0
$595$	1.61803			1.61803
$596$	0.618034	−	1.90211i	0.618034	−	1.90211i
$597$		0			0
$598$		0			0
$599$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$599$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$600$		0			0
$601$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$601$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.61803			−1.61803
$606$		0			0
$607$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$607$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$613$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$613$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.00000			2.00000			1.00000			$0$
$617$	2.00000			2.00000			1.00000			$0$
$618$		0			0
$619$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$619$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$	−1.61803			−1.61803
$629$		0			0
$630$		0			0
$631$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$631$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$641$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$642$		0			0
$643$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$643$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$644$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$645$		0			0
$646$		0			0
$647$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$647$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	1.30902	−	0.951057i	1.30902	−	0.951057i
$653$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$653$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$654$		0			0
$655$	0.809017	+	2.48990i	0.809017	+	2.48990i
$656$		0			0
$657$	−1.61803	+	1.17557i	−1.61803	+	1.17557i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.809017	+	2.48990i	0.809017	+	2.48990i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$672$		0			0
$673$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$673$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$674$		0			0
$675$		0			0
$676$	0.309017	+	0.951057i	0.309017	+	0.951057i
$677$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$677$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	0.309017	−	0.951057i	0.309017	−	0.951057i
$685$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$686$		0			0
$687$		0			0
$688$	0.190983	+	0.587785i	0.190983	+	0.587785i
$689$		0			0
$690$		0			0
$691$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$691$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$692$		0			0
$693$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$694$		0			0
$695$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	2.11803	+	1.53884i	2.11803	+	1.53884i
$701$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$701$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$702$		0			0
$703$		0			0
$704$	0.309017	+	0.951057i	0.309017	+	0.951057i
$705$		0			0
$706$		0			0
$707$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$708$		0			0
$709$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$709$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$719$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$720$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$727$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$728$		0			0
$729$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$730$		0			0
$731$	0.118034	+	0.363271i	0.118034	+	0.363271i
$732$		0			0
$733$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$733$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$739$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$743$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$744$		0			0
$745$	2.61803	−	1.90211i	2.61803	−	1.90211i
$746$		0			0
$747$	−1.61803			−1.61803
$748$	0.190983	+	0.587785i	0.190983	+	0.587785i
$749$		0			0
$750$		0			0
$751$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$751$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$752$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$757$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$761$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$762$		0			0
