LMFDB - Embedded newform 344.1.h.a.85.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [344,1,Mod(85,344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("344.85");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 344 = 2^{3} \cdot 43 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	344.h (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:	$0.171678364346$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.118336.1

Embedding invariants

Embedding label			85.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	344.85

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -0.618034 q^{5} -1.61803 q^{6} -1.00000 q^{8} +1.61803 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -0.618034 q^{5} -1.61803 q^{6} -1.00000 q^{8} +1.61803 q^{9} +0.618034 q^{10} +1.61803 q^{12} -1.00000 q^{15} +1.00000 q^{16} -1.61803 q^{17} -1.61803 q^{18} -0.618034 q^{19} -0.618034 q^{20} +0.618034 q^{23} -1.61803 q^{24} -0.618034 q^{25} +1.00000 q^{27} +1.61803 q^{29} +1.00000 q^{30} -1.61803 q^{31} -1.00000 q^{32} +1.61803 q^{34} +1.61803 q^{36} +1.61803 q^{37} +0.618034 q^{38} +0.618034 q^{40} +0.618034 q^{41} -1.00000 q^{43} -1.00000 q^{45} -0.618034 q^{46} -1.61803 q^{47} +1.61803 q^{48} +1.00000 q^{49} +0.618034 q^{50} -2.61803 q^{51} -1.00000 q^{54} -1.00000 q^{57} -1.61803 q^{58} -1.00000 q^{60} -2.00000 q^{61} +1.61803 q^{62} +1.00000 q^{64} -1.61803 q^{68} +1.00000 q^{69} -1.61803 q^{72} -1.61803 q^{74} -1.00000 q^{75} -0.618034 q^{76} +0.618034 q^{79} -0.618034 q^{80} -0.618034 q^{82} +1.00000 q^{85} +1.00000 q^{86} +2.61803 q^{87} +1.00000 q^{90} +0.618034 q^{92} -2.61803 q^{93} +1.61803 q^{94} +0.381966 q^{95} -1.61803 q^{96} +0.618034 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + q^{3} + 2 q^{4} + q^{5} - q^{6} - 2 q^{8} + q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + q^3 + 2 * q^4 + q^5 - q^6 - 2 * q^8 + q^9	$ 2 q - 2 q^{2} + q^{3} + 2 q^{4} + q^{5} - q^{6} - 2 q^{8} + q^{9} - q^{10} + q^{12} - 2 q^{15} + 2 q^{16} - q^{17} - q^{18} + q^{19} + q^{20} - q^{23} - q^{24} + q^{25} + 2 q^{27} + q^{29} + 2 q^{30} - q^{31} - 2 q^{32} + q^{34} + q^{36} + q^{37} - q^{38} - q^{40} - q^{41} - 2 q^{43} - 2 q^{45} + q^{46} - q^{47} + q^{48} + 2 q^{49} - q^{50} - 3 q^{51} - 2 q^{54} - 2 q^{57} - q^{58} - 2 q^{60} - 4 q^{61} + q^{62} + 2 q^{64} - q^{68} + 2 q^{69} - q^{72} - q^{74} - 2 q^{75} + q^{76} - q^{79} + q^{80} + q^{82} + 2 q^{85} + 2 q^{86} + 3 q^{87} + 2 q^{90} - q^{92} - 3 q^{93} + q^{94} + 3 q^{95} - q^{96} - q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + q^3 + 2 * q^4 + q^5 - q^6 - 2 * q^8 + q^9 - q^10 + q^12 - 2 * q^15 + 2 * q^16 - q^17 - q^18 + q^19 + q^20 - q^23 - q^24 + q^25 + 2 * q^27 + q^29 + 2 * q^30 - q^31 - 2 * q^32 + q^34 + q^36 + q^37 - q^38 - q^40 - q^41 - 2 * q^43 - 2 * q^45 + q^46 - q^47 + q^48 + 2 * q^49 - q^50 - 3 * q^51 - 2 * q^54 - 2 * q^57 - q^58 - 2 * q^60 - 4 * q^61 + q^62 + 2 * q^64 - q^68 + 2 * q^69 - q^72 - q^74 - 2 * q^75 + q^76 - q^79 + q^80 + q^82 + 2 * q^85 + 2 * q^86 + 3 * q^87 + 2 * q^90 - q^92 - 3 * q^93 + q^94 + 3 * q^95 - q^96 - q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/344\mathbb{Z}\right)^\times$.

