LMFDB - Embedded newform 119.1.d.a.118.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [119,1,Mod(118,119)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(119, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("119.118");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 119 = 7 \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	119.d (of order $2$, degree $1$, 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:	yes
Analytic conductor:	$0.0593887365033$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.14161.1
Artin image:	$D_5$
Artin field:	Galois closure of 5.1.14161.1

Embedding invariants

Embedding label			118.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	119.118

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{2} +0.618034 q^{3} -0.618034 q^{4} -1.61803 q^{5} +0.381966 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.618034 q^{9} +O(q^{10})$	$q+0.618034 q^{2} +0.618034 q^{3} -0.618034 q^{4} -1.61803 q^{5} +0.381966 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.618034 q^{9} -1.00000 q^{10} -0.381966 q^{12} +0.618034 q^{14} -1.00000 q^{15} +1.00000 q^{17} -0.381966 q^{18} +1.00000 q^{20} +0.618034 q^{21} -0.618034 q^{24} +1.61803 q^{25} -1.00000 q^{27} -0.618034 q^{28} -0.618034 q^{30} -1.61803 q^{31} +1.00000 q^{32} +0.618034 q^{34} -1.61803 q^{35} +0.381966 q^{36} +1.61803 q^{40} +0.618034 q^{41} +0.381966 q^{42} +0.618034 q^{43} +1.00000 q^{45} +1.00000 q^{49} +1.00000 q^{50} +0.618034 q^{51} -1.61803 q^{53} -0.618034 q^{54} -1.00000 q^{56} +0.618034 q^{60} +0.618034 q^{61} -1.00000 q^{62} -0.618034 q^{63} +0.618034 q^{64} -1.61803 q^{67} -0.618034 q^{68} -1.00000 q^{70} +0.618034 q^{72} +0.618034 q^{73} +1.00000 q^{75} +0.381966 q^{82} -0.381966 q^{84} -1.61803 q^{85} +0.381966 q^{86} +0.618034 q^{90} -1.00000 q^{93} +0.618034 q^{96} -1.61803 q^{97} +0.618034 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{3} + q^{4} - q^{5} + 3 q^{6} + 2 q^{7} - 2 q^{8} + q^{9}+O(q^{10}) $ 2 * q - q^2 - q^3 + q^4 - q^5 + 3 * q^6 + 2 * q^7 - 2 * q^8 + q^9	$ 2 q - q^{2} - q^{3} + q^{4} - q^{5} + 3 q^{6} + 2 q^{7} - 2 q^{8} + q^{9} - 2 q^{10} - 3 q^{12} - q^{14} - 2 q^{15} + 2 q^{17} - 3 q^{18} + 2 q^{20} - q^{21} + q^{24} + q^{25} - 2 q^{27} + q^{28} + q^{30} - q^{31} + 2 q^{32} - q^{34} - q^{35} + 3 q^{36} + q^{40} - q^{41} + 3 q^{42} - q^{43} + 2 q^{45} + 2 q^{49} + 2 q^{50} - q^{51} - q^{53} + q^{54} - 2 q^{56} - q^{60} - q^{61} - 2 q^{62} + q^{63} - q^{64} - q^{67} + q^{68} - 2 q^{70} - q^{72} - q^{73} + 2 q^{75} + 3 q^{82} - 3 q^{84} - q^{85} + 3 q^{86} - q^{90} - 2 q^{93} - q^{96} - q^{97} - q^{98}+O(q^{100}) $ 2 * q - q^2 - q^3 + q^4 - q^5 + 3 * q^6 + 2 * q^7 - 2 * q^8 + q^9 - 2 * q^10 - 3 * q^12 - q^14 - 2 * q^15 + 2 * q^17 - 3 * q^18 + 2 * q^20 - q^21 + q^24 + q^25 - 2 * q^27 + q^28 + q^30 - q^31 + 2 * q^32 - q^34 - q^35 + 3 * q^36 + q^40 - q^41 + 3 * q^42 - q^43 + 2 * q^45 + 2 * q^49 + 2 * q^50 - q^51 - q^53 + q^54 - 2 * q^56 - q^60 - q^61 - 2 * q^62 + q^63 - q^64 - q^67 + q^68 - 2 * q^70 - q^72 - q^73 + 2 * q^75 + 3 * q^82 - 3 * q^84 - q^85 + 3 * q^86 - q^90 - 2 * q^93 - q^96 - q^97 - q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/119\mathbb{Z}\right)^\times$.

