LMFDB - Newform orbit 1444.1.j.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1444,1,Mod(333,1444)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1444, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 17]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1444.333");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1444 = 2^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1444.j (of order $18$, degree $6$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$0.720649878242$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{18})$
Coefficient field:	12.0.101559956668416.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 8x^{6} + 64 $ x^12 - 8*x^6 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.27436.1

$q$-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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{11} - \beta_{5}) q^{3} + ( - \beta_{8} + \beta_{2}) q^{5} + (\beta_{6} - 1) q^{7} + ( - \beta_{10} + \beta_{4}) q^{9}+O(q^{10}) $ q + (b11 - b5) * q^3 + (-b8 + b2) * q^5 + (b6 - 1) * q^7 + (-b10 + b4) * q^9	$ q + (\beta_{11} - \beta_{5}) q^{3} + ( - \beta_{8} + \beta_{2}) q^{5} + (\beta_{6} - 1) q^{7} + ( - \beta_{10} + \beta_{4}) q^{9} + \beta_{6} q^{11} + (\beta_{7} - \beta_1) q^{15} - \beta_{2} q^{17} - \beta_{11} q^{21} + \beta_{7} q^{29} - \beta_{5} q^{33} + \beta_{8} q^{35} + \beta_{9} q^{37} + ( - \beta_{11} + \beta_{5}) q^{41} + (\beta_{8} - \beta_{2}) q^{43} + ( - \beta_{6} + 1) q^{45} + ( - \beta_{10} + \beta_{4}) q^{47} + \beta_1 q^{51} + \beta_{2} q^{55} + \beta_{11} q^{59} - \beta_{4} q^{61} + \beta_{10} q^{63} + \beta_{5} q^{71} + \beta_{8} q^{73} - q^{77} + (\beta_{8} - \beta_{2}) q^{81} + (\beta_{10} - \beta_{4}) q^{85} - 2 \beta_{6} q^{87} + \beta_{4} q^{99}+O(q^{100}) $ q + (b11 - b5) * q^3 + (-b8 + b2) * q^5 + (b6 - 1) * q^7 + (-b10 + b4) * q^9 + b6 * q^11 + (b7 - b1) * q^15 - b2 * q^17 - b11 * q^21 + b7 * q^29 - b5 * q^33 + b8 * q^35 + b9 * q^37 + (-b11 + b5) * q^41 + (b8 - b2) * q^43 + (-b6 + 1) * q^45 + (-b10 + b4) * q^47 + b1 * q^51 + b2 * q^55 + b11 * q^59 - b4 * q^61 + b10 * q^63 + b5 * q^71 + b8 * q^73 - q^77 + (b8 - b2) * q^81 + (b10 - b4) * q^85 - 2b6 q^87 + b4 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 6 q^{7}+O(q^{10}) $ 12 * q - 6 * q^7	$ 12 q - 6 q^{7} + 6 q^{11} + 6 q^{45} - 12 q^{77} - 12 q^{87}+O(q^{100}) $ 12 * q - 6 * q^7 + 6 * q^11 + 6 * q^45 - 12 * q^77 - 12 * q^87

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 8x^{6} + 64 $ x^12 - 8*x^6 + 64 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{2} ) / 2 $ (v^2) / 2
$\beta_{3}$	$=$	$ ( \nu^{3} ) / 2 $ (v^3) / 2
$\beta_{4}$	$=$	$ ( \nu^{4} ) / 4 $ (v^4) / 4
$\beta_{5}$	$=$	$ ( \nu^{5} ) / 4 $ (v^5) / 4
$\beta_{6}$	$=$	$ ( \nu^{6} ) / 8 $ (v^6) / 8
$\beta_{7}$	$=$	$ ( \nu^{7} ) / 8 $ (v^7) / 8
$\beta_{8}$	$=$	$ ( \nu^{8} ) / 16 $ (v^8) / 16
$\beta_{9}$	$=$	$ ( \nu^{9} ) / 16 $ (v^9) / 16
$\beta_{10}$	$=$	$ ( \nu^{10} ) / 32 $ (v^10) / 32
$\beta_{11}$	$=$	$ ( \nu^{11} ) / 32 $ (v^11) / 32

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ 2\beta_{2} $ 2*b2
$\nu^{3}$	$=$	$ 2\beta_{3} $ 2*b3
$\nu^{4}$	$=$	$ 4\beta_{4} $ 4*b4
$\nu^{5}$	$=$	$ 4\beta_{5} $ 4*b5
$\nu^{6}$	$=$	$ 8\beta_{6} $ 8*b6
$\nu^{7}$	$=$	$ 8\beta_{7} $ 8*b7
$\nu^{8}$	$=$	$ 16\beta_{8} $ 16*b8
$\nu^{9}$	$=$	$ 16\beta_{9} $ 16*b9
$\nu^{10}$	$=$	$ 32\beta_{10} $ 32*b10
$\nu^{11}$	$=$	$ 32\beta_{11} $ 32*b11

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times$.