$763$		0			0
$764$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$765$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$769$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$773$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$774$		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
$784$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$785$	−2.11803	−	1.53884i	−2.11803	−	1.53884i
$786$		0			0
$787$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$787$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$788$	−1.61803	−	1.17557i	−1.61803	−	1.17557i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	1.30902	+	0.951057i	1.30902	+	0.951057i
$797$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$797$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$798$		0			0
$799$	−0.309017	−	0.224514i	−0.309017	−	0.224514i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.00000			2.00000
$804$		0			0
$805$	0.500000	−	1.53884i	0.500000	−	1.53884i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$809$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$810$		0			0
$811$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$811$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.61803			2.61803
$816$		0			0
$817$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$821$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$822$		0			0
$823$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$823$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$827$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$828$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$829$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$829$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$834$		0			0
$835$		0			0
$836$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$837$		0			0
$838$		0			0
$839$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$839$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$840$		0			0
$841$	0.309017	+	0.951057i	0.309017	+	0.951057i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$846$		0			0
$847$	1.30902	−	0.951057i	1.30902	−	0.951057i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$853$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$854$		0			0
$855$	1.30902	−	0.951057i	1.30902	−	0.951057i
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$859$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$860$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$861$		0			0
$862$		0			0
$863$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$863$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.500000	+	1.53884i	0.500000	+	1.53884i
$876$		0			0
$877$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$877$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$878$		0			0
$879$		0			0
$880$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$881$	2.00000			2.00000			1.00000			$0$
$881$	2.00000			2.00000			1.00000			$0$
$882$		0			0
$883$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$883$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$887$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1.00000			1.00000
$892$		0			0
$893$	0.190983	−	0.587785i	0.190983	−	0.587785i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.500000	−	1.53884i	0.500000	−	1.53884i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$907$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$908$		0			0
$909$	1.30902	+	0.951057i	1.30902	+	0.951057i
$910$		0			0
$911$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$911$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$912$		0			0
$913$	1.30902	+	0.951057i	1.30902	+	0.951057i
$914$		0			0
$915$		0			0
$916$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$917$	−2.11803	−	1.53884i	−2.11803	−	1.53884i
$918$		0			0
$919$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$919$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$920$		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
