LMFDB - Newform orbit 459.2.e.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [459,2,Mod(154,459)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(459, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("459.154");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$459 = 3^{3} \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	459.e (of order $3$ , degree $2$ , 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:	$3.66513345278$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.152695449.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 2x^{7} + 3x^{5} - 5x^{4} + 6x^{3} - 16x + 16$ x^8 - 2x^7 + 3x^5 - 5x^4 + 6x^3 - 16*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$3^{2}$
Twist minimal:	no (minimal twist has level 153)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/459\mathbb{Z}\right)^\times$ .

$n$	$137$	$190$
$\chi(n)$	$-1 - \beta_{4}$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

154.1

−0.0915132	−	1.41125i
1.40702	+	0.142460i
1.05924	+	0.937022i
−1.37475	+	0.331768i
−0.0915132	+	1.41125i
1.40702	−	0.142460i
1.05924	−	0.937022i
−1.37475	−	0.331768i

−1.17642

2.03762i

−1.76793

−

3.06215i

1.17642

2.03762i

1.76793

−

3.06215i

3.61366

−5.53587

154.2

−0.580136

1.00483i

0.326884

0.566179i

0.580136

1.00483i

−0.326884

0.566179i

−3.07909

−1.34623

154.3

0.281864

−

0.488204i

0.841105

1.45684i

−0.281864

−

0.488204i

−0.841105

1.45684i

2.07577

−0.317790

154.4

0.974693

−

1.68822i

−0.900054

−

1.55894i

−0.974693

−

1.68822i

0.900054

−

1.55894i

0.389667

−3.80011

307.1

−1.17642

−

2.03762i

−1.76793

3.06215i

1.17642

−

2.03762i

1.76793

3.06215i

3.61366

−5.53587

307.2

−0.580136

−

1.00483i

0.326884

−

0.566179i

0.580136

−

1.00483i

−0.326884

−

0.566179i

−3.07909

−1.34623

307.3

0.281864

0.488204i

0.841105

−

1.45684i

−0.281864

0.488204i

−0.841105

−

1.45684i

2.07577

−0.317790

307.4

0.974693

1.68822i

−0.900054

1.55894i

−0.974693

1.68822i

0.900054

1.55894i

0.389667

−3.80011

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	459.2.e.b		8
3.b	odd	2	1		153.2.e.b	✓	8
9.c	even	3	1	inner	459.2.e.b		8
9.c	even	3	1		1377.2.a.f		4
9.d	odd	6	1		153.2.e.b	✓	8
9.d	odd	6	1		1377.2.a.e		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
153.2.e.b	✓	8	3.b	odd	2	1
153.2.e.b	✓	8	9.d	odd	6	1
459.2.e.b		8	1.a	even	1	1	trivial
459.2.e.b		8	9.c	even	3	1	inner
1377.2.a.e		4	9.d	odd	6	1
1377.2.a.f		4	9.c	even	3	1