$n$	$52$	$71$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$2$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$3$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$3$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$4$	−0.618034	−0.618034
$5$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$5$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$6$	0.381966	0.381966
$7$	1.00000	1.00000
$7$	1.00000	1.00000
$8$	−1.00000	−1.00000
$9$	−0.618034	−0.618034
$10$	−1.00000	−1.00000
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−0.381966	−0.381966
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0.618034	0.618034
$15$	−1.00000	−1.00000
$16$	0	0
$17$	1.00000	1.00000
$17$	1.00000	1.00000
$18$	−0.381966	−0.381966
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	1.00000	1.00000
$21$	0.618034	0.618034
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−0.618034	−0.618034
$25$	1.61803	1.61803
$26$	0	0
$27$	−1.00000	−1.00000
$28$	−0.618034	−0.618034
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−0.618034	−0.618034
$31$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$31$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$32$	1.00000	1.00000
$33$	0	0
$34$	0.618034	0.618034
$35$	−1.61803	−1.61803
$36$	0.381966	0.381966
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	1.61803	1.61803
$41$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$41$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$42$	0.381966	0.381966
$43$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$43$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$44$	0	0
$45$	1.00000	1.00000
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	1.00000	1.00000
$50$	1.00000	1.00000
$51$	0.618034	0.618034
$52$	0	0
$53$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$53$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$54$	−0.618034	−0.618034
$55$	0	0
$56$	−1.00000	−1.00000
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0.618034	0.618034
$61$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$61$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$62$	−1.00000	−1.00000
$63$	−0.618034	−0.618034
$64$	0.618034	0.618034
$65$	0	0
$66$	0	0
$67$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$67$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$68$	−0.618034	−0.618034
$69$	0	0
$70$	−1.00000	−1.00000
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0.618034	0.618034
$73$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$73$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$74$	0	0
$75$	1.00000	1.00000
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0	0
$82$	0.381966	0.381966
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−0.381966	−0.381966
$85$	−1.61803	−1.61803
$86$	0.381966	0.381966
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0.618034	0.618034
$91$	0	0
$92$	0	0
$93$	−1.00000	−1.00000
$94$	0	0
$95$	0	0
$96$	0.618034	0.618034
$97$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$97$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$98$	0.618034	0.618034
$99$	0	0
$100$	−1.00000	−1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0.381966	0.381966
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	−1.00000	−1.00000
$106$	−1.00000	−1.00000
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0.618034	0.618034
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.00000	1.00000
$120$	1.00000	1.00000
$121$	1.00000	1.00000
$122$	0.381966	0.381966
$123$	0.381966	0.381966
$124$	1.00000	1.00000
$125$	−1.00000	−1.00000
$126$	−0.381966	−0.381966
$127$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$127$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$128$	−0.618034	−0.618034
$129$	0.381966	0.381966
$130$	0	0
$131$	2.00000	2.00000	1.00000			$0$
$131$	2.00000	2.00000	1.00000			$0$
$132$	0	0
$133$	0	0
$134$	−1.00000	−1.00000
$135$	1.61803	1.61803
$136$	−1.00000	−1.00000
$137$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$137$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$138$	0	0
$139$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$139$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$140$	1.00000	1.00000
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0.381966	0.381966
$147$	0.618034	0.618034
$148$	0	0