$929$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$929$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$930$		0			0
$931$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$932$	1.30902	−	0.951057i	1.30902	−	0.951057i
$933$		0			0
$934$		0			0
$935$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$936$		0			0
$937$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$937$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$938$		0			0
$939$		0			0
$940$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$941$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$941$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$947$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$953$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$954$		0			0
$955$	0.809017	−	2.48990i	0.809017	−	2.48990i
$956$	0.618034			0.618034
$957$		0			0
$958$		0			0
$959$	0.809017	−	2.48990i	0.809017	−	2.48990i
$960$		0			0
$961$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$967$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$971$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$972$		0			0
$973$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$974$		0			0
$975$		0			0
$976$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$977$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$977$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$978$		0			0
$979$		0			0
$980$	−2.61803			−2.61803
$981$		0			0
$982$		0			0
$983$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$983$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$984$		0			0
$985$	−1.00000	−	3.07768i	−1.00000	−	3.07768i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.381966			0.381966
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.809017	+	2.48990i	0.809017	+	2.48990i
$996$		0			0
$997$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$997$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	209.1.m.a.75.1	✓	4
3.2	odd	2		1881.1.bv.a.1747.1		4
4.3	odd	2		3344.1.bx.a.1329.1		4
11.2	odd	10		2299.1.m.c.1576.1		4
11.3	even	5		2299.1.m.b.1291.1		4
11.4	even	5		2299.1.b.a.1937.1		2
11.5	even	5	inner	209.1.m.a.170.1	yes	4
11.6	odd	10		2299.1.m.a.1842.1		4
11.7	odd	10		2299.1.b.b.1937.1		2
11.8	odd	10		2299.1.m.c.1291.1		4
11.9	even	5		2299.1.m.b.1576.1		4
11.10	odd	2		2299.1.m.a.493.1		4
19.2	odd	18		3971.1.bc.a.262.1		24
19.3	odd	18		3971.1.bc.a.3221.1		24
19.4	even	9		3971.1.bc.a.2473.1		24
19.5	even	9		3971.1.bc.a.1571.1		24
19.6	even	9		3971.1.bc.a.1560.1		24
19.7	even	3		3971.1.s.a.3903.1		8
19.8	odd	6		3971.1.s.a.2957.1		8
19.9	even	9		3971.1.bc.a.1021.1		24
19.10	odd	18		3971.1.bc.a.1021.1		24
19.11	even	3		3971.1.s.a.2957.1		8
19.12	odd	6		3971.1.s.a.3903.1		8
19.13	odd	18		3971.1.bc.a.1560.1		24
19.14	odd	18		3971.1.bc.a.1571.1		24
19.15	odd	18		3971.1.bc.a.2473.1		24
19.16	even	9		3971.1.bc.a.3221.1		24
19.17	even	9		3971.1.bc.a.262.1		24
19.18	odd	2	CM	209.1.m.a.75.1	✓	4
33.5	odd	10		1881.1.bv.a.379.1		4
44.27	odd	10		3344.1.bx.a.1633.1		4
57.56	even	2		1881.1.bv.a.1747.1		4
76.75	even	2		3344.1.bx.a.1329.1		4
209.5	even	45		3971.1.bc.a.2293.1		24
209.16	even	45		3971.1.bc.a.3943.1		24
209.18	even	10		2299.1.b.b.1937.1		2
209.27	odd	30		3971.1.s.a.3679.1		8
209.37	odd	10		2299.1.b.a.1937.1		2
209.49	even	15		3971.1.s.a.3679.1		8
209.60	odd	90		3971.1.bc.a.3943.1		24
209.71	odd	90		3971.1.bc.a.2293.1		24
209.75	odd	10		2299.1.m.b.1576.1		4
209.82	even	45		3971.1.bc.a.2282.1		24
209.93	even	45		3971.1.bc.a.984.1		24
209.94	even	10		2299.1.m.a.1842.1		4
209.104	even	45		3971.1.bc.a.1743.1		24
209.113	odd	10		2299.1.m.b.1291.1		4
209.126	odd	30		3971.1.s.a.654.1		8
209.137	even	45		3971.1.bc.a.3195.1		24
209.148	odd	90		3971.1.bc.a.3195.1		24
209.151	even	10		2299.1.m.c.1291.1		4
209.159	even	15		3971.1.s.a.654.1		8
209.170	odd	10	inner	209.1.m.a.170.1	yes	4
209.181	odd	90		3971.1.bc.a.1743.1		24
209.189	even	10		2299.1.m.c.1576.1		4
209.192	odd	90		3971.1.bc.a.984.1		24
209.203	odd	90		3971.1.bc.a.2282.1		24
209.208	even	2		2299.1.m.a.493.1		4
627.170	even	10		1881.1.bv.a.379.1		4
