Properties

Label 2800.1.v.a
Level $2800$
Weight $1$
Character orbit 2800.v
Analytic conductor $1.397$
Analytic rank $0$
Dimension $4$
Projective image $D_{2}$
CM/RM discs -20, -35, 28
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2800,1,Mod(1007,2800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2800, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 1, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2800.1007");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2800.v (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: \(1.39738203537\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-5}, \sqrt{7})\)
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.

\(f(q)\) \(=\) \( q - \zeta_{8} q^{3} + \zeta_{8}^{3} q^{7} + 3 \zeta_{8}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8} q^{3} + \zeta_{8}^{3} q^{7} + 3 \zeta_{8}^{2} q^{9} + 2 q^{21} - 4 \zeta_{8}^{3} q^{27} - \zeta_{8}^{2} q^{29} - \zeta_{8}^{3} q^{47} - \zeta_{8}^{2} q^{49} - 3 \zeta_{8} q^{63} + 3 q^{81} + \zeta_{8} q^{83} + 4 \zeta_{8}^{3} q^{87} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{21} - 20 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(801\) \(2101\) \(2577\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-\zeta_{8}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1007.1
0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
0 −1.41421 + 1.41421i 0 0 0 −0.707107 0.707107i 0 3.00000i 0
1007.2 0 1.41421 1.41421i 0 0 0 0.707107 + 0.707107i 0 3.00000i 0
1343.1 0 −1.41421 1.41421i 0 0 0 −0.707107 + 0.707107i 0 3.00000i 0
1343.2 0 1.41421 + 1.41421i 0 0 0 0.707107 0.707107i 0 3.00000i 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
28.d even 2 1 RM by \(\Q(\sqrt{7}) \)
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
7.b odd 2 1 inner
20.e even 4 2 inner
35.f even 4 2 inner
140.c even 2 1 inner
140.j odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2800.1.v.a 4
4.b odd 2 1 inner 2800.1.v.a 4
5.b even 2 1 inner 2800.1.v.a 4
5.c odd 4 2 inner 2800.1.v.a 4
7.b odd 2 1 inner 2800.1.v.a 4
20.d odd 2 1 CM 2800.1.v.a 4
20.e even 4 2 inner 2800.1.v.a 4
28.d even 2 1 RM 2800.1.v.a 4
35.c odd 2 1 CM 2800.1.v.a 4
35.f even 4 2 inner 2800.1.v.a 4
140.c even 2 1 inner 2800.1.v.a 4
140.j odd 4 2 inner 2800.1.v.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2800.1.v.a 4 1.a even 1 1 trivial
2800.1.v.a 4 4.b odd 2 1 inner
2800.1.v.a 4 5.b even 2 1 inner
2800.1.v.a 4 5.c odd 4 2 inner
2800.1.v.a 4 7.b odd 2 1 inner
2800.1.v.a 4 20.d odd 2 1 CM
2800.1.v.a 4 20.e even 4 2 inner
2800.1.v.a 4 28.d even 2 1 RM
2800.1.v.a 4 35.c odd 2 1 CM
2800.1.v.a 4 35.f even 4 2 inner
2800.1.v.a 4 140.c even 2 1 inner
2800.1.v.a 4 140.j odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 16 \) acting on \(S_{1}^{\mathrm{new}}(2800, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 1 \) 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
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 4)^{2} \) 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
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 16 \) 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
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 16 \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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