$149$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$149$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$150$	0.618034	0.618034
$151$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$151$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$152$	0	0
$153$	−0.618034	−0.618034
$154$	0	0
$155$	2.61803	2.61803
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	−1.00000	−1.00000
$160$	−1.61803	−1.61803
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	−0.381966	−0.381966
$165$	0	0
$166$	0	0
$167$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$167$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$168$	−0.618034	−0.618034
$169$	1.00000	1.00000
$170$	−1.00000	−1.00000
$171$	0	0
$172$	−0.381966	−0.381966
$173$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$173$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$174$	0	0
$175$	1.61803	1.61803
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$179$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$180$	−0.618034	−0.618034
$181$	2.00000	2.00000	1.00000			$0$
$181$	2.00000	2.00000	1.00000			$0$
$182$	0	0
$183$	0.381966	0.381966
$184$	0	0
$185$	0	0
$186$	−0.618034	−0.618034
$187$	0	0
$188$	0	0
$189$	−1.00000	−1.00000
$190$	0	0
$191$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$191$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$192$	0.381966	0.381966
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	−1.00000	−1.00000
$195$	0	0
$196$	−0.618034	−0.618034
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$199$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$200$	−1.61803	−1.61803
$201$	−1.00000	−1.00000
$202$	0	0
$203$	0	0
$204$	−0.381966	−0.381966
$205$	−1.00000	−1.00000
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	−0.618034	−0.618034
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	1.00000	1.00000
$213$	0	0
$214$	0	0
$215$	−1.00000	−1.00000
$216$	1.00000	1.00000
$217$	−1.61803	−1.61803
$218$	0	0
$219$	0.381966	0.381966
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	1.00000	1.00000
$225$	−1.00000	−1.00000
$226$	0	0
$227$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$227$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0.618034	0.618034
$239$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$239$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$240$	0	0
$241$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$241$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$242$	0.618034	0.618034
$243$	1.00000	1.00000
$244$	−0.381966	−0.381966
$245$	−1.61803	−1.61803
$246$	0.236068	0.236068
$247$	0	0
$248$	1.61803	1.61803
$249$	0	0
$250$	−0.618034	−0.618034
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0.381966	0.381966
$253$	0	0
$254$	−1.00000	−1.00000
$255$	−1.00000	−1.00000
$256$	−1.00000	−1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0.236068	0.236068
$259$	0	0
$260$	0	0
$261$	0	0
$262$	1.23607	1.23607
$263$	2.00000	2.00000	1.00000			$0$
$263$	2.00000	2.00000	1.00000			$0$
$264$	0	0
$265$	2.61803	2.61803
$266$	0	0
$267$	0	0
$268$	1.00000	1.00000
$269$	2.00000	2.00000	1.00000			$0$
$269$	2.00000	2.00000	1.00000			$0$
$270$	1.00000	1.00000
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0.381966	0.381966
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−1.00000	−1.00000
$279$	1.00000	1.00000
$280$	1.61803	1.61803
$281$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$281$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$282$	0	0
$283$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$283$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.618034	0.618034
$288$	−0.618034	−0.618034
$289$	1.00000	1.00000
$290$	0	0
$291$	−1.00000	−1.00000
$292$	−0.381966	−0.381966
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0.381966	0.381966
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0.381966	0.381966
$299$	0	0
$300$	−0.618034	−0.618034
$301$	0.618034	0.618034
$302$	0.381966	0.381966
$303$	0	0
$304$	0	0
$305$	−1.00000	−1.00000
$306$	−0.381966	−0.381966
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	1.61803	1.61803
$311$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$311$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$312$	0	0
$313$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$313$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$314$	0	0
$315$	1.00000	1.00000
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	−0.618034	−0.618034
