LMFDB - Newform orbit 1728.4.c.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1728,4,Mod(1727,1728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1728.1727"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,-38] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$101.955300490$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 432)
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 $\beta = \sqrt{-3}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta q^{7} - 19 q^{13} - 17 \beta q^{19} + 125 q^{25} + 90 \beta q^{31} - 323 q^{37} - 126 \beta q^{43} + 340 q^{49} - 719 q^{61} - 629 \beta q^{67} + 919 q^{73} - 127 \beta q^{79} - 19 \beta q^{91} + \cdots + 523 q^{97} +O(q^{100}) $ q + b * q^7 - 19 * q^13 - 17b q^19 + 125 * q^25 + 90b q^31 - 323 * q^37 - 126b q^43 + 340 * q^49 - 719 * q^61 - 629b q^67 + 919 * q^73 - 127b q^79 - 19b q^91 + 523 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 38 q^{13} + 250 q^{25} - 646 q^{37} + 680 q^{49} - 1438 q^{61} + 1838 q^{73} + 1046 q^{97}+O(q^{100}) $ 2 * q - 38 * q^13 + 250 * q^25 - 646 * q^37 + 680 * q^49 - 1438 * q^61 + 1838 * q^73 + 1046 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$-1$	$-1$

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} $

1727.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−

1.73205i

1727.2

1.73205i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1728.4.c.b		2
3.b	odd	2	1	CM	1728.4.c.b		2
4.b	odd	2	1	inner	1728.4.c.b		2
8.b	even	2	1		432.4.c.c	✓	2
8.d	odd	2	1		432.4.c.c	✓	2
12.b	even	2	1	inner	1728.4.c.b		2
24.f	even	2	1		432.4.c.c	✓	2
24.h	odd	2	1		432.4.c.c	✓	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
432.4.c.c	✓	2	8.b	even	2	1
432.4.c.c	✓	2	8.d	odd	2	1
432.4.c.c	✓	2	24.f	even	2	1
432.4.c.c	✓	2	24.h	odd	2	1
1728.4.c.b		2	1.a	even	1	1	trivial
1728.4.c.b		2	3.b	odd	2	1	CM
1728.4.c.b		2	4.b	odd	2	1	inner
1728.4.c.b		2	12.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(1728, [\chi])$:

$ T_{5} $

T5

$ T_{7}^{2} + 3 $

T7^2 + 3

$ T_{11} $

T11

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} $ T^2
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 3 $ T^2 + 3
$11$	$ T^{2} $ T^2
$13$	$ (T + 19)^{2} $ (T + 19)^2
$17$	$ T^{2} $ T^2
$19$	$ T^{2} + 867 $ T^2 + 867
$23$	$ T^{2} $ T^2
$29$	$ T^{2} $ T^2
$31$	$ T^{2} + 24300 $ T^2 + 24300
$37$	$ (T + 323)^{2} $ (T + 323)^2
$41$	$ T^{2} $ T^2
$43$	$ T^{2} + 47628 $ T^2 + 47628
$47$	$ T^{2} $ T^2
$53$	$ T^{2} $ T^2
$59$	$ T^{2} $ T^2
$61$	$ (T + 719)^{2} $ (T + 719)^2
$67$	$ T^{2} + 1186923 $ T^2 + 1186923
$71$	$ T^{2} $ T^2
$73$	$ (T - 919)^{2} $ (T - 919)^2
$79$	$ T^{2} + 48387 $ T^2 + 48387
$83$	$ T^{2} $ T^2
$89$	$ T^{2} $ T^2
$97$	$ (T - 523)^{2} $ (T - 523)^2
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Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta q^{7} - 19 q^{13} - 17 \beta q^{19} + 125 q^{25} + 90 \beta q^{31} - 323 q^{37} - 126 \beta q^{43} + 340 q^{49} - 719 q^{61} - 629 \beta q^{67} + 919 q^{73} - 127 \beta q^{79} - 19 \beta q^{91} + \cdots + 523 q^{97} +O(q^{100}) \) q + b * q^7 - 19 * q^13 - 17b q^19 + 125 * q^25 + 90b q^31 - 323 * q^37 - 126b q^43 + 340 * q^49 - 719 * q^61 - 629b q^67 + 919 * q^73 - 127b q^79 - 19b q^91 + 523 * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 38 q^{13} + 250 q^{25} - 646 q^{37} + 680 q^{49} - 1438 q^{61} + 1838 q^{73} + 1046 q^{97}+O(q^{100}) \) 2 * q - 38 * q^13 + 250 * q^25 - 646 * q^37 + 680 * q^49 - 1438 * q^61 + 1838 * q^73 + 1046 * q^97

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