$n$	$87$	$89$	$173$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	−1.00000	−1.00000
$2$	−1.00000	−1.00000
$3$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$3$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$4$	1.00000	1.00000
$5$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$5$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$6$	−1.61803	−1.61803
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−1.00000	−1.00000
$9$	1.61803	1.61803
$10$	0.618034	0.618034
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	1.61803	1.61803
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	−1.00000	−1.00000
$16$	1.00000	1.00000
$17$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$17$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$18$	−1.61803	−1.61803
$19$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$19$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$20$	−0.618034	−0.618034
$21$	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	$0.400000\pi$
$24$	−1.61803	−1.61803
$25$	−0.618034	−0.618034
$26$	0	0
$27$	1.00000	1.00000
$28$	0	0
$29$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$29$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$30$	1.00000	1.00000
$31$	−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
$34$	1.61803	1.61803
$35$	0	0
$36$	1.61803	1.61803
$37$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$37$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$38$	0.618034	0.618034
$39$	0	0
$40$	0.618034	0.618034
$41$	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	0
$43$	−1.00000	−1.00000
$43$	−1.00000	−1.00000
$44$	0	0
$45$	−1.00000	−1.00000
$46$	−0.618034	−0.618034
$47$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$47$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$48$	1.61803	1.61803
$49$	1.00000	1.00000
$50$	0.618034	0.618034
$51$	−2.61803	−2.61803
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−1.00000	−1.00000
$55$	0	0
$56$	0	0
$57$	−1.00000	−1.00000
$58$	−1.61803	−1.61803
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−1.00000	−1.00000
$61$	−2.00000	−2.00000	−1.00000			$\pi$
$61$	−2.00000	−2.00000	−1.00000			$\pi$
$62$	1.61803	1.61803
$63$	0	0
$64$	1.00000	1.00000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−1.61803	−1.61803
$69$	1.00000	1.00000
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−1.61803	−1.61803
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−1.61803	−1.61803
$75$	−1.00000	−1.00000
$76$	−0.618034	−0.618034
$77$	0	0
$78$	0	0
$79$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$79$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$80$	−0.618034	−0.618034
$81$	0	0
$82$	−0.618034	−0.618034
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	1.00000	1.00000
$86$	1.00000	1.00000
$87$	2.61803	2.61803
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	1.00000	1.00000
$91$	0	0
$92$	0.618034	0.618034
$93$	−2.61803	−2.61803
$94$	1.61803	1.61803
$95$	0.381966	0.381966
$96$	−1.61803	−1.61803
$97$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$97$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$98$	−1.00000	−1.00000
$99$	0	0
$100$	−0.618034	−0.618034
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	2.61803	2.61803
$103$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$103$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	1.00000	1.00000
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	2.61803	2.61803
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	1.00000	1.00000
$115$	−0.381966	−0.381966
$116$	1.61803	1.61803
$117$	0	0
$118$	0	0
$119$	0	0
$120$	1.00000	1.00000
$121$	1.00000	1.00000
$122$	2.00000	2.00000
$123$	1.00000	1.00000
$124$	−1.61803	−1.61803
$125$	1.00000	1.00000
$126$	0	0
$127$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$127$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$128$	−1.00000	−1.00000
$129$	−1.61803	−1.61803
$130$	0	0