$319$	0	0
$320$	−1.00000	−1.00000
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	−0.618034	−0.618034
$329$	0	0
$330$	0	0
$331$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$331$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$332$	0	0
$333$	0	0
$334$	0.381966	0.381966
$335$	2.61803	2.61803
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0.618034	0.618034
$339$	0	0
$340$	1.00000	1.00000
$341$	0	0
$342$	0	0
$343$	1.00000	1.00000
$344$	−0.618034	−0.618034
$345$	0	0
$346$	0.381966	0.381966
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	1.00000	1.00000
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.618034	0.618034
$358$	0.381966	0.381966
$359$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$359$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$360$	−1.00000	−1.00000
$361$	1.00000	1.00000
$362$	1.23607	1.23607
$363$	0.618034	0.618034
$364$	0	0
$365$	−1.00000	−1.00000
$366$	0.236068	0.236068
$367$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$367$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$368$	0	0
$369$	−0.381966	−0.381966
$370$	0	0
$371$	−1.61803	−1.61803
$372$	0.618034	0.618034
$373$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$373$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$374$	0	0
$375$	−0.618034	−0.618034
$376$	0	0
$377$	0	0
$378$	−0.618034	−0.618034
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−1.00000	−1.00000
$382$	−1.00000	−1.00000
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−0.381966	−0.381966
$385$	0	0
$386$	0	0
$387$	−0.381966	−0.381966
$388$	1.00000	1.00000
$389$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$389$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$390$	0	0
$391$	0	0
$392$	−1.00000	−1.00000
$393$	1.23607	1.23607
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$397$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$398$	0.381966	0.381966
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	−0.618034	−0.618034
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	−0.618034	−0.618034
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	−0.618034	−0.618034
$411$	0.381966	0.381966
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1.00000	−1.00000
$418$	0	0
$419$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$419$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$420$	0.618034	0.618034
$421$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$421$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$422$	0	0
$423$	0	0
$424$	1.61803	1.61803
$425$	1.61803	1.61803
$426$	0	0
$427$	0.618034	0.618034
$428$	0	0
$429$	0	0
$430$	−0.618034	−0.618034
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	−1.00000	−1.00000
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0.236068	0.236068
$439$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$439$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$440$	0	0
$441$	−0.618034	−0.618034
$442$	0	0
$443$	2.00000	2.00000	1.00000			$0$
$443$	2.00000	2.00000	1.00000			$0$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0.381966	0.381966
$448$	0.618034	0.618034
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−0.618034	−0.618034
$451$	0	0
$452$	0	0
$453$	0.381966	0.381966
$454$	−1.00000	−1.00000
$455$	0	0
$456$	0	0
$457$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$457$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$458$	0	0
$459$	−1.00000	−1.00000
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$463$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$464$	0	0
$465$	1.61803	1.61803
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	−1.61803	−1.61803
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−0.618034	−0.618034
$477$	1.00000	1.00000
$478$	0.381966	0.381966
$479$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$479$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$480$	−1.00000	−1.00000
$481$	0	0
$482$	−1.00000	−1.00000
$483$	0	0
$484$	−0.618034	−0.618034
$485$	2.61803	2.61803
$486$	0.618034	0.618034
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	−0.618034	−0.618034
$489$	0	0
$490$	−1.00000	−1.00000
$491$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$491$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$492$	−0.236068	−0.236068
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0.618034	0.618034