$n$	$723$	$1085$
$\chi(n)$	$1$	$-\beta_{4}$

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} $

333.1

0.909039	−	1.08335i
−0.909039	+	1.08335i
0.909039	+	1.08335i
−0.909039	−	1.08335i
0.483690	+	1.32893i
−0.483690	−	1.32893i
1.39273	−	0.245576i
−1.39273	+	0.245576i
0.483690	−	1.32893i
−0.483690	+	1.32893i
1.39273	+	0.245576i
−1.39273	−	0.245576i

−0.909039

−

1.08335i

−0.939693

−

0.342020i

−0.500000

0.866025i

−0.173648

0.984808i

333.2

0.909039

1.08335i

−0.939693

−

0.342020i

−0.500000

0.866025i

−0.173648

0.984808i

477.1

−0.909039

1.08335i

−0.939693

0.342020i

−0.500000

−

0.866025i

−0.173648

−

0.984808i

477.2

0.909039

−

1.08335i

−0.939693

0.342020i

−0.500000

−

0.866025i

−0.173648

−

0.984808i

849.1

−0.483690

1.32893i

0.173648

0.984808i

−0.500000

0.866025i

−0.766044

−

0.642788i

849.2

0.483690

−

1.32893i

0.173648

0.984808i

−0.500000

0.866025i

−0.766044

−

0.642788i

1021.1

−1.39273

−

0.245576i

0.766044

0.642788i

−0.500000

−

0.866025i

0.939693

0.342020i

1021.2

1.39273

0.245576i

0.766044

0.642788i

−0.500000

−

0.866025i

0.939693

0.342020i

1029.1

−0.483690

−

1.32893i

0.173648

−

0.984808i

−0.500000

−

0.866025i

−0.766044

0.642788i

1029.2

0.483690

1.32893i

0.173648

−

0.984808i

−0.500000

−

0.866025i

−0.766044

0.642788i

1345.1

−1.39273

0.245576i

0.766044

−

0.642788i

−0.500000

0.866025i

0.939693

−

0.342020i

1345.2

1.39273

−

0.245576i

0.766044

−

0.642788i

−0.500000

0.866025i

0.939693

−

0.342020i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.b	odd	2	1	inner
19.c	even	3	2	inner
19.d	odd	6	2	inner
19.e	even	9	3	inner
19.f	odd	18	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1444.1.j.b		12
19.b	odd	2	1	inner	1444.1.j.b		12
19.c	even	3	2	inner	1444.1.j.b		12
19.d	odd	6	2	inner	1444.1.j.b		12
19.e	even	9	1		1444.1.c.a	✓	2
19.e	even	9	2		1444.1.h.b		4
19.e	even	9	3	inner	1444.1.j.b		12
19.f	odd	18	1		1444.1.c.a	✓	2
19.f	odd	18	2		1444.1.h.b		4
19.f	odd	18	3	inner	1444.1.j.b		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1444.1.c.a	✓	2	19.e	even	9	1
1444.1.c.a	✓	2	19.f	odd	18	1
1444.1.h.b		4	19.e	even	9	2
1444.1.h.b		4	19.f	odd	18	2
1444.1.j.b		12	1.a	even	1	1	trivial
1444.1.j.b		12	19.b	odd	2	1	inner
1444.1.j.b		12	19.c	even	3	2	inner
1444.1.j.b		12	19.d	odd	6	2	inner
1444.1.j.b		12	19.e	even	9	3	inner
1444.1.j.b		12	19.f	odd	18	3	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{12} - 8T_{3}^{6} + 64 $ T3^12 - 8*T3^6 + 64 acting on $S_{1}^{\mathrm{new}}(1444, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} - 8T^{6} + 64 $ T^12 - 8*T^6 + 64
