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Properties

Label 1024.2.e.h
Level $1024$
Weight $2$
Character orbit 1024.e
Analytic conductor $8.177$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1024,2,Mod(257,1024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1024, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1024.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.e (of order \(4\), degree \(2\), 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: \(8.17668116698\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 512)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{3} - \beta_1 q^{5} + ( - \beta_{3} - \beta_1) q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{3} - \beta_1 q^{5} + ( - \beta_{3} - \beta_1) q^{7} + \beta_{2} q^{9} + (3 \beta_{2} + 3) q^{11} + 3 \beta_{3} q^{13} + ( - \beta_{3} + \beta_1) q^{15} + ( - 3 \beta_{2} + 3) q^{19} + 2 \beta_1 q^{21} + ( - 3 \beta_{3} - 3 \beta_1) q^{23} - \beta_{2} q^{25} + ( - 4 \beta_{2} - 4) q^{27} - \beta_{3} q^{29} + ( - 2 \beta_{3} + 2 \beta_1) q^{31} - 6 q^{33} + (4 \beta_{2} - 4) q^{35} - 3 \beta_1 q^{37} + ( - 3 \beta_{3} - 3 \beta_1) q^{39} - 6 \beta_{2} q^{41} + ( - 3 \beta_{2} - 3) q^{43} - \beta_{3} q^{45} - q^{49} - \beta_1 q^{53} + ( - 3 \beta_{3} - 3 \beta_1) q^{55} + 6 \beta_{2} q^{57} + ( - \beta_{2} - 1) q^{59} - 3 \beta_{3} q^{61} + ( - \beta_{3} + \beta_1) q^{63} + 12 q^{65} + ( - 9 \beta_{2} + 9) q^{67} + 6 \beta_1 q^{69} + ( - 3 \beta_{3} - 3 \beta_1) q^{71} + 12 \beta_{2} q^{73} + (\beta_{2} + 1) q^{75} - 6 \beta_{3} q^{77} + ( - 2 \beta_{3} + 2 \beta_1) q^{79} + 5 q^{81} + (3 \beta_{2} - 3) q^{83} + (\beta_{3} + \beta_1) q^{87} - 12 \beta_{2} q^{89} + (12 \beta_{2} + 12) q^{91} + 4 \beta_{3} q^{93} + (3 \beta_{3} - 3 \beta_1) q^{95} - 8 q^{97} + (3 \beta_{2} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 12 q^{11} + 12 q^{19} - 16 q^{27} - 24 q^{33} - 16 q^{35} - 12 q^{43} - 4 q^{49} - 4 q^{59} + 48 q^{65} + 36 q^{67} + 4 q^{75} + 20 q^{81} - 12 q^{83} + 48 q^{91} - 32 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{8}^{3} \) Copy content Toggle raw display

Display \(\zeta_{8}^j\) in terms of \(\beta_i\)

\(\zeta_{8}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{8}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
0.707107 − 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
−0.707107 − 0.707107i
0 −1.00000 − 1.00000i 0 −1.41421 + 1.41421i 0 2.82843i 0 − 1.00000i 0
257.2 0 −1.00000 − 1.00000i 0 1.41421 − 1.41421i 0 − 2.82843i 0 − 1.00000i 0
769.1 0 −1.00000 + 1.00000i 0 −1.41421 − 1.41421i 0 − 2.82843i 0 1.00000i 0
769.2 0 −1.00000 + 1.00000i 0 1.41421 + 1.41421i 0 2.82843i 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
8.d odd 2 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1024.2.e.h 4
4.b odd 2 1 1024.2.e.n 4
8.b even 2 1 1024.2.e.n 4
8.d odd 2 1 inner 1024.2.e.h 4
16.e even 4 1 inner 1024.2.e.h 4
16.e even 4 1 1024.2.e.n 4
16.f odd 4 1 inner 1024.2.e.h 4
16.f odd 4 1 1024.2.e.n 4
32.g even 8 1 512.2.a.b ✓ 2
32.g even 8 1 512.2.a.e yes 2
32.g even 8 2 512.2.b.e 4
32.h odd 8 1 512.2.a.b ✓ 2
32.h odd 8 1 512.2.a.e yes 2
32.h odd 8 2 512.2.b.e 4
96.o even 8 1 4608.2.a.c 2
96.o even 8 1 4608.2.a.p 2
96.o even 8 2 4608.2.d.j 4
96.p odd 8 1 4608.2.a.c 2
96.p odd 8 1 4608.2.a.p 2
96.p odd 8 2 4608.2.d.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.a.b ✓ 2 32.g even 8 1
512.2.a.b ✓ 2 32.h odd 8 1
512.2.a.e yes 2 32.g even 8 1
512.2.a.e yes 2 32.h odd 8 1
512.2.b.e 4 32.g even 8 2
512.2.b.e 4 32.h odd 8 2
1024.2.e.h 4 1.a even 1 1 trivial
1024.2.e.h 4 8.d odd 2 1 inner
1024.2.e.h 4 16.e even 4 1 inner
1024.2.e.h 4 16.f odd 4 1 inner
1024.2.e.n 4 4.b odd 2 1
1024.2.e.n 4 8.b even 2 1
1024.2.e.n 4 16.e even 4 1
1024.2.e.n 4 16.f odd 4 1
4608.2.a.c 2 96.o even 8 1
4608.2.a.c 2 96.p odd 8 1
4608.2.a.p 2 96.o even 8 1
4608.2.a.p 2 96.p odd 8 1
4608.2.d.j 4 96.o even 8 2
4608.2.d.j 4 96.p odd 8 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1024, [\chi])\):

\( T_{3}^{2} + 2T_{3} + 2 \) Copy content Toggle raw display
\( T_{5}^{4} + 16 \) Copy content Toggle raw display
\( T_{47} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 16 \) Copy content Toggle raw display
$7$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 1296 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 16 \) Copy content Toggle raw display
$31$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 1296 \) Copy content Toggle raw display
$41$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 16 \) Copy content Toggle raw display
$59$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 1296 \) Copy content Toggle raw display
$67$ \( (T^{2} - 18 T + 162)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$97$ \( (T + 8)^{4} \) Copy content Toggle raw display
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