$501$	0.381966	0.381966
$502$	0	0
$503$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$503$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$504$	0.618034	0.618034
$505$	0	0
$506$	0	0
$507$	0.618034	0.618034
$508$	1.00000	1.00000
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	−0.618034	−0.618034
$511$	0.618034	0.618034
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	−0.236068	−0.236068
$517$	0	0
$518$	0	0
$519$	0.381966	0.381966
$520$	0	0
$521$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$521$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−1.23607	−1.23607
$525$	1.00000	1.00000
$526$	1.23607	1.23607
$527$	−1.61803	−1.61803
$528$	0	0
$529$	1.00000	1.00000
$530$	1.61803	1.61803
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	1.61803	1.61803
$537$	0.381966	0.381966
$538$	1.23607	1.23607
$539$	0	0
$540$	−1.00000	−1.00000
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	1.23607	1.23607
$544$	1.00000	1.00000
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	−0.381966	−0.381966
$549$	−0.381966	−0.381966
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	1.00000	1.00000
$557$	2.00000	2.00000	1.00000			$0$
$557$	2.00000	2.00000	1.00000			$0$
$558$	0.618034	0.618034
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−1.00000	−1.00000
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	−1.00000	−1.00000
$567$	0	0
$568$	0	0
$569$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$569$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	−1.00000	−1.00000
$574$	0.381966	0.381966
$575$	0	0
$576$	−0.381966	−0.381966
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0.618034	0.618034
$579$	0	0
$580$	0	0
$581$	0	0
$582$	−0.618034	−0.618034
$583$	0	0
$584$	−0.618034	−0.618034
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−0.381966	−0.381966
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	−1.61803	−1.61803
$596$	−0.381966	−0.381966
$597$	0.381966	0.381966
$598$	0	0
$599$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$599$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$600$	−1.00000	−1.00000
$601$	2.00000	2.00000	1.00000			$0$
$601$	2.00000	2.00000	1.00000			$0$
$602$	0.381966	0.381966
$603$	1.00000	1.00000
$604$	−0.381966	−0.381966
$605$	−1.61803	−1.61803
$606$	0	0
$607$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$607$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$608$	0	0
$609$	0	0
$610$	−0.618034	−0.618034
$611$	0	0
$612$	0.381966	0.381966
$613$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$613$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$614$	0	0
$615$	−0.618034	−0.618034
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	2.00000	2.00000	1.00000			$0$
$619$	2.00000	2.00000	1.00000			$0$
$620$	−1.61803	−1.61803
$621$	0	0
$622$	−1.00000	−1.00000
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−1.00000	−1.00000
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0.618034	0.618034
$631$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$631$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.61803	2.61803
$636$	0.618034	0.618034
$637$	0	0
$638$	0	0
$639$	0	0
$640$	1.00000	1.00000
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$643$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$644$	0	0
$645$	−0.618034	−0.618034
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−1.00000	−1.00000
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	−3.23607	−3.23607
$656$	0	0
$657$	−0.381966	−0.381966
$658$	0	0
$659$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$659$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−1.00000	−1.00000
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−0.381966	−0.381966
$669$	0	0
$670$	1.61803	1.61803
$671$	0	0
$672$	0.618034	0.618034
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	−1.61803	−1.61803
$676$	−0.618034	−0.618034
$677$	2.00000	2.00000	1.00000			$0$
$677$	2.00000	2.00000	1.00000			$0$
$678$	0	0
$679$	−1.61803	−1.61803
$680$	1.61803	1.61803
$681$	−1.00000	−1.00000
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	−1.00000	−1.00000
$686$	0.618034	0.618034
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$691$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$692$	−0.381966	−0.381966
$693$	0	0
$694$	0	0
$695$	2.61803	2.61803
$696$	0	0
$697$	0.618034	0.618034
$698$	0	0
$699$	0	0
