LMFDB - Embedded newform 2975.1.b.b.2974.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2975,1,Mod(2974,2975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2975, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("2975.2974");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2975 = 5^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2975.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:	$1.48471841258$
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 119)
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.14161.1
Artin image:	$C_4\times D_5$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{20} - \cdots)$

Embedding invariants

Embedding label			2974.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	2975.2974
Dual form			2975.1.b.b.2974.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803i q^{2} +1.61803i q^{3} -1.61803 q^{4} +2.61803 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.61803 q^{9} +O(q^{10})$	$q-1.61803i q^{2} +1.61803i q^{3} -1.61803 q^{4} +2.61803 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.61803 q^{9} -2.61803i q^{12} +1.61803 q^{14} +1.00000i q^{17} +2.61803i q^{18} -1.61803 q^{21} -1.61803 q^{24} -1.00000i q^{27} -1.61803i q^{28} +0.618034 q^{31} +1.00000i q^{32} +1.61803 q^{34} +2.61803 q^{36} -1.61803 q^{41} +2.61803i q^{42} +1.61803i q^{43} -1.00000 q^{49} -1.61803 q^{51} -0.618034i q^{53} -1.61803 q^{54} -1.00000 q^{56} -1.61803 q^{61} -1.00000i q^{62} -1.61803i q^{63} +1.61803 q^{64} +0.618034i q^{67} -1.61803i q^{68} -1.61803i q^{72} +1.61803i q^{73} +2.61803i q^{82} +2.61803 q^{84} +2.61803 q^{86} +1.00000i q^{93} -1.61803 q^{96} +0.618034i q^{97} +1.61803i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{4} + 6 q^{6} - 2 q^{9}+O(q^{10}) $ 4 * q - 2 * q^4 + 6 * q^6 - 2 * q^9	$ 4 q - 2 q^{4} + 6 q^{6} - 2 q^{9} + 2 q^{14} - 2 q^{21} - 2 q^{24} - 2 q^{31} + 2 q^{34} + 6 q^{36} - 2 q^{41} - 4 q^{49} - 2 q^{51} - 2 q^{54} - 4 q^{56} - 2 q^{61} + 2 q^{64} + 6 q^{84} + 6 q^{86} - 2 q^{96}+O(q^{100}) $ 4 * q - 2 * q^4 + 6 * q^6 - 2 * q^9 + 2 * q^14 - 2 * q^21 - 2 * q^24 - 2 * q^31 + 2 * q^34 + 6 * q^36 - 2 * q^41 - 4 * q^49 - 2 * q^51 - 2 * q^54 - 4 * q^56 - 2 * q^61 + 2 * q^64 + 6 * q^84 + 6 * q^86 - 2 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2975\mathbb{Z}\right)^\times$.

$n$	$477$	$2451$	$2551$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$2$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$3$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$3$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$4$	−1.61803	−1.61803
$5$	0	0
$5$	0	0
$6$	2.61803	2.61803
$7$	1.00000i	1.00000i
$7$	1.00000i	1.00000i
$8$	1.00000i	1.00000i
$9$	−1.61803	−1.61803
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	− 2.61803i	− 2.61803i
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	1.61803	1.61803
$15$	0	0
$16$	0	0
$17$	1.00000i	1.00000i
$17$	1.00000i	1.00000i
$18$	2.61803i	2.61803i
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	−1.61803	−1.61803
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.61803	−1.61803
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 1.00000i
$28$	− 1.61803i	− 1.61803i
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$31$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$32$	1.00000i	1.00000i
$33$	0	0
$34$	1.61803	1.61803
$35$	0	0
$36$	2.61803	2.61803
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$41$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$42$	2.61803i	2.61803i
$43$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$43$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−1.00000	−1.00000
$50$	0	0
$51$	−1.61803	−1.61803
$52$	0	0
$53$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$53$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$54$	−1.61803	−1.61803
$55$	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	0
$61$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$61$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$62$	− 1.00000i	− 1.00000i
$63$	− 1.61803i	− 1.61803i
$64$	1.61803	1.61803
$65$	0	0
$66$	0	0
