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Properties

Label 2400.1.cb.a
Level $2400$
Weight $1$
Character orbit 2400.cb
Analytic conductor $1.198$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -15
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2400,1,Mod(107,2400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2400, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 5, 4, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2400.107");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2400.cb (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: \(1.19775603032\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.21743271936000.8

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{16}^{7} q^{2} - \zeta_{16}^{7} q^{3} - \zeta_{16}^{6} q^{4} - \zeta_{16}^{6} q^{6} - \zeta_{16}^{5} q^{8} - \zeta_{16}^{6} q^{9} - \zeta_{16}^{5} q^{12} - \zeta_{16}^{4} q^{16} + (\zeta_{16}^{3} - \zeta_{16}) q^{17} + \cdots + \zeta_{16}^{7} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{19} - 8 q^{34} - 8 q^{49} - 8 q^{51} + 8 q^{61} - 8 q^{76}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(1951\)
\(\chi(n)\) \(-\zeta_{16}^{4}\) \(-\zeta_{16}^{6}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−0.382683 − 0.923880i
0.382683 + 0.923880i
−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 −0.707107 − 0.707107i 0 −0.707107 − 0.707107i 0 0.923880 − 0.382683i −0.707107 − 0.707107i 0
107.2 0.382683 − 0.923880i 0.382683 − 0.923880i −0.707107 − 0.707107i 0 −0.707107 − 0.707107i 0 −0.923880 + 0.382683i −0.707107 − 0.707107i 0
1043.1 −0.923880 + 0.382683i −0.923880 + 0.382683i 0.707107 − 0.707107i 0 0.707107 − 0.707107i 0 −0.382683 + 0.923880i 0.707107 − 0.707107i 0
1043.2 0.923880 − 0.382683i 0.923880 − 0.382683i 0.707107 − 0.707107i 0 0.707107 − 0.707107i 0 0.382683 − 0.923880i 0.707107 − 0.707107i 0
1307.1 −0.923880 − 0.382683i −0.923880 − 0.382683i 0.707107 + 0.707107i 0 0.707107 + 0.707107i 0 −0.382683 − 0.923880i 0.707107 + 0.707107i 0
1307.2 0.923880 + 0.382683i 0.923880 + 0.382683i 0.707107 + 0.707107i 0 0.707107 + 0.707107i 0 0.382683 + 0.923880i 0.707107 + 0.707107i 0
2243.1 −0.382683 − 0.923880i −0.382683 − 0.923880i −0.707107 + 0.707107i 0 −0.707107 + 0.707107i 0 0.923880 + 0.382683i −0.707107 + 0.707107i 0
2243.2 0.382683 + 0.923880i 0.382683 + 0.923880i −0.707107 + 0.707107i 0 −0.707107 + 0.707107i 0 −0.923880 − 0.382683i −0.707107 + 0.707107i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
160.u even 8 1 inner
160.ba even 8 1 inner
480.bq odd 8 1 inner
480.ca odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2400.1.cb.a yes 8
3.b odd 2 1 inner 2400.1.cb.a yes 8
5.b even 2 1 inner 2400.1.cb.a yes 8
5.c odd 4 2 2400.1.br.a ✓ 8
15.d odd 2 1 CM 2400.1.cb.a yes 8
15.e even 4 2 2400.1.br.a ✓ 8
32.h odd 8 1 2400.1.br.a ✓ 8
96.o even 8 1 2400.1.br.a ✓ 8
160.u even 8 1 inner 2400.1.cb.a yes 8
160.y odd 8 1 2400.1.br.a ✓ 8
160.ba even 8 1 inner 2400.1.cb.a yes 8
480.bq odd 8 1 inner 2400.1.cb.a yes 8
480.bs even 8 1 2400.1.br.a ✓ 8
480.ca odd 8 1 inner 2400.1.cb.a yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2400.1.br.a ✓ 8 5.c odd 4 2
2400.1.br.a ✓ 8 15.e even 4 2
2400.1.br.a ✓ 8 32.h odd 8 1
2400.1.br.a ✓ 8 96.o even 8 1
2400.1.br.a ✓ 8 160.y odd 8 1
2400.1.br.a ✓ 8 480.bs even 8 1
2400.1.cb.a yes 8 1.a even 1 1 trivial
2400.1.cb.a yes 8 3.b odd 2 1 inner
2400.1.cb.a yes 8 5.b even 2 1 inner
2400.1.cb.a yes 8 15.d odd 2 1 CM
2400.1.cb.a yes 8 160.u even 8 1 inner
2400.1.cb.a yes 8 160.ba even 8 1 inner
2400.1.cb.a yes 8 480.bq odd 8 1 inner
2400.1.cb.a yes 8 480.ca odd 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2400, [\chi])\).

Hecke characteristic polynomials

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