$131$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$131$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.618034	−0.618034
$136$	1.61803	1.61803
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	−1.00000	−1.00000
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	−2.61803	−2.61803
$142$	0	0
$143$	0	0
$144$	1.61803	1.61803
$145$	−1.00000	−1.00000
$146$	0	0
$147$	1.61803	1.61803
$148$	1.61803	1.61803
$149$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$149$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$150$	1.00000	1.00000
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0.618034	0.618034
$153$	−2.61803	−2.61803
$154$	0	0
$155$	1.00000	1.00000
$156$	0	0
$157$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$157$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$158$	−0.618034	−0.618034
$159$	0	0
$160$	0.618034	0.618034
$161$	0	0
$162$	0	0
$163$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$163$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$164$	0.618034	0.618034
$165$	0	0
$166$	0	0
$167$	2.00000	2.00000	1.00000			$0$
$167$	2.00000	2.00000	1.00000			$0$
$168$	0	0
$169$	1.00000	1.00000
$170$	−1.00000	−1.00000
$171$	−1.00000	−1.00000
$172$	−1.00000	−1.00000
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	−2.61803	−2.61803
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$179$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$180$	−1.00000	−1.00000
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	−3.23607	−3.23607
$184$	−0.618034	−0.618034
$185$	−1.00000	−1.00000
$186$	2.61803	2.61803
$187$	0	0
$188$	−1.61803	−1.61803
$189$	0	0
$190$	−0.381966	−0.381966
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	1.61803	1.61803
$193$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$193$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$194$	−0.618034	−0.618034
$195$	0	0
$196$	1.00000	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0.618034	0.618034
$201$	0	0
$202$	0	0
$203$	0	0
$204$	−2.61803	−2.61803
$205$	−0.381966	−0.381966
$206$	−0.618034	−0.618034
$207$	1.00000	1.00000
$208$	0	0
$209$	0	0
$210$	0	0
$211$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$211$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.618034	0.618034
$216$	−1.00000	−1.00000
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	−2.61803	−2.61803
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−1.00000	−1.00000
$226$	0	0
$227$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$227$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$228$	−1.00000	−1.00000
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0.381966	0.381966
$231$	0	0
$232$	−1.61803	−1.61803
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	1.00000	1.00000
$236$	0	0
$237$	1.00000	1.00000
$238$	0	0
$239$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$239$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$240$	−1.00000	−1.00000
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−1.00000	−1.00000
$243$	−1.00000	−1.00000
$244$	−2.00000	−2.00000
$245$	−0.618034	−0.618034
$246$	−1.00000	−1.00000
$247$	0	0
$248$	1.61803	1.61803
$249$	0	0
$250$	−1.00000	−1.00000
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	1.61803	1.61803
$255$	1.61803	1.61803
$256$	1.00000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	1.61803	1.61803
$259$	0	0
$260$	0	0
$261$	2.61803	2.61803
$262$	−1.61803	−1.61803
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0.618034	0.618034
$271$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$271$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$272$	−1.61803	−1.61803
$273$	0	0
$274$	0	0
$275$	0	0
$276$	1.00000	1.00000
$277$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$277$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$278$	0	0
$279$	−2.61803	−2.61803
$280$	0	0
$281$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$281$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$282$	2.61803	2.61803
