LMFDB - Newform orbit 3060.2.k.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3060,2,Mod(1189,3060)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3060, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3060.1189"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3060.k (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$24.4342230185$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 5x^{2} + 4x + 16 $ x^4 - x^3 + 5x^2 + 4x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{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 + ( - \beta_{2} - \beta_1) q^{5} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{11} + ( - \beta_{2} - 2 \beta_1 + 1) q^{13} + ( - 2 \beta_{2} + 1) q^{17} + ( - 3 \beta_{2} - 1) q^{19} + (\beta_{2} + 7) q^{23}+ \cdots + ( - 3 \beta_{3} + 4 \beta_{2} + \cdots + 6) q^{95}+O(q^{100}) $ q + (-b2 - b1) * q^5 + (b3 - b2 - 2b1 + 1) q^11 + (-b2 - 2b1 + 1) q^13 + (-2b2 + 1) q^17 + (-3b2 - 1) q^19 + (b2 + 7) * q^23 + (-b3 + 2b2 + b1 - 3) q^25 + 2b3 q^29 + (-2b3 + b2 + 2b1 - 1) * q^41 + (-3b3 + b2 + 2b1 - 1) * q^43 - 7 * q^49 + (-2b3 + 3b1 - 6) * q^55 + (-b3 + 2b2 + b1 - 8) q^65 + (b3 - 4b2 - 8b1 + 4) * q^67 - b3 * q^71 + (-2b3 + b2 - b1 + 4) q^85 + (-3b3 + 4b2 + b1 + 6) * q^95
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{5} - 10 q^{19} + 30 q^{23} - 7 q^{25} - 28 q^{49} - 21 q^{55} - 27 q^{65} + 17 q^{85} + 33 q^{95}+O(q^{100}) $ 4 * q - 3 * q^5 - 10 * q^19 + 30 * q^23 - 7 * q^25 - 28 * q^49 - 21 * q^55 - 27 * q^65 + 17 * q^85 + 33 * q^95

Basis of coefficient ring in terms of a root $\nu$ of $ x^{4} - x^{3} + 5x^{2} + 4x + 16 $ x^4 - x^3 + 5*x^2 + 4*x + 16 :

$\beta_{1}$	$=$	$ ( -\nu^{3} + 5\nu^{2} - 25\nu + 16 ) / 20 $ (-v^3 + 5v^2 - 25v + 16) / 20
$\beta_{2}$	$=$	$ ( -\nu^{3} - 4 ) / 5 $ (-v^3 - 4) / 5
$\beta_{3}$	$=$	$ ( -\nu^{3} + 5\nu^{2} - 5\nu + 6 ) / 5 $ (-v^3 + 5v^2 - 5v + 6) / 5

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

$\nu$	$=$	$ ( \beta_{3} - 4\beta _1 + 2 ) / 4 $ (b3 - 4*b1 + 2) / 4
$\nu^{2}$	$=$	$ ( 5\beta_{3} - 4\beta_{2} - 4\beta _1 - 6 ) / 4 $ (5b3 - 4b2 - 4*b1 - 6) / 4
$\nu^{3}$	$=$	$ -5\beta_{2} - 4 $ -5*b2 - 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3060\mathbb{Z}\right)^\times$.

$n$	$1261$	$1361$	$1531$	$1837$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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} $

1189.1

1.28078	−	2.21837i
1.28078	+	2.21837i
−0.780776	−	1.35234i
−0.780776	+	1.35234i

−1.78078

−

1.35234i

1189.2

−1.78078

1.35234i

1189.3

0.280776

−

2.21837i

1189.4

0.280776

2.21837i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
51.c	odd	2	1	CM by $\Q(\sqrt{-51}) $
15.d	odd	2	1	inner
85.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3060.2.k.d	✓	4
3.b	odd	2	1		3060.2.k.e	yes	4
5.b	even	2	1		3060.2.k.e	yes	4
15.d	odd	2	1	inner	3060.2.k.d	✓	4
17.b	even	2	1		3060.2.k.e	yes	4
51.c	odd	2	1	CM	3060.2.k.d	✓	4
85.c	even	2	1	inner	3060.2.k.d	✓	4
255.h	odd	2	1		3060.2.k.e	yes	4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3060.2.k.d	✓	4	1.a	even	1	1	trivial
3060.2.k.d	✓	4	15.d	odd	2	1	inner
3060.2.k.d	✓	4	51.c	odd	2	1	CM
3060.2.k.d	✓	4	85.c	even	2	1	inner
3060.2.k.e	yes	4	3.b	odd	2	1
3060.2.k.e	yes	4	5.b	even	2	1
3060.2.k.e	yes	4	17.b	even	2	1
3060.2.k.e	yes	4	255.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3060, [\chi])$:

$ T_{7} $

T7

$ T_{23}^{2} - 15T_{23} + 52 $

T23^2 - 15*T23 + 52

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} $ T^4
$5$	$ T^{4} + 3 T^{3} + \cdots + 25 $ T^4 + 3T^3 + 8T^2 + 15*T + 25
$7$	$ T^{4} $ T^4
$11$	$ T^{4} + 39T^{2} + 36 $ T^4 + 39*T^2 + 36
$13$	$ T^{4} + 27T^{2} + 144 $ T^4 + 27*T^2 + 144
$17$	$ (T^{2} - 17)^{2} $ (T^2 - 17)^2
$19$	$ (T^{2} + 5 T - 32)^{2} $ (T^2 + 5*T - 32)^2
$23$	$ (T^{2} - 15 T + 52)^{2} $ (T^2 - 15*T + 52)^2
$29$	$ (T^{2} + 48)^{2} $ (T^2 + 48)^2
$31$	$ T^{4} $ T^4
$37$	$ T^{4} $ T^4
$41$	$ T^{4} + 99T^{2} + 576 $ T^4 + 99*T^2 + 576
$43$	$ T^{4} + 207T^{2} + 6084 $ T^4 + 207*T^2 + 6084
$47$	$ T^{4} $ T^4
$53$	$ T^{4} $ T^4
$59$	$ T^{4} $ T^4
$61$	$ T^{4} $ T^4
$67$	$ (T^{2} + 204)^{2} $ (T^2 + 204)^2
$71$	$ (T^{2} + 12)^{2} $ (T^2 + 12)^2
$73$	$ T^{4} $ T^4
$79$	$ T^{4} $ T^4
$83$	$ T^{4} $ T^4
$89$	$ T^{4} $ T^4
$97$	$ T^{4} $ T^4
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Level:	\( N \)	\(=\)	\( 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3060.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{2} - \beta_1) q^{5} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{11} + ( - \beta_{2} - 2 \beta_1 + 1) q^{13} + ( - 2 \beta_{2} + 1) q^{17} + ( - 3 \beta_{2} - 1) q^{19} + (\beta_{2} + 7) q^{23}+ \cdots + ( - 3 \beta_{3} + 4 \beta_{2} + \cdots + 6) q^{95}+O(q^{100}) \) q + (-b2 - b1) * q^5 + (b3 - b2 - 2b1 + 1) q^11 + (-b2 - 2b1 + 1) q^13 + (-2b2 + 1) q^17 + (-3b2 - 1) q^19 + (b2 + 7) * q^23 + (-b3 + 2b2 + b1 - 3) q^25 + 2b3 q^29 + (-2b3 + b2 + 2b1 - 1) * q^41 + (-3b3 + b2 + 2b1 - 1) * q^43 - 7 * q^49 + (-2b3 + 3b1 - 6) * q^55 + (-b3 + 2b2 + b1 - 8) q^65 + (b3 - 4b2 - 8b1 + 4) * q^67 - b3 * q^71 + (-2b3 + b2 - b1 + 4) q^85 + (-3b3 + 4b2 + b1 + 6) * q^95
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{5} - 10 q^{19} + 30 q^{23} - 7 q^{25} - 28 q^{49} - 21 q^{55} - 27 q^{65} + 17 q^{85} + 33 q^{95}+O(q^{100}) \) 4 * q - 3 * q^5 - 10 * q^19 + 30 * q^23 - 7 * q^25 - 28 * q^49 - 21 * q^55 - 27 * q^65 + 17 * q^85 + 33 * q^95

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