LMFDB - Newform orbit 1440.3.j.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,3,Mod(1279,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1440.1279");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1440.j (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$39.2371580679$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.1827904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 14x^{2} + 1 $ x^6 + 9x^4 + 14x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{4} q^{5} + ( - \beta_{4} + \beta_1 + 2) q^{7}+O(q^{10}) $ q - b4 * q^5 + (-b4 + b1 + 2) * q^7	$ q - \beta_{4} q^{5} + ( - \beta_{4} + \beta_1 + 2) q^{7} + ( - 2 \beta_{4} + 2 \beta_{3} + \cdots - 2) q^{11}+ \cdots + ( - \beta_{5} + 5 \beta_{4} - 5 \beta_{3} + \cdots + 5) q^{97}+O(q^{100}) $ q - b4 * q^5 + (-b4 + b1 + 2) * q^7 + (-2b4 + 2b3 - b2 - 2b1 - 2) q^11 - b5 * q^13 + (b5 - b4 + b3 - 4b2 - b1 - 1) q^17 + (-2b5 - b2) q^19 + (3b4 - 2b3 - 3b1 + 12) q^23 + (-b5 - 2b4 + 6b3 - 4b2 - 2b1 - 5) * q^25 + (8b3 - 10) q^29 + (2b5 - 2b4 + 2b3 - 4b2 - 2b1 - 2) q^31 + (-4b4 + 5b3 - 5b2 + 15) q^35 + (2b5 - b4 + b3 + 8b2 - b1 - 1) * q^37 + (-b4 - 5b3 + b1 + 13) q^41 + (6b4 + b3 - 6b1 - 13) * q^43 + (b4 - 8b3 - b1 - 42) q^47 + (-7b4 + 5b3 + 7b1 - 12) q^49 + (3b5 + 8b4 - 8b3 + 8b2 + 8b1 + 8) q^53 + (-b5 - 3b4 + 11b3 - 9b2 - 7b1 - 55) * q^55 + (-2b5 + 15b2) * q^59 + (-7b4 + b3 + 7b1 - 17) * q^61 + (2b5 - b4 + 13b3 + 8b2 - b1 - 5) q^65 + (2b4 + 13b3 - 2b1 + 47) q^67 + (2b5 - 6b4 + 6b3 + 2b2 - 6b1 - 6) q^71 + (3b5 - b4 + b3 - 4b2 - b1 - 1) * q^73 + (b5 - 11b4 + 11b3 - 24b2 - 11b1 - 11) * q^77 + (2b5 + 6b4 - 6b3 + 24b2 + 6b1 + 6) q^79 + (14b4 + 3b3 - 14b1 + 33) q^83 + (b5 + 3b4 - 11b3 - 16b2 - 13b1 - 5) * q^85 + (-16b4 - 8b3 + 16b1 - 10) q^89 + (2b5 + 2b4 - 2b3 + 10b2 + 2b1 + 2) q^91 + (5b5 - b4 + 25b3 + 15b2 - 5b1 - 5) * q^95 + (-b5 + 5b4 - 5b3 + 12b2 + 5b1 + 5) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5} + 12 q^{7}+O(q^{10}) $ 6 * q + 2 * q^5 + 12 * q^7	$ 6 q + 2 q^{5} + 12 q^{7} + 68 q^{23} - 10 q^{25} - 44 q^{29} + 108 q^{35} + 68 q^{41} - 76 q^{43} - 268 q^{47} - 62 q^{49} - 288 q^{55} - 100 q^{61} + 308 q^{67} + 204 q^{83} - 32 q^{85} - 76 q^{89} + 32 q^{95}+O(q^{100}) $ 6 * q + 2 * q^5 + 12 * q^7 + 68 * q^23 - 10 * q^25 - 44 * q^29 + 108 * q^35 + 68 * q^41 - 76 * q^43 - 268 * q^47 - 62 * q^49 - 288 * q^55 - 100 * q^61 + 308 * q^67 + 204 * q^83 - 32 * q^85 - 76 * q^89 + 32 * q^95

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} + 9x^{4} + 14x^{2} + 1 $ x^6 + 9*x^4 + 14*x^2 + 1 :

