LMFDB - Embedded newform 1175.1.b.b.1174.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1175,1,Mod(1174,1175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1175, 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("1175.1174");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1175 = 5^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1175.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$0.586401389844$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 47)
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.2209.1
Artin image:	$C_4\times D_5$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{20} - \cdots)$

Embedding invariants

Embedding label			1174.4
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	1175.1174
Dual form			1175.1.b.b.1174.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803i q^{2} +0.618034i q^{3} -1.61803 q^{4} -1.00000 q^{6} +1.61803i q^{7} -1.00000i q^{8} +0.618034 q^{9} +O(q^{10})$	$q+1.61803i q^{2} +0.618034i q^{3} -1.61803 q^{4} -1.00000 q^{6} +1.61803i q^{7} -1.00000i q^{8} +0.618034 q^{9} -1.00000i q^{12} -2.61803 q^{14} -0.618034i q^{17} +1.00000i q^{18} -1.00000 q^{21} +0.618034 q^{24} +1.00000i q^{27} -2.61803i q^{28} -1.00000i q^{32} +1.00000 q^{34} -1.00000 q^{36} -0.618034i q^{37} -1.61803i q^{42} -1.00000i q^{47} -1.61803 q^{49} +0.381966 q^{51} -1.61803i q^{53} -1.61803 q^{54} +1.61803 q^{56} +1.61803 q^{59} -1.61803 q^{61} +1.00000i q^{63} +1.61803 q^{64} +1.00000i q^{68} +0.618034 q^{71} -0.618034i q^{72} +1.00000 q^{74} -0.618034 q^{79} +2.00000i q^{83} +1.61803 q^{84} +1.61803 q^{89} +1.61803 q^{94} +0.618034 q^{96} +1.61803i q^{97} -2.61803i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) $ 4 * q - 2 * q^4 - 4 * q^6 - 2 * q^9	$ 4 q - 2 q^{4} - 4 q^{6} - 2 q^{9} - 6 q^{14} - 4 q^{21} - 2 q^{24} + 4 q^{34} - 4 q^{36} - 2 q^{49} + 6 q^{51} - 2 q^{54} + 2 q^{56} + 2 q^{59} - 2 q^{61} + 2 q^{64} - 2 q^{71} + 4 q^{74} + 2 q^{79} + 2 q^{84} + 2 q^{89} + 2 q^{94} - 2 q^{96}+O(q^{100}) $ 4 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 - 6 * q^14 - 4 * q^21 - 2 * q^24 + 4 * q^34 - 4 * q^36 - 2 * q^49 + 6 * q^51 - 2 * q^54 + 2 * q^56 + 2 * q^59 - 2 * q^61 + 2 * q^64 - 2 * q^71 + 4 * q^74 + 2 * q^79 + 2 * q^84 + 2 * q^89 + 2 * q^94 - 2 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1175\mathbb{Z}\right)^\times$.

$n$	$377$	$851$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$2$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$3$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$3$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$4$	−1.61803	−1.61803
$5$	0	0
$5$	0	0
$6$	−1.00000	−1.00000
$7$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$7$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$8$	− 1.00000i	− 1.00000i
$9$	0.618034	0.618034
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	− 1.00000i	− 1.00000i
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−2.61803	−2.61803
$15$	0	0
$16$	0	0
$17$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$17$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$18$	1.00000i	1.00000i
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	−1.00000	−1.00000
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0.618034	0.618034
$25$	0	0
$26$	0	0
$27$	1.00000i	1.00000i
$28$	− 2.61803i	− 2.61803i
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	− 1.00000i	− 1.00000i
$33$	0	0
$34$	1.00000	1.00000
$35$	0	0
$36$	−1.00000	−1.00000
$37$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$37$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	− 1.61803i	− 1.61803i
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1.00000i	− 1.00000i
$47$	− 1.00000i	− 1.00000i
$48$	0	0
$49$	−1.61803	−1.61803
$50$	0	0
$51$	0.381966	0.381966
$52$	0	0
$53$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$53$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$54$	−1.61803	−1.61803
$55$	0	0
$56$	1.61803	1.61803
$57$	0	0
$58$	0	0
$59$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$59$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$60$	0	0
$61$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$61$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$62$	0	0
$63$	1.00000i	1.00000i
$64$	1.61803	1.61803
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	1.00000i	1.00000i