$700$	−1.00000	−1.00000
$701$	2.00000	2.00000	1.00000			$0$
$701$	2.00000	2.00000	1.00000			$0$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0.381966	0.381966
$715$	0	0
$716$	−0.381966	−0.381966
$717$	0.381966	0.381966
$718$	−1.00000	−1.00000
$719$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$719$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$720$	0	0
$721$	0	0
$722$	0.618034	0.618034
$723$	−1.00000	−1.00000
$724$	−1.23607	−1.23607
$725$	0	0
$726$	0.381966	0.381966
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0.618034	0.618034
$730$	−0.618034	−0.618034
$731$	0.618034	0.618034
$732$	−0.236068	−0.236068
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0.381966	0.381966
$735$	−1.00000	−1.00000
$736$	0	0
$737$	0	0
$738$	−0.236068	−0.236068
$739$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$739$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$740$	0	0
$741$	0	0
$742$	−1.00000	−1.00000
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	1.00000	1.00000
$745$	−1.00000	−1.00000
$746$	0.381966	0.381966
$747$	0	0
$748$	0	0
$749$	0	0
$750$	−0.381966	−0.381966
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.00000	−1.00000
$756$	0.618034	0.618034
$757$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$757$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	−0.618034	−0.618034
$763$	0	0
$764$	1.00000	1.00000
$765$	1.00000	1.00000
$766$	0	0
$767$	0	0
$768$	−0.618034	−0.618034
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−0.236068	−0.236068
$775$	−2.61803	−2.61803
$776$	1.61803	1.61803
$777$	0	0
$778$	−1.00000	−1.00000
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0.763932	0.763932
$787$	2.00000	2.00000	1.00000			$0$
$787$	2.00000	2.00000	1.00000			$0$
$788$	0	0
$789$	1.23607	1.23607
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	−1.00000	−1.00000
$795$	1.61803	1.61803
$796$	−0.381966	−0.381966
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	1.61803	1.61803
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0.618034	0.618034
$805$	0	0
$806$	0	0
$807$	1.23607	1.23607
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$811$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0.618034	0.618034
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0.236068	0.236068
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.00000	1.00000
$834$	−0.618034	−0.618034
$835$	−1.00000	−1.00000
$836$	0	0
$837$	1.61803	1.61803
$838$	−1.00000	−1.00000
$839$	2.00000	2.00000	1.00000			$0$
$839$	2.00000	2.00000	1.00000			$0$
$840$	1.00000	1.00000
$841$	1.00000	1.00000
$842$	0.381966	0.381966
$843$	−1.00000	−1.00000
$844$	0	0
$845$	−1.61803	−1.61803
$846$	0	0
$847$	1.00000	1.00000
$848$	0	0
$849$	−1.00000	−1.00000
$850$	1.00000	1.00000
$851$	0	0
$852$	0	0
$853$	2.00000	2.00000	1.00000			$0$
$853$	2.00000	2.00000	1.00000			$0$
$854$	0.381966	0.381966
$855$	0	0
$856$	0	0
$857$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$857$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0.618034	0.618034
$861$	0.381966	0.381966
$862$	0	0
$863$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$863$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$864$	−1.00000	−1.00000
$865$	−1.00000	−1.00000
$866$	0	0
$867$	0.618034	0.618034
$868$	1.00000	1.00000
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	1.00000	1.00000
$874$	0	0
$875$	−1.00000	−1.00000
$876$	−0.236068	−0.236068
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0.381966	0.381966
$879$	0	0
$880$	0	0
$881$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$881$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$882$	−0.381966	−0.381966
$883$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$883$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$884$	0	0
$885$	0	0
$886$	1.23607	1.23607
$887$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$887$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$888$	0	0
$889$	−1.61803	−1.61803
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0.236068	0.236068
$895$	−1.00000	−1.00000
$896$	−0.618034	−0.618034
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0.618034	0.618034
$901$	−1.61803	−1.61803
$902$	0	0
$903$	0.381966	0.381966
$904$	0	0
$905$	−3.23607	−3.23607
$906$	0.236068	0.236068
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	1.00000	1.00000