$5$	$ (T^{6} + T^{3} + 1)^{2} $ (T^6 + T^3 + 1)^2
$7$	$ (T^{2} + T + 1)^{6} $ (T^2 + T + 1)^6
$11$	$ (T^{2} - T + 1)^{6} $ (T^2 - T + 1)^6
$13$	$ T^{12} $ T^12
$17$	$ (T^{6} + T^{3} + 1)^{2} $ (T^6 + T^3 + 1)^2
$19$	$ T^{12} $ T^12
$23$	$ T^{12} $ T^12
$29$	$ T^{12} - 8T^{6} + 64 $ T^12 - 8*T^6 + 64
$31$	$ T^{12} $ T^12
$37$	$ (T^{2} + 2)^{6} $ (T^2 + 2)^6
$41$	$ T^{12} - 8T^{6} + 64 $ T^12 - 8*T^6 + 64
$43$	$ (T^{6} - T^{3} + 1)^{2} $ (T^6 - T^3 + 1)^2
$47$	$ (T^{6} - T^{3} + 1)^{2} $ (T^6 - T^3 + 1)^2
$53$	$ T^{12} $ T^12
$59$	$ T^{12} - 8T^{6} + 64 $ T^12 - 8*T^6 + 64
$61$	$ (T^{6} - T^{3} + 1)^{2} $ (T^6 - T^3 + 1)^2
$67$	$ T^{12} $ T^12
$71$	$ T^{12} - 8T^{6} + 64 $ T^12 - 8*T^6 + 64
$73$	$ (T^{6} + T^{3} + 1)^{2} $ (T^6 + T^3 + 1)^2
$79$	$ T^{12} $ T^12
$83$	$ T^{12} $ T^12
$89$	$ T^{12} $ T^12
$97$	$ T^{12} $ T^12
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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:	\( N \)	\(=\)	\( 1444 = 2^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1444.j (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{11} - \beta_{5}) q^{3} + ( - \beta_{8} + \beta_{2}) q^{5} + (\beta_{6} - 1) q^{7} + ( - \beta_{10} + \beta_{4}) q^{9}+O(q^{10}) \) q + (b11 - b5) * q^3 + (-b8 + b2) * q^5 + (b6 - 1) * q^7 + (-b10 + b4) * q^9	\( q + (\beta_{11} - \beta_{5}) q^{3} + ( - \beta_{8} + \beta_{2}) q^{5} + (\beta_{6} - 1) q^{7} + ( - \beta_{10} + \beta_{4}) q^{9} + \beta_{6} q^{11} + (\beta_{7} - \beta_1) q^{15} - \beta_{2} q^{17} - \beta_{11} q^{21} + \beta_{7} q^{29} - \beta_{5} q^{33} + \beta_{8} q^{35} + \beta_{9} q^{37} + ( - \beta_{11} + \beta_{5}) q^{41} + (\beta_{8} - \beta_{2}) q^{43} + ( - \beta_{6} + 1) q^{45} + ( - \beta_{10} + \beta_{4}) q^{47} + \beta_1 q^{51} + \beta_{2} q^{55} + \beta_{11} q^{59} - \beta_{4} q^{61} + \beta_{10} q^{63} + \beta_{5} q^{71} + \beta_{8} q^{73} - q^{77} + (\beta_{8} - \beta_{2}) q^{81} + (\beta_{10} - \beta_{4}) q^{85} - 2 \beta_{6} q^{87} + \beta_{4} q^{99}+O(q^{100}) \) q + (b11 - b5) * q^3 + (-b8 + b2) * q^5 + (b6 - 1) * q^7 + (-b10 + b4) * q^9 + b6 * q^11 + (b7 - b1) * q^15 - b2 * q^17 - b11 * q^21 + b7 * q^29 - b5 * q^33 + b8 * q^35 + b9 * q^37 + (-b11 + b5) * q^41 + (b8 - b2) * q^43 + (-b6 + 1) * q^45 + (-b10 + b4) * q^47 + b1 * q^51 + b2 * q^55 + b11 * q^59 - b4 * q^61 + b10 * q^63 + b5 * q^71 + b8 * q^73 - q^77 + (b8 - b2) * q^81 + (b10 - b4) * q^85 - 2b6 q^87 + b4 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{7}+O(q^{10}) \) 12 * q - 6 * q^7	\( 12 q - 6 q^{7} + 6 q^{11} + 6 q^{45} - 12 q^{77} - 12 q^{87}+O(q^{100}) \) 12 * q - 6 * q^7 + 6 * q^11 + 6 * q^45 - 12 * q^77 - 12 * q^87