$69$	0	0
$70$	0	0
$71$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$71$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$72$	− 0.618034i	− 0.618034i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	1.00000	1.00000
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$79$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$83$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$84$	1.61803	1.61803
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$89$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	1.61803	1.61803
$95$	0	0
$96$	0.618034	0.618034
$97$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$97$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$98$	− 2.61803i	− 2.61803i
$99$	0	0
$100$	0	0
$101$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$101$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$102$	0.618034i	0.618034i
$103$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$103$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$104$	0	0
$105$	0	0
$106$	2.61803	2.61803
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	− 1.61803i	− 1.61803i
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0.381966	0.381966
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	2.61803i	2.61803i
$119$	1.00000	1.00000
$120$	0	0
$121$	1.00000	1.00000
$122$	− 2.61803i	− 2.61803i
$123$	0	0
$124$	0	0
$125$	0	0
$126$	−1.61803	−1.61803
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	1.61803i	1.61803i
$129$	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
$133$	0	0
$134$	0	0
$135$	0	0
$136$	−0.618034	−0.618034
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0.618034	0.618034
$142$	1.00000i	1.00000i
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.00000i	− 1.00000i
$148$	1.00000i	1.00000i
$149$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$149$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	− 0.381966i	− 0.381966i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$157$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$158$	− 1.00000i	− 1.00000i
$159$	1.00000	1.00000
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	−3.23607	−3.23607
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	1.00000i	1.00000i
$169$	−1.00000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$173$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.00000i	1.00000i
$178$	2.61803i	2.61803i
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	− 1.00000i	− 1.00000i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	1.61803i	1.61803i
$189$	−1.61803	−1.61803
$190$	0	0
$191$	2.00000	2.00000	1.00000			$0$
$191$	2.00000	2.00000	1.00000			$0$
$192$	1.00000i	1.00000i
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	−2.61803	−2.61803
$195$	0	0
$196$	2.61803	2.61803
$197$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$197$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	1.00000i	1.00000i
$203$	0	0
$204$	−0.618034	−0.618034
$205$	0	0
$206$	−1.00000	−1.00000
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	2.61803i	2.61803i
$213$	0.381966i	0.381966i
$214$	0	0
$215$	0	0
$216$	1.00000	1.00000
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0.618034i	0.618034i
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	1.61803	1.61803
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	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		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	−2.61803	−2.61803
$237$	− 0.381966i	− 0.381966i
$238$	1.61803i	1.61803i
$239$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$239$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$240$	0	0
$241$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$241$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$242$	1.61803i	1.61803i
$243$	1.00000i	1.00000i
$244$	2.61803	2.61803
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−1.23607	−1.23607
$250$	0	0
$251$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$251$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$252$	− 1.61803i	− 1.61803i
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−1.00000	−1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	1.00000	1.00000
$260$	0	0
$261$	0	0
$262$	− 2.61803i	− 2.61803i
$263$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$263$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.00000i	1.00000i
