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Properties

Label 1536.2.c.f
Level $1536$
Weight $2$
Character orbit 1536.c
Analytic conductor $12.265$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1536,2,Mod(1535,1536)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1536, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("1536.1535");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1536 = 2^{9} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1536.c (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: \(12.2650217505\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{3} + (\beta_{3} + 2 \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{3} + (\beta_{3} + 2 \beta_{2} - \beta_1) q^{9} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots + 2) q^{11}+ \cdots + (2 \beta_{3} + 4 \beta_{2} + \beta_1 - 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 8 q^{11} + 20 q^{25} - 4 q^{27} + 4 q^{33} + 28 q^{49} - 24 q^{51} + 20 q^{57} + 40 q^{59} - 20 q^{75} + 28 q^{81} + 8 q^{83} - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{8}^{2} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} \) Copy content Toggle raw display

Display \(\zeta_{8}^j\) in terms of \(\beta_i\)

\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_{3} - 2\beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} - \beta_{2} + 2\beta_1 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{8}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(517\) \(1025\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1535.1
0.707107 − 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
−0.707107 − 0.707107i
0 −1.70711 − 0.292893i 0 0 0 0 0 2.82843 + 1.00000i 0
1535.2 0 −1.70711 + 0.292893i 0 0 0 0 0 2.82843 − 1.00000i 0
1535.3 0 −0.292893 − 1.70711i 0 0 0 0 0 −2.82843 + 1.00000i 0
1535.4 0 −0.292893 + 1.70711i 0 0 0 0 0 −2.82843 − 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
12.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1536.2.c.f ✓ 4
3.b odd 2 1 1536.2.c.j yes 4
4.b odd 2 1 1536.2.c.j yes 4
8.b even 2 1 1536.2.c.j yes 4
8.d odd 2 1 CM 1536.2.c.f ✓ 4
12.b even 2 1 inner 1536.2.c.f ✓ 4
16.e even 4 1 1536.2.f.a 4
16.e even 4 1 1536.2.f.j 4
16.f odd 4 1 1536.2.f.a 4
16.f odd 4 1 1536.2.f.j 4
24.f even 2 1 1536.2.c.j yes 4
24.h odd 2 1 inner 1536.2.c.f ✓ 4
48.i odd 4 1 1536.2.f.a 4
48.i odd 4 1 1536.2.f.j 4
48.k even 4 1 1536.2.f.a 4
48.k even 4 1 1536.2.f.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.c.f ✓ 4 1.a even 1 1 trivial
1536.2.c.f ✓ 4 8.d odd 2 1 CM
1536.2.c.f ✓ 4 12.b even 2 1 inner
1536.2.c.f ✓ 4 24.h odd 2 1 inner
1536.2.c.j yes 4 3.b odd 2 1
1536.2.c.j yes 4 4.b odd 2 1
1536.2.c.j yes 4 8.b even 2 1
1536.2.c.j yes 4 24.f even 2 1
1536.2.f.a 4 16.e even 4 1
1536.2.f.a 4 16.f odd 4 1
1536.2.f.a 4 48.i odd 4 1
1536.2.f.a 4 48.k even 4 1
1536.2.f.j 4 16.e even 4 1
1536.2.f.j 4 16.f odd 4 1
1536.2.f.j 4 48.i odd 4 1
1536.2.f.j 4 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1536, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} - 14 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{23} \) Copy content Toggle raw display
\( T_{59}^{2} - 20T_{59} + 82 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T - 14)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 76T^{2} + 1156 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 128)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 172T^{2} + 196 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 20 T + 82)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 268T^{2} + 3844 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 288)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 4 T - 158)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 324)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 288)^{2} \) Copy content Toggle raw display
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