LMFDB - Embedded newform 1000.1.e.b.499.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1000,1,Mod(499,1000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1000, 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("1000.499");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1000 = 2^{3} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1000.e (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:	yes
Analytic conductor:	$0.499065012633$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.1000000.1
Artin image:	$D_5$
Artin field:	Galois closure of 5.1.1000000.1

Embedding invariants

Embedding label			499.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	1000.499

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +0.618034 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +0.618034 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.61803 q^{11} -1.61803 q^{13} +0.618034 q^{14} +1.00000 q^{16} +1.00000 q^{18} +0.618034 q^{19} -1.61803 q^{22} -1.61803 q^{23} -1.61803 q^{26} +0.618034 q^{28} +1.00000 q^{32} +1.00000 q^{36} +0.618034 q^{37} +0.618034 q^{38} +0.618034 q^{41} -1.61803 q^{44} -1.61803 q^{46} -1.61803 q^{47} -0.618034 q^{49} -1.61803 q^{52} +0.618034 q^{53} +0.618034 q^{56} +0.618034 q^{59} +0.618034 q^{63} +1.00000 q^{64} +1.00000 q^{72} +0.618034 q^{74} +0.618034 q^{76} -1.00000 q^{77} +1.00000 q^{81} +0.618034 q^{82} -1.61803 q^{88} -1.61803 q^{89} -1.00000 q^{91} -1.61803 q^{92} -1.61803 q^{94} -0.618034 q^{98} -1.61803 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - q^7 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{4} - q^{7} + 2 q^{8} + 2 q^{9} - q^{11} - q^{13} - q^{14} + 2 q^{16} + 2 q^{18} - q^{19} - q^{22} - q^{23} - q^{26} - q^{28} + 2 q^{32} + 2 q^{36} - q^{37} - q^{38} - q^{41} - q^{44} - q^{46} - q^{47} + q^{49} - q^{52} - q^{53} - q^{56} - q^{59} - q^{63} + 2 q^{64} + 2 q^{72} - q^{74} - q^{76} - 2 q^{77} + 2 q^{81} - q^{82} - q^{88} - q^{89} - 2 q^{91} - q^{92} - q^{94} + q^{98} - q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - q^7 + 2 * q^8 + 2 * q^9 - q^11 - q^13 - q^14 + 2 * q^16 + 2 * q^18 - q^19 - q^22 - q^23 - q^26 - q^28 + 2 * q^32 + 2 * q^36 - q^37 - q^38 - q^41 - q^44 - q^46 - q^47 + q^49 - q^52 - q^53 - q^56 - q^59 - q^63 + 2 * q^64 + 2 * q^72 - q^74 - q^76 - 2 * q^77 + 2 * q^81 - q^82 - q^88 - q^89 - 2 * q^91 - q^92 - q^94 + q^98 - q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times$.

$n$	$377$	$501$	$751$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	1.00000	1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$7$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$8$	1.00000	1.00000
$9$	1.00000	1.00000
$10$	0	0
$11$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$11$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$12$	0	0
$13$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$13$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$14$	0.618034	0.618034
$15$	0	0
$16$	1.00000	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	1.00000	1.00000
$19$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$19$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$20$	0	0
$21$	0	0
$22$	−1.61803	−1.61803
$23$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$23$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$24$	0	0
$25$	0	0
$26$	−1.61803	−1.61803
$27$	0	0
$28$	0.618034	0.618034
$29$	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			$\pi$
$32$	1.00000	1.00000
$33$	0	0
$34$	0	0
$35$	0	0
$36$	1.00000	1.00000
$37$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$37$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$38$	0.618034	0.618034
$39$	0	0
$40$	0	0
$41$	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
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	−1.61803	−1.61803
$45$	0	0
$46$	−1.61803	−1.61803
$47$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$47$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$48$	0	0
$49$	−0.618034	−0.618034
$50$	0	0
$51$	0	0
$52$	−1.61803	−1.61803
$53$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$53$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$54$	0	0
$55$	0	0
$56$	0.618034	0.618034
$57$	0	0
$58$	0	0
$59$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$59$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0.618034	0.618034
$64$	1.00000	1.00000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	1.00000	1.00000
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0.618034	0.618034
$75$	0	0
$76$	0.618034	0.618034
$77$	−1.00000	−1.00000
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	1.00000	1.00000
$82$	0.618034	0.618034
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	−1.61803	−1.61803
$89$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$89$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$90$	0	0
$91$	−1.00000	−1.00000
$92$	−1.61803	−1.61803
$93$	0	0
$94$	−1.61803	−1.61803
