Properties

Label 4096.2.a.e
Level $4096$
Weight $2$
Character orbit 4096.a
Self dual yes
Analytic conductor $32.707$
Analytic rank $1$
Dimension $4$
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] = [4096,2,Mod(1,4096)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4096, 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("4096.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4096 = 2^{12} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4096.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: \(32.7067246679\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
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 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - 2 \beta_{3} - \beta_1) q^{5} - \beta_{2} q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - 2 \beta_{3} - \beta_1) q^{5} - \beta_{2} q^{7} + (\beta_{2} - 1) q^{9} + (2 \beta_{3} - \beta_1) q^{11} + \beta_1 q^{13} + ( - 3 \beta_{2} - 2) q^{15} + 2 \beta_{2} q^{17} + (3 \beta_{3} + 2 \beta_1) q^{19} + ( - \beta_{3} - \beta_1) q^{21} + (3 \beta_{2} - 4) q^{23} + (\beta_{2} + 5) q^{25} + (\beta_{3} - 3 \beta_1) q^{27} + (\beta_{3} - 2 \beta_1) q^{29} + 4 q^{31} + (\beta_{2} - 2) q^{33} + ( - \beta_{3} + 3 \beta_1) q^{35} - \beta_1 q^{37} + (\beta_{2} + 2) q^{39} + ( - 3 \beta_{2} - 4) q^{41} + (4 \beta_{3} + 3 \beta_1) q^{43} + (3 \beta_{3} - 2 \beta_1) q^{45} + ( - 4 \beta_{2} - 6) q^{47} - 5 q^{49} + (2 \beta_{3} + 2 \beta_1) q^{51} + ( - \beta_{3} - 4 \beta_1) q^{53} + (5 \beta_{2} - 6) q^{55} + (5 \beta_{2} + 4) q^{57} + (\beta_{3} - 4 \beta_1) q^{59} + \beta_{3} q^{61} + (\beta_{2} - 2) q^{63} + ( - 3 \beta_{2} - 2) q^{65} + ( - 2 \beta_{3} + 3 \beta_1) q^{67} + (3 \beta_{3} - \beta_1) q^{69} + (3 \beta_{2} - 4) q^{71} - 7 \beta_{2} q^{73} + (\beta_{3} + 6 \beta_1) q^{75} + (3 \beta_{3} - \beta_1) q^{77} - 6 q^{79} + ( - 5 \beta_{2} - 3) q^{81} + (\beta_{3} - 4 \beta_1) q^{83} + (2 \beta_{3} - 6 \beta_1) q^{85} + ( - \beta_{2} - 4) q^{87} + (3 \beta_{2} - 8) q^{89} + ( - \beta_{3} - \beta_1) q^{91} + 4 \beta_1 q^{93} + ( - 3 \beta_{2} - 16) q^{95} + ( - 6 \beta_{2} + 10) q^{97} + ( - 5 \beta_{3} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} - 8 q^{15} - 16 q^{23} + 20 q^{25} + 16 q^{31} - 8 q^{33} + 8 q^{39} - 16 q^{41} - 24 q^{47} - 20 q^{49} - 24 q^{55} + 16 q^{57} - 8 q^{63} - 8 q^{65} - 16 q^{71} - 24 q^{79} - 12 q^{81} - 16 q^{87} - 32 q^{89} - 64 q^{95} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{16} + \zeta_{16}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) 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
−1.84776
−0.765367
0.765367
1.84776
0 −1.84776 0 3.37849 0 −1.41421 0 0.414214 0
1.2 0 −0.765367 0 −2.93015 0 1.41421 0 −2.41421 0
1.3 0 0.765367 0 2.93015 0 1.41421 0 −2.41421 0
1.4 0 1.84776 0 −3.37849 0 −1.41421 0 0.414214 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4096.2.a.e 4
4.b odd 2 1 4096.2.a.f 4
8.b even 2 1 inner 4096.2.a.e 4
8.d odd 2 1 4096.2.a.f 4
64.i even 16 2 32.2.g.a 4
64.i even 16 2 256.2.g.b 4
64.i even 16 2 512.2.g.a 4
64.i even 16 2 512.2.g.d 4
64.j odd 16 2 128.2.g.a 4
64.j odd 16 2 256.2.g.a 4
64.j odd 16 2 512.2.g.b 4
64.j odd 16 2 512.2.g.c 4
192.q odd 16 2 288.2.v.a 4
192.s even 16 2 1152.2.v.a 4
320.bc odd 16 2 800.2.ba.a 4
320.bf even 16 2 800.2.y.a 4
320.bi odd 16 2 800.2.ba.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.g.a 4 64.i even 16 2
128.2.g.a 4 64.j odd 16 2
256.2.g.a 4 64.j odd 16 2
256.2.g.b 4 64.i even 16 2
288.2.v.a 4 192.q odd 16 2
512.2.g.a 4 64.i even 16 2
512.2.g.b 4 64.j odd 16 2
512.2.g.c 4 64.j odd 16 2
512.2.g.d 4 64.i even 16 2
800.2.y.a 4 320.bf even 16 2
800.2.ba.a 4 320.bc odd 16 2
800.2.ba.b 4 320.bi odd 16 2
1152.2.v.a 4 192.s even 16 2
4096.2.a.e 4 1.a even 1 1 trivial
4096.2.a.e 4 8.b even 2 1 inner
4096.2.a.f 4 4.b odd 2 1
4096.2.a.f 4 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4096))\):

\( T_{3}^{4} - 4T_{3}^{2} + 2 \) Copy content Toggle raw display
\( T_{5}^{4} - 20T_{5}^{2} + 98 \) Copy content Toggle raw display
\( T_{7}^{2} - 2 \) Copy content Toggle raw display
\( T_{23}^{2} + 8T_{23} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 4T^{2} + 2 \) Copy content Toggle raw display
$5$ \( T^{4} - 20T^{2} + 98 \) Copy content Toggle raw display
$7$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 20T^{2} + 2 \) Copy content Toggle raw display
$13$ \( T^{4} - 4T^{2} + 2 \) Copy content Toggle raw display
$17$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 52T^{2} + 578 \) Copy content Toggle raw display
$23$ \( (T^{2} + 8 T - 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 20T^{2} + 98 \) Copy content Toggle raw display
$31$ \( (T - 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 4T^{2} + 2 \) Copy content Toggle raw display
$41$ \( (T^{2} + 8 T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 100T^{2} + 1922 \) Copy content Toggle raw display
$47$ \( (T^{2} + 12 T + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 68T^{2} + 98 \) Copy content Toggle raw display
$59$ \( T^{4} - 68T^{2} + 1058 \) Copy content Toggle raw display
$61$ \( T^{4} - 4T^{2} + 2 \) Copy content Toggle raw display
$67$ \( T^{4} - 52T^{2} + 578 \) Copy content Toggle raw display
$71$ \( (T^{2} + 8 T - 2)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 98)^{2} \) Copy content Toggle raw display
$79$ \( (T + 6)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - 68T^{2} + 1058 \) Copy content Toggle raw display
$89$ \( (T^{2} + 16 T + 46)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 20 T + 28)^{2} \) Copy content Toggle raw display
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