Properties

Label 2624.1.bi.a
Level $2624$
Weight $1$
Character orbit 2624.bi
Analytic conductor $1.310$
Analytic rank $0$
Dimension $4$
Projective image $D_{8}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2624,1,Mod(577,2624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2624, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2624.577");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2624 = 2^{6} \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2624.bi (of order \(8\), degree \(4\), 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: \(1.30954659315\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1312)
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.199428376454144.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{8}^{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8}^{3} q^{9} + (\zeta_{8}^{2} - \zeta_{8}) q^{13} + (\zeta_{8} + 1) q^{17} + \zeta_{8}^{2} q^{25} + (\zeta_{8}^{3} - 1) q^{29} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{37} + q^{41} + \zeta_{8}^{3} q^{49} + (\zeta_{8}^{3} + 1) q^{53} + (\zeta_{8}^{2} + 1) q^{61} - \zeta_{8}^{2} q^{81} + (\zeta_{8}^{2} - \zeta_{8}) q^{89} + (\zeta_{8} - 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{17} - 4 q^{29} + 4 q^{41} + 4 q^{53} + 4 q^{61} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2624\mathbb{Z}\right)^\times\).

\(n\) \(129\) \(575\) \(1477\)
\(\chi(n)\) \(\zeta_{8}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
−0.707107 + 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
0 0 0 0 0 0 0 0.707107 + 0.707107i 0
1473.1 0 0 0 0 0 0 0 −0.707107 0.707107i 0
2241.1 0 0 0 0 0 0 0 −0.707107 + 0.707107i 0
2433.1 0 0 0 0 0 0 0 0.707107 0.707107i 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 CM by \(\Q(\sqrt{-1}) \)
41.e odd 8 1 inner
164.i even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2624.1.bi.a 4
4.b odd 2 1 CM 2624.1.bi.a 4
8.b even 2 1 1312.1.bi.a 4
8.d odd 2 1 1312.1.bi.a 4
41.e odd 8 1 inner 2624.1.bi.a 4
164.i even 8 1 inner 2624.1.bi.a 4
328.n odd 8 1 1312.1.bi.a 4
328.p even 8 1 1312.1.bi.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1312.1.bi.a 4 8.b even 2 1
1312.1.bi.a 4 8.d odd 2 1
1312.1.bi.a 4 328.n odd 8 1
1312.1.bi.a 4 328.p even 8 1
2624.1.bi.a 4 1.a even 1 1 trivial
2624.1.bi.a 4 4.b odd 2 1 CM
2624.1.bi.a 4 41.e odd 8 1 inner
2624.1.bi.a 4 164.i even 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2624, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) 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
$13$ \( T^{4} + 2 T^{2} + 4 T + 2 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$41$ \( (T - 1)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 2 T + 2)^{2} \) 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
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} + 2 T^{2} + 4 T + 2 \) Copy content Toggle raw display
$97$ \( T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2 \) Copy content Toggle raw display
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