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−0.618034	−0.618034
$99$	−1.61803	−1.61803
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$103$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$104$	−1.61803	−1.61803
$105$	0	0
$106$	0.618034	0.618034
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0.618034	0.618034
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.61803	−1.61803
$118$	0.618034	0.618034
$119$	0	0
$120$	0	0
$121$	1.61803	1.61803
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0.618034	0.618034
$127$	2.00000	2.00000	1.00000			$0$
$127$	2.00000	2.00000	1.00000			$0$
$128$	1.00000	1.00000
$129$	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
$133$	0.381966	0.381966
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$139$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.61803	2.61803
$144$	1.00000	1.00000
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0.618034	0.618034
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0.618034	0.618034
$153$	0	0
$154$	−1.00000	−1.00000
$155$	0	0
$156$	0	0
$157$	2.00000	2.00000	1.00000			$0$
$157$	2.00000	2.00000	1.00000			$0$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.00000	−1.00000
$162$	1.00000	1.00000
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0.618034	0.618034
$165$	0	0
$166$	0	0
$167$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$167$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$168$	0	0
$169$	1.61803	1.61803
$170$	0	0
$171$	0.618034	0.618034
$172$	0	0
$173$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$173$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$174$	0	0
$175$	0	0
$176$	−1.61803	−1.61803
$177$	0	0
$178$	−1.61803	−1.61803
$179$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$179$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	−1.00000	−1.00000
$183$	0	0
$184$	−1.61803	−1.61803
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−1.61803	−1.61803
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	−0.618034	−0.618034
$197$	2.00000	2.00000	1.00000			$0$
$197$	2.00000	2.00000	1.00000			$0$
$198$	−1.61803	−1.61803
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−1.61803	−1.61803
$207$	−1.61803	−1.61803
$208$	−1.61803	−1.61803
$209$	−1.00000	−1.00000
$210$	0	0
$211$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$211$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$212$	0.618034	0.618034
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	2.00000	2.00000	1.00000			$0$
$223$	2.00000	2.00000	1.00000			$0$
$224$	0.618034	0.618034
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	−1.61803	−1.61803
$235$	0	0
$236$	0.618034	0.618034
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$241$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$242$	1.61803	1.61803
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.00000	−1.00000
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.00000	2.00000	1.00000			$0$
$251$	2.00000	2.00000	1.00000			$0$
$252$	0.618034	0.618034
$253$	2.61803	2.61803
$254$	2.00000	2.00000
$255$	0	0
$256$	1.00000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0.381966	0.381966
$260$	0	0
$261$	0	0
$262$	−1.61803	−1.61803
$263$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$263$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$264$	0	0
$265$	0	0
$266$	0.381966	0.381966
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$277$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$278$	−1.61803	−1.61803
$279$	0	0
$280$	0	0
$281$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$281$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	2.61803	2.61803
$287$	0.381966	0.381966
$288$	1.00000	1.00000
$289$	1.00000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$293$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$294$	0	0
$295$	0	0
$296$	0.618034	0.618034
$297$	0	0
$298$	0	0
$299$	2.61803	2.61803
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0.618034	0.618034
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	−1.00000	−1.00000
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	2.00000	2.00000
$315$	0	0
$316$	0	0
$317$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$317$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	−1.00000	−1.00000
$323$	0	0
$324$	1.00000	1.00000
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0.618034	0.618034
$329$	−1.00000	−1.00000
$330$	0	0
$331$	2.00000	2.00000	1.00000			$0$
$331$	2.00000	2.00000	1.00000			$0$
$332$	0	0
$333$	0.618034	0.618034
$334$	0.618034	0.618034
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	1.61803	1.61803
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0.618034	0.618034
$343$	−1.00000	−1.00000
$344$	0	0
$345$	0	0
$346$	0.618034	0.618034
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	−1.61803	−1.61803
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	−1.61803	−1.61803
$357$	0	0
$358$	0.618034	0.618034