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0.618034	0.618034
$286$	0	0
$287$	0	0
$288$	−1.61803	−1.61803
$289$	1.61803	1.61803
$290$	1.00000	1.00000
$291$	1.00000	1.00000
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−1.61803	−1.61803
$295$	0	0
$296$	−1.61803	−1.61803
$297$	0	0
$298$	−1.61803	−1.61803
$299$	0	0
$300$	−1.00000	−1.00000
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−0.618034	−0.618034
$305$	1.23607	1.23607
$306$	2.61803	2.61803
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	1.00000	1.00000
$310$	−1.00000	−1.00000
$311$	2.00000	2.00000	1.00000			$0$
$311$	2.00000	2.00000	1.00000			$0$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0.618034	0.618034
$315$	0	0
$316$	0.618034	0.618034
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	−0.618034	−0.618034
$321$	0	0
$322$	0	0
$323$	1.00000	1.00000
$324$	0	0
$325$	0	0
$326$	0.618034	0.618034
$327$	0	0
$328$	−0.618034	−0.618034
$329$	0	0
$330$	0	0
$331$	−2.00000	−2.00000	−1.00000			$\pi$
$331$	−2.00000	−2.00000	−1.00000			$\pi$
$332$	0	0
$333$	2.61803	2.61803
$334$	−2.00000	−2.00000
$335$	0	0
$336$	0	0
$337$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$337$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$338$	−1.00000	−1.00000
$339$	0	0
$340$	1.00000	1.00000
$341$	0	0
$342$	1.00000	1.00000
$343$	0	0
$344$	1.00000	1.00000
$345$	−0.618034	−0.618034
$346$	0	0
$347$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$347$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$348$	2.61803	2.61803
$349$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$349$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.00000	2.00000	1.00000			$0$
$353$	2.00000	2.00000	1.00000			$0$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0.618034	0.618034
$359$	−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
$361$	−0.618034	−0.618034
$362$	0	0
$363$	1.61803	1.61803
$364$	0	0
$365$	0	0
$366$	3.23607	3.23607
$367$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$367$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$368$	0.618034	0.618034
$369$	1.00000	1.00000
$370$	1.00000	1.00000
$371$	0	0
$372$	−2.61803	−2.61803
$373$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$373$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$374$	0	0
$375$	1.61803	1.61803
$376$	1.61803	1.61803
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0.381966	0.381966
$381$	−2.61803	−2.61803
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−1.61803	−1.61803
$385$	0	0
$386$	−0.618034	−0.618034
$387$	−1.61803	−1.61803
$388$	0.618034	0.618034
$389$	−2.00000	−2.00000	−1.00000			$\pi$
$389$	−2.00000	−2.00000	−1.00000			$\pi$
$390$	0	0
$391$	−1.00000	−1.00000
$392$	−1.00000	−1.00000
$393$	2.61803	2.61803
$394$	0	0
$395$	−0.381966	−0.381966
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	−0.618034	−0.618034
$401$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$401$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	2.61803	2.61803
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0.381966	0.381966
$411$	0	0
$412$	0.618034	0.618034
$413$	0	0
$414$	−1.00000	−1.00000
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.00000	−2.00000	−1.00000			$\pi$
$419$	−2.00000	−2.00000	−1.00000			$\pi$
$420$	0	0
$421$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$421$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$422$	−1.61803	−1.61803
$423$	−2.61803	−2.61803
$424$	0	0
$425$	1.00000	1.00000
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−0.618034	−0.618034
$431$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$431$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$432$	1.00000	1.00000
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	−1.61803	−1.61803
$436$	0	0
$437$	−0.381966	−0.381966
$438$	0	0
$439$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$439$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$440$	0	0
$441$	1.61803	1.61803