$268$	0	0
$269$	−2.00000	−2.00000	−1.00000			$\pi$
$269$	−2.00000	−2.00000	−1.00000			$\pi$
$270$	0	0
$271$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$271$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$277$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	1.00000i	1.00000i
$283$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$283$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$284$	−1.00000	−1.00000
$285$	0	0
$286$	0	0
$287$	0	0
$288$	− 0.618034i	− 0.618034i
$289$	0.618034	0.618034
$290$	0	0
$291$	−1.00000	−1.00000
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	1.61803	1.61803
$295$	0	0
$296$	−0.618034	−0.618034
$297$	0	0
$298$	− 1.00000i	− 1.00000i
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0.381966i	0.381966i
$304$	0	0
$305$	0	0
$306$	0.618034	0.618034
$307$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$307$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$308$	0	0
$309$	−0.381966	−0.381966
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	−2.61803	−2.61803
$315$	0	0
$316$	1.00000	1.00000
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	1.61803i	1.61803i
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.61803	1.61803
$330$	0	0
$331$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$331$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$332$	− 3.23607i	− 3.23607i
$333$	− 0.381966i	− 0.381966i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$337$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$338$	− 1.61803i	− 1.61803i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 1.00000i	− 1.00000i
$344$	0	0
$345$	0	0
$346$	2.61803	2.61803
$347$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$347$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$353$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$354$	−1.61803	−1.61803
$355$	0	0
$356$	−2.61803	−2.61803
$357$	0.618034i	0.618034i
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	1.00000	1.00000
$362$	0	0
$363$	0.618034i	0.618034i
$364$	0	0
$365$	0	0
$366$	1.61803	1.61803
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.61803	2.61803
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	−1.00000	−1.00000
$377$	0	0
$378$	− 2.61803i	− 2.61803i
$379$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$379$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$380$	0	0
$381$	0	0
$382$	3.23607i	3.23607i
$383$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$383$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$384$	−1.00000	−1.00000
$385$	0	0
$386$	0	0
$387$	0	0
$388$	− 2.61803i	− 2.61803i
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	1.61803i	1.61803i
$393$	− 1.00000i	− 1.00000i
$394$	3.23607	3.23607
$395$	0	0
$396$	0	0
$397$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$397$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$401$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$402$	0	0
$403$	0	0
$404$	−1.00000	−1.00000
$405$	0	0
$406$	0	0
$407$	0	0
$408$	− 0.381966i	− 0.381966i
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	− 1.00000i	− 1.00000i
$413$	2.61803i	2.61803i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	− 0.618034i	− 0.618034i
$424$	−1.61803	−1.61803
$425$	0	0
$426$	−0.618034	−0.618034
$427$	− 2.61803i	− 2.61803i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$431$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−2.00000	−2.00000	−1.00000			$\pi$
$439$	−2.00000	−2.00000	−1.00000			$\pi$
$440$	0	0
$441$	−1.00000	−1.00000
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	−0.618034	−0.618034
$445$	0	0
$446$	0	0
$447$	− 0.381966i	− 0.381966i
$448$	2.61803i	2.61803i
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$457$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$458$	0	0
$459$	0.618034	0.618034
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1.00000	−1.00000
$472$	− 1.61803i	− 1.61803i
$473$	0	0
$474$	0.618034	0.618034
$475$	0	0
$476$	−1.61803	−1.61803
$477$	− 1.00000i	− 1.00000i
$478$	− 1.00000i	− 1.00000i
$479$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$479$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$480$	0	0
$481$	0	0
$482$	1.00000i	1.00000i
$483$	0	0
$484$	−1.61803	−1.61803
$485$	0	0
$486$	−1.61803	−1.61803