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{8} + T_{2}^{7} + 6T_{2}^{6} + T_{2}^{5} + 25T_{2}^{4} + 9T_{2}^{3} + 24T_{2}^{2} - 9T_{2} + 9$ T2^8 + T2^7 + 6*T2^6 + T2^5 + 25*T2^4 + 9*T2^3 + 24*T2^2 - 9*T2 + 9 acting on $S_{2}^{\mathrm{new}}(459, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8} + T^{7} + 6 T^{6} + \cdots + 9$ T^8 + T^7 + 6T^6 + T^5 + 25T^4 + 9T^3 + 24T^2 - 9*T + 9
$3$	$T^{8}$ T^8
$5$	$T^{8} - T^{7} + 6 T^{6} + \cdots + 9$ T^8 - T^7 + 6T^6 - T^5 + 25T^4 - 9T^3 + 24T^2 + 9*T + 9
$7$	$T^{8} - 3 T^{7} + \cdots + 49$ T^8 - 3T^7 + 14T^6 - 3T^5 + 45T^4 - 3T^3 + 116T^2 + 63*T + 49
$11$	$T^{8} - 3 T^{7} + \cdots + 81$ T^8 - 3T^7 + 18T^6 - 27T^5 + 171T^4 - 297T^3 + 648T^2 - 243*T + 81
$13$	$T^{8} - 9 T^{7} + \cdots + 529$ T^8 - 9T^7 + 65T^6 - 192T^5 + 495T^4 - 30T^3 + 944T^2 - 552*T + 529
$17$	$(T - 1)^{8}$ (T - 1)^8
$19$	$(T^{4} + 7 T^{3} + \cdots + 451)^{2}$ (T^4 + 7T^3 - 42T^2 - 218*T + 451)^2
$23$	$T^{8} - 10 T^{7} + \cdots + 25281$ T^8 - 10T^7 + 93T^6 - 274T^5 + 1228T^4 - 2466T^3 + 11517T^2 - 16218*T + 25281
$29$	$T^{8} + 15 T^{7} + \cdots + 729$ T^8 + 15T^7 + 159T^6 + 810T^5 + 2979T^4 + 5130T^3 + 6318T^2 + 2430*T + 729
$31$	$T^{8} - 15 T^{7} + \cdots + 9409$ T^8 - 15T^7 + 149T^6 - 840T^5 + 3429T^4 - 8490T^3 + 15128T^2 - 14550*T + 9409
$37$	$(T^{4} + 12 T^{3} + \cdots - 1223)^{2}$ (T^4 + 12T^3 - 35T^2 - 576*T - 1223)^2
$41$	$T^{8} + 6 T^{7} + \cdots + 23409$ T^8 + 6T^7 + 81T^6 - 234T^5 + 1980T^4 - 1026T^3 + 7209T^2 - 2754*T + 23409
$43$	$T^{8} - 18 T^{7} + \cdots + 2611456$ T^8 - 18T^7 + 320T^6 - 1944T^5 + 18480T^4 - 54432T^3 + 882560T^2 - 1512576*T + 2611456
$47$	$T^{8} - 12 T^{7} + \cdots + 29241$ T^8 - 12T^7 + 108T^6 - 486T^5 + 1791T^4 - 3132T^3 + 6885T^2 - 4617*T + 29241
$53$	$(T^{4} + 12 T^{3} + \cdots - 81)^{2}$ (T^4 + 12T^3 + 15T^2 - 54*T - 81)^2
$59$	$T^{8} + 114 T^{6} + \cdots + 263169$ T^8 + 114T^6 + 576T^5 + 12483T^4 + 32832T^3 + 141426T^2 - 147744T + 263169
$61$	$T^{8} + 5 T^{7} + \cdots + 33674809$ T^8 + 5T^7 + 184T^6 + 191T^5 + 21943T^4 + 20357T^3 + 1165726T^2 - 2860879*T + 33674809
$67$	$T^{8} - 6 T^{7} + \cdots + 790321$ T^8 - 6T^7 + 113T^6 + 330T^5 + 5436T^4 + 5586T^3 + 72809T^2 + 58674*T + 790321
$71$	$(T^{4} - 25 T^{3} + \cdots - 1107)^{2}$ (T^4 - 25T^3 + 163T^2 - 81*T - 1107)^2
$73$	$(T^{4} - 4 T^{3} - 95 T^{2} + \cdots - 27)^{2}$ (T^4 - 4T^3 - 95T^2 + 252*T - 27)^2
$79$	$T^{8} - 8 T^{7} + \cdots + 720801$ T^8 - 8T^7 + 186T^6 - 566T^5 + 21901T^4 - 107646T^3 + 490863T^2 - 654579*T + 720801
$83$	$T^{8} + 7 T^{7} + \cdots + 53361$ T^8 + 7T^7 + 57T^6 + 220T^5 + 1261T^4 + 4338T^3 + 17196T^2 + 31878*T + 53361
$89$	$(T^{4} - 14 T^{3} + \cdots - 4131)^{2}$ (T^4 - 14T^3 - 134T^2 + 2241*T - 4131)^2
$97$	$T^{8} - 16 T^{7} + \cdots + 20241001$ T^8 - 16T^7 + 334T^6 - 1888T^5 + 35671T^4 - 266272T^3 + 2107702T^2 - 7054432*T + 20241001
