Properties

Label 512.2.a.e
Level $512$
Weight $2$
Character orbit 512.a
Self dual yes
Analytic conductor $4.088$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [512,2,Mod(1,512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(512, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("512.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.a (trivial)

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: yes
Analytic conductor: \(4.08834058349\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
0 −1.41421 0 2.00000 0 −2.82843 0 −1.00000 0
1.2 0 1.41421 0 2.00000 0 2.82843 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 512.2.a.e yes 2
3.b odd 2 1 4608.2.a.c 2
4.b odd 2 1 inner 512.2.a.e yes 2
8.b even 2 1 512.2.a.b 2
8.d odd 2 1 512.2.a.b 2
12.b even 2 1 4608.2.a.c 2
16.e even 4 2 512.2.b.e 4
16.f odd 4 2 512.2.b.e 4
24.f even 2 1 4608.2.a.p 2
24.h odd 2 1 4608.2.a.p 2
32.g even 8 2 1024.2.e.h 4
32.g even 8 2 1024.2.e.n 4
32.h odd 8 2 1024.2.e.h 4
32.h odd 8 2 1024.2.e.n 4
48.i odd 4 2 4608.2.d.j 4
48.k even 4 2 4608.2.d.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.a.b 2 8.b even 2 1
512.2.a.b 2 8.d odd 2 1
512.2.a.e yes 2 1.a even 1 1 trivial
512.2.a.e yes 2 4.b odd 2 1 inner
512.2.b.e 4 16.e even 4 2
512.2.b.e 4 16.f odd 4 2
1024.2.e.h 4 32.g even 8 2
1024.2.e.h 4 32.h odd 8 2
1024.2.e.n 4 32.g even 8 2
1024.2.e.n 4 32.h odd 8 2
4608.2.a.c 2 3.b odd 2 1
4608.2.a.c 2 12.b even 2 1
4608.2.a.p 2 24.f even 2 1
4608.2.a.p 2 24.h odd 2 1
4608.2.d.j 4 48.i odd 4 2
4608.2.d.j 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(512))\):

\( T_{3}^{2} - 2 \) Copy content Toggle raw display
\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2 \) Copy content Toggle raw display
$5$ \( (T - 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 8 \) Copy content Toggle raw display
$11$ \( T^{2} - 18 \) Copy content Toggle raw display
$13$ \( (T - 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 18 \) Copy content Toggle raw display
$23$ \( T^{2} - 72 \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 32 \) Copy content Toggle raw display
$37$ \( (T - 6)^{2} \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 18 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 2 \) Copy content Toggle raw display
$61$ \( (T - 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 162 \) Copy content Toggle raw display
$71$ \( T^{2} - 72 \) Copy content Toggle raw display
$73$ \( (T + 12)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 32 \) Copy content Toggle raw display
$83$ \( T^{2} - 18 \) Copy content Toggle raw display
$89$ \( (T + 12)^{2} \) Copy content Toggle raw display
$97$ \( (T + 8)^{2} \) Copy content Toggle raw display
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