LMFDB - Newform orbit 1476.2.f.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1476,2,Mod(901,1476)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1476, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1476.901");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1476 = 2^{2} \cdot 3^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1476.f (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:	$11.7859193383$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{41})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 21x^{2} + 100 $ x^4 + 21*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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 $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q+O(q^{10}) $ q	$ q + \beta_1 q^{11} + ( - 2 \beta_{2} + \beta_1) q^{17} - 5 q^{25} + (2 \beta_{2} + \beta_1) q^{29} + ( - \beta_{3} + 1) q^{31} + (\beta_{3} - 3) q^{37} + ( - \beta_{2} + 2 \beta_1) q^{41} + (\beta_{3} + 3) q^{43} + ( - 4 \beta_{2} + \beta_1) q^{47} + 7 q^{49} + ( - 2 \beta_{2} + 4 \beta_1) q^{53} + ( - \beta_{3} - 5) q^{61} + (4 \beta_{2} + \beta_1) q^{71} + ( - \beta_{3} + 7) q^{73} + ( - 2 \beta_{2} + 4 \beta_1) q^{89}+O(q^{100}) $ q + b1 * q^11 + (-2b2 + b1) q^17 - 5 * q^25 + (2b2 + b1) q^29 + (-b3 + 1) * q^31 + (b3 - 3) * q^37 + (-b2 + 2b1) q^41 + (b3 + 3) * q^43 + (-4b2 + b1) q^47 + 7 * q^49 + (-2b2 + 4b1) * q^53 + (-b3 - 5) * q^61 + (4b2 + b1) q^71 + (-b3 + 7) * q^73 + (-2b2 + 4b1) * q^89
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 20 q^{25} + 2 q^{31} - 10 q^{37} + 14 q^{43} + 28 q^{49} - 22 q^{61} + 26 q^{73}+O(q^{100}) $ 4 * q - 20 * q^25 + 2 * q^31 - 10 * q^37 + 14 * q^43 + 28 * q^49 - 22 * q^61 + 26 * q^73

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{4} + 21x^{2} + 100 $ x^4 + 21*x^2 + 100 :

$\beta_{1}$	$=$	$ ( \nu^{3} + \nu ) / 10 $ (v^3 + v) / 10
$\beta_{2}$	$=$	$ ( 3\nu^{3} + 33\nu ) / 10 $ (3v^3 + 33v) / 10
$\beta_{3}$	$=$	$ 3\nu^{2} + 32 $ 3*v^2 + 32

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

$\nu$	$=$	$ ( \beta_{2} - 3\beta_1 ) / 3 $ (b2 - 3*b1) / 3
$\nu^{2}$	$=$	$ ( \beta_{3} - 32 ) / 3 $ (b3 - 32) / 3
$\nu^{3}$	$=$	$ ( -\beta_{2} + 33\beta_1 ) / 3 $ (-b2 + 33*b1) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1476\mathbb{Z}\right)^\times$.

$n$	$739$	$821$	$1441$
$\chi(n)$	$1$	$1$	$-1$

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} $

901.1

		3.70156i
		2.70156i
	−	2.70156i
	−	3.70156i

901.2

901.3

901.4

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
123.b	odd	2	1	CM by $\Q(\sqrt{-123}) $
3.b	odd	2	1	inner
41.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1476.2.f.c	✓	4
3.b	odd	2	1	inner	1476.2.f.c	✓	4
41.b	even	2	1	inner	1476.2.f.c	✓	4
123.b	odd	2	1	CM	1476.2.f.c	✓	4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1476.2.f.c	✓	4	1.a	even	1	1	trivial
1476.2.f.c	✓	4	3.b	odd	2	1	inner
1476.2.f.c	✓	4	41.b	even	2	1	inner
1476.2.f.c	✓	4	123.b	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} $ T5 acting on $S_{2}^{\mathrm{new}}(1476, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} $ T^4
$5$	$ T^{4} $ T^4
$7$	$ T^{4} $ T^4
$11$	$ T^{4} + 25T^{2} + 64 $ T^4 + 25*T^2 + 64
$13$	$ T^{4} $ T^4
$17$	$ T^{4} + 61T^{2} + 100 $ T^4 + 61*T^2 + 100
$19$	$ T^{4} $ T^4
$23$	$ T^{4} $ T^4
$29$	$ T^{4} + 133T^{2} + 2116 $ T^4 + 133*T^2 + 2116
$31$	$ (T^{2} - T - 92)^{2} $ (T^2 - T - 92)^2
$37$	$ (T^{2} + 5 T - 86)^{2} $ (T^2 + 5*T - 86)^2
$41$	$ (T^{2} + 41)^{2} $ (T^2 + 41)^2
$43$	$ (T^{2} - 7 T - 80)^{2} $ (T^2 - 7*T - 80)^2
$47$	$ T^{4} + 241 T^{2} + 10000 $ T^4 + 241*T^2 + 10000
$53$	$ (T^{2} + 164)^{2} $ (T^2 + 164)^2
$59$	$ T^{4} $ T^4
$61$	$ (T^{2} + 11 T - 62)^{2} $ (T^2 + 11*T - 62)^2
$67$	$ T^{4} $ T^4
$71$	$ T^{4} + 385 T^{2} + 29584 $ T^4 + 385*T^2 + 29584
$73$	$ (T^{2} - 13 T - 50)^{2} $ (T^2 - 13*T - 50)^2
$79$	$ T^{4} $ T^4
$83$	$ T^{4} $ T^4
$89$	$ (T^{2} + 164)^{2} $ (T^2 + 164)^2
$97$	$ T^{4} $ T^4
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Level:	\( N \)	\(=\)	\( 1476 = 2^{2} \cdot 3^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1476.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q+O(q^{10}) \) q	\( q + \beta_1 q^{11} + ( - 2 \beta_{2} + \beta_1) q^{17} - 5 q^{25} + (2 \beta_{2} + \beta_1) q^{29} + ( - \beta_{3} + 1) q^{31} + (\beta_{3} - 3) q^{37} + ( - \beta_{2} + 2 \beta_1) q^{41} + (\beta_{3} + 3) q^{43} + ( - 4 \beta_{2} + \beta_1) q^{47} + 7 q^{49} + ( - 2 \beta_{2} + 4 \beta_1) q^{53} + ( - \beta_{3} - 5) q^{61} + (4 \beta_{2} + \beta_1) q^{71} + ( - \beta_{3} + 7) q^{73} + ( - 2 \beta_{2} + 4 \beta_1) q^{89}+O(q^{100}) \) q + b1 * q^11 + (-2b2 + b1) q^17 - 5 * q^25 + (2b2 + b1) q^29 + (-b3 + 1) * q^31 + (b3 - 3) * q^37 + (-b2 + 2b1) q^41 + (b3 + 3) * q^43 + (-4b2 + b1) q^47 + 7 * q^49 + (-2b2 + 4b1) * q^53 + (-b3 - 5) * q^61 + (4b2 + b1) q^71 + (-b3 + 7) * q^73 + (-2b2 + 4b1) * q^89
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 20 q^{25} + 2 q^{31} - 10 q^{37} + 14 q^{43} + 28 q^{49} - 22 q^{61} + 26 q^{73}+O(q^{100}) \) 4 * q - 20 * q^25 + 2 * q^31 - 10 * q^37 + 14 * q^43 + 28 * q^49 - 22 * q^61 + 26 * q^73

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