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$f(q)$	$=$	$q + ( - \beta_{3} - \beta_{2}) q^{2} + ( - \beta_{7} - \beta_{4} - \beta_1 - 1) q^{4} - \beta_{3} q^{5} + ( - \beta_{7} - \beta_{4}) q^{7} + ( - \beta_{6} + \beta_1 + 1) q^{8} + ( - \beta_1 - 3) q^{10}+ \cdots + ( - 2 \beta_{6} - \beta_{2} + 4 \beta_1 + 5) q^{98}+O(q^{100})$ q + (-b3 - b2) * q^2 + (-b7 - b4 - b1 - 1) * q^4 - b3 * q^5 + (-b7 - b4) * q^7 + (-b6 + b1 + 1) * q^8 + (-b1 - 3) * q^10 + (-b7 - b6 - b5 - b4) * q^11 + (b5 + 2b4 - b3 + 2) q^13 + (b7 + b5 + b4 - 2b3 + b1 + 1) q^14 + (b6 + b5 - b4 - b3 - b2) * q^16 + q^17 + (-2b6 + 2b2 + b1 - 2) * q^19 + (-b7 - b6 - b5 - b4 + 2b3 + 2b2) * q^20 + (2b7 + b5 + b4 - b3 + 2b1 + 1) * q^22 + (-2b7 - b5 + 2b4 - 2b1 + 2) q^23 + (b7 - 2b4) q^25 + (-b2 - 2b1 - 3) q^26 + (b6 - b2 - 2b1 - 5) q^28 + (b6 + b5 + 4b4 + b3 + b2) q^29 + (b7 + 4b4 + b1 + 4) q^31 + (2b5 - b4 - 2b3 - 1) * q^32 + (-b3 - b2) * q^34 + (-b6 + 2b2 + b1 + 1) q^35 + (b6 - 4b2 - 2) q^37 + (5b7 + b6 + b5 + 7b4 + 3b3 + 3b2) * q^38 + (2b7 + b5 + b4 - b3 + 2b1 + 1) * q^40 + (2b7 - b5 - b4 + 2b1 - 1) * q^41 + (-2b7 - 2b6 - 2b5 - 6b4 - 4b3 - 4b2) * q^43 + (-2b2 - 2b1 - 3) * q^44 + (-2b6 - b2 + 3b1 + 2) * q^46 + (b7 - 3b4 - b3 - b2) q^47 + (-2b7 - b5 + 2b4 + b3 - 2b1 + 2) q^49 + (-b7 - b5 - b4 - b3 - b1 - 1) * q^50 + (-3b7 - b4 + 3b3 + 3b2) q^52 + (-b6 - 2b2 + 2b1 - 2) * q^53 + (-b6 + b2 + 2b1 + 1) q^55 + (-2b7 - 3b4 + 2b3 + 2b2) * q^56 + (3b4 + 3b3 + 3) * q^58 + (-2b5 + b4 + 4b3 + 1) * q^59 + (5b7 + b6 + b5 + 3b4 + 2b3 + 2b2) * q^61 + (b6 - 5b2 - b1 - 1) q^62 + (2b6 + b2 - 4b1 - 4) * q^64 + (2b7 + 3b4 - b3 - b2) * q^65 + (-2b7 - 3b5 + b4 - 2b1 + 1) q^67 + (-b7 - b4 - b1 - 1) * q^68 + (4b7 + b6 + b5 + 7b4 - b3 - b2) * q^70 + (2b6 - 2b2 + b1 + 7) * q^71 + (b6 + 2b2 + 2b1 + 1) * q^73 + (-5b7 - 12b4 + b3 + b2) * q^74 + (-b7 - b5 + 7b3 - b1) q^76 + (-2b7 - 3b4 + 2b3 - 2b1 - 3) * q^77 + (3b7 + b6 + b5 + 5b3 + 5b2) q^79 + (2b2 - 2b1 - 3) * q^80 + (2b6 - 2b2 - b1 - 2) * q^82 + (b7 - b6 - b5 + 2b4) q^83 - b3 * q^85 + (2b5 - 10b4 - 6b3 - 10) q^86 + (-6b4 + 3b3 + 3b2) q^88 + (3b6 + 3b2 + b1 + 3) * q^89 + (3b2 + 3b1 + 1) * q^91 + (b6 + b5 + 4b4 - 3b3 - 3b2) q^92 + (-2b7 - b5 - 4b4 - 2b3 - 2b1 - 4) * q^94 + (5b7 + b5 + 7b4 + 3b3 + 5b1 + 7) * q^95 + (-2b7 - 4b6 - 4b5 - 5b4 - 2b3 - 2b2) * q^97 + (-2b6 - b2 + 4b1 + 5) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$8 q - q^{2} - 3 q^{4} + q^{5} + 3 q^{7} + 6 q^{8} - 22 q^{10} + 3 q^{11} + 9 q^{13} + 5 q^{14} + 3 q^{16} + 8 q^{17} - 14 q^{19} + 5 q^{20} + 3 q^{22} + 10 q^{23} + 9 q^{25} - 22 q^{26} - 38 q^{28} - 15 q^{29}+ \cdots + 30 q^{98}+O(q^{100})$ 8 * q - q^2 - 3 * q^4 + q^5 + 3 * q^7 + 6 * q^8 - 22 * q^10 + 3 * q^11 + 9 * q^13 + 5 * q^14 + 3 * q^16 + 8 * q^17 - 14 * q^19 + 5 * q^20 + 3 * q^22 + 10 * q^23 + 9 * q^25 - 22 * q^26 - 38 * q^28 - 15 * q^29 + 15 * q^31 - 2 * q^32 - q^34 + 10 * q^35 - 24 * q^37 - 20 * q^38 + 3 * q^40 - 6 * q^41 + 18 * q^43 - 24 * q^44 + 8 * q^46 + 12 * q^47 + 9 * q^49 - 2 * q^50 + 4 * q^52 - 24 * q^53 + 6 * q^55 + 12 * q^56 + 9 * q^58 - 5 * q^61 - 16 * q^62 - 22 * q^64 - 11 * q^65 + 6 * q^67 - 3 * q^68 - 25 * q^70 + 50 * q^71 + 8 * q^73 + 44 * q^74 - 6 * q^76 - 12 * q^77 + 8 * q^79 - 16 * q^80 - 18 * q^82 - 7 * q^83 + q^85 - 34 * q^86 + 27 * q^88 + 28 * q^89 + 8 * q^91 - 19 * q^92 - 12 * q^94 + 20 * q^95 + 16 * q^97 + 30 * q^98