$\beta_{1}$	$=$	$ ( 2\nu^{5} + 2\nu^{4} + 20\nu^{3} + 10\nu^{2} + 48\nu - 7 ) / 5 $ (2v^5 + 2v^4 + 20v^3 + 10v^2 + 48*v - 7) / 5
$\beta_{2}$	$=$	$ ( -4\nu^{5} - 40\nu^{3} - 76\nu ) / 5 $ (-4v^5 - 40v^3 - 76*v) / 5
$\beta_{3}$	$=$	$ ( 4\nu^{4} + 30\nu^{2} + 21 ) / 5 $ (4v^4 + 30v^2 + 21) / 5
$\beta_{4}$	$=$	$ ( 2\nu^{5} + 2\nu^{4} + 20\nu^{3} + 20\nu^{2} + 48\nu + 23 ) / 5 $ (2v^5 + 2v^4 + 20v^3 + 20v^2 + 48*v + 23) / 5
$\beta_{5}$	$=$	$ ( -28\nu^{5} - 240\nu^{3} - 312\nu ) / 5 $ (-28v^5 - 240v^3 - 312*v) / 5

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 4 $ (b4 - b3 + b2 + b1 + 1) / 4
$\nu^{2}$	$=$	$ ( \beta_{4} - \beta _1 - 6 ) / 2 $ (b4 - b1 - 6) / 2
$\nu^{3}$	$=$	$ ( \beta_{5} - 11\beta_{4} + 11\beta_{3} - 18\beta_{2} - 11\beta _1 - 11 ) / 8 $ (b5 - 11b4 + 11b3 - 18b2 - 11b1 - 11) / 8
$\nu^{4}$	$=$	$ ( -15\beta_{4} + 5\beta_{3} + 15\beta _1 + 69 ) / 4 $ (-15b4 + 5b3 + 15*b1 + 69) / 4
$\nu^{5}$	$=$	$ ( -5\beta_{5} + 36\beta_{4} - 36\beta_{3} + 66\beta_{2} + 36\beta _1 + 36 ) / 4 $ (-5b5 + 36b4 - 36b3 + 66b2 + 36*b1 + 36) / 4

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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} $

1279.1

		0.273891i
	−	0.273891i
		1.37720i
	−	1.37720i
		2.65109i
	−	2.65109i

−4.30219

−

2.54778i

−3.84997

1279.2

−4.30219

2.54778i

−3.84997

1279.3

1.54778

−

4.75441i

−0.206625

1279.4

1.54778

4.75441i

−0.206625

1279.5

3.75441

−

3.30219i

10.0566

1279.6

3.75441

3.30219i

10.0566

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1440.3.j.b		6
3.b	odd	2	1		160.3.h.b	yes	6
4.b	odd	2	1		1440.3.j.a		6
5.b	even	2	1		1440.3.j.a		6
12.b	even	2	1		160.3.h.a	✓	6
15.d	odd	2	1		160.3.h.a	✓	6
15.e	even	4	1		800.3.b.h		6
15.e	even	4	1		800.3.b.i		6
20.d	odd	2	1	inner	1440.3.j.b		6
24.f	even	2	1		320.3.h.g		6
24.h	odd	2	1		320.3.h.f		6
48.i	odd	4	1		1280.3.e.f		6
48.i	odd	4	1		1280.3.e.h		6
48.k	even	4	1		1280.3.e.g		6
48.k	even	4	1		1280.3.e.i		6
60.h	even	2	1		160.3.h.b	yes	6
60.l	odd	4	1		800.3.b.h		6
60.l	odd	4	1		800.3.b.i		6
120.i	odd	2	1		320.3.h.g		6
120.m	even	2	1		320.3.h.f		6
120.q	odd	4	1		1600.3.b.v		6
120.q	odd	4	1		1600.3.b.w		6
120.w	even	4	1		1600.3.b.v		6
120.w	even	4	1		1600.3.b.w		6
240.t	even	4	1		1280.3.e.f		6
240.t	even	4	1		1280.3.e.h		6
240.bm	odd	4	1		1280.3.e.g		6
240.bm	odd	4	1		1280.3.e.i		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
160.3.h.a	✓	6	12.b	even	2	1
160.3.h.a	✓	6	15.d	odd	2	1
160.3.h.b	yes	6	3.b	odd	2	1
160.3.h.b	yes	6	60.h	even	2	1
320.3.h.f		6	24.h	odd	2	1
320.3.h.f		6	120.m	even	2	1
320.3.h.g		6	24.f	even	2	1
320.3.h.g		6	120.i	odd	2	1
800.3.b.h		6	15.e	even	4	1
800.3.b.h		6	60.l	odd	4	1
800.3.b.i		6	15.e	even	4	1
800.3.b.i		6	60.l	odd	4	1
1280.3.e.f		6	48.i	odd	4	1
1280.3.e.f		6	240.t	even	4	1
1280.3.e.g		6	48.k	even	4	1
1280.3.e.g		6	240.bm	odd	4	1
1280.3.e.h		6	48.i	odd	4	1
1280.3.e.h		6	240.t	even	4	1
1280.3.e.i		6	48.k	even	4	1
1280.3.e.i		6	240.bm	odd	4	1
1440.3.j.a		6	4.b	odd	2	1
1440.3.j.a		6	5.b	even	2	1
1440.3.j.b		6	1.a	even	1	1	trivial
1440.3.j.b		6	20.d	odd	2	1	inner
1600.3.b.v		6	120.q	odd	4	1
1600.3.b.v		6	120.w	even	4	1
1600.3.b.w		6	120.q	odd	4	1
1600.3.b.w		6	120.w	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(1440, [\chi])$:

$ T_{7}^{3} - 6T_{7}^{2} - 40T_{7} - 8 $

$ T_{23}^{3} - 34T_{23}^{2} - 152T_{23} + 9160 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} - 2 T^{5} + \cdots + 15625 $ T^6 - 2T^5 + 7T^4 + 100T^3 + 175T^2 - 1250*T + 15625
$7$	$ (T^{3} - 6 T^{2} - 40 T - 8)^{2} $ (T^3 - 6T^2 - 40T - 8)^2
$11$	$ T^{6} + 560 T^{4} + \cdots + 2560000 $ T^6 + 560T^4 + 86784T^2 + 2560000
$13$	$ T^{6} + 416 T^{4} + \cdots + 692224 $ T^6 + 416T^4 + 43264T^2 + 692224
$17$	$ T^{6} + 1088 T^{4} + \cdots + 6553600 $ T^6 + 1088T^4 + 270336T^2 + 6553600
$19$	$ T^{6} + 1712 T^{4} + \cdots + 11505664 $ T^6 + 1712T^4 + 533248T^2 + 11505664
$23$	$ (T^{3} - 34 T^{2} + \cdots + 9160)^{2} $ (T^3 - 34T^2 - 152T + 9160)^2
$29$	$ (T^{3} + 22 T^{2} + \cdots + 2120)^{2} $ (T^3 + 22T^2 - 948T + 2120)^2
$31$	$ T^{6} + 2560 T^{4} + \cdots + 419430400 $ T^6 + 2560T^4 + 1900544T^2 + 419430400
$37$	$ T^{6} + 5408 T^{4} + \cdots + 432640000 $ T^6 + 5408T^4 + 2985216T^2 + 432640000
$41$	$ (T^{3} - 34 T^{2} + \cdots + 5000)^{2} $ (T^3 - 34T^2 - 100T + 5000)^2
$43$	$ (T^{3} + 38 T^{2} + \cdots - 6280)^{2} $ (T^3 + 38T^2 - 1408T - 6280)^2
$47$	$ (T^{3} + 134 T^{2} + \cdots + 22216)^{2} $ (T^3 + 134T^2 + 4824T + 22216)^2
$53$	$ T^{6} + \cdots + 50319462400 $ T^6 + 12960T^4 + 49973504T^2 + 50319462400
$59$	$ T^{6} + \cdots + 25416011776 $ T^6 + 12464T^4 + 41968384T^2 + 25416011776
$61$	$ (T^{3} + 50 T^{2} + \cdots - 81544)^{2} $ (T^3 + 50T^2 - 1732T - 81544)^2
$67$	$ (T^{3} - 154 T^{2} + \cdots + 24760)^{2} $ (T^3 - 154T^2 + 4768T + 24760)^2
$71$	$ T^{6} + \cdots + 14231535616 $ T^6 + 8384T^4 + 19738624T^2 + 14231535616
$73$	$ T^{6} + \cdots + 3114532864 $ T^6 + 4416T^4 + 6447104T^2 + 3114532864
$79$	$ T^{6} + \cdots + 37060870144 $ T^6 + 25856T^4 + 159514624T^2 + 37060870144
$83$	$ (T^{3} - 102 T^{2} + \cdots + 483400)^{2} $ (T^3 - 102T^2 - 6880T + 483400)^2
$89$	$ (T^{3} + 38 T^{2} + \cdots - 155000)^{2} $ (T^3 + 38T^2 - 13940T - 155000)^2
$97$	$ T^{6} + 7488 T^{4} + \cdots + 44302336 $ T^6 + 7488T^4 + 1384448T^2 + 44302336
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Overview	Random