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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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	1444.1.j.b

Level	$1444$
Weight	$1$
Character orbit	1444.j
Analytic conductor	$0.721$
Analytic rank	$0$
Dimension	$12$
Projective image	$S_{4}$
CM/RM	no
Inner twists	$12$

Self dual:	no
Analytic conductor:	\(0.720649878242\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{18})\)
Coefficient field:	12.0.101559956668416.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 8x^{6} + 64 \) x^12 - 8*x^6 + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(S_{4}\)
Projective field:	Galois closure of 4.2.27436.1

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 2 \) (v^2) / 2
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} ) / 2 \) (v^3) / 2
\(\beta_{4}\)	\(=\)	\( ( \nu^{4} ) / 4 \) (v^4) / 4
\(\beta_{5}\)	\(=\)	\( ( \nu^{5} ) / 4 \) (v^5) / 4
\(\beta_{6}\)	\(=\)	\( ( \nu^{6} ) / 8 \) (v^6) / 8
\(\beta_{7}\)	\(=\)	\( ( \nu^{7} ) / 8 \) (v^7) / 8
\(\beta_{8}\)	\(=\)	\( ( \nu^{8} ) / 16 \) (v^8) / 16
\(\beta_{9}\)	\(=\)	\( ( \nu^{9} ) / 16 \) (v^9) / 16
\(\beta_{10}\)	\(=\)	\( ( \nu^{10} ) / 32 \) (v^10) / 32
\(\beta_{11}\)	\(=\)	\( ( \nu^{11} ) / 32 \) (v^11) / 32

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( 2\beta_{2} \) 2*b2
\(\nu^{3}\)	\(=\)	\( 2\beta_{3} \) 2*b3
\(\nu^{4}\)	\(=\)	\( 4\beta_{4} \) 4*b4
\(\nu^{5}\)	\(=\)	\( 4\beta_{5} \) 4*b5
\(\nu^{6}\)	\(=\)	\( 8\beta_{6} \) 8*b6
\(\nu^{7}\)	\(=\)	\( 8\beta_{7} \) 8*b7
\(\nu^{8}\)	\(=\)	\( 16\beta_{8} \) 16*b8
\(\nu^{9}\)	\(=\)	\( 16\beta_{9} \) 16*b9
\(\nu^{10}\)	\(=\)	\( 32\beta_{10} \) 32*b10
\(\nu^{11}\)	\(=\)	\( 32\beta_{11} \) 32*b11

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 333.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} - 8T^{6} + 64 \) T^12 - 8*T^6 + 64
$5$	\( (T^{6} + T^{3} + 1)^{2} \) (T^6 + T^3 + 1)^2
$7$	\( (T^{2} + T + 1)^{6} \) (T^2 + T + 1)^6
$11$	\( (T^{2} - T + 1)^{6} \) (T^2 - T + 1)^6
$13$	\( T^{12} \) T^12
$17$	\( (T^{6} + T^{3} + 1)^{2} \) (T^6 + T^3 + 1)^2
$19$	\( T^{12} \) T^12
$23$	\( T^{12} \) T^12
$29$	\( T^{12} - 8T^{6} + 64 \) T^12 - 8*T^6 + 64
$31$	\( T^{12} \) T^12
$37$	\( (T^{2} + 2)^{6} \) (T^2 + 2)^6
$41$	\( T^{12} - 8T^{6} + 64 \) T^12 - 8*T^6 + 64
$43$	\( (T^{6} - T^{3} + 1)^{2} \) (T^6 - T^3 + 1)^2
$47$	\( (T^{6} - T^{3} + 1)^{2} \) (T^6 - T^3 + 1)^2
$53$	\( T^{12} \) T^12
$59$	\( T^{12} - 8T^{6} + 64 \) T^12 - 8*T^6 + 64
$61$	\( (T^{6} - T^{3} + 1)^{2} \) (T^6 - T^3 + 1)^2
$67$	\( T^{12} \) T^12
$71$	\( T^{12} - 8T^{6} + 64 \) T^12 - 8*T^6 + 64
$73$	\( (T^{6} + T^{3} + 1)^{2} \) (T^6 + T^3 + 1)^2
$79$	\( T^{12} \) T^12
$83$	\( T^{12} \) T^12
$89$	\( T^{12} \) T^12
$97$	\( T^{12} \) T^12
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\(n\)	\(723\)	\(1085\)
\(\chi(n)\)	\(1\)	\(-\beta_{4}\)