$\beta_{1}$	$=$	$( \nu^{7} - 4\nu^{6} + 11\nu^{4} - 3\nu^{3} - 4\nu^{2} - 16\nu - 24 ) / 24$ (v^7 - 4v^6 + 11v^4 - 3v^3 - 4v^2 - 16*v - 24) / 24
$\beta_{2}$	$=$	$( -\nu^{7} + \nu^{6} + \nu^{4} + 6\nu^{3} - 11\nu^{2} + 4\nu + 12 ) / 12$ (-v^7 + v^6 + v^4 + 6v^3 - 11v^2 + 4*v + 12) / 12
$\beta_{3}$	$=$	$( -3\nu^{7} - 2\nu^{6} + 8\nu^{5} - 9\nu^{4} - \nu^{3} + 6\nu^{2} - 16\nu + 32 ) / 24$ (-3v^7 - 2v^6 + 8v^5 - 9v^4 - v^3 + 6v^2 - 16v + 32) / 24
$\beta_{4}$	$=$	$( 5\nu^{7} - 4\nu^{6} - 4\nu^{5} + 7\nu^{4} - 7\nu^{3} + 16\nu^{2} + 12\nu - 64 ) / 24$ (5v^7 - 4v^6 - 4v^5 + 7v^4 - 7v^3 + 16v^2 + 12*v - 64) / 24
$\beta_{5}$	$=$	$( -5\nu^{7} - 4\nu^{5} - 7\nu^{4} + 11\nu^{3} - 4\nu^{2} + 4\nu + 32 ) / 24$ (-5v^7 - 4v^5 - 7v^4 + 11v^3 - 4v^2 + 4v + 32) / 24
$\beta_{6}$	$=$	$( -7\nu^{7} + 4\nu^{6} + 12\nu^{5} - 5\nu^{4} + 21\nu^{3} - 8\nu^{2} - 44\nu + 72 ) / 24$ (-7v^7 + 4v^6 + 12v^5 - 5v^4 + 21v^3 - 8v^2 - 44*v + 72) / 24
$\beta_{7}$	$=$	$( -9\nu^{7} + 10\nu^{6} + 8\nu^{5} - 27\nu^{4} + 29\nu^{3} - 30\nu^{2} - 40\nu + 128 ) / 24$ (-9v^7 + 10v^6 + 8v^5 - 27v^4 + 29v^3 - 30v^2 - 40*v + 128) / 24