$67$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$67$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$68$	− 1.61803i	− 1.61803i
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	− 1.61803i	− 1.61803i
$73$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$73$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$74$	0	0
$75$	0	0
$76$	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
$82$	2.61803i	2.61803i
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	2.61803	2.61803
$85$	0	0
$86$	2.61803	2.61803
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.00000i	1.00000i
$94$	0	0
$95$	0	0
$96$	−1.61803	−1.61803
$97$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$97$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$98$	1.61803i	1.61803i
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	2.61803i	2.61803i
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	−1.00000	−1.00000
$107$	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	0
$112$	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
$118$	0	0
$119$	−1.00000	−1.00000
$120$	0	0
$121$	1.00000	1.00000
$122$	2.61803i	2.61803i
$123$	− 2.61803i	− 2.61803i
$124$	−1.00000	−1.00000
$125$	0	0
$126$	−2.61803	−2.61803
$127$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$127$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$128$	− 1.61803i	− 1.61803i
$129$	−2.61803	−2.61803
$130$	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
$135$	0	0
$136$	−1.00000	−1.00000
$137$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$137$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$138$	0	0
$139$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$139$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	2.61803	2.61803
$147$	− 1.61803i	− 1.61803i
$148$	0	0
$149$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$149$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$150$	0	0
$151$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$151$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$152$	0	0
$153$	− 1.61803i	− 1.61803i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	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	$0.5\pi$
$164$	2.61803	2.61803
$165$	0	0
$166$	0	0
$167$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$167$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$168$	− 1.61803i	− 1.61803i
$169$	−1.00000	−1.00000
$170$	0	0
$171$	0	0
$172$	− 2.61803i	− 2.61803i
$173$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$173$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$179$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$180$	0	0
$181$	2.00000	2.00000	1.00000			$0$
$181$	2.00000	2.00000	1.00000			$0$
$182$	0	0
$183$	− 2.61803i	− 2.61803i
$184$	0	0
$185$	0	0
$186$	1.61803	1.61803
$187$	0	0
$188$	0	0
$189$	1.00000	1.00000
$190$	0	0
$191$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$191$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$192$	2.61803i	2.61803i
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	1.00000	1.00000
$195$	0	0
$196$	1.61803	1.61803
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$199$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$200$	0	0
$201$	−1.00000	−1.00000
$202$	0	0
$203$	0	0
$204$	2.61803	2.61803
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	1.00000i	1.00000i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	1.00000	1.00000
$217$	0.618034i	0.618034i
$218$	0	0
$219$	−2.61803	−2.61803
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	−1.00000	−1.00000
$225$	0	0
$226$	0	0
$227$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$227$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$228$	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
$236$	0	0
$237$	0	0
$238$	1.61803i	1.61803i
$239$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$239$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$240$	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
$246$	−4.23607	−4.23607
$247$	0	0
$248$	0.618034i	0.618034i
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	2.61803i	2.61803i
$253$	0	0
$254$	1.00000	1.00000