$487$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$487$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$488$	1.61803i	1.61803i
$489$	0	0
$490$	0	0
$491$	−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	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.00000i	1.00000i
$498$	− 2.00000i	− 2.00000i
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	1.00000i	1.00000i
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	1.00000	1.00000
$505$	0	0
$506$	0	0
$507$	− 0.618034i	− 0.618034i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	1.61803i	1.61803i
$519$	1.00000	1.00000
$520$	0	0
$521$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$521$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$522$	0	0
$523$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$523$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$524$	2.61803	2.61803
$525$	0	0
$526$	2.61803	2.61803
$527$	0	0
$528$	0	0
$529$	−1.00000	−1.00000
$530$	0	0
$531$	1.00000	1.00000
$532$	0	0
$533$	0	0
$534$	−1.61803	−1.61803
$535$	0	0
$536$	0	0
$537$	0	0
$538$	− 3.23607i	− 3.23607i
$539$	0	0
$540$	0	0
$541$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$541$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$542$	1.00000i	1.00000i
$543$	0	0
$544$	−0.618034	−0.618034
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	−1.00000	−1.00000
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 1.00000i	− 1.00000i
$554$	1.00000	1.00000
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	−1.00000	−1.00000
$565$	0	0
$566$	2.61803	2.61803
$567$	0	0
$568$	− 0.618034i	− 0.618034i
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$571$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$572$	0	0
$573$	1.23607i	1.23607i
$574$	0	0
$575$	0	0
$576$	1.00000	1.00000
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	1.00000i	1.00000i
$579$	0	0
$580$	0	0
$581$	−3.23607	−3.23607
$582$	− 1.61803i	− 1.61803i
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	1.61803i	1.61803i
$589$	0	0
$590$	0	0
$591$	1.23607	1.23607
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	1.00000	1.00000
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$601$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	−0.618034	−0.618034
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0.618034i	0.618034i
$613$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$613$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$614$	1.00000	1.00000
$615$	0	0
$616$	0	0
$617$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$617$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$618$	− 0.618034i	− 0.618034i
$619$	−2.00000	−2.00000	−1.00000			$\pi$
$619$	−2.00000	−2.00000	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.61803i	2.61803i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	− 2.61803i	− 2.61803i
$629$	−0.381966	−0.381966
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0.618034i	0.618034i
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−1.61803	−1.61803
$637$	0	0
$638$	0	0
$639$	0.381966	0.381966
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$643$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$647$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$653$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	2.61803i	2.61803i
$659$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$659$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$660$	0	0
$661$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$661$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$662$	− 2.61803i	− 2.61803i
$663$	0	0
$664$	2.00000	2.00000
$665$	0	0
$666$	0.618034	0.618034
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	1.00000i	1.00000i
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	−2.61803	−2.61803
$675$	0	0
$676$	1.61803	1.61803
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	−2.61803	−2.61803
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$683$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$684$	0	0
$685$	0	0
$686$	1.61803	1.61803
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	2.61803i	2.61803i
$693$	0	0
$694$	1.00000	1.00000
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	2.61803	2.61803
$707$	1.00000i	1.00000i
$708$	− 1.61803i	− 1.61803i
