↑

Properties

Label 4096.2.a.q
Level $4096$
Weight $2$
Character orbit 4096.a
Self dual yes
Analytic conductor $32.707$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.4848615424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 16x^{4} - 8x^{2} + 1 \) 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 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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_{3}) q^{3} + \beta_{4} q^{5} + ( - \beta_{2} + \beta_1 + 1) q^{7} + (\beta_{6} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_{3}) q^{3} + \beta_{4} q^{5} + ( - \beta_{2} + \beta_1 + 1) q^{7} + (\beta_{6} + \beta_1 + 1) q^{9} + (\beta_{7} + \beta_{3}) q^{11} + ( - \beta_{4} + 2 \beta_{3}) q^{13} + (\beta_{6} + 1) q^{15} + ( - \beta_{6} - \beta_{2} + \beta_1) q^{17} + (\beta_{5} - \beta_{3}) q^{19} + ( - 3 \beta_{7} - \beta_{4} + 2 \beta_{3}) q^{21} + (\beta_{2} + \beta_1 + 1) q^{23} + ( - \beta_{2} - 3) q^{25} + ( - \beta_{7} - \beta_{5} + 2 \beta_{4}) q^{27} + (\beta_{7} + 2 \beta_{5}) q^{29} + (2 \beta_{2} + 4) q^{31} + (\beta_{6} - 2 \beta_{2} - \beta_1 + 2) q^{33} + (\beta_{7} - \beta_{5} + 2 \beta_{4}) q^{35} + ( - 2 \beta_{7} - 2 \beta_{5} + \cdots + 2 \beta_{3}) q^{37}+ \cdots + (3 \beta_{5} + 4 \beta_{4} - \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{7} + 8 q^{9} + 8 q^{15} + 8 q^{23} - 24 q^{25} + 32 q^{31} + 16 q^{33} + 40 q^{39} + 16 q^{41} + 16 q^{47} + 8 q^{49} + 8 q^{55} - 16 q^{57} + 40 q^{63} + 40 q^{71} + 16 q^{79} - 8 q^{81} + 8 q^{87} + 32 q^{89} - 8 q^{95} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 8x^{6} + 16x^{4} - 8x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{6} - 7\nu^{4} + 9\nu^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{6} - 8\nu^{4} + 15\nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{7} + 8\nu^{5} - 15\nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{7} - 8\nu^{5} + 16\nu^{3} - 7\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{7} + 8\nu^{5} - 16\nu^{3} + 9\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2\nu^{6} - 15\nu^{4} + 26\nu^{2} - 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\nu^{7} - 15\nu^{5} + 25\nu^{3} - 6\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - \beta_{2} - \beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{5} + 5\beta_{4} + 2\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{6} - 4\beta_{2} - 2\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{7} + 13\beta_{5} + 23\beta_{4} + 14\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 33\beta_{6} - 47\beta_{2} - 17\beta _1 + 76 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 16\beta_{7} + 63\beta_{5} + 113\beta_{4} + 80\beta_{3} ) / 2 \) 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
0.688039
−2.28533
−0.437573
1.45341
−1.45341
0.437573
2.28533
−0.688039
0 −2.82079 0 −0.765367 0 2.38372 0 4.95687 0
1.2 0 −2.46658 0 −1.84776 0 0.940588 0 3.08402 0
1.3 0 −1.38419 0 1.84776 0 3.88784 0 −1.08402 0
1.4 0 −0.207667 0 0.765367 0 −3.21215 0 −2.95687 0
1.5 0 0.207667 0 −0.765367 0 −3.21215 0 −2.95687 0
1.6 0 1.38419 0 −1.84776 0 3.88784 0 −1.08402 0
1.7 0 2.46658 0 1.84776 0 0.940588 0 3.08402 0
1.8 0 2.82079 0 0.765367 0 2.38372 0 4.95687 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.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.q 8
4.b odd 2 1 4096.2.a.k 8
8.b even 2 1 inner 4096.2.a.q 8
8.d odd 2 1 4096.2.a.k 8
64.i even 16 2 128.2.g.b 8
64.i even 16 2 256.2.g.c 8
64.i even 16 2 512.2.g.f 8
64.i even 16 2 512.2.g.g 8
64.j odd 16 2 32.2.g.b ✓ 8
64.j odd 16 2 256.2.g.d 8
64.j odd 16 2 512.2.g.e 8
64.j odd 16 2 512.2.g.h 8
192.q odd 16 2 1152.2.v.b 8
192.s even 16 2 288.2.v.b 8
320.bd even 16 2 800.2.ba.d 8
320.bh odd 16 2 800.2.y.b 8
320.bj even 16 2 800.2.ba.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.g.b ✓ 8 64.j odd 16 2
128.2.g.b 8 64.i even 16 2
256.2.g.c 8 64.i even 16 2
256.2.g.d 8 64.j odd 16 2
288.2.v.b 8 192.s even 16 2
512.2.g.e 8 64.j odd 16 2
512.2.g.f 8 64.i even 16 2
512.2.g.g 8 64.i even 16 2
512.2.g.h 8 64.j odd 16 2
800.2.y.b 8 320.bh odd 16 2
800.2.ba.c 8 320.bj even 16 2
800.2.ba.d 8 320.bd even 16 2
1152.2.v.b 8 192.q odd 16 2
4096.2.a.k 8 4.b odd 2 1
4096.2.a.k 8 8.d odd 2 1
4096.2.a.q 8 1.a even 1 1 trivial
4096.2.a.q 8 8.b even 2 1 inner

Hecke kernels

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

\( T_{3}^{8} - 16T_{3}^{6} + 76T_{3}^{4} - 96T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{5}^{4} - 4T_{5}^{2} + 2 \) Copy content Toggle raw display
\( T_{7}^{4} - 4T_{7}^{3} - 8T_{7}^{2} + 40T_{7} - 28 \) Copy content Toggle raw display
\( T_{23}^{4} - 4T_{23}^{3} - 8T_{23}^{2} + 8T_{23} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 16 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{4} - 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 4 T^{3} - 8 T^{2} + \cdots - 28)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 32 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( T^{8} - 56 T^{6} + \cdots + 6724 \) Copy content Toggle raw display
$17$ \( (T^{4} - 32 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 32 T^{6} + \cdots + 196 \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{3} - 8 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 104 T^{6} + \cdots + 188356 \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T + 8)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} - 152 T^{6} + \cdots + 64516 \) Copy content Toggle raw display
$41$ \( (T^{4} - 8 T^{3} + \cdots + 164)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 80 T^{6} + \cdots + 31684 \) Copy content Toggle raw display
$47$ \( (T^{4} - 8 T^{3} + 32 T + 16)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 152 T^{6} + \cdots + 158404 \) Copy content Toggle raw display
$59$ \( T^{8} - 176 T^{6} + \cdots + 643204 \) Copy content Toggle raw display
$61$ \( T^{8} - 248 T^{6} + \cdots + 42436 \) Copy content Toggle raw display
$67$ \( T^{8} - 288 T^{6} + \cdots + 1285956 \) Copy content Toggle raw display
$71$ \( (T^{4} - 20 T^{3} + \cdots - 4604)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 92 T^{2} + \cdots + 196)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + \cdots - 9968)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 464 T^{6} + \cdots + 138250564 \) Copy content Toggle raw display
$89$ \( (T^{4} - 16 T^{3} + \cdots - 4124)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 16 T^{3} + \cdots - 992)^{2} \) Copy content Toggle raw display
show more
show less