$255$	0	0
$256$	−1.00000	−1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	4.23607i	4.23607i
$259$	0	0
$260$	0	0
$261$	0	0
$262$	− 3.23607i	− 3.23607i
$263$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$263$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	− 1.00000i	− 1.00000i
$269$	−2.00000	−2.00000	−1.00000			$\pi$
$269$	−2.00000	−2.00000	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	−2.61803	−2.61803
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	1.00000i	1.00000i
$279$	−1.00000	−1.00000
$280$	0	0
$281$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$281$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$282$	0	0
$283$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$283$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.61803i	− 1.61803i
$288$	− 1.61803i	− 1.61803i
$289$	−1.00000	−1.00000
$290$	0	0
$291$	−1.00000	−1.00000
$292$	− 2.61803i	− 2.61803i
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	−2.61803	−2.61803
$295$	0	0
$296$	0	0
$297$	0	0
$298$	− 2.61803i	− 2.61803i
$299$	0	0
$300$	0	0
$301$	−1.61803	−1.61803
$302$	2.61803i	2.61803i
$303$	0	0
$304$	0	0
$305$	0	0
$306$	−2.61803	−2.61803
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$311$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$312$	0	0
$313$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$313$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	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
$328$	− 1.61803i	− 1.61803i
$329$	0	0
$330$	0	0
$331$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$331$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$332$	0	0
$333$	0	0
$334$	−2.61803	−2.61803
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	1.61803i	1.61803i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 1.00000i	− 1.00000i
$344$	−1.61803	−1.61803
$345$	0	0
$346$	2.61803	2.61803
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1.61803i	− 1.61803i
$358$	− 2.61803i	− 2.61803i
$359$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$359$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$360$	0	0
$361$	1.00000	1.00000
$362$	− 3.23607i	− 3.23607i
$363$	1.61803i	1.61803i
$364$	0	0
$365$	0	0
$366$	−4.23607	−4.23607
$367$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$367$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$368$	0	0
$369$	2.61803	2.61803
$370$	0	0
$371$	0.618034	0.618034
$372$	− 1.61803i	− 1.61803i
$373$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$373$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	− 1.61803i	− 1.61803i
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−1.00000	−1.00000
$382$	− 1.00000i	− 1.00000i
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	2.61803	2.61803
$385$	0	0
$386$	0	0
$387$	− 2.61803i	− 2.61803i
$388$	− 1.00000i	− 1.00000i
$389$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$389$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$390$	0	0
$391$	0	0
$392$	− 1.00000i	− 1.00000i
$393$	3.23607i	3.23607i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$397$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$398$	− 2.61803i	− 2.61803i
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	1.61803i	1.61803i
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	− 1.61803i	− 1.61803i
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	2.61803	2.61803
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1.00000i	− 1.00000i
$418$	0	0
$419$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$419$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$420$	0	0
$421$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$421$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$422$	0	0
$423$	0	0
$424$	0.618034	0.618034
$425$	0	0
$426$	0	0
$427$	− 1.61803i	− 1.61803i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	1.00000	1.00000
$435$	0	0
$436$	0	0
$437$	0	0
$438$	4.23607i	4.23607i
$439$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$439$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$440$	0	0
$441$	1.61803	1.61803
$442$	0	0
