Properties

Label 1536.2.j.d
Level $1536$
Weight $2$
Character orbit 1536.j
Analytic conductor $12.265$
Analytic rank $0$
Dimension $4$
CM no
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(385,1536)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1536, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1536.385");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1536 = 2^{9} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1536.j (of order \(4\), degree \(2\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
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: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8}^{3} q^{3} + (\zeta_{8}^{2} + 1) q^{5} - \zeta_{8}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8}^{3} q^{3} + (\zeta_{8}^{2} + 1) q^{5} - \zeta_{8}^{2} q^{9} + 4 \zeta_{8} q^{11} + (\zeta_{8}^{2} - 1) q^{13} + (\zeta_{8}^{3} - \zeta_{8}) q^{15} + 4 \zeta_{8}^{3} q^{19} + (4 \zeta_{8}^{3} + 4 \zeta_{8}) q^{23} - 3 \zeta_{8}^{2} q^{25} + \zeta_{8} q^{27} + ( - 5 \zeta_{8}^{2} + 5) q^{29} + (4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{31} - 4 q^{33} + (3 \zeta_{8}^{2} + 3) q^{37} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{39} + 12 \zeta_{8} q^{43} + ( - \zeta_{8}^{2} + 1) q^{45} + (8 \zeta_{8}^{3} - 8 \zeta_{8}) q^{47} + 7 q^{49} + ( - \zeta_{8}^{2} - 1) q^{53} + (4 \zeta_{8}^{3} + 4 \zeta_{8}) q^{55} - 4 \zeta_{8}^{2} q^{57} + 4 \zeta_{8} q^{59} + (9 \zeta_{8}^{2} - 9) q^{61} - 2 q^{65} - 4 \zeta_{8}^{3} q^{67} + ( - 4 \zeta_{8}^{2} - 4) q^{69} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{71} + 10 \zeta_{8}^{2} q^{73} + 3 \zeta_{8} q^{75} + (12 \zeta_{8}^{3} - 12 \zeta_{8}) q^{79} - q^{81} + 12 \zeta_{8}^{3} q^{83} + (5 \zeta_{8}^{3} + 5 \zeta_{8}) q^{87} + 6 \zeta_{8}^{2} q^{89} + ( - 4 \zeta_{8}^{2} + 4) q^{93} + (4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{95} + 8 q^{97} - 4 \zeta_{8}^{3} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{5} - 4 q^{13} + 20 q^{29} - 16 q^{33} + 12 q^{37} + 4 q^{45} + 28 q^{49} - 4 q^{53} - 36 q^{61} - 8 q^{65} - 16 q^{69} - 4 q^{81} + 16 q^{93} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(511\) \(517\) \(1025\)
\(\chi(n)\) \(1\) \(\zeta_{8}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
385.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0 −0.707107 + 0.707107i 0 1.00000 + 1.00000i 0 0 0 1.00000i 0
385.2 0 0.707107 0.707107i 0 1.00000 + 1.00000i 0 0 0 1.00000i 0
1153.1 0 −0.707107 0.707107i 0 1.00000 1.00000i 0 0 0 1.00000i 0
1153.2 0 0.707107 + 0.707107i 0 1.00000 1.00000i 0 0 0 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
4.b odd 2 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1536.2.j.d yes 4
3.b odd 2 1 4608.2.k.z 4
4.b odd 2 1 inner 1536.2.j.d yes 4
8.b even 2 1 1536.2.j.a 4
8.d odd 2 1 1536.2.j.a 4
12.b even 2 1 4608.2.k.z 4
16.e even 4 1 1536.2.j.a 4
16.e even 4 1 inner 1536.2.j.d yes 4
16.f odd 4 1 1536.2.j.a 4
16.f odd 4 1 inner 1536.2.j.d yes 4
24.f even 2 1 4608.2.k.ba 4
24.h odd 2 1 4608.2.k.ba 4
32.g even 8 1 3072.2.a.c 2
32.g even 8 1 3072.2.a.e 2
32.g even 8 2 3072.2.d.b 4
32.h odd 8 1 3072.2.a.c 2
32.h odd 8 1 3072.2.a.e 2
32.h odd 8 2 3072.2.d.b 4
48.i odd 4 1 4608.2.k.z 4
48.i odd 4 1 4608.2.k.ba 4
48.k even 4 1 4608.2.k.z 4
48.k even 4 1 4608.2.k.ba 4
96.o even 8 1 9216.2.a.f 2
96.o even 8 1 9216.2.a.r 2
96.p odd 8 1 9216.2.a.f 2
96.p odd 8 1 9216.2.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.j.a 4 8.b even 2 1
1536.2.j.a 4 8.d odd 2 1
1536.2.j.a 4 16.e even 4 1
1536.2.j.a 4 16.f odd 4 1
1536.2.j.d yes 4 1.a even 1 1 trivial
1536.2.j.d yes 4 4.b odd 2 1 inner
1536.2.j.d yes 4 16.e even 4 1 inner
1536.2.j.d yes 4 16.f odd 4 1 inner
3072.2.a.c 2 32.g even 8 1
3072.2.a.c 2 32.h odd 8 1
3072.2.a.e 2 32.g even 8 1
3072.2.a.e 2 32.h odd 8 1
3072.2.d.b 4 32.g even 8 2
3072.2.d.b 4 32.h odd 8 2
4608.2.k.z 4 3.b odd 2 1
4608.2.k.z 4 12.b even 2 1
4608.2.k.z 4 48.i odd 4 1
4608.2.k.z 4 48.k even 4 1
4608.2.k.ba 4 24.f even 2 1
4608.2.k.ba 4 24.h odd 2 1
4608.2.k.ba 4 48.i odd 4 1
4608.2.k.ba 4 48.k even 4 1
9216.2.a.f 2 96.o even 8 1
9216.2.a.f 2 96.p odd 8 1
9216.2.a.r 2 96.o even 8 1
9216.2.a.r 2 96.p odd 8 1

Hecke kernels

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

\( T_{5}^{2} - 2T_{5} + 2 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} + 2 \) Copy content Toggle raw display
\( T_{19}^{4} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 256 \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 256 \) Copy content Toggle raw display
$23$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 10 T + 50)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 20736 \) Copy content Toggle raw display
$47$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 256 \) Copy content Toggle raw display
$61$ \( (T^{2} + 18 T + 162)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 256 \) Copy content Toggle raw display
$71$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 288)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 20736 \) Copy content Toggle raw display
$89$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$97$ \( (T - 8)^{4} \) Copy content Toggle raw display
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