836.379	even	10		3344.1.bx.a.1633.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.1.m.a.75.1	✓	4	1.1	even	1	trivial
209.1.m.a.75.1	✓	4	19.18	odd	2	CM
209.1.m.a.170.1	yes	4	11.5	even	5	inner
209.1.m.a.170.1	yes	4	209.170	odd	10	inner
1881.1.bv.a.379.1		4	33.5	odd	10
1881.1.bv.a.379.1		4	627.170	even	10
1881.1.bv.a.1747.1		4	3.2	odd	2
1881.1.bv.a.1747.1		4	57.56	even	2
2299.1.b.a.1937.1		2	11.4	even	5
2299.1.b.a.1937.1		2	209.37	odd	10
2299.1.b.b.1937.1		2	11.7	odd	10
2299.1.b.b.1937.1		2	209.18	even	10
2299.1.m.a.493.1		4	11.10	odd	2
2299.1.m.a.493.1		4	209.208	even	2
2299.1.m.a.1842.1		4	11.6	odd	10
2299.1.m.a.1842.1		4	209.94	even	10
2299.1.m.b.1291.1		4	11.3	even	5
2299.1.m.b.1291.1		4	209.113	odd	10
2299.1.m.b.1576.1		4	11.9	even	5
2299.1.m.b.1576.1		4	209.75	odd	10
2299.1.m.c.1291.1		4	11.8	odd	10
2299.1.m.c.1291.1		4	209.151	even	10
2299.1.m.c.1576.1		4	11.2	odd	10
2299.1.m.c.1576.1		4	209.189	even	10
3344.1.bx.a.1329.1		4	4.3	odd	2
3344.1.bx.a.1329.1		4	76.75	even	2
3344.1.bx.a.1633.1		4	44.27	odd	10
3344.1.bx.a.1633.1		4	836.379	even	10
3971.1.s.a.654.1		8	209.126	odd	30
3971.1.s.a.654.1		8	209.159	even	15
3971.1.s.a.2957.1		8	19.8	odd	6
3971.1.s.a.2957.1		8	19.11	even	3
3971.1.s.a.3679.1		8	209.27	odd	30
3971.1.s.a.3679.1		8	209.49	even	15
3971.1.s.a.3903.1		8	19.7	even	3
3971.1.s.a.3903.1		8	19.12	odd	6
3971.1.bc.a.262.1		24	19.2	odd	18
3971.1.bc.a.262.1		24	19.17	even	9
3971.1.bc.a.984.1		24	209.93	even	45
3971.1.bc.a.984.1		24	209.192	odd	90
3971.1.bc.a.1021.1		24	19.9	even	9
3971.1.bc.a.1021.1		24	19.10	odd	18
3971.1.bc.a.1560.1		24	19.6	even	9
3971.1.bc.a.1560.1		24	19.13	odd	18
3971.1.bc.a.1571.1		24	19.5	even	9
3971.1.bc.a.1571.1		24	19.14	odd	18
3971.1.bc.a.1743.1		24	209.104	even	45
3971.1.bc.a.1743.1		24	209.181	odd	90
3971.1.bc.a.2282.1		24	209.82	even	45
3971.1.bc.a.2282.1		24	209.203	odd	90
3971.1.bc.a.2293.1		24	209.5	even	45
3971.1.bc.a.2293.1		24	209.71	odd	90
3971.1.bc.a.2473.1		24	19.4	even	9
3971.1.bc.a.2473.1		24	19.15	odd	18
3971.1.bc.a.3195.1		24	209.137	even	45
3971.1.bc.a.3195.1		24	209.148	odd	90
3971.1.bc.a.3221.1		24	19.3	odd	18
3971.1.bc.a.3221.1		24	19.16	even	9
3971.1.bc.a.3943.1		24	209.16	even	45
3971.1.bc.a.3943.1		24	209.60	odd	90

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:	\( N \)	\(=\)	\( 209 = 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	209.m (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.809017 - 0.587785i) q^{4} +(-0.500000 - 1.53884i) q^{5} +(1.30902 + 0.951057i) q^{7} +(0.309017 - 0.951057i) q^{9} +(-0.809017 + 0.587785i) q^{11} +(0.309017 + 0.951057i) q^{16} +(0.190983 + 0.587785i) q^{17} +(-0.809017 + 0.587785i) q^{19} +(-0.500000 + 1.53884i) q^{20} +0.618034 q^{23} +(-1.30902 + 0.951057i) q^{25} +(-0.500000 - 1.53884i) q^{28} +(0.809017 - 2.48990i) q^{35} +(-0.809017 + 0.587785i) q^{36} +0.618034 q^{43} +1.00000 q^{44} -1.61803 q^{45} +(-0.500000 + 0.363271i) q^{47} +(0.500000 + 1.53884i) q^{49} +(1.30902 + 0.951057i) q^{55} +(0.190983 + 0.587785i) q^{61} +(1.30902 - 0.951057i) q^{63} +(0.309017 - 0.951057i) q^{64} +(0.190983 - 0.587785i) q^{68} +(-1.61803 - 1.17557i) q^{73} +1.00000 q^{76} -1.61803 q^{77} +(1.30902 - 0.951057i) q^{80} +(-0.809017 - 0.587785i) q^{81} +(-0.500000 - 1.53884i) q^{83} +(0.809017 - 0.587785i) q^{85} +(-0.500000 - 0.363271i) q^{92} +(1.30902 + 0.951057i) q^{95} +(0.309017 + 0.951057i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{4} - 2 q^{5} + 3 q^{7} - q^{9} - q^{11} - q^{16} + 3 q^{17} - q^{19} - 2 q^{20} - 2 q^{23} - 3 q^{25} - 2 q^{28} + q^{35} - q^{36} - 2 q^{43} + 4 q^{44} - 2 q^{45} - 2 q^{47} + 2 q^{49}+ \cdots - q^{99}+O(q^{100}) \) 4 * q - q^4 - 2 * q^5 + 3 * q^7 - q^9 - q^11 - q^16 + 3 * q^17 - q^19 - 2 * q^20 - 2 * q^23 - 3 * q^25 - 2 * q^28 + q^35 - q^36 - 2 * q^43 + 4 * q^44 - 2 * q^45 - 2 * q^47 + 2 * q^49 + 3 * q^55 + 3 * q^61 + 3 * q^63 - q^64 + 3 * q^68 - 2 * q^73 + 4 * q^76 - 2 * q^77 + 3 * q^80 - q^81 - 2 * q^83 + q^85 - 2 * q^92 + 3 * q^95 - q^99

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