$443$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$443$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.61803i	2.61803i
$448$	1.61803i	1.61803i
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 2.61803i	− 2.61803i
$454$	1.00000	1.00000
$455$	0	0
$456$	0	0
$457$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$457$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$458$	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
$463$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$463$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$464$	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
$469$	−0.618034	−0.618034
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	1.61803	1.61803
$477$	1.00000i	1.00000i
$478$	− 2.61803i	− 2.61803i
$479$	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
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	− 1.61803i	− 1.61803i
$489$	0	0
$490$	0	0
$491$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$491$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$492$	4.23607i	4.23607i
$493$	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	0
$501$	2.61803	2.61803
$502$	0	0
$503$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$503$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$504$	1.61803	1.61803
$505$	0	0
$506$	0	0
$507$	− 1.61803i	− 1.61803i
$508$	− 1.00000i	− 1.00000i
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	−1.61803	−1.61803
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	4.23607	4.23607
$517$	0	0
$518$	0	0
$519$	−2.61803	−2.61803
$520$	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
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	−3.23607	−3.23607
$525$	0	0
$526$	−3.23607	−3.23607
$527$	0.618034i	0.618034i
$528$	0	0
$529$	−1.00000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−0.618034	−0.618034
$537$	2.61803i	2.61803i
$538$	3.23607i	3.23607i
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	3.23607i	3.23607i
$544$	−1.00000	−1.00000
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	2.61803i	2.61803i
$549$	2.61803	2.61803
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	1.00000	1.00000
$557$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$557$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$558$	1.61803i	1.61803i
$559$	0	0
$560$	0	0
$561$	0	0
$562$	− 1.00000i	− 1.00000i
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	−1.00000	−1.00000
$567$	0	0
$568$	0	0
$569$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$569$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	1.00000i	1.00000i
$574$	−2.61803	−2.61803
$575$	0	0
$576$	−2.61803	−2.61803
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	1.61803i	1.61803i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	1.61803i	1.61803i
$583$	0	0
$584$	−1.61803	−1.61803
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	2.61803i	2.61803i
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	−2.61803	−2.61803
$597$	2.61803i	2.61803i
$598$	0	0
$599$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$599$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$600$	0	0
$601$	2.00000	2.00000	1.00000			$0$
$601$	2.00000	2.00000	1.00000			$0$
$602$	2.61803i	2.61803i
$603$	− 1.00000i	− 1.00000i
$604$	2.61803	2.61803
$605$	0	0
$606$	0	0
$607$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$607$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	2.61803i	2.61803i
$613$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$613$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−2.00000	−2.00000	−1.00000			$\pi$
$619$	−2.00000	−2.00000	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	− 1.00000i	− 1.00000i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−1.00000	−1.00000
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$631$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−1.61803	−1.61803
$637$	0	0
$638$	0	0
$639$	0	0
$640$	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	$0.300000\pi$
$643$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−1.00000	−1.00000
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.61803i	− 2.61803i