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−0.618034	−0.618034
$362$	0	0
$363$	0	0
$364$	−1.00000	−1.00000
$365$	0	0
$366$	0	0
$367$	2.00000	2.00000	1.00000			$0$
$367$	2.00000	2.00000	1.00000			$0$
$368$	−1.61803	−1.61803
$369$	0.618034	0.618034
$370$	0	0
$371$	0.381966	0.381966
$372$	0	0
$373$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$373$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$374$	0	0
$375$	0	0
$376$	−1.61803	−1.61803
$377$	0	0
$378$	0	0
$379$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$379$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$383$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−0.618034	−0.618034
$393$	0	0
$394$	2.00000	2.00000
$395$	0	0
$396$	−1.61803	−1.61803
$397$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$397$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$398$	0	0
$399$	0	0
$400$	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
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.00000	−1.00000
$408$	0	0
$409$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$409$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$410$	0	0
$411$	0	0
$412$	−1.61803	−1.61803
$413$	0.381966	0.381966
$414$	−1.61803	−1.61803
$415$	0	0
$416$	−1.61803	−1.61803
$417$	0	0
$418$	−1.00000	−1.00000
$419$	2.00000	2.00000	1.00000			$0$
$419$	2.00000	2.00000	1.00000			$0$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0.618034	0.618034
$423$	−1.61803	−1.61803
$424$	0.618034	0.618034
$425$	0	0
$426$	0	0
$427$	0	0
$428$	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	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.00000	−1.00000
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−0.618034	−0.618034
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	2.00000	2.00000
$447$	0	0
$448$	0.618034	0.618034
$449$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$449$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$450$	0	0
$451$	−1.00000	−1.00000
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	2.00000	2.00000	1.00000			$0$
$463$	2.00000	2.00000	1.00000			$0$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−1.61803	−1.61803
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0.618034	0.618034
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.618034	0.618034
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	−1.00000	−1.00000
$482$	−1.61803	−1.61803
$483$	0	0
$484$	1.61803	1.61803
$485$	0	0
$486$	0	0
$487$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$487$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$488$	0	0
$489$	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	$0.400000\pi$
$492$	0	0
$493$	0	0
$494$	−1.00000	−1.00000
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$499$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$500$	0	0
$501$	0	0
$502$	2.00000	2.00000
$503$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$503$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$504$	0.618034	0.618034
$505$	0	0
$506$	2.61803	2.61803
$507$	0	0
$508$	2.00000	2.00000
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	1.00000
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.61803	2.61803
$518$	0.381966	0.381966
$519$	0	0
$520$	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	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−1.61803	−1.61803
$525$	0	0
$526$	0.618034	0.618034
$527$	0	0
$528$	0	0
$529$	1.61803	1.61803
$530$	0	0
$531$	0.618034	0.618034
$532$	0.381966	0.381966
$533$	−1.00000	−1.00000
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.00000	1.00000
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−1.61803	−1.61803
$555$	0	0
$556$	−1.61803	−1.61803
$557$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$557$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−1.61803	−1.61803
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.618034	0.618034
$568$	0	0
$569$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$569$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$570$	0	0
$571$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$571$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$572$	2.61803	2.61803
$573$	0	0
$574$	0.381966	0.381966
$575$	0	0
$576$	1.00000	1.00000
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	1.00000	1.00000
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−1.00000	−1.00000
$584$	0	0
$585$	0	0
$586$	−1.61803	−1.61803
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0.618034	0.618034
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	2.61803	2.61803
$599$	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	0
$607$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$607$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$608$	0.618034	0.618034
$609$	0	0
$610$	0	0
$611$	2.61803	2.61803
$612$	0	0
$613$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$613$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$614$	0	0
