Properties

Label 128.4.b.c
Level $128$
Weight $4$
Character orbit 128.b
Analytic conductor $7.552$
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] = [128,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.65");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(7.55224448073\)
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.

\(f(q)\) \(=\) \( q + \beta q^{5} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{5} + 27 q^{9} + 23 \beta q^{13} + 94 q^{17} + 109 q^{25} + 71 \beta q^{29} - 99 \beta q^{37} - 230 q^{41} + 27 \beta q^{45} - 343 q^{49} - 143 \beta q^{53} - 117 \beta q^{61} - 368 q^{65} - 1098 q^{73} + 729 q^{81} + 94 \beta q^{85} + 1670 q^{89} - 594 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 54 q^{9} + 188 q^{17} + 218 q^{25} - 460 q^{41} - 686 q^{49} - 736 q^{65} - 2196 q^{73} + 1458 q^{81} + 3340 q^{89} - 1188 q^{97}+O(q^{100}) \) 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 27.0000 0
65.2 0 0 0 4.00000i 0 0 0 27.0000 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 128.4.b.c 2
3.b odd 2 1 1152.4.d.d 2
4.b odd 2 1 CM 128.4.b.c 2
8.b even 2 1 inner 128.4.b.c 2
8.d odd 2 1 inner 128.4.b.c 2
12.b even 2 1 1152.4.d.d 2
16.e even 4 1 256.4.a.d 1
16.e even 4 1 256.4.a.e 1
16.f odd 4 1 256.4.a.d 1
16.f odd 4 1 256.4.a.e 1
24.f even 2 1 1152.4.d.d 2
24.h odd 2 1 1152.4.d.d 2
48.i odd 4 1 2304.4.a.g 1
48.i odd 4 1 2304.4.a.j 1
48.k even 4 1 2304.4.a.g 1
48.k even 4 1 2304.4.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.4.b.c 2 1.a even 1 1 trivial
128.4.b.c 2 4.b odd 2 1 CM
128.4.b.c 2 8.b even 2 1 inner
128.4.b.c 2 8.d odd 2 1 inner
256.4.a.d 1 16.e even 4 1
256.4.a.d 1 16.f odd 4 1
256.4.a.e 1 16.e even 4 1
256.4.a.e 1 16.f odd 4 1
1152.4.d.d 2 3.b odd 2 1
1152.4.d.d 2 12.b even 2 1
1152.4.d.d 2 24.f even 2 1
1152.4.d.d 2 24.h odd 2 1
2304.4.a.g 1 48.i odd 4 1
2304.4.a.g 1 48.k even 4 1
2304.4.a.j 1 48.i odd 4 1
2304.4.a.j 1 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(128, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7} \) 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
$13$ \( T^{2} + 8464 \) Copy content Toggle raw display
$17$ \( (T - 94)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 80656 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 156816 \) Copy content Toggle raw display
$41$ \( (T + 230)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 327184 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 219024 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 1098)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 1670)^{2} \) Copy content Toggle raw display
$97$ \( (T + 594)^{2} \) Copy content Toggle raw display
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