Properties

Label 3072.1.i.g
Level $3072$
Weight $1$
Character orbit 3072.i
Analytic conductor $1.533$
Analytic rank $0$
Dimension $4$
Projective image $D_{2}$
CM/RM discs -3, -8, 24
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3072,1,Mod(257,3072)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3072, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3072.257");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3072.i (of order \(4\), degree \(2\), not 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: \(4\)
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: no (minimal twist has level 192)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-2}, \sqrt{-3})\)
Artin image: $\OD_{16}:C_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{8} q^{3} + \zeta_{8}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8} q^{3} + \zeta_{8}^{2} q^{9} - 2 \zeta_{8} q^{19} - \zeta_{8}^{2} q^{25} - \zeta_{8}^{3} q^{27} - 2 \zeta_{8}^{3} q^{43} + q^{49} + 2 \zeta_{8}^{2} q^{57} - 2 \zeta_{8} q^{67} - 2 \zeta_{8}^{2} q^{73} + \zeta_{8}^{3} q^{75} - q^{81} - 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{49} - 4 q^{81} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1025\) \(2047\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(-\zeta_{8}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0 −0.707107 0.707107i 0 0 0 0 0 1.00000i 0
257.2 0 0.707107 + 0.707107i 0 0 0 0 0 1.00000i 0
1793.1 0 −0.707107 + 0.707107i 0 0 0 0 0 1.00000i 0
1793.2 0 0.707107 0.707107i 0 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
24.f even 2 1 RM by \(\Q(\sqrt{6}) \)
4.b odd 2 1 inner
8.b even 2 1 inner
12.b even 2 1 inner
16.e even 4 2 inner
16.f odd 4 2 inner
24.h odd 2 1 inner
48.i odd 4 2 inner
48.k even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3072.1.i.g 4
3.b odd 2 1 CM 3072.1.i.g 4
4.b odd 2 1 inner 3072.1.i.g 4
8.b even 2 1 inner 3072.1.i.g 4
8.d odd 2 1 CM 3072.1.i.g 4
12.b even 2 1 inner 3072.1.i.g 4
16.e even 4 2 inner 3072.1.i.g 4
16.f odd 4 2 inner 3072.1.i.g 4
24.f even 2 1 RM 3072.1.i.g 4
24.h odd 2 1 inner 3072.1.i.g 4
32.g even 8 2 192.1.h.a 2
32.g even 8 1 768.1.e.a 1
32.g even 8 1 768.1.e.b 1
32.h odd 8 2 192.1.h.a 2
32.h odd 8 1 768.1.e.a 1
32.h odd 8 1 768.1.e.b 1
48.i odd 4 2 inner 3072.1.i.g 4
48.k even 4 2 inner 3072.1.i.g 4
96.o even 8 2 192.1.h.a 2
96.o even 8 1 768.1.e.a 1
96.o even 8 1 768.1.e.b 1
96.p odd 8 2 192.1.h.a 2
96.p odd 8 1 768.1.e.a 1
96.p odd 8 1 768.1.e.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.1.h.a 2 32.g even 8 2
192.1.h.a 2 32.h odd 8 2
192.1.h.a 2 96.o even 8 2
192.1.h.a 2 96.p odd 8 2
768.1.e.a 1 32.g even 8 1
768.1.e.a 1 32.h odd 8 1
768.1.e.a 1 96.o even 8 1
768.1.e.a 1 96.p odd 8 1
768.1.e.b 1 32.g even 8 1
768.1.e.b 1 32.h odd 8 1
768.1.e.b 1 96.o even 8 1
768.1.e.b 1 96.p odd 8 1
3072.1.i.g 4 1.a even 1 1 trivial
3072.1.i.g 4 3.b odd 2 1 CM
3072.1.i.g 4 4.b odd 2 1 inner
3072.1.i.g 4 8.b even 2 1 inner
3072.1.i.g 4 8.d odd 2 1 CM
3072.1.i.g 4 12.b even 2 1 inner
3072.1.i.g 4 16.e even 4 2 inner
3072.1.i.g 4 16.f odd 4 2 inner
3072.1.i.g 4 24.f even 2 1 RM
3072.1.i.g 4 24.h odd 2 1 inner
3072.1.i.g 4 48.i odd 4 2 inner
3072.1.i.g 4 48.k even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3072, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{19}^{4} + 16 \) 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
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 16 \) 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
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 16 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 16 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T + 2)^{4} \) Copy content Toggle raw display
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