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	−1.00000	−1.00000
$915$	−0.618034	−0.618034
$916$	0	0
$917$	2.00000	2.00000
$918$	−0.618034	−0.618034
$919$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$919$	0.618034	0.618034	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
$926$	0.381966	0.381966
$927$	0	0
$928$	0	0
$929$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$929$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$930$	1.00000	1.00000
$931$	0	0
$932$	0	0
$933$	−1.00000	−1.00000
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	−1.00000	−1.00000
$939$	−1.00000	−1.00000
$940$	0	0
$941$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$941$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	1.61803	1.61803
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	−1.00000	−1.00000
$953$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$953$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$954$	0.618034	0.618034
$955$	2.61803	2.61803
$956$	−0.381966	−0.381966
$957$	0	0
$958$	0.381966	0.381966
$959$	0.618034	0.618034
$960$	−0.618034	−0.618034
$961$	1.61803	1.61803
$962$	0	0
$963$	0	0
$964$	1.00000	1.00000
$965$	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	$0.400000\pi$
$968$	−1.00000	−1.00000
$969$	0	0
$970$	1.61803	1.61803
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−0.618034	−0.618034
$973$	−1.61803	−1.61803
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$977$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$978$	0	0
$979$	0	0
$980$	1.00000	1.00000
$981$	0	0
$982$	−1.00000	−1.00000
$983$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$983$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$984$	−0.381966	−0.381966
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	−1.61803	−1.61803
$993$	−1.00000	−1.00000
$994$	0	0
$995$	−1.00000	−1.00000
$996$	0	0
$997$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$997$	0.618034	0.618034	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	119.1.d.a.118.2	✓	2
3.2	odd	2		1071.1.h.b.118.1		2
4.3	odd	2		1904.1.n.b.1665.1		2
5.2	odd	4		2975.1.b.b.2974.3		4
5.3	odd	4		2975.1.b.b.2974.2		4
5.4	even	2		2975.1.h.d.951.1		2
7.2	even	3		833.1.h.b.815.1		4
7.3	odd	6		833.1.h.a.509.1		4
7.4	even	3		833.1.h.b.509.1		4
7.5	odd	6		833.1.h.a.815.1		4
7.6	odd	2		119.1.d.b.118.2	yes	2
17.2	even	8		2023.1.f.b.251.4		8
17.3	odd	16		2023.1.l.b.1266.4		16
17.4	even	4		2023.1.c.e.1735.1		4
17.5	odd	16		2023.1.l.b.468.1		16
17.6	odd	16		2023.1.l.b.1868.4		16
17.7	odd	16		2023.1.l.b.1889.1		16
17.8	even	8		2023.1.f.b.1483.1		8
17.9	even	8		2023.1.f.b.1483.2		8
17.10	odd	16		2023.1.l.b.1889.2		16
17.11	odd	16		2023.1.l.b.1868.3		16
17.12	odd	16		2023.1.l.b.468.2		16
17.13	even	4		2023.1.c.e.1735.2		4
17.14	odd	16		2023.1.l.b.1266.3		16
17.15	even	8		2023.1.f.b.251.3		8
17.16	even	2		119.1.d.b.118.2	yes	2
21.20	even	2		1071.1.h.a.118.1		2
28.27	even	2		1904.1.n.a.1665.2		2
35.13	even	4		2975.1.b.a.2974.2		4
35.27	even	4		2975.1.b.a.2974.3		4
35.34	odd	2		2975.1.h.c.951.1		2
51.50	odd	2		1071.1.h.a.118.1		2
68.67	odd	2		1904.1.n.a.1665.2		2
85.33	odd	4		2975.1.b.a.2974.2		4
85.67	odd	4		2975.1.b.a.2974.3		4
85.84	even	2		2975.1.h.c.951.1		2
119.6	even	16		2023.1.l.b.1868.3		16
119.13	odd	4		2023.1.c.e.1735.1		4
119.16	even	6		833.1.h.a.815.1		4
119.20	even	16		2023.1.l.b.1266.3		16
119.27	even	16		2023.1.l.b.1889.1		16
119.33	odd	6		833.1.h.b.815.1		4
119.41	even	16		2023.1.l.b.1889.2		16
119.48	even	16		2023.1.l.b.1266.4		16
119.55	odd	4		2023.1.c.e.1735.2		4
119.62	even	16		2023.1.l.b.1868.4		16
119.67	even	6		833.1.h.a.509.1		4
119.76	odd	8		2023.1.f.b.1483.2		8
119.83	odd	8		2023.1.f.b.251.4		8
119.90	even	16		2023.1.l.b.468.2		16
119.97	even	16		2023.1.l.b.468.1		16
119.101	odd	6		833.1.h.b.509.1		4
119.104	odd	8		2023.1.f.b.251.3		8
119.111	odd	8		2023.1.f.b.1483.1		8
119.118	odd	2	CM	119.1.d.a.118.2	✓	2
357.356	even	2		1071.1.h.b.118.1		2
476.475	even	2		1904.1.n.b.1665.1		2
595.118	even	4		2975.1.b.b.2974.2		4
595.237	even	4		2975.1.b.b.2974.3		4
595.594	odd	2		2975.1.h.d.951.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
119.1.d.a.118.2	✓	2	1.1	even	1	trivial
119.1.d.a.118.2	✓	2	119.118	odd	2	CM
119.1.d.b.118.2	yes	2	7.6	odd	2
119.1.d.b.118.2	yes	2	17.16	even	2
833.1.h.a.509.1		4	7.3	odd	6
833.1.h.a.509.1		4	119.67	even	6
833.1.h.a.815.1		4	7.5	odd	6
833.1.h.a.815.1		4	119.16	even	6
