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Properties

Label 4096.2.a.n
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: 8.8.89405784064.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 16x^{6} + 68x^{4} - 104x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{7} + \beta_{2}) q^{5} + ( - \beta_{7} - \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{6} + \beta_{5} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{7} + \beta_{2}) q^{5} + ( - \beta_{7} - \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{6} + \beta_{5} + 2) q^{9} + ( - \beta_1 - 2) q^{11} + ( - \beta_{4} + 2 \beta_{3} + 2 \beta_{2}) q^{13} + (\beta_{4} + \beta_{3} - 4 \beta_{2}) q^{15} + ( - 2 \beta_{6} - \beta_{5} - 1) q^{17} + (2 \beta_{6} - \beta_{5} - \beta_1 - 3) q^{19} + ( - \beta_{3} + 3 \beta_{2}) q^{21} + ( - \beta_{7} - 3 \beta_{3} - \beta_{2}) q^{23} + (3 \beta_{6} - 2 \beta_1 + 3) q^{25} + (4 \beta_{6} - \beta_{5} - 3) q^{27} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{29} + 2 \beta_{2} q^{31} + (\beta_{6} - \beta_{5} - 2 \beta_1 - 5) q^{33} + ( - 4 \beta_{6} + \beta_{5} + 2 \beta_1 - 7) q^{35} + (2 \beta_{7} + \beta_{4} + \cdots - 2 \beta_{2}) q^{37}+ \cdots + ( - 2 \beta_{6} - \beta_{5} - 3 \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{9} - 16 q^{11} - 8 q^{17} - 24 q^{19} + 24 q^{25} - 24 q^{27} - 40 q^{33} - 56 q^{35} - 32 q^{41} + 32 q^{43} + 8 q^{49} - 8 q^{57} - 40 q^{59} + 16 q^{65} + 8 q^{73} - 56 q^{75} - 24 q^{83} + 8 q^{89} - 40 q^{91} - 8 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 16x^{6} + 68x^{4} - 104x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} - 11\nu^{2} + 14 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{7} - 41\nu^{5} + 106\nu^{3} - 46\nu ) / 21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{7} - 32\nu^{5} + 129\nu^{3} - 131\nu ) / 21 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{7} + 57\nu^{5} - 181\nu^{3} + 206\nu ) / 21 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 13\nu^{4} - 27\nu^{2} + 4 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - 14\nu^{4} + 41\nu^{2} - 30 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 14\nu^{5} + 41\nu^{3} - 30\nu ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{4} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{7} + 5\beta_{4} - 2\beta_{3} - 13\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11\beta_{6} + 11\beta_{5} + 14\beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 91\beta_{7} + 41\beta_{4} - 34\beta_{3} - 135\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 116\beta_{6} + 113\beta_{5} + 155\beta _1 + 286 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 941\beta_{7} + 399\beta_{4} - 394\beta_{3} - 1387\beta_{2} ) / 2 \) 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
1.69883
−1.69883
−1.37128
1.37128
0.933460
−0.933460
−3.21904
3.21904
0 −3.13902 0 −3.16787 0 1.32011 0 6.85346 0
1.2 0 −3.13902 0 3.16787 0 −1.32011 0 6.85346 0
1.3 0 −1.04953 0 −3.78704 0 4.55241 0 −1.89848 0
1.4 0 −1.04953 0 3.78704 0 −4.55241 0 −1.89848 0
1.5 0 1.72481 0 −0.554745 0 2.40250 0 −0.0250345 0
1.6 0 1.72481 0 0.554745 0 −2.40250 0 −0.0250345 0
1.7 0 2.46375 0 −2.70465 0 1.93929 0 3.07005 0
1.8 0 2.46375 0 2.70465 0 −1.93929 0 3.07005 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.n 8
4.b odd 2 1 4096.2.a.o 8
8.b even 2 1 4096.2.a.o 8
8.d odd 2 1 inner 4096.2.a.n 8
64.i even 16 2 1024.2.g.b ✓ 16
64.i even 16 2 1024.2.g.c yes 16
64.i even 16 2 1024.2.g.e yes 16
64.i even 16 2 1024.2.g.h yes 16
64.j odd 16 2 1024.2.g.b ✓ 16
64.j odd 16 2 1024.2.g.c yes 16
64.j odd 16 2 1024.2.g.e yes 16
64.j odd 16 2 1024.2.g.h yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1024.2.g.b ✓ 16 64.i even 16 2
1024.2.g.b ✓ 16 64.j odd 16 2
1024.2.g.c yes 16 64.i even 16 2
1024.2.g.c yes 16 64.j odd 16 2
1024.2.g.e yes 16 64.i even 16 2
1024.2.g.e yes 16 64.j odd 16 2
1024.2.g.h yes 16 64.i even 16 2
1024.2.g.h yes 16 64.j odd 16 2
4096.2.a.n 8 1.a even 1 1 trivial
4096.2.a.n 8 8.d odd 2 1 inner
4096.2.a.o 8 4.b odd 2 1
4096.2.a.o 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4096))\):

\( T_{3}^{4} - 10T_{3}^{2} + 4T_{3} + 14 \) Copy content Toggle raw display
\( T_{5}^{8} - 32T_{5}^{6} + 332T_{5}^{4} - 1152T_{5}^{2} + 324 \) Copy content Toggle raw display
\( T_{7}^{8} - 32T_{7}^{6} + 272T_{7}^{4} - 832T_{7}^{2} + 784 \) Copy content Toggle raw display
\( T_{23}^{8} - 80T_{23}^{6} + 1808T_{23}^{4} - 8064T_{23}^{2} + 1296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 10 T^{2} + \cdots + 14)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 32 T^{6} + \cdots + 324 \) Copy content Toggle raw display
$7$ \( T^{8} - 32 T^{6} + \cdots + 784 \) Copy content Toggle raw display
$11$ \( (T^{4} + 8 T^{3} + 14 T^{2} + \cdots - 18)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 80 T^{6} + \cdots + 56644 \) Copy content Toggle raw display
$17$ \( (T^{4} + 4 T^{3} - 28 T^{2} + \cdots - 72)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 12 T^{3} + \cdots - 434)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 80 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$29$ \( T^{8} - 64 T^{6} + \cdots + 324 \) Copy content Toggle raw display
$31$ \( (T^{4} - 16 T^{2} + 32)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 192 T^{6} + \cdots + 1110916 \) Copy content Toggle raw display
$41$ \( (T^{4} + 16 T^{3} + \cdots + 324)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 16 T^{3} + \cdots - 82)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 128 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$53$ \( T^{8} - 224 T^{6} + \cdots + 93636 \) Copy content Toggle raw display
$59$ \( (T^{4} + 20 T^{3} + \cdots - 1314)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} - 288 T^{6} + \cdots + 3740356 \) Copy content Toggle raw display
$67$ \( (T^{4} - 122 T^{2} + \cdots - 162)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 400 T^{6} + \cdots + 70358544 \) Copy content Toggle raw display
$73$ \( (T^{4} - 4 T^{3} + \cdots - 412)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 512 T^{6} + \cdots + 3268864 \) Copy content Toggle raw display
$83$ \( (T^{4} + 12 T^{3} + \cdots + 126)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 4 T^{3} + \cdots + 1764)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 4 T^{3} - 36 T^{2} + \cdots + 56)^{2} \) Copy content Toggle raw display
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