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	459.2.e.b

Level	$459$
Weight	$2$
Character orbit	459.e
Analytic conductor	$3.665$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$

$\nu$	$=$	$( -\beta_{7} + \beta_{3} + 2\beta_{2} - 2\beta_1 ) / 3$ (-b7 + b3 + 2b2 - 2b1) / 3
$\nu^{2}$	$=$	$( -\beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2\beta_{2} - 2\beta _1 + 2 ) / 3$ (-b7 + 2b6 + b5 + b4 - b3 - 2b2 - 2*b1 + 2) / 3
$\nu^{3}$	$=$	$( \beta_{7} + 2\beta_{6} + \beta_{5} + 7\beta_{4} + 3\beta_{2} - \beta _1 + 2 ) / 3$ (b7 + 2b6 + b5 + 7b4 + 3*b2 - b1 + 2) / 3
$\nu^{4}$	$=$	$-\beta_{7} + \beta_{6} - \beta_{4} - \beta_{3} + 1$ -b7 + b6 - b4 - b3 + 1
$\nu^{5}$	$=$	$( -2\beta_{7} + 2\beta_{6} - 5\beta_{5} + \beta_{4} + 6\beta_{3} + 6\beta_{2} - 4\beta _1 - 4 ) / 3$ (-2b7 + 2b6 - 5b5 + b4 + 6b3 + 6b2 - 4b1 - 4) / 3
$\nu^{6}$	$=$	$( -2\beta_{7} + 4\beta_{6} - 4\beta_{5} - 10\beta_{4} - 11\beta_{3} - 7\beta_{2} - 7\beta _1 - 8 ) / 3$ (-2b7 + 4b6 - 4b5 - 10b4 - 11b3 - 7b2 - 7*b1 - 8) / 3
$\nu^{7}$	$=$	$( 8\beta_{7} - 3\beta_{6} - 9\beta_{5} + 18\beta_{4} + \beta_{3} + 5\beta_{2} + \beta _1 + 21 ) / 3$ (8b7 - 3b6 - 9b5 + 18b4 + b3 + 5*b2 + b1 + 21) / 3

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 154.4
Significant digits:
Format: