Properties

Label 128.2.b.b
Level $128$
Weight $2$
Character orbit 128.b
Analytic conductor $1.022$
Analytic rank $0$
Dimension $2$
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] = [128,2,Mod(65,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 128.b (of order \(2\), degree \(1\), 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: \(1.02208514587\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
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: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{5} + 3 q^{9} - \beta q^{13} - 2 q^{17} - 11 q^{25} - \beta q^{29} - 3 \beta q^{37} + 10 q^{41} + 3 \beta q^{45} - 7 q^{49} + \beta q^{53} + 3 \beta q^{61} + 16 q^{65} + 6 q^{73} + 9 q^{81} - 2 \beta q^{85} - 10 q^{89} - 18 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{9} - 4 q^{17} - 22 q^{25} + 20 q^{41} - 14 q^{49} + 32 q^{65} + 12 q^{73} + 18 q^{81} - 20 q^{89} - 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
1.00000i
1.00000i
0 0 0 4.00000i 0 0 0 3.00000 0
65.2 0 0 0 4.00000i 0 0 0 3.00000 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 \(\Q(\sqrt{-1}) \)
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.2.b.b 2
3.b odd 2 1 1152.2.d.d 2
4.b odd 2 1 CM 128.2.b.b 2
5.b even 2 1 3200.2.d.e 2
5.c odd 4 1 3200.2.f.c 2
5.c odd 4 1 3200.2.f.d 2
8.b even 2 1 inner 128.2.b.b 2
8.d odd 2 1 inner 128.2.b.b 2
12.b even 2 1 1152.2.d.d 2
16.e even 4 1 256.2.a.b 1
16.e even 4 1 256.2.a.c 1
16.f odd 4 1 256.2.a.b 1
16.f odd 4 1 256.2.a.c 1
20.d odd 2 1 3200.2.d.e 2
20.e even 4 1 3200.2.f.c 2
20.e even 4 1 3200.2.f.d 2
24.f even 2 1 1152.2.d.d 2
24.h odd 2 1 1152.2.d.d 2
32.g even 8 4 1024.2.e.k 4
32.h odd 8 4 1024.2.e.k 4
40.e odd 2 1 3200.2.d.e 2
40.f even 2 1 3200.2.d.e 2
40.i odd 4 1 3200.2.f.c 2
40.i odd 4 1 3200.2.f.d 2
40.k even 4 1 3200.2.f.c 2
40.k even 4 1 3200.2.f.d 2
48.i odd 4 1 2304.2.a.a 1
48.i odd 4 1 2304.2.a.p 1
48.k even 4 1 2304.2.a.a 1
48.k even 4 1 2304.2.a.p 1
80.k odd 4 1 6400.2.a.l 1
80.k odd 4 1 6400.2.a.m 1
80.q even 4 1 6400.2.a.l 1
80.q even 4 1 6400.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.b.b 2 1.a even 1 1 trivial
128.2.b.b 2 4.b odd 2 1 CM
128.2.b.b 2 8.b even 2 1 inner
128.2.b.b 2 8.d odd 2 1 inner
256.2.a.b 1 16.e even 4 1
256.2.a.b 1 16.f odd 4 1
256.2.a.c 1 16.e even 4 1
256.2.a.c 1 16.f odd 4 1
1024.2.e.k 4 32.g even 8 4
1024.2.e.k 4 32.h odd 8 4
1152.2.d.d 2 3.b odd 2 1
1152.2.d.d 2 12.b even 2 1
1152.2.d.d 2 24.f even 2 1
1152.2.d.d 2 24.h odd 2 1
2304.2.a.a 1 48.i odd 4 1
2304.2.a.a 1 48.k even 4 1
2304.2.a.p 1 48.i odd 4 1
2304.2.a.p 1 48.k even 4 1
3200.2.d.e 2 5.b even 2 1
3200.2.d.e 2 20.d odd 2 1
3200.2.d.e 2 40.e odd 2 1
3200.2.d.e 2 40.f even 2 1
3200.2.f.c 2 5.c odd 4 1
3200.2.f.c 2 20.e even 4 1
3200.2.f.c 2 40.i odd 4 1
3200.2.f.c 2 40.k even 4 1
3200.2.f.d 2 5.c odd 4 1
3200.2.f.d 2 20.e even 4 1
3200.2.f.d 2 40.i odd 4 1
3200.2.f.d 2 40.k even 4 1
6400.2.a.l 1 80.k odd 4 1
6400.2.a.l 1 80.q even 4 1
6400.2.a.m 1 80.k odd 4 1
6400.2.a.m 1 80.q even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(128, [\chi])\). 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
$5$ \( T^{2} + 16 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 16 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 144 \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 16 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 144 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 6)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( (T + 18)^{2} \) Copy content Toggle raw display
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