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	2.61803	2.61803
$445$	0	0
$446$	0	0
$447$	2.61803	2.61803
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	1.00000	1.00000
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0.618034	0.618034
$455$	0	0
$456$	1.00000	1.00000
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	−1.61803	−1.61803
$460$	−0.381966	−0.381966
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	1.61803	1.61803
$465$	1.61803	1.61803
$466$	0	0
$467$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$467$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$468$	0	0
$469$	0	0
$470$	−1.00000	−1.00000
$471$	−1.00000	−1.00000
$472$	0	0
$473$	0	0
$474$	−1.00000	−1.00000
$475$	0.381966	0.381966
$476$	0	0
$477$	0	0
$478$	−0.618034	−0.618034
$479$	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
$482$	0	0
$483$	0	0
$484$	1.00000	1.00000
$485$	−0.381966	−0.381966
$486$	1.00000	1.00000
$487$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$487$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$488$	2.00000	2.00000
$489$	−1.00000	−1.00000
$490$	0.618034	0.618034
$491$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$491$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$492$	1.00000	1.00000
$493$	−2.61803	−2.61803
$494$	0	0
$495$	0	0
$496$	−1.61803	−1.61803
$497$	0	0
$498$	0	0
$499$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$499$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$500$	1.00000	1.00000
$501$	3.23607	3.23607
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.61803	1.61803
$508$	−1.61803	−1.61803
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	−1.61803	−1.61803
$511$	0	0
$512$	−1.00000	−1.00000
$513$	−0.618034	−0.618034
$514$	0	0
$515$	−0.381966	−0.381966
$516$	−1.61803	−1.61803
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−2.61803	−2.61803
$523$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$523$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$524$	1.61803	1.61803
$525$	0	0
$526$	0	0
$527$	2.61803	2.61803
$528$	0	0
$529$	−0.618034	−0.618034
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.00000	−1.00000
$538$	0	0
$539$	0	0
$540$	−0.618034	−0.618034
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−0.618034	−0.618034
$543$	0	0
$544$	1.61803	1.61803
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	−3.23607	−3.23607
$550$	0	0
$551$	−1.00000	−1.00000
$552$	−1.00000	−1.00000
$553$	0	0
$554$	0.618034	0.618034
$555$	−1.61803	−1.61803
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	2.61803	2.61803
$559$	0	0
$560$	0	0
$561$	0	0
$562$	1.61803	1.61803
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	−2.61803	−2.61803
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.00000	2.00000	1.00000			$0$
$569$	2.00000	2.00000	1.00000			$0$
$570$	−0.618034	−0.618034
$571$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$571$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.381966	−0.381966
$576$	1.61803	1.61803
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−1.61803	−1.61803
$579$	1.00000	1.00000
$580$	−1.00000	−1.00000
$581$	0	0
$582$	−1.00000	−1.00000
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.00000	−2.00000	−1.00000			$\pi$
$587$	−2.00000	−2.00000	−1.00000			$\pi$
$588$	1.61803	1.61803
$589$	1.00000	1.00000
$590$	0	0
$591$	0	0
$592$	1.61803	1.61803
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	1.61803	1.61803
$597$	0	0
$598$	0	0
$599$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$599$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$600$	1.00000	1.00000
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−0.618034	−0.618034
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0.618034	0.618034
$609$	0	0
$610$	−1.23607	−1.23607
$611$	0	0
$612$	−2.61803	−2.61803
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	−0.618034	−0.618034
$616$	0	0
$617$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$617$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$618$	−1.00000	−1.00000