$658$	0	0
$659$	−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	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	− 1.00000i	− 1.00000i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	2.61803i	2.61803i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	− 1.61803i	− 1.61803i
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	1.61803	1.61803
$677$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$677$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$678$	0	0
$679$	−0.618034	−0.618034
$680$	0	0
$681$	−1.00000	−1.00000
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	−1.61803	−1.61803
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$691$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$692$	− 2.61803i	− 2.61803i
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.61803i	− 1.61803i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	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
$714$	−2.61803	−2.61803
$715$	0	0
$716$	−2.61803	−2.61803
$717$	2.61803i	2.61803i
$718$	1.00000i	1.00000i
$719$	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
$721$	0	0
$722$	− 1.61803i	− 1.61803i
$723$	1.00000i	1.00000i
$724$	−3.23607	−3.23607
$725$	0	0
$726$	2.61803	2.61803
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	1.61803	1.61803
$730$	0	0
$731$	−1.61803	−1.61803
$732$	4.23607i	4.23607i
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	−2.61803	−2.61803
$735$	0	0
$736$	0	0
$737$	0	0
$738$	− 4.23607i	− 4.23607i
$739$	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
$742$	− 1.00000i	− 1.00000i
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	−1.00000	−1.00000
$745$	0	0
$746$	2.61803	2.61803
$747$	0	0
$748$	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	0
$754$	0	0
$755$	0	0
$756$	−1.61803	−1.61803
$757$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$757$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	1.61803i	1.61803i
$763$	0	0
$764$	−1.00000	−1.00000
$765$	0	0
$766$	0	0
$767$	0	0
$768$	− 1.61803i	− 1.61803i
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	−4.23607	−4.23607
$775$	0	0
$776$	−0.618034	−0.618034
$777$	0	0
$778$	1.00000i	1.00000i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	5.23607	5.23607
$787$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$787$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$788$	0	0
$789$	3.23607	3.23607
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	1.00000	1.00000
$795$	0	0
$796$	−2.61803	−2.61803
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	1.61803	1.61803
$805$	0	0
$806$	0	0
$807$	− 3.23607i	− 3.23607i
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$811$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$812$	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	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	− 4.23607i	− 4.23607i
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 1.00000i	− 1.00000i
$834$	−1.61803	−1.61803
$835$	0	0
$836$	0	0
$837$	− 0.618034i	− 0.618034i
$838$	1.00000i	1.00000i
$839$	−2.00000	−2.00000	−1.00000			$\pi$
$839$	−2.00000	−2.00000	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	2.61803i	2.61803i
$843$	1.00000i	1.00000i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.00000i	1.00000i
$848$	0	0
$849$	1.00000	1.00000
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$853$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$854$	−2.61803	−2.61803
$855$	0	0
$856$	0	0
$857$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$857$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	2.61803	2.61803
$862$	0	0
$863$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$863$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$864$	1.00000	1.00000
$865$	0	0
$866$	0	0
$867$	− 1.61803i	− 1.61803i
$868$	− 1.00000i	− 1.00000i
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 1.00000i	− 1.00000i
$874$	0	0
$875$	0	0
$876$	4.23607	4.23607
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	− 2.61803i	− 2.61803i
$879$	0	0
$880$	0	0
$881$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$881$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$882$	− 2.61803i	− 2.61803i