$615$	0	0
$616$	−1.00000	−1.00000
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$619$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.00000	−1.00000
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	2.00000	2.00000
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	−1.61803	−1.61803
$635$	0	0
$636$	0	0
$637$	1.00000	1.00000
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$641$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	−1.00000	−1.00000
$645$	0	0
$646$	0	0
$647$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$647$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$648$	1.00000	1.00000
$649$	−1.00000	−1.00000
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$653$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$654$	0	0
$655$	0	0
$656$	0.618034	0.618034
$657$	0	0
$658$	−1.00000	−1.00000
$659$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$659$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	2.00000	2.00000
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0.618034	0.618034
$667$	0	0
$668$	0.618034	0.618034
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	1.61803	1.61803
$677$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$677$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0.618034	0.618034
$685$	0	0
$686$	−1.00000	−1.00000
$687$	0	0
$688$	0	0
$689$	−1.00000	−1.00000
$690$	0	0
$691$	2.00000	2.00000	1.00000			$0$
$691$	2.00000	2.00000	1.00000			$0$
$692$	0.618034	0.618034
$693$	−1.00000	−1.00000
$694$	0	0
$695$	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
$703$	0.381966	0.381966
$704$	−1.61803	−1.61803
$705$	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
$712$	−1.61803	−1.61803
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0.618034	0.618034
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	−1.00000	−1.00000
$722$	−0.618034	−0.618034
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$727$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$728$	−1.00000	−1.00000
$729$	1.00000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$733$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$734$	2.00000	2.00000
$735$	0	0
$736$	−1.61803	−1.61803
$737$	0	0
$738$	0.618034	0.618034
$739$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$739$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$740$	0	0
$741$	0	0
$742$	0.381966	0.381966
$743$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$743$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$744$	0	0
$745$	0	0
$746$	0.618034	0.618034
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−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	$-0.800000\pi$
$757$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$758$	−1.61803	−1.61803
$759$	0	0
$760$	0	0
$761$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$761$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0.618034	0.618034
$767$	−1.00000	−1.00000
$768$	0	0
$769$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$769$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.00000	2.00000	1.00000			$0$
$773$	2.00000	2.00000	1.00000			$0$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.381966	0.381966
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−0.618034	−0.618034
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	2.00000	2.00000
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−1.61803	−1.61803
$793$	0	0
$794$	0.618034	0.618034
$795$	0	0
$796$	0	0
$797$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$797$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−1.61803	−1.61803
$802$	0.618034	0.618034
$803$	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
$811$	−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
$814$	−1.00000	−1.00000
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0.618034	0.618034
$819$	−1.00000	−1.00000
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$823$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$824$	−1.61803	−1.61803
$825$	0	0
$826$	0.381966	0.381966
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−1.61803	−1.61803
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	−1.61803	−1.61803
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−1.00000	−1.00000
$837$	0	0
$838$	2.00000	2.00000
$839$	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
$844$	0.618034	0.618034
$845$	0	0
$846$	−1.61803	−1.61803
$847$	1.00000	1.00000
$848$	0.618034	0.618034
$849$	0	0
$850$	0	0
$851$	−1.00000	−1.00000
$852$	0	0
$853$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$853$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$859$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$863$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	−1.00000	−1.00000
$875$	0	0
$876$	0	0
$877$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$877$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$878$	0	0