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	1.00000	1.00000
$621$	0.618034	0.618034
$622$	−2.00000	−2.00000
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	−0.618034	−0.618034
$629$	−2.61803	−2.61803
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	−0.618034	−0.618034
$633$	2.61803	2.61803
$634$	0	0
$635$	1.00000	1.00000
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0.618034	0.618034
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	1.00000	1.00000
$646$	−1.00000	−1.00000
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−0.618034	−0.618034
$653$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$653$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$654$	0	0
$655$	−1.00000	−1.00000
$656$	0.618034	0.618034
$657$	0	0
$658$	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			$\pi$
$662$	2.00000	2.00000
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−2.61803	−2.61803
$667$	1.00000	1.00000
$668$	2.00000	2.00000
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	1.61803	1.61803
$675$	−0.618034	−0.618034
$676$	1.00000	1.00000
$677$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$677$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$678$	0	0
$679$	0	0
$680$	−1.00000	−1.00000
$681$	−1.00000	−1.00000
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	−1.00000	−1.00000
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−1.00000	−1.00000
$689$	0	0
$690$	0.618034	0.618034
$691$	−2.00000	−2.00000	−1.00000			$\pi$
$691$	−2.00000	−2.00000	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0.618034	0.618034
$695$	0	0
$696$	−2.61803	−2.61803
$697$	−1.00000	−1.00000
$698$	−1.61803	−1.61803
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	−1.00000	−1.00000
$704$	0	0
$705$	1.61803	1.61803
$706$	−2.00000	−2.00000
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	1.00000	1.00000
$712$	0	0
$713$	−1.00000	−1.00000
$714$	0	0
$715$	0	0
$716$	−0.618034	−0.618034
$717$	1.00000	1.00000
$718$	1.61803	1.61803
$719$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$719$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$720$	−1.00000	−1.00000
$721$	0	0
$722$	0.618034	0.618034
$723$	0	0
$724$	0	0
$725$	−1.00000	−1.00000
$726$	−1.61803	−1.61803
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−1.61803	−1.61803
$730$	0	0
$731$	1.61803	1.61803
$732$	−3.23607	−3.23607
$733$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$733$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$734$	1.61803	1.61803
$735$	−1.00000	−1.00000
$736$	−0.618034	−0.618034
$737$	0	0
$738$	−1.00000	−1.00000
$739$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$739$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$740$	−1.00000	−1.00000
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	2.61803	2.61803
$745$	−1.00000	−1.00000
$746$	0.618034	0.618034
$747$	0	0
$748$	0	0
$749$	0	0
$750$	−1.61803	−1.61803
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−1.61803	−1.61803
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$757$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$758$	0	0
$759$	0	0
$760$	−0.381966	−0.381966
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	2.61803	2.61803
$763$	0	0
$764$	0	0
$765$	1.61803	1.61803
$766$	0	0
$767$	0	0
$768$	1.61803	1.61803
$769$	−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
$772$	0.618034	0.618034
$773$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$773$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$774$	1.61803	1.61803
$775$	1.00000	1.00000
$776$	−0.618034	−0.618034
$777$	0	0
$778$	2.00000	2.00000
$779$	−0.381966	−0.381966
$780$	0	0
$781$	0	0
$782$	1.00000	1.00000
$783$	1.61803	1.61803
$784$	1.00000	1.00000
$785$	0.381966	0.381966
$786$	−2.61803	−2.61803
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0.381966	0.381966
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	2.61803	2.61803
