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Properties

Label 3072.1.p.c
Level $3072$
Weight $1$
Character orbit 3072.p
Analytic conductor $1.533$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -3
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3072,1,Mod(641,3072)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3072, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3072.641");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3072.p (of order \(8\), degree \(4\), 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: \(1.53312771881\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.57982058496.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{16} q^{3} + (\zeta_{16}^{3} - \zeta_{16}) q^{7} + \zeta_{16}^{2} q^{9} + (\zeta_{16}^{6} + \zeta_{16}^{4}) q^{13} + ( - \zeta_{16}^{7} - \zeta_{16}^{3}) q^{19} + (\zeta_{16}^{4} - \zeta_{16}^{2}) q^{21}+ \cdots + ( - \zeta_{16}^{6} + \zeta_{16}^{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{37} + 8 q^{57} - 8 q^{61} + 8 q^{73} + 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times\).

\(n\) \(1025\) \(2047\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(\zeta_{16}^{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
641.1
−0.923880 − 0.382683i
0.923880 + 0.382683i
−0.923880 + 0.382683i
0.923880 − 0.382683i
−0.382683 + 0.923880i
0.382683 − 0.923880i
−0.382683 − 0.923880i
0.382683 + 0.923880i
0 −0.923880 − 0.382683i 0 0 0 0.541196 − 0.541196i 0 0.707107 + 0.707107i 0
641.2 0 0.923880 + 0.382683i 0 0 0 −0.541196 + 0.541196i 0 0.707107 + 0.707107i 0
1409.1 0 −0.923880 + 0.382683i 0 0 0 0.541196 + 0.541196i 0 0.707107 − 0.707107i 0
1409.2 0 0.923880 − 0.382683i 0 0 0 −0.541196 − 0.541196i 0 0.707107 − 0.707107i 0
2177.1 0 −0.382683 + 0.923880i 0 0 0 1.30656 − 1.30656i 0 −0.707107 − 0.707107i 0
2177.2 0 0.382683 − 0.923880i 0 0 0 −1.30656 + 1.30656i 0 −0.707107 − 0.707107i 0
2945.1 0 −0.382683 − 0.923880i 0 0 0 1.30656 + 1.30656i 0 −0.707107 + 0.707107i 0
2945.2 0 0.382683 + 0.923880i 0 0 0 −1.30656 − 1.30656i 0 −0.707107 + 0.707107i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 641.2
Significant digits:
Format:

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
32.g even 8 1 inner
32.h odd 8 1 inner
96.o even 8 1 inner
96.p odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3072.1.p.c yes 8
3.b odd 2 1 CM 3072.1.p.c yes 8
4.b odd 2 1 inner 3072.1.p.c yes 8
8.b even 2 1 3072.1.p.b yes 8
8.d odd 2 1 3072.1.p.b yes 8
12.b even 2 1 inner 3072.1.p.c yes 8
16.e even 4 1 3072.1.p.a ✓ 8
16.e even 4 1 3072.1.p.d yes 8
16.f odd 4 1 3072.1.p.a ✓ 8
16.f odd 4 1 3072.1.p.d yes 8
24.f even 2 1 3072.1.p.b yes 8
24.h odd 2 1 3072.1.p.b yes 8
32.g even 8 1 3072.1.p.a ✓ 8
32.g even 8 1 3072.1.p.b yes 8
32.g even 8 1 inner 3072.1.p.c yes 8
32.g even 8 1 3072.1.p.d yes 8
32.h odd 8 1 3072.1.p.a ✓ 8
32.h odd 8 1 3072.1.p.b yes 8
32.h odd 8 1 inner 3072.1.p.c yes 8
32.h odd 8 1 3072.1.p.d yes 8
48.i odd 4 1 3072.1.p.a ✓ 8
48.i odd 4 1 3072.1.p.d yes 8
48.k even 4 1 3072.1.p.a ✓ 8
48.k even 4 1 3072.1.p.d yes 8
96.o even 8 1 3072.1.p.a ✓ 8
96.o even 8 1 3072.1.p.b yes 8
96.o even 8 1 inner 3072.1.p.c yes 8
96.o even 8 1 3072.1.p.d yes 8
96.p odd 8 1 3072.1.p.a ✓ 8
96.p odd 8 1 3072.1.p.b yes 8
96.p odd 8 1 inner 3072.1.p.c yes 8
96.p odd 8 1 3072.1.p.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3072.1.p.a ✓ 8 16.e even 4 1
3072.1.p.a ✓ 8 16.f odd 4 1
3072.1.p.a ✓ 8 32.g even 8 1
3072.1.p.a ✓ 8 32.h odd 8 1
3072.1.p.a ✓ 8 48.i odd 4 1
3072.1.p.a ✓ 8 48.k even 4 1
3072.1.p.a ✓ 8 96.o even 8 1
3072.1.p.a ✓ 8 96.p odd 8 1
3072.1.p.b yes 8 8.b even 2 1
3072.1.p.b yes 8 8.d odd 2 1
3072.1.p.b yes 8 24.f even 2 1
3072.1.p.b yes 8 24.h odd 2 1
3072.1.p.b yes 8 32.g even 8 1
3072.1.p.b yes 8 32.h odd 8 1
3072.1.p.b yes 8 96.o even 8 1
3072.1.p.b yes 8 96.p odd 8 1
3072.1.p.c yes 8 1.a even 1 1 trivial
3072.1.p.c yes 8 3.b odd 2 1 CM
3072.1.p.c yes 8 4.b odd 2 1 inner
3072.1.p.c yes 8 12.b even 2 1 inner
3072.1.p.c yes 8 32.g even 8 1 inner
3072.1.p.c yes 8 32.h odd 8 1 inner
3072.1.p.c yes 8 96.o even 8 1 inner
3072.1.p.c yes 8 96.p odd 8 1 inner
3072.1.p.d yes 8 16.e even 4 1
3072.1.p.d yes 8 16.f odd 4 1
3072.1.p.d yes 8 32.g even 8 1
3072.1.p.d yes 8 32.h odd 8 1
3072.1.p.d yes 8 48.i odd 4 1
3072.1.p.d yes 8 48.k even 4 1
3072.1.p.d yes 8 96.o even 8 1
3072.1.p.d yes 8 96.p odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{4} + 2T_{13}^{2} - 4T_{13} + 2 \) acting on \(S_{1}^{\mathrm{new}}(3072, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2 T^{2} - 4 T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + 16 \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 4 T^{3} + 6 T^{2} + \cdots + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} + 16 \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} + 4 T^{3} + 6 T^{2} + \cdots + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T + 2)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
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