$883$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$883$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$884$	0	0
$885$	0	0
$886$	−3.23607	−3.23607
$887$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$887$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$888$	0	0
$889$	−0.618034	−0.618034
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	4.23607	4.23607
$895$	0	0
$896$	1.61803	1.61803
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0.618034	0.618034
$902$	0	0
$903$	− 2.61803i	− 2.61803i
$904$	0	0
$905$	0	0
$906$	−4.23607	−4.23607
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	− 1.00000i	− 1.00000i
$909$	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	0
$916$	0	0
$917$	2.00000i	2.00000i
$918$	− 1.61803i	− 1.61803i
$919$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$919$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	2.61803	2.61803
$927$	0	0
$928$	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
$933$	1.00000i	1.00000i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	1.00000i	1.00000i
$939$	1.00000	1.00000
$940$	0	0
$941$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$941$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	− 1.00000i	− 1.00000i
$953$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$953$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$954$	1.61803	1.61803
$955$	0	0
$956$	−2.61803	−2.61803
$957$	0	0
$958$	− 2.61803i	− 2.61803i
$959$	1.61803	1.61803
$960$	0	0
$961$	−0.618034	−0.618034
$962$	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	$-0.700000\pi$
$967$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$968$	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
$973$	− 0.618034i	− 0.618034i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$977$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	− 1.00000i	− 1.00000i
$983$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$983$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$984$	2.61803	2.61803
$985$	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			$\pi$
$992$	0.618034i	0.618034i
$993$	1.00000i	1.00000i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$997$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2975.1.b.b.2974.1		4
5.2	odd	4		2975.1.h.d.951.2		2
5.3	odd	4		119.1.d.a.118.1	✓	2
5.4	even	2	inner	2975.1.b.b.2974.4		4
7.6	odd	2		2975.1.b.a.2974.1		4
15.8	even	4		1071.1.h.b.118.2		2
17.16	even	2		2975.1.b.a.2974.1		4
20.3	even	4		1904.1.n.b.1665.2		2
35.3	even	12		833.1.h.a.509.2		4
35.13	even	4		119.1.d.b.118.1	yes	2
35.18	odd	12		833.1.h.b.509.2		4
35.23	odd	12		833.1.h.b.815.2		4
35.27	even	4		2975.1.h.c.951.2		2
35.33	even	12		833.1.h.a.815.2		4
35.34	odd	2		2975.1.b.a.2974.4		4
85.3	even	16		2023.1.l.b.1266.1		16
85.8	odd	8		2023.1.f.b.1483.4		8
85.13	odd	4		2023.1.c.e.1735.3		4
85.23	even	16		2023.1.l.b.1868.1		16
85.28	even	16		2023.1.l.b.1868.2		16
85.33	odd	4		119.1.d.b.118.1	yes	2
85.38	odd	4		2023.1.c.e.1735.4		4
85.43	odd	8		2023.1.f.b.1483.3		8
85.48	even	16		2023.1.l.b.1266.2		16
85.53	odd	8		2023.1.f.b.251.1		8
85.58	even	16		2023.1.l.b.1889.4		16
85.63	even	16		2023.1.l.b.468.3		16
85.67	odd	4		2975.1.h.c.951.2		2
85.73	even	16		2023.1.l.b.468.4		16
85.78	even	16		2023.1.l.b.1889.3		16
85.83	odd	8		2023.1.f.b.251.2		8
85.84	even	2		2975.1.b.a.2974.4		4
105.83	odd	4		1071.1.h.a.118.2		2
119.118	odd	2	CM	2975.1.b.b.2974.1		4
140.83	odd	4		1904.1.n.a.1665.1		2
255.203	even	4		1071.1.h.a.118.2		2
340.203	even	4		1904.1.n.a.1665.1		2
595.13	even	4		2023.1.c.e.1735.4		4
595.33	even	12		833.1.h.b.815.2		4
595.48	odd	16		2023.1.l.b.1266.1		16
595.83	even	8		2023.1.f.b.251.1		8
595.118	even	4		119.1.d.a.118.1	✓	2
595.223	even	8		2023.1.f.b.251.2		8
595.237	even	4		2975.1.h.d.951.2		2
595.258	odd	16		2023.1.l.b.1266.2		16
595.293	even	4		2023.1.c.e.1735.3		4
595.328	odd	16		2023.1.l.b.468.3		16
595.363	odd	16		2023.1.l.b.1868.2		16
595.373	odd	12		833.1.h.a.815.2		4
595.398	odd	16		2023.1.l.b.1889.3		16
595.433	even	8		2023.1.f.b.1483.3		8
595.458	even	12		833.1.h.b.509.2		4