$800$	0.618034	0.618034
$801$	0	0
$802$	1.61803	1.61803
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$809$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$810$	0	0
$811$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$811$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$812$	0	0
$813$	1.00000	1.00000
$814$	0	0
$815$	0.381966	0.381966
$816$	−2.61803	−2.61803
$817$	0.618034	0.618034
$818$	0	0
$819$	0	0
$820$	−0.381966	−0.381966
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	2.00000	2.00000	1.00000			$0$
$823$	2.00000	2.00000	1.00000			$0$
$824$	−0.618034	−0.618034
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	1.00000	1.00000
$829$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$829$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$830$	0	0
$831$	−1.00000	−1.00000
$832$	0	0
$833$	−1.61803	−1.61803
$834$	0	0
$835$	−1.23607	−1.23607
$836$	0	0
$837$	−1.61803	−1.61803
$838$	2.00000	2.00000
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.61803	1.61803
$842$	0.618034	0.618034
$843$	−2.61803	−2.61803
$844$	1.61803	1.61803
$845$	−0.618034	−0.618034
$846$	2.61803	2.61803
$847$	0	0
$848$	0	0
$849$	0	0
$850$	−1.00000	−1.00000
$851$	1.00000	1.00000
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0.618034	0.618034
$856$	0	0
$857$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$857$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$858$	0	0
$859$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$859$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$860$	0.618034	0.618034
$861$	0	0
$862$	1.61803	1.61803
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−1.00000	−1.00000
$865$	0	0
$866$	0	0
$867$	2.61803	2.61803
$868$	0	0
$869$	0	0
$870$	1.61803	1.61803
$871$	0	0
$872$	0	0
$873$	1.00000	1.00000
$874$	0.381966	0.381966
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	1.61803	1.61803
$879$	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	$0.800000\pi$
$882$	−1.61803	−1.61803
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	−2.61803	−2.61803
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.00000	1.00000
$894$	−2.61803	−2.61803
$895$	0.381966	0.381966
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.61803	−2.61803
$900$	−1.00000	−1.00000
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	−0.618034	−0.618034
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−1.00000	−1.00000
$913$	0	0
$914$	0	0
$915$	2.00000	2.00000
$916$	0	0
$917$	0	0
$918$	1.61803	1.61803
$919$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$919$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$920$	0.381966	0.381966
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−1.00000	−1.00000
$926$	0	0
$927$	1.00000	1.00000
$928$	−1.61803	−1.61803
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	−1.61803	−1.61803
$931$	−0.618034	−0.618034
$932$	0	0
$933$	3.23607	3.23607
$934$	−1.61803	−1.61803
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	1.00000	1.00000
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	1.00000	1.00000
$943$	0.381966	0.381966
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	1.00000	1.00000
$949$	0	0
$950$	−0.381966	−0.381966
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0.618034	0.618034
$957$	0	0
$958$	−0.618034	−0.618034
$959$	0	0
$960$	−1.00000	−1.00000
$961$	1.61803	1.61803
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−0.381966	−0.381966
$966$	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
$969$	1.61803	1.61803
$970$	0.381966	0.381966
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−1.00000	−1.00000
$973$	0	0
$974$	−0.618034	−0.618034
$975$	0	0
$976$	−2.00000	−2.00000
$977$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$977$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$978$	1.00000	1.00000
$979$	0	0
$980$	−0.618034	−0.618034
$981$	0	0
$982$	0.618034	0.618034
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−1.00000	−1.00000
