Properties

Label 128.4.b.e
Level $128$
Weight $4$
Character orbit 128.b
Analytic conductor $7.552$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{3} q^{5} + \beta_{2} q^{7} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{3} q^{5} + \beta_{2} q^{7} - 13 q^{9} + 7 \beta_1 q^{11} - \beta_{3} q^{13} + 5 \beta_{2} q^{15} + 70 q^{17} - 13 \beta_1 q^{19} - 8 \beta_{3} q^{21} + 7 \beta_{2} q^{23} - 195 q^{25} - 14 \beta_1 q^{27} + 7 \beta_{3} q^{29} + 280 q^{33} + 64 \beta_1 q^{35} + 21 \beta_{3} q^{37} - 5 \beta_{2} q^{39} - 182 q^{41} - 21 \beta_1 q^{43} - 13 \beta_{3} q^{45} - 14 \beta_{2} q^{47} + 169 q^{49} - 70 \beta_1 q^{51} - 7 \beta_{3} q^{53} - 35 \beta_{2} q^{55} - 520 q^{57} - 13 \beta_1 q^{59} - 13 \beta_{3} q^{61} - 13 \beta_{2} q^{63} + 320 q^{65} + 35 \beta_1 q^{67} - 56 \beta_{3} q^{69} + 5 \beta_{2} q^{71} + 910 q^{73} + 195 \beta_1 q^{75} + 56 \beta_{3} q^{77} - 30 \beta_{2} q^{79} - 911 q^{81} - 113 \beta_1 q^{83} + 70 \beta_{3} q^{85} + 35 \beta_{2} q^{87} - 546 q^{89} - 64 \beta_1 q^{91} + 65 \beta_{2} q^{95} - 490 q^{97} - 91 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 52 q^{9} + 280 q^{17} - 780 q^{25} + 1120 q^{33} - 728 q^{41} + 676 q^{49} - 2080 q^{57} + 1280 q^{65} + 3640 q^{73} - 3644 q^{81} - 2184 q^{89} - 1960 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} + 14\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -16\nu^{3} - 16\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 8\nu^{2} + 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 8\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 16 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{2} - 8\beta_1 ) / 32 \) 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
−0.707107 + 1.58114i
0.707107 + 1.58114i
0.707107 1.58114i
−0.707107 1.58114i
0 6.32456i 0 17.8885i 0 −22.6274 0 −13.0000 0
65.2 0 6.32456i 0 17.8885i 0 22.6274 0 −13.0000 0
65.3 0 6.32456i 0 17.8885i 0 22.6274 0 −13.0000 0
65.4 0 6.32456i 0 17.8885i 0 −22.6274 0 −13.0000 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 inner
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.e 4
3.b odd 2 1 1152.4.d.j 4
4.b odd 2 1 inner 128.4.b.e 4
8.b even 2 1 inner 128.4.b.e 4
8.d odd 2 1 inner 128.4.b.e 4
12.b even 2 1 1152.4.d.j 4
16.e even 4 2 256.4.a.n 4
16.f odd 4 2 256.4.a.n 4
24.f even 2 1 1152.4.d.j 4
24.h odd 2 1 1152.4.d.j 4
48.i odd 4 2 2304.4.a.bz 4
48.k even 4 2 2304.4.a.bz 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.4.b.e 4 1.a even 1 1 trivial
128.4.b.e 4 4.b odd 2 1 inner
128.4.b.e 4 8.b even 2 1 inner
128.4.b.e 4 8.d odd 2 1 inner
256.4.a.n 4 16.e even 4 2
256.4.a.n 4 16.f odd 4 2
1152.4.d.j 4 3.b odd 2 1
1152.4.d.j 4 12.b even 2 1
1152.4.d.j 4 24.f even 2 1
1152.4.d.j 4 24.h odd 2 1
2304.4.a.bz 4 48.i odd 4 2
2304.4.a.bz 4 48.k even 4 2

Hecke kernels

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

\( T_{3}^{2} + 40 \) Copy content Toggle raw display
\( T_{7}^{2} - 512 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 40)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 320)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 512)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1960)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 320)^{2} \) Copy content Toggle raw display
$17$ \( (T - 70)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 6760)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 25088)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 15680)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 141120)^{2} \) Copy content Toggle raw display
$41$ \( (T + 182)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 17640)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 100352)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 15680)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 6760)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 54080)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 49000)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 12800)^{2} \) Copy content Toggle raw display
$73$ \( (T - 910)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 460800)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 510760)^{2} \) Copy content Toggle raw display
$89$ \( (T + 546)^{4} \) Copy content Toggle raw display
$97$ \( (T + 490)^{4} \) Copy content Toggle raw display
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