833.1.h.b.509.1		4	7.4	even	3
833.1.h.b.509.1		4	119.101	odd	6
833.1.h.b.815.1		4	7.2	even	3
833.1.h.b.815.1		4	119.33	odd	6
1071.1.h.a.118.1		2	21.20	even	2
1071.1.h.a.118.1		2	51.50	odd	2
1071.1.h.b.118.1		2	3.2	odd	2
1071.1.h.b.118.1		2	357.356	even	2
1904.1.n.a.1665.2		2	28.27	even	2
1904.1.n.a.1665.2		2	68.67	odd	2
1904.1.n.b.1665.1		2	4.3	odd	2
1904.1.n.b.1665.1		2	476.475	even	2
2023.1.c.e.1735.1		4	17.4	even	4
2023.1.c.e.1735.1		4	119.13	odd	4
2023.1.c.e.1735.2		4	17.13	even	4
2023.1.c.e.1735.2		4	119.55	odd	4
2023.1.f.b.251.3		8	17.15	even	8
2023.1.f.b.251.3		8	119.104	odd	8
2023.1.f.b.251.4		8	17.2	even	8
2023.1.f.b.251.4		8	119.83	odd	8
2023.1.f.b.1483.1		8	17.8	even	8
2023.1.f.b.1483.1		8	119.111	odd	8
2023.1.f.b.1483.2		8	17.9	even	8
2023.1.f.b.1483.2		8	119.76	odd	8
2023.1.l.b.468.1		16	17.5	odd	16
2023.1.l.b.468.1		16	119.97	even	16
2023.1.l.b.468.2		16	17.12	odd	16
2023.1.l.b.468.2		16	119.90	even	16
2023.1.l.b.1266.3		16	17.14	odd	16
2023.1.l.b.1266.3		16	119.20	even	16
2023.1.l.b.1266.4		16	17.3	odd	16
2023.1.l.b.1266.4		16	119.48	even	16
2023.1.l.b.1868.3		16	17.11	odd	16
2023.1.l.b.1868.3		16	119.6	even	16
2023.1.l.b.1868.4		16	17.6	odd	16
2023.1.l.b.1868.4		16	119.62	even	16
2023.1.l.b.1889.1		16	17.7	odd	16
2023.1.l.b.1889.1		16	119.27	even	16
2023.1.l.b.1889.2		16	17.10	odd	16
2023.1.l.b.1889.2		16	119.41	even	16
2975.1.b.a.2974.2		4	35.13	even	4
2975.1.b.a.2974.2		4	85.33	odd	4
2975.1.b.a.2974.3		4	35.27	even	4
2975.1.b.a.2974.3		4	85.67	odd	4
2975.1.b.b.2974.2		4	5.3	odd	4
2975.1.b.b.2974.2		4	595.118	even	4
2975.1.b.b.2974.3		4	5.2	odd	4
2975.1.b.b.2974.3		4	595.237	even	4
2975.1.h.c.951.1		2	35.34	odd	2
2975.1.h.c.951.1		2	85.84	even	2
2975.1.h.d.951.1		2	5.4	even	2
2975.1.h.d.951.1		2	595.594	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 119 = 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	119.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} +0.618034 q^{3} -0.618034 q^{4} -1.61803 q^{5} +0.381966 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.618034 q^{9} +O(q^{10})\)	\(q+0.618034 q^{2} +0.618034 q^{3} -0.618034 q^{4} -1.61803 q^{5} +0.381966 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.618034 q^{9} -1.00000 q^{10} -0.381966 q^{12} +0.618034 q^{14} -1.00000 q^{15} +1.00000 q^{17} -0.381966 q^{18} +1.00000 q^{20} +0.618034 q^{21} -0.618034 q^{24} +1.61803 q^{25} -1.00000 q^{27} -0.618034 q^{28} -0.618034 q^{30} -1.61803 q^{31} +1.00000 q^{32} +0.618034 q^{34} -1.61803 q^{35} +0.381966 q^{36} +1.61803 q^{40} +0.618034 q^{41} +0.381966 q^{42} +0.618034 q^{43} +1.00000 q^{45} +1.00000 q^{49} +1.00000 q^{50} +0.618034 q^{51} -1.61803 q^{53} -0.618034 q^{54} -1.00000 q^{56} +0.618034 q^{60} +0.618034 q^{61} -1.00000 q^{62} -0.618034 q^{63} +0.618034 q^{64} -1.61803 q^{67} -0.618034 q^{68} -1.00000 q^{70} +0.618034 q^{72} +0.618034 q^{73} +1.00000 q^{75} +0.381966 q^{82} -0.381966 q^{84} -1.61803 q^{85} +0.381966 q^{86} +0.618034 q^{90} -1.00000 q^{93} +0.618034 q^{96} -1.61803 q^{97} +0.618034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{3} + q^{4} - q^{5} + 3 q^{6} + 2 q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) 2 * q - q^2 - q^3 + q^4 - q^5 + 3 * q^6 + 2 * q^7 - 2 * q^8 + q^9	\( 2 q - q^{2} - q^{3} + q^{4} - q^{5} + 3 q^{6} + 2 q^{7} - 2 q^{8} + q^{9} - 2 q^{10} - 3 q^{12} - q^{14} - 2 q^{15} + 2 q^{17} - 3 q^{18} + 2 q^{20} - q^{21} + q^{24} + q^{25} - 2 q^{27} + q^{28} + q^{30} - q^{31} + 2 q^{32} - q^{34} - q^{35} + 3 q^{36} + q^{40} - q^{41} + 3 q^{42} - q^{43} + 2 q^{45} + 2 q^{49} + 2 q^{50} - q^{51} - q^{53} + q^{54} - 2 q^{56} - q^{60} - q^{61} - 2 q^{62} + q^{63} - q^{64} - q^{67} + q^{68} - 2 q^{70} - q^{72} - q^{73} + 2 q^{75} + 3 q^{82} - 3 q^{84} - q^{85} + 3 q^{86} - q^{90} - 2 q^{93} - q^{96} - q^{97} - q^{98}+O(q^{100}) \) 2 * q - q^2 - q^3 + q^4 - q^5 + 3 * q^6 + 2 * q^7 - 2 * q^8 + q^9 - 2 * q^10 - 3 * q^12 - q^14 - 2 * q^15 + 2 * q^17 - 3 * q^18 + 2 * q^20 - q^21 + q^24 + q^25 - 2 * q^27 + q^28 + q^30 - q^31 + 2 * q^32 - q^34 - q^35 + 3 * q^36 + q^40 - q^41 + 3 * q^42 - q^43 + 2 * q^45 + 2 * q^49 + 2 * q^50 - q^51 - q^53 + q^54 - 2 * q^56 - q^60 - q^61 - 2 * q^62 + q^63 - q^64 - q^67 + q^68 - 2 * q^70 - q^72 - q^73 + 2 * q^75 + 3 * q^82 - 3 * q^84 - q^85 + 3 * q^86 - q^90 - 2 * q^93 - q^96 - q^97 - q^98

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