$985$	0	0
$986$	2.61803	2.61803
$987$	0	0
$988$	0	0
$989$	−0.618034	−0.618034
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	1.61803	1.61803
$993$	−3.23607	−3.23607
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$997$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$998$	−1.61803	−1.61803
$999$	1.61803	1.61803

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	344.1.h.a.85.2	✓	2
3.2	odd	2		3096.1.k.b.1117.2		2
4.3	odd	2		1376.1.h.a.945.1		2
8.3	odd	2		1376.1.h.b.945.2		2
8.5	even	2		344.1.h.b.85.1	yes	2
24.5	odd	2		3096.1.k.a.1117.1		2
43.42	odd	2		344.1.h.b.85.1	yes	2
129.128	even	2		3096.1.k.a.1117.1		2
172.171	even	2		1376.1.h.b.945.2		2
344.85	odd	2	CM	344.1.h.a.85.2	✓	2
344.171	even	2		1376.1.h.a.945.1		2
1032.773	even	2		3096.1.k.b.1117.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
344.1.h.a.85.2	✓	2	1.1	even	1	trivial
344.1.h.a.85.2	✓	2	344.85	odd	2	CM
344.1.h.b.85.1	yes	2	8.5	even	2
344.1.h.b.85.1	yes	2	43.42	odd	2
1376.1.h.a.945.1		2	4.3	odd	2
1376.1.h.a.945.1		2	344.171	even	2
1376.1.h.b.945.2		2	8.3	odd	2
1376.1.h.b.945.2		2	172.171	even	2
3096.1.k.a.1117.1		2	24.5	odd	2
3096.1.k.a.1117.1		2	129.128	even	2
3096.1.k.b.1117.2		2	3.2	odd	2
3096.1.k.b.1117.2		2	1032.773	even	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:	\( N \)	\(=\)	\( 344 = 2^{3} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	344.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -0.618034 q^{5} -1.61803 q^{6} -1.00000 q^{8} +1.61803 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -0.618034 q^{5} -1.61803 q^{6} -1.00000 q^{8} +1.61803 q^{9} +0.618034 q^{10} +1.61803 q^{12} -1.00000 q^{15} +1.00000 q^{16} -1.61803 q^{17} -1.61803 q^{18} -0.618034 q^{19} -0.618034 q^{20} +0.618034 q^{23} -1.61803 q^{24} -0.618034 q^{25} +1.00000 q^{27} +1.61803 q^{29} +1.00000 q^{30} -1.61803 q^{31} -1.00000 q^{32} +1.61803 q^{34} +1.61803 q^{36} +1.61803 q^{37} +0.618034 q^{38} +0.618034 q^{40} +0.618034 q^{41} -1.00000 q^{43} -1.00000 q^{45} -0.618034 q^{46} -1.61803 q^{47} +1.61803 q^{48} +1.00000 q^{49} +0.618034 q^{50} -2.61803 q^{51} -1.00000 q^{54} -1.00000 q^{57} -1.61803 q^{58} -1.00000 q^{60} -2.00000 q^{61} +1.61803 q^{62} +1.00000 q^{64} -1.61803 q^{68} +1.00000 q^{69} -1.61803 q^{72} -1.61803 q^{74} -1.00000 q^{75} -0.618034 q^{76} +0.618034 q^{79} -0.618034 q^{80} -0.618034 q^{82} +1.00000 q^{85} +1.00000 q^{86} +2.61803 q^{87} +1.00000 q^{90} +0.618034 q^{92} -2.61803 q^{93} +1.61803 q^{94} +0.381966 q^{95} -1.61803 q^{96} +0.618034 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + q^{3} + 2 q^{4} + q^{5} - q^{6} - 2 q^{8} + q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + q^3 + 2 * q^4 + q^5 - q^6 - 2 * q^8 + q^9	\( 2 q - 2 q^{2} + q^{3} + 2 q^{4} + q^{5} - q^{6} - 2 q^{8} + q^{9} - q^{10} + q^{12} - 2 q^{15} + 2 q^{16} - q^{17} - q^{18} + q^{19} + q^{20} - q^{23} - q^{24} + q^{25} + 2 q^{27} + q^{29} + 2 q^{30} - q^{31} - 2 q^{32} + q^{34} + q^{36} + q^{37} - q^{38} - q^{40} - q^{41} - 2 q^{43} - 2 q^{45} + q^{46} - q^{47} + q^{48} + 2 q^{49} - q^{50} - 3 q^{51} - 2 q^{54} - 2 q^{57} - q^{58} - 2 q^{60} - 4 q^{61} + q^{62} + 2 q^{64} - q^{68} + 2 q^{69} - q^{72} - q^{74} - 2 q^{75} + q^{76} - q^{79} + q^{80} + q^{82} + 2 q^{85} + 2 q^{86} + 3 q^{87} + 2 q^{90} - q^{92} - 3 q^{93} + q^{94} + 3 q^{95} - q^{96} - q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + q^3 + 2 * q^4 + q^5 - q^6 - 2 * q^8 + q^9 - q^10 + q^12 - 2 * q^15 + 2 * q^16 - q^17 - q^18 + q^19 + q^20 - q^23 - q^24 + q^25 + 2 * q^27 + q^29 + 2 * q^30 - q^31 - 2 * q^32 + q^34 + q^36 + q^37 - q^38 - q^40 - q^41 - 2 * q^43 - 2 * q^45 + q^46 - q^47 + q^48 + 2 * q^49 - q^50 - 3 * q^51 - 2 * q^54 - 2 * q^57 - q^58 - 2 * q^60 - 4 * q^61 + q^62 + 2 * q^64 - q^68 + 2 * q^69 - q^72 - q^74 - 2 * q^75 + q^76 - q^79 + q^80 + q^82 + 2 * q^85 + 2 * q^86 + 3 * q^87 + 2 * q^90 - q^92 - 3 * q^93 + q^94 + 3 * q^95 - q^96 - q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more