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Properties

Label 4096.2.a.i
Level $4096$
Weight $2$
Character orbit 4096.a
Self dual yes
Analytic conductor $32.707$
Analytic rank $1$
Dimension $8$
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: \(8\)
Coefficient field: \(\Q(\zeta_{48})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 20x^{4} - 16x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1024)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} - 1) q^{3} + (\beta_{5} + \beta_{2}) q^{5} + (\beta_{4} + 2 \beta_{2}) q^{7} + ( - 2 \beta_{6} - \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - 1) q^{3} + (\beta_{5} + \beta_{2}) q^{5} + (\beta_{4} + 2 \beta_{2}) q^{7} + ( - 2 \beta_{6} - \beta_{3}) q^{9} + (\beta_{6} - 2 \beta_1 - 1) q^{11} + (\beta_{4} - \beta_{2}) q^{13} + ( - \beta_{5} - 2 \beta_{2}) q^{15} + ( - 3 \beta_{6} + \beta_{3} - \beta_1 + 1) q^{17} + (2 \beta_{6} - \beta_{3} + 3 \beta_1) q^{19} + ( - 3 \beta_{7} - 2 \beta_{5} + \cdots - 3 \beta_{2}) q^{21}+ \cdots + (6 \beta_{6} + 3 \beta_{3} + 5 \beta_1 - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{3} - 8 q^{11} + 8 q^{17} - 8 q^{25} - 8 q^{27} + 40 q^{33} + 8 q^{35} + 32 q^{41} - 56 q^{43} + 8 q^{49} - 48 q^{51} + 8 q^{57} - 32 q^{59} - 16 q^{65} - 24 q^{67} - 8 q^{73} + 16 q^{81} - 48 q^{83} - 8 q^{89} - 8 q^{91} + 8 q^{97} - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of \(\nu = \zeta_{48} + \zeta_{48}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 4\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - 6\nu^{3} + 7\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 6\nu^{3} + 9\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} - 6\nu^{4} + 8\nu^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - 7\nu^{5} + 14\nu^{3} - 8\nu \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{5} - 3\beta_{4} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 4\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11\beta_{5} - 9\beta_{4} + 12\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} + 6\beta_{3} + 16\beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2\beta_{7} + 43\beta_{5} - 29\beta_{4} + 56\beta_{2} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
1.58671
−1.58671
−1.98289
1.98289
−0.261052
0.261052
−1.21752
1.21752
0 −2.93185 0 −0.396183 0 −4.33496 0 5.59575 0
1.2 0 −2.93185 0 0.396183 0 4.33496 0 5.59575 0
1.3 0 −1.51764 0 −3.56960 0 −1.45158 0 −0.696775 0
1.4 0 −1.51764 0 3.56960 0 1.45158 0 −0.696775 0
1.5 0 −0.482362 0 −1.47858 0 −0.191104 0 −2.76733 0
1.6 0 −0.482362 0 1.47858 0 0.191104 0 −2.76733 0
1.7 0 0.931852 0 −0.956470 0 3.32633 0 −2.13165 0
1.8 0 0.931852 0 0.956470 0 −3.32633 0 −2.13165 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 4096.2.a.i 8
4.b odd 2 1 4096.2.a.s 8
8.b even 2 1 4096.2.a.s 8
8.d odd 2 1 inner 4096.2.a.i 8
64.i even 16 2 1024.2.g.a ✓ 16
64.i even 16 2 1024.2.g.d yes 16
64.i even 16 2 1024.2.g.f yes 16
64.i even 16 2 1024.2.g.g yes 16
64.j odd 16 2 1024.2.g.a ✓ 16
64.j odd 16 2 1024.2.g.d yes 16
64.j odd 16 2 1024.2.g.f yes 16
64.j odd 16 2 1024.2.g.g yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1024.2.g.a ✓ 16 64.i even 16 2
1024.2.g.a ✓ 16 64.j odd 16 2
1024.2.g.d yes 16 64.i even 16 2
1024.2.g.d yes 16 64.j odd 16 2
1024.2.g.f yes 16 64.i even 16 2
1024.2.g.f yes 16 64.j odd 16 2
1024.2.g.g yes 16 64.i even 16 2
1024.2.g.g yes 16 64.j odd 16 2
4096.2.a.i 8 1.a even 1 1 trivial
4096.2.a.i 8 8.d odd 2 1 inner
4096.2.a.s 8 4.b odd 2 1
4096.2.a.s 8 8.b even 2 1

Hecke kernels

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

\( T_{3}^{4} + 4T_{3}^{3} + 2T_{3}^{2} - 4T_{3} - 2 \) Copy content Toggle raw display
\( T_{5}^{8} - 16T_{5}^{6} + 44T_{5}^{4} - 32T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{8} - 32T_{7}^{6} + 272T_{7}^{4} - 448T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{23}^{8} - 112T_{23}^{6} + 3920T_{23}^{4} - 43904T_{23}^{2} + 80656 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 4 T^{3} + 2 T^{2} + \cdots - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 16 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{8} - 32 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{4} + 4 T^{3} + \cdots + 142)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 32 T^{6} + \cdots + 2116 \) Copy content Toggle raw display
$17$ \( (T^{4} - 4 T^{3} + \cdots - 200)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 34 T^{2} + \cdots + 46)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 112 T^{6} + \cdots + 80656 \) Copy content Toggle raw display
$29$ \( T^{8} - 176 T^{6} + \cdots + 1119364 \) Copy content Toggle raw display
$31$ \( T^{8} - 224 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$37$ \( T^{8} - 112 T^{6} + \cdots + 8836 \) Copy content Toggle raw display
$41$ \( (T^{4} - 16 T^{3} + \cdots - 92)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 28 T^{3} + \cdots + 1150)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 256 T^{6} + \cdots + 1364224 \) Copy content Toggle raw display
$53$ \( T^{8} - 208 T^{6} + \cdots + 21316 \) Copy content Toggle raw display
$59$ \( (T^{4} + 16 T^{3} + \cdots - 2450)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} - 208 T^{6} + \cdots + 2500 \) Copy content Toggle raw display
$67$ \( (T^{4} + 12 T^{3} + \cdots - 2978)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 176 T^{6} + \cdots + 595984 \) Copy content Toggle raw display
$73$ \( (T^{4} + 4 T^{3} + \cdots - 1436)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 128 T^{6} + \cdots + 160000 \) Copy content Toggle raw display
$83$ \( (T^{4} + 24 T^{3} + \cdots - 2162)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 4 T^{3} + \cdots + 292)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 4 T^{3} + \cdots + 376)^{2} \) Copy content Toggle raw display
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