595.468	even	8		2023.1.f.b.1483.4		8
595.503	odd	16		2023.1.l.b.1889.4		16
595.538	odd	16		2023.1.l.b.1868.1		16
595.543	odd	12		833.1.h.a.509.2		4
595.573	odd	16		2023.1.l.b.468.4		16
595.594	odd	2	inner	2975.1.b.b.2974.4		4
1785.713	odd	4		1071.1.h.b.118.2		2
2380.1903	odd	4		1904.1.n.b.1665.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
119.1.d.a.118.1	✓	2	5.3	odd	4
119.1.d.a.118.1	✓	2	595.118	even	4
119.1.d.b.118.1	yes	2	35.13	even	4
119.1.d.b.118.1	yes	2	85.33	odd	4
833.1.h.a.509.2		4	35.3	even	12
833.1.h.a.509.2		4	595.543	odd	12
833.1.h.a.815.2		4	35.33	even	12
833.1.h.a.815.2		4	595.373	odd	12
833.1.h.b.509.2		4	35.18	odd	12
833.1.h.b.509.2		4	595.458	even	12
833.1.h.b.815.2		4	35.23	odd	12
833.1.h.b.815.2		4	595.33	even	12
1071.1.h.a.118.2		2	105.83	odd	4
1071.1.h.a.118.2		2	255.203	even	4
1071.1.h.b.118.2		2	15.8	even	4
1071.1.h.b.118.2		2	1785.713	odd	4
1904.1.n.a.1665.1		2	140.83	odd	4
1904.1.n.a.1665.1		2	340.203	even	4
1904.1.n.b.1665.2		2	20.3	even	4
1904.1.n.b.1665.2		2	2380.1903	odd	4
2023.1.c.e.1735.3		4	85.13	odd	4
2023.1.c.e.1735.3		4	595.293	even	4
2023.1.c.e.1735.4		4	85.38	odd	4
2023.1.c.e.1735.4		4	595.13	even	4
2023.1.f.b.251.1		8	85.53	odd	8
2023.1.f.b.251.1		8	595.83	even	8
2023.1.f.b.251.2		8	85.83	odd	8
2023.1.f.b.251.2		8	595.223	even	8
2023.1.f.b.1483.3		8	85.43	odd	8
2023.1.f.b.1483.3		8	595.433	even	8
2023.1.f.b.1483.4		8	85.8	odd	8
2023.1.f.b.1483.4		8	595.468	even	8
2023.1.l.b.468.3		16	85.63	even	16
2023.1.l.b.468.3		16	595.328	odd	16
2023.1.l.b.468.4		16	85.73	even	16
2023.1.l.b.468.4		16	595.573	odd	16
2023.1.l.b.1266.1		16	85.3	even	16
2023.1.l.b.1266.1		16	595.48	odd	16
2023.1.l.b.1266.2		16	85.48	even	16
2023.1.l.b.1266.2		16	595.258	odd	16
2023.1.l.b.1868.1		16	85.23	even	16
2023.1.l.b.1868.1		16	595.538	odd	16
2023.1.l.b.1868.2		16	85.28	even	16
2023.1.l.b.1868.2		16	595.363	odd	16
2023.1.l.b.1889.3		16	85.78	even	16
2023.1.l.b.1889.3		16	595.398	odd	16
2023.1.l.b.1889.4		16	85.58	even	16
2023.1.l.b.1889.4		16	595.503	odd	16
2975.1.b.a.2974.1		4	7.6	odd	2
2975.1.b.a.2974.1		4	17.16	even	2
2975.1.b.a.2974.4		4	35.34	odd	2
2975.1.b.a.2974.4		4	85.84	even	2
2975.1.b.b.2974.1		4	1.1	even	1	trivial
2975.1.b.b.2974.1		4	119.118	odd	2	CM
2975.1.b.b.2974.4		4	5.4	even	2	inner
2975.1.b.b.2974.4		4	595.594	odd	2	inner
2975.1.h.c.951.2		2	35.27	even	4
2975.1.h.c.951.2		2	85.67	odd	4
2975.1.h.d.951.2		2	5.2	odd	4
2975.1.h.d.951.2		2	595.237	even	4

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:	\( N \)	\(=\)	\( 2975 = 5^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2975.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.61803i q^{2} +1.61803i q^{3} -1.61803 q^{4} +2.61803 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.61803 q^{9} +O(q^{10})\)	\(q-1.61803i q^{2} +1.61803i q^{3} -1.61803 q^{4} +2.61803 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.61803 q^{9} -2.61803i q^{12} +1.61803 q^{14} +1.00000i q^{17} +2.61803i q^{18} -1.61803 q^{21} -1.61803 q^{24} -1.00000i q^{27} -1.61803i q^{28} +0.618034 q^{31} +1.00000i q^{32} +1.61803 q^{34} +2.61803 q^{36} -1.61803 q^{41} +2.61803i q^{42} +1.61803i q^{43} -1.00000 q^{49} -1.61803 q^{51} -0.618034i q^{53} -1.61803 q^{54} -1.00000 q^{56} -1.61803 q^{61} -1.00000i q^{62} -1.61803i q^{63} +1.61803 q^{64} +0.618034i q^{67} -1.61803i q^{68} -1.61803i q^{72} +1.61803i q^{73} +2.61803i q^{82} +2.61803 q^{84} +2.61803 q^{86} +1.00000i q^{93} -1.61803 q^{96} +0.618034i q^{97} +1.61803i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{4} + 6 q^{6} - 2 q^{9}+O(q^{10}) \) 4 * q - 2 * q^4 + 6 * q^6 - 2 * q^9	\( 4 q - 2 q^{4} + 6 q^{6} - 2 q^{9} + 2 q^{14} - 2 q^{21} - 2 q^{24} - 2 q^{31} + 2 q^{34} + 6 q^{36} - 2 q^{41} - 4 q^{49} - 2 q^{51} - 2 q^{54} - 4 q^{56} - 2 q^{61} + 2 q^{64} + 6 q^{84} + 6 q^{86} - 2 q^{96}+O(q^{100}) \) 4 * q - 2 * q^4 + 6 * q^6 - 2 * q^9 + 2 * q^14 - 2 * q^21 - 2 * q^24 - 2 * q^31 + 2 * q^34 + 6 * q^36 - 2 * q^41 - 4 * q^49 - 2 * q^51 - 2 * q^54 - 4 * q^56 - 2 * q^61 + 2 * q^64 + 6 * q^84 + 6 * q^86 - 2 * q^96

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