$879$	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	$0.400000\pi$
$882$	−0.618034	−0.618034
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$887$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$888$	0	0
$889$	1.23607	1.23607
$890$	0	0
$891$	−1.61803	−1.61803
$892$	2.00000	2.00000
$893$	−1.00000	−1.00000
$894$	0	0
$895$	0	0
$896$	0.618034	0.618034
$897$	0	0
$898$	0.618034	0.618034
$899$	0	0
$900$	0	0
$901$	0	0
$902$	−1.00000	−1.00000
$903$	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	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.00000	−1.00000
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	2.00000	2.00000
$927$	−1.61803	−1.61803
$928$	0	0
$929$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$929$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$930$	0	0
$931$	−0.381966	−0.381966
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	−1.61803	−1.61803
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	−1.00000	−1.00000
$944$	0.618034	0.618034
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0.618034	0.618034
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.00000	1.00000
$962$	−1.00000	−1.00000
$963$	0	0
$964$	−1.61803	−1.61803
$965$	0	0
$966$	0	0
$967$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$967$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$968$	1.61803	1.61803
$969$	0	0
$970$	0	0
$971$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$971$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$972$	0	0
$973$	−1.00000	−1.00000
$974$	−1.61803	−1.61803
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	2.61803	2.61803
$980$	0	0
$981$	0	0
$982$	0.618034	0.618034
$983$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$983$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	−1.00000	−1.00000
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$997$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$998$	−1.61803	−1.61803
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1000.1.e.b.499.2		2
4.3	odd	2		4000.1.e.b.1999.1		2
5.2	odd	4		1000.1.g.a.251.4	yes	4
5.3	odd	4		1000.1.g.a.251.1	✓	4
5.4	even	2		1000.1.e.a.499.1		2
8.3	odd	2		1000.1.e.a.499.1		2
8.5	even	2		4000.1.e.a.1999.2		2
20.3	even	4		4000.1.g.a.751.3		4
20.7	even	4		4000.1.g.a.751.2		4
20.19	odd	2		4000.1.e.a.1999.2		2
40.3	even	4		1000.1.g.a.251.4	yes	4
40.13	odd	4		4000.1.g.a.751.2		4
40.19	odd	2	CM	1000.1.e.b.499.2		2
40.27	even	4		1000.1.g.a.251.1	✓	4
40.29	even	2		4000.1.e.b.1999.1		2
40.37	odd	4		4000.1.g.a.751.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1000.1.e.a.499.1		2	5.4	even	2
1000.1.e.a.499.1		2	8.3	odd	2
1000.1.e.b.499.2		2	1.1	even	1	trivial
1000.1.e.b.499.2		2	40.19	odd	2	CM
1000.1.g.a.251.1	✓	4	5.3	odd	4
1000.1.g.a.251.1	✓	4	40.27	even	4
1000.1.g.a.251.4	yes	4	5.2	odd	4
1000.1.g.a.251.4	yes	4	40.3	even	4
4000.1.e.a.1999.2		2	8.5	even	2
4000.1.e.a.1999.2		2	20.19	odd	2
4000.1.e.b.1999.1		2	4.3	odd	2
4000.1.e.b.1999.1		2	40.29	even	2
4000.1.g.a.751.2		4	20.7	even	4
4000.1.g.a.751.2		4	40.13	odd	4
4000.1.g.a.751.3		4	20.3	even	4
4000.1.g.a.751.3		4	40.37	odd	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:	\( N \)	\(=\)	\( 1000 = 2^{3} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1000.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.618034 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.618034 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.61803 q^{11} -1.61803 q^{13} +0.618034 q^{14} +1.00000 q^{16} +1.00000 q^{18} +0.618034 q^{19} -1.61803 q^{22} -1.61803 q^{23} -1.61803 q^{26} +0.618034 q^{28} +1.00000 q^{32} +1.00000 q^{36} +0.618034 q^{37} +0.618034 q^{38} +0.618034 q^{41} -1.61803 q^{44} -1.61803 q^{46} -1.61803 q^{47} -0.618034 q^{49} -1.61803 q^{52} +0.618034 q^{53} +0.618034 q^{56} +0.618034 q^{59} +0.618034 q^{63} +1.00000 q^{64} +1.00000 q^{72} +0.618034 q^{74} +0.618034 q^{76} -1.00000 q^{77} +1.00000 q^{81} +0.618034 q^{82} -1.61803 q^{88} -1.61803 q^{89} -1.00000 q^{91} -1.61803 q^{92} -1.61803 q^{94} -0.618034 q^{98} -1.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{4} - q^{7} + 2 q^{8} + 2 q^{9} - q^{11} - q^{13} - q^{14} + 2 q^{16} + 2 q^{18} - q^{19} - q^{22} - q^{23} - q^{26} - q^{28} + 2 q^{32} + 2 q^{36} - q^{37} - q^{38} - q^{41} - q^{44} - q^{46} - q^{47} + q^{49} - q^{52} - q^{53} - q^{56} - q^{59} - q^{63} + 2 q^{64} + 2 q^{72} - q^{74} - q^{76} - 2 q^{77} + 2 q^{81} - q^{82} - q^{88} - q^{89} - 2 q^{91} - q^{92} - q^{94} + q^{98} - q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - q^7 + 2 * q^8 + 2 * q^9 - q^11 - q^13 - q^14 + 2 * q^16 + 2 * q^18 - q^19 - q^22 - q^23 - q^26 - q^28 + 2 * q^32 + 2 * q^36 - q^37 - q^38 - q^41 - q^44 - q^46 - q^47 + q^49 - q^52 - q^53 - q^56 - q^59 - q^63 + 2 * q^64 + 2 * q^72 - q^74 - q^76 - 2 * q^77 + 2 * q^81 - q^82 - q^88 - q^89 - 2 * q^91 - q^92 - q^94 + q^98 - q^99

Introduction

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