$709$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$709$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$710$	0	0
$711$	−0.381966	−0.381966
$712$	− 1.61803i	− 1.61803i
$713$	0	0
$714$	−1.00000	−1.00000
$715$	0	0
$716$	0	0
$717$	− 0.381966i	− 0.381966i
$718$	0	0
$719$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$719$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$720$	0	0
$721$	−1.00000	−1.00000
$722$	1.61803i	1.61803i
$723$	0.381966i	0.381966i
$724$	0	0
$725$	0	0
$726$	−1.00000	−1.00000
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−0.618034	−0.618034
$730$	0	0
$731$	0	0
$732$	1.61803i	1.61803i
$733$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$733$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$739$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$740$	0	0
$741$	0	0
$742$	4.23607i	4.23607i
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.23607i	1.23607i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0.381966i	0.381966i
$754$	0	0
$755$	0	0
$756$	2.61803	2.61803
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	2.61803i	2.61803i
$759$	0	0
$760$	0	0
$761$	2.00000	2.00000	1.00000			$0$
$761$	2.00000	2.00000	1.00000			$0$
$762$	0	0
$763$	0	0
$764$	−3.23607	−3.23607
$765$	0	0
$766$	−1.00000	−1.00000
$767$	0	0
$768$	− 0.618034i	− 0.618034i
$769$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$769$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$773$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$774$	0	0
$775$	0	0
$776$	1.61803	1.61803
$777$	0.618034i	0.618034i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	1.61803	1.61803
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	3.23607i	3.23607i
$789$	1.00000	1.00000
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	1.00000	1.00000
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	−0.618034	−0.618034
$800$	0	0
$801$	1.00000	1.00000
$802$	1.00000i	1.00000i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 1.23607i	− 1.23607i
$808$	− 0.618034i	− 0.618034i
$809$	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
$813$	0.381966i	0.381966i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$823$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$824$	0.618034	0.618034
$825$	0	0
$826$	−4.23607	−4.23607
$827$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$827$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0.381966	0.381966
$832$	0	0
$833$	1.00000i	1.00000i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	1.00000	1.00000
$847$	1.61803i	1.61803i
$848$	0	0
$849$	1.00000	1.00000
$850$	0	0
$851$	0	0
$852$	− 0.618034i	− 0.618034i
$853$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$853$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$854$	4.23607	4.23607
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	− 2.61803i	− 2.61803i
$863$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$863$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$864$	1.00000	1.00000
$865$	0	0
$866$	0	0
$867$	0.381966i	0.381966i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	1.00000i	1.00000i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	− 3.23607i	− 3.23607i
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	− 1.61803i	− 1.61803i
$883$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$883$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	− 0.381966i	− 0.381966i
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0.618034	0.618034
$895$	0	0
$896$	−2.61803	−2.61803
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−1.00000	−1.00000
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$907$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$908$	0	0
$909$	0.381966	0.381966
$910$	0	0
$911$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$911$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$912$	0	0
$913$	0	0
$914$	1.00000	1.00000
$915$	0	0
$916$	0	0
$917$	− 2.61803i	− 2.61803i
$918$	1.00000i	1.00000i
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0.381966	0.381966
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.381966i	0.381966i
$928$	0	0
$929$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$929$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$941$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$942$	− 1.61803i	− 1.61803i
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$947$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$948$	0.618034i	0.618034i
$949$	0	0
$950$	0	0
$951$	0	0
$952$	− 1.00000i	− 1.00000i
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	1.61803	1.61803
$955$	0	0
$956$	1.00000	1.00000
$957$	0	0
$958$	2.61803i	2.61803i
$959$	0	0
$960$	0	0
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	−1.00000	−1.00000
$965$	0	0
$966$	0	0
$967$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$967$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$968$	− 1.00000i	− 1.00000i
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	− 1.61803i	− 1.61803i
$973$	0	0
$974$	3.23607	3.23607
$975$	0	0
$976$	0	0
$977$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$977$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	− 2.61803i	− 2.61803i
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.00000i	1.00000i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$991$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$992$	0	0
$993$	− 1.00000i	− 1.00000i
$994$	−1.61803	−1.61803
$995$	0	0
$996$	2.00000	2.00000
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	0.618034	0.618034

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1175.1.b.b.1174.4		4
5.2	odd	4		47.1.b.a.46.1	✓	2
5.3	odd	4		1175.1.d.c.751.2		2
5.4	even	2	inner	1175.1.b.b.1174.1		4
15.2	even	4		423.1.d.a.46.2		2
20.7	even	4		752.1.g.a.657.1		2
35.2	odd	12		2303.1.f.c.704.2		4
35.12	even	12		2303.1.f.b.704.2		4
35.17	even	12		2303.1.f.b.422.2		4
35.27	even	4		2303.1.d.c.2255.1		2
35.32	odd	12		2303.1.f.c.422.2		4
40.27	even	4		3008.1.g.a.1409.2		2
40.37	odd	4		3008.1.g.b.1409.1		2
45.2	even	12		3807.1.f.a.3430.1		4
45.7	odd	12		3807.1.f.b.3430.2		4
45.22	odd	12		3807.1.f.b.2161.2		4
45.32	even	12		3807.1.f.a.2161.1		4
47.46	odd	2	CM	1175.1.b.b.1174.4		4
235.2	odd	92		2209.1.d.a.1124.1		44
235.7	odd	92		2209.1.d.a.280.2		44
235.12	odd	92		2209.1.d.a.655.1		44
235.17	odd	92		2209.1.d.a.1121.2		44
235.22	even	92		2209.1.d.a.2007.1		44
235.27	odd	92		2209.1.d.a.2138.2		44
235.32	odd	92		2209.1.d.a.339.1		44
235.37	odd	92		2209.1.d.a.2156.2		44
235.42	odd	92		2209.1.d.a.116.2		44
235.52	even	92		2209.1.d.a.116.2		44
235.57	even	92		2209.1.d.a.2156.2		44
235.62	even	92		2209.1.d.a.339.1		44
235.67	even	92		2209.1.d.a.2138.2		44
235.72	odd	92		2209.1.d.a.2007.1		44
235.77	even	92		2209.1.d.a.1121.2		44
235.82	even	92		2209.1.d.a.655.1		44
235.87	even	92		2209.1.d.a.280.2		44
235.92	even	92		2209.1.d.a.1124.1		44
235.93	even	4		1175.1.d.c.751.2		2
235.97	odd	92		2209.1.d.a.1730.1		44
235.102	odd	92		2209.1.d.a.1064.1		44
235.107	even	92		2209.1.d.a.1335.2		44
235.112	odd	92		2209.1.d.a.1979.2		44
235.117	even	92		2209.1.d.a.1586.1		44
235.122	odd	92		2209.1.d.a.438.2		44
235.127	even	92		2209.1.d.a.1167.2		44
235.132	even	92		2209.1.d.a.295.2		44
235.137	even	92		2209.1.d.a.172.1		44
235.147	odd	92		2209.1.d.a.1609.1		44
235.152	even	92		2209.1.d.a.67.2		44
235.157	odd	92		2209.1.d.a.1342.1		44
235.162	odd	92		2209.1.d.a.1580.2		44
235.167	even	92		2209.1.d.a.1580.2		44
235.172	even	92		2209.1.d.a.1342.1		44
235.177	odd	92		2209.1.d.a.67.2		44
235.182	even	92		2209.1.d.a.1609.1		44
235.187	even	4		47.1.b.a.46.1	✓	2
235.192	odd	92		2209.1.d.a.172.1		44
235.197	odd	92		2209.1.d.a.295.2		44
235.202	odd	92		2209.1.d.a.1167.2		44
235.207	even	92		2209.1.d.a.438.2		44
235.212	odd	92		2209.1.d.a.1586.1		44
235.217	even	92		2209.1.d.a.1979.2		44
235.222	odd	92		2209.1.d.a.1335.2		44
235.227	even	92		2209.1.d.a.1064.1		44
235.232	even	92		2209.1.d.a.1730.1		44
235.234	odd	2	inner	1175.1.b.b.1174.1		4
705.422	odd	4		423.1.d.a.46.2		2
940.187	odd	4		752.1.g.a.657.1		2
1645.187	odd	12		2303.1.f.b.704.2		4
1645.422	even	12		2303.1.f.c.704.2		4
1645.657	odd	4		2303.1.d.c.2255.1		2
1645.892	odd	12		2303.1.f.b.422.2		4
1645.1362	even	12		2303.1.f.c.422.2		4
1880.187	odd	4		3008.1.g.a.1409.2		2
1880.1597	even	4		3008.1.g.b.1409.1		2
2115.187	even	12		3807.1.f.b.3430.2		4
2115.1127	odd	12		3807.1.f.a.3430.1		4
2115.1597	even	12		3807.1.f.b.2161.2		4
2115.1832	odd	12		3807.1.f.a.2161.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
47.1.b.a.46.1	✓	2	5.2	odd	4
47.1.b.a.46.1	✓	2	235.187	even	4
423.1.d.a.46.2		2	15.2	even	4
423.1.d.a.46.2		2	705.422	odd	4
752.1.g.a.657.1		2	20.7	even	4
752.1.g.a.657.1		2	940.187	odd	4
1175.1.b.b.1174.1		4	5.4	even	2	inner
1175.1.b.b.1174.1		4	235.234	odd	2	inner
1175.1.b.b.1174.4		4	1.1	even	1	trivial
1175.1.b.b.1174.4		4	47.46	odd	2	CM
1175.1.d.c.751.2		2	5.3	odd	4
1175.1.d.c.751.2		2	235.93	even	4
2209.1.d.a.67.2		44	235.152	even	92
2209.1.d.a.67.2		44	235.177	odd	92
2209.1.d.a.116.2		44	235.42	odd	92
2209.1.d.a.116.2		44	235.52	even	92
2209.1.d.a.172.1		44	235.137	even	92
2209.1.d.a.172.1		44	235.192	odd	92
2209.1.d.a.280.2		44	235.7	odd	92
2209.1.d.a.280.2		44	235.87	even	92
2209.1.d.a.295.2		44	235.132	even	92
2209.1.d.a.295.2		44	235.197	odd	92
2209.1.d.a.339.1		44	235.32	odd	92
2209.1.d.a.339.1		44	235.62	even	92
2209.1.d.a.438.2		44	235.122	odd	92
2209.1.d.a.438.2		44	235.207	even	92
2209.1.d.a.655.1		44	235.12	odd	92
2209.1.d.a.655.1		44	235.82	even	92
2209.1.d.a.1064.1		44	235.102	odd	92
2209.1.d.a.1064.1		44	235.227	even	92
2209.1.d.a.1121.2		44	235.17	odd	92
2209.1.d.a.1121.2		44	235.77	even	92
2209.1.d.a.1124.1		44	235.2	odd	92
2209.1.d.a.1124.1		44	235.92	even	92
2209.1.d.a.1167.2		44	235.127	even	92
2209.1.d.a.1167.2		44	235.202	odd	92
2209.1.d.a.1335.2		44	235.107	even	92
2209.1.d.a.1335.2		44	235.222	odd	92
2209.1.d.a.1342.1		44	235.157	odd	92
2209.1.d.a.1342.1		44	235.172	even	92
2209.1.d.a.1580.2		44	235.162	odd	92
2209.1.d.a.1580.2		44	235.167	even	92
2209.1.d.a.1586.1		44	235.117	even	92
2209.1.d.a.1586.1		44	235.212	odd	92
2209.1.d.a.1609.1		44	235.147	odd	92
2209.1.d.a.1609.1		44	235.182	even	92
2209.1.d.a.1730.1		44	235.97	odd	92
2209.1.d.a.1730.1		44	235.232	even	92
2209.1.d.a.1979.2		44	235.112	odd	92
2209.1.d.a.1979.2		44	235.217	even	92
2209.1.d.a.2007.1		44	235.22	even	92
2209.1.d.a.2007.1		44	235.72	odd	92
2209.1.d.a.2138.2		44	235.27	odd	92
2209.1.d.a.2138.2		44	235.67	even	92
2209.1.d.a.2156.2		44	235.37	odd	92
2209.1.d.a.2156.2		44	235.57	even	92
2303.1.d.c.2255.1		2	35.27	even	4
2303.1.d.c.2255.1		2	1645.657	odd	4
2303.1.f.b.422.2		4	35.17	even	12
2303.1.f.b.422.2		4	1645.892	odd	12
2303.1.f.b.704.2		4	35.12	even	12
2303.1.f.b.704.2		4	1645.187	odd	12
2303.1.f.c.422.2		4	35.32	odd	12
2303.1.f.c.422.2		4	1645.1362	even	12
2303.1.f.c.704.2		4	35.2	odd	12
2303.1.f.c.704.2		4	1645.422	even	12
3008.1.g.a.1409.2		2	40.27	even	4
3008.1.g.a.1409.2		2	1880.187	odd	4
3008.1.g.b.1409.1		2	40.37	odd	4
3008.1.g.b.1409.1		2	1880.1597	even	4
3807.1.f.a.2161.1		4	45.32	even	12
3807.1.f.a.2161.1		4	2115.1832	odd	12
3807.1.f.a.3430.1		4	45.2	even	12
3807.1.f.a.3430.1		4	2115.1127	odd	12
3807.1.f.b.2161.2		4	45.22	odd	12
3807.1.f.b.2161.2		4	2115.1597	even	12
3807.1.f.b.3430.2		4	45.7	odd	12
3807.1.f.b.3430.2		4	2115.187	even	12

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:	\( N \)	\(=\)	\( 1175 = 5^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1175.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.61803i q^{2} +0.618034i q^{3} -1.61803 q^{4} -1.00000 q^{6} +1.61803i q^{7} -1.00000i q^{8} +0.618034 q^{9} +O(q^{10})\)	\(q+1.61803i q^{2} +0.618034i q^{3} -1.61803 q^{4} -1.00000 q^{6} +1.61803i q^{7} -1.00000i q^{8} +0.618034 q^{9} -1.00000i q^{12} -2.61803 q^{14} -0.618034i q^{17} +1.00000i q^{18} -1.00000 q^{21} +0.618034 q^{24} +1.00000i q^{27} -2.61803i q^{28} -1.00000i q^{32} +1.00000 q^{34} -1.00000 q^{36} -0.618034i q^{37} -1.61803i q^{42} -1.00000i q^{47} -1.61803 q^{49} +0.381966 q^{51} -1.61803i q^{53} -1.61803 q^{54} +1.61803 q^{56} +1.61803 q^{59} -1.61803 q^{61} +1.00000i q^{63} +1.61803 q^{64} +1.00000i q^{68} +0.618034 q^{71} -0.618034i q^{72} +1.00000 q^{74} -0.618034 q^{79} +2.00000i q^{83} +1.61803 q^{84} +1.61803 q^{89} +1.61803 q^{94} +0.618034 q^{96} +1.61803i q^{97} -2.61803i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) \) 4 * q - 2 * q^4 - 4 * q^6 - 2 * q^9	\( 4 q - 2 q^{4} - 4 q^{6} - 2 q^{9} - 6 q^{14} - 4 q^{21} - 2 q^{24} + 4 q^{34} - 4 q^{36} - 2 q^{49} + 6 q^{51} - 2 q^{54} + 2 q^{56} + 2 q^{59} - 2 q^{61} + 2 q^{64} - 2 q^{71} + 4 q^{74} + 2 q^{79} + 2 q^{84} + 2 q^{89} + 2 q^{94} - 2 q^{96}+O(q^{100}) \) 4 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 - 6 * q^14 - 4 * q^21 - 2 * q^24 + 4 * q^34 - 4 * q^36 - 2 * q^49 + 6 * q^51 - 2 * q^54 + 2 * q^56 + 2 * q^59 - 2 * q^61 + 2 * q^64 - 2 * q^71 + 4 * q^74 + 2 * q^79 + 2 * q^84 + 2 * q^89 + 2 * q^94 - 2 * q^96

Introduction

L-functions

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