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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:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1440.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{4} q^{5} + ( - \beta_{4} + \beta_1 + 2) q^{7}+O(q^{10}) \) q - b4 * q^5 + (-b4 + b1 + 2) * q^7	\( q - \beta_{4} q^{5} + ( - \beta_{4} + \beta_1 + 2) q^{7} + ( - 2 \beta_{4} + 2 \beta_{3} + \cdots - 2) q^{11}+ \cdots + ( - \beta_{5} + 5 \beta_{4} - 5 \beta_{3} + \cdots + 5) q^{97}+O(q^{100}) \) q - b4 * q^5 + (-b4 + b1 + 2) * q^7 + (-2b4 + 2b3 - b2 - 2b1 - 2) q^11 - b5 * q^13 + (b5 - b4 + b3 - 4b2 - b1 - 1) q^17 + (-2b5 - b2) q^19 + (3b4 - 2b3 - 3b1 + 12) q^23 + (-b5 - 2b4 + 6b3 - 4b2 - 2b1 - 5) * q^25 + (8b3 - 10) q^29 + (2b5 - 2b4 + 2b3 - 4b2 - 2b1 - 2) q^31 + (-4b4 + 5b3 - 5b2 + 15) q^35 + (2b5 - b4 + b3 + 8b2 - b1 - 1) * q^37 + (-b4 - 5b3 + b1 + 13) q^41 + (6b4 + b3 - 6b1 - 13) * q^43 + (b4 - 8b3 - b1 - 42) q^47 + (-7b4 + 5b3 + 7b1 - 12) q^49 + (3b5 + 8b4 - 8b3 + 8b2 + 8b1 + 8) q^53 + (-b5 - 3b4 + 11b3 - 9b2 - 7b1 - 55) * q^55 + (-2b5 + 15b2) * q^59 + (-7b4 + b3 + 7b1 - 17) * q^61 + (2b5 - b4 + 13b3 + 8b2 - b1 - 5) q^65 + (2b4 + 13b3 - 2b1 + 47) q^67 + (2b5 - 6b4 + 6b3 + 2b2 - 6b1 - 6) q^71 + (3b5 - b4 + b3 - 4b2 - b1 - 1) * q^73 + (b5 - 11b4 + 11b3 - 24b2 - 11b1 - 11) * q^77 + (2b5 + 6b4 - 6b3 + 24b2 + 6b1 + 6) q^79 + (14b4 + 3b3 - 14b1 + 33) q^83 + (b5 + 3b4 - 11b3 - 16b2 - 13b1 - 5) * q^85 + (-16b4 - 8b3 + 16b1 - 10) q^89 + (2b5 + 2b4 - 2b3 + 10b2 + 2b1 + 2) q^91 + (5b5 - b4 + 25b3 + 15b2 - 5b1 - 5) * q^95 + (-b5 + 5b4 - 5b3 + 12b2 + 5b1 + 5) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} + 12 q^{7}+O(q^{10}) \) 6 * q + 2 * q^5 + 12 * q^7	\( 6 q + 2 q^{5} + 12 q^{7} + 68 q^{23} - 10 q^{25} - 44 q^{29} + 108 q^{35} + 68 q^{41} - 76 q^{43} - 268 q^{47} - 62 q^{49} - 288 q^{55} - 100 q^{61} + 308 q^{67} + 204 q^{83} - 32 q^{85} - 76 q^{89} + 32 q^{95}+O(q^{100}) \) 6 * q + 2 * q^5 + 12 * q^7 + 68 * q^23 - 10 * q^25 - 44 * q^29 + 108 * q^35 + 68 * q^41 - 76 * q^43 - 268 * q^47 - 62 * q^49 - 288 * q^55 - 100 * q^61 + 308 * q^67 + 204 * q^83 - 32 * q^85 - 76 * q^89 + 32 * q^95

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