Properties

Label 640.1.m.b
Level $640$
Weight $1$
Character orbit 640.m
Analytic conductor $0.319$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [640,1,Mod(193,640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(640, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("640.193");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 640.m (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: \(0.319401608085\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.32000.1
Artin image: $C_4\wr C_2$
Artin field: Galois closure of 8.0.131072000.1

$q$-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.

\(f(q)\) \(=\) \( q + i q^{5} - i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{5} - i q^{9} + (i + 1) q^{13} + (i + 1) q^{17} - q^{25} - 2 q^{29} + ( - i + 1) q^{37} + q^{45} - i q^{49} + ( - i - 1) q^{53} - 2 i q^{61} + (i - 1) q^{65} + (i - 1) q^{73} - q^{81} + (i - 1) q^{85} + ( - i - 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{13} + 2 q^{17} - 2 q^{25} - 4 q^{29} + 2 q^{37} + 2 q^{45} - 2 q^{53} - 2 q^{65} - 2 q^{73} - 2 q^{81} - 2 q^{85} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/640\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(i\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
1.00000i
1.00000i
0 0 0 1.00000i 0 0 0 1.00000i 0
577.1 0 0 0 1.00000i 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
40.i odd 4 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 640.1.m.b yes 2
4.b odd 2 1 CM 640.1.m.b yes 2
5.b even 2 1 3200.1.m.a 2
5.c odd 4 1 640.1.m.a 2
5.c odd 4 1 3200.1.m.b 2
8.b even 2 1 640.1.m.a 2
8.d odd 2 1 640.1.m.a 2
16.e even 4 1 1280.1.p.a 2
16.e even 4 1 1280.1.p.b 2
16.f odd 4 1 1280.1.p.a 2
16.f odd 4 1 1280.1.p.b 2
20.d odd 2 1 3200.1.m.a 2
20.e even 4 1 640.1.m.a 2
20.e even 4 1 3200.1.m.b 2
40.e odd 2 1 3200.1.m.b 2
40.f even 2 1 3200.1.m.b 2
40.i odd 4 1 inner 640.1.m.b yes 2
40.i odd 4 1 3200.1.m.a 2
40.k even 4 1 inner 640.1.m.b yes 2
40.k even 4 1 3200.1.m.a 2
80.i odd 4 1 1280.1.p.b 2
80.j even 4 1 1280.1.p.a 2
80.s even 4 1 1280.1.p.b 2
80.t odd 4 1 1280.1.p.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.1.m.a 2 5.c odd 4 1
640.1.m.a 2 8.b even 2 1
640.1.m.a 2 8.d odd 2 1
640.1.m.a 2 20.e even 4 1
640.1.m.b yes 2 1.a even 1 1 trivial
640.1.m.b yes 2 4.b odd 2 1 CM
640.1.m.b yes 2 40.i odd 4 1 inner
640.1.m.b yes 2 40.k even 4 1 inner
1280.1.p.a 2 16.e even 4 1
1280.1.p.a 2 16.f odd 4 1
1280.1.p.a 2 80.j even 4 1
1280.1.p.a 2 80.t odd 4 1
1280.1.p.b 2 16.e even 4 1
1280.1.p.b 2 16.f odd 4 1
1280.1.p.b 2 80.i odd 4 1
1280.1.p.b 2 80.s even 4 1
3200.1.m.a 2 5.b even 2 1
3200.1.m.a 2 20.d odd 2 1
3200.1.m.a 2 40.i odd 4 1
3200.1.m.a 2 40.k even 4 1
3200.1.m.b 2 5.c odd 4 1
3200.1.m.b 2 20.e even 4 1
3200.1.m.b 2 40.e odd 2 1
3200.1.m.b 2 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{2} - 2T_{13} + 2 \) acting on \(S_{1}^{\mathrm{new}}(640, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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