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Properties

Label 4864.2.a.bm
Level $4864$
Weight $2$
Character orbit 4864.a
Self dual yes
Analytic conductor $38.839$
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] = [4864,2,Mod(1,4864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4864, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4864.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.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: \(38.8392355432\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.34309996544.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 8x^{6} + 28x^{5} + 31x^{4} - 36x^{3} - 22x^{2} + 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 2432)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{6} q^{5} + \beta_{4} q^{7} + (\beta_{7} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{6} q^{5} + \beta_{4} q^{7} + (\beta_{7} - \beta_1) q^{9} + ( - \beta_{7} + \beta_1 - 2) q^{11} + (\beta_{6} + \beta_{5} + \cdots - \beta_{2}) q^{13}+ \cdots + ( - 3 \beta_{7} + 2 \beta_{3} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 4 q^{9} - 20 q^{11} - 8 q^{17} + 8 q^{19} + 20 q^{25} - 4 q^{27} + 24 q^{33} - 12 q^{35} + 8 q^{41} - 28 q^{43} + 8 q^{49} - 12 q^{51} - 4 q^{57} - 36 q^{59} + 8 q^{65} - 28 q^{67} - 8 q^{73} - 68 q^{75} - 32 q^{81} - 40 q^{83} - 8 q^{89} + 12 q^{91} + 40 q^{97} - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 8x^{6} + 28x^{5} + 31x^{4} - 36x^{3} - 22x^{2} + 12x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -5\nu^{7} + 37\nu^{6} - 39\nu^{5} - 205\nu^{4} + 256\nu^{3} + 412\nu^{2} - 126\nu - 162 ) / 52 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{7} + 37\nu^{6} - 39\nu^{5} - 205\nu^{4} + 308\nu^{3} + 308\nu^{2} - 438\nu - 58 ) / 52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -9\nu^{7} + 25\nu^{6} + 117\nu^{5} - 161\nu^{4} - 600\nu^{3} - 28\nu^{2} + 522\nu + 10 ) / 52 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} + 24\nu^{6} + 13\nu^{5} - 114\nu^{4} - 30\nu^{3} - 4\nu^{2} - 22\nu + 98 ) / 26 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\nu^{7} - 59\nu^{6} - 91\nu^{5} + 251\nu^{4} + 428\nu^{3} + 220\nu^{2} - 142\nu - 138 ) / 52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10\nu^{7} - 35\nu^{6} - 104\nu^{5} + 267\nu^{4} + 424\nu^{3} - 304\nu^{2} - 268\nu + 38 ) / 26 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 23\nu^{7} - 87\nu^{6} - 195\nu^{5} + 579\nu^{4} + 736\nu^{3} - 460\nu^{2} - 294\nu + 38 ) / 52 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + \beta _1 + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} - 2\beta_{4} + \beta_{3} + \beta _1 + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -10\beta_{7} + 10\beta_{6} + 3\beta_{5} - 4\beta_{4} + 5\beta_{3} - \beta_{2} + 3\beta _1 + 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -15\beta_{7} + 16\beta_{6} + 3\beta_{5} - 10\beta_{4} + 10\beta_{3} + \beta_{2} + 5\beta _1 + 48 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -116\beta_{7} + 124\beta_{6} + 35\beta_{5} - 62\beta_{4} + 83\beta_{3} + 11\beta_{2} + 31\beta _1 + 323 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -199\beta_{7} + 222\beta_{6} + 60\beta_{5} - 120\beta_{4} + 167\beta_{3} + 38\beta_{2} + 54\beta _1 + 611 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -731\beta_{7} + 829\beta_{6} + 255\beta_{5} - 421\beta_{4} + 665\beta_{3} + 178\beta_{2} + 175\beta _1 + 2202 \) 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
−1.58857
−0.139869
0.545005
3.79159
−0.836706
3.15093
0.904430
−1.82681
0 −2.63640 0 −2.63403 0 −3.88603 0 3.95063 0
1.2 0 −2.63640 0 2.63403 0 3.88603 0 3.95063 0
1.3 0 −1.65222 0 −4.30533 0 2.73423 0 −0.270160 0
1.4 0 −1.65222 0 4.30533 0 −2.73423 0 −0.270160 0
1.5 0 0.222191 0 −1.55081 0 3.06334 0 −2.95063 0
1.6 0 0.222191 0 1.55081 0 −3.06334 0 −2.95063 0
1.7 0 2.06644 0 −1.45635 0 −0.196723 0 1.27016 0
1.8 0 2.06644 0 1.45635 0 0.196723 0 1.27016 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
\(2\) \(-1\)
\(19\) \(-1\)

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 4864.2.a.bm 8
4.b odd 2 1 4864.2.a.br 8
8.b even 2 1 4864.2.a.br 8
8.d odd 2 1 inner 4864.2.a.bm 8
16.e even 4 2 2432.2.c.i ✓ 16
16.f odd 4 2 2432.2.c.i ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2432.2.c.i ✓ 16 16.e even 4 2
2432.2.c.i ✓ 16 16.f odd 4 2
4864.2.a.bm 8 1.a even 1 1 trivial
4864.2.a.bm 8 8.d odd 2 1 inner
4864.2.a.br 8 4.b odd 2 1
4864.2.a.br 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4864))\):

\( T_{3}^{4} + 2T_{3}^{3} - 5T_{3}^{2} - 8T_{3} + 2 \) Copy content Toggle raw display
\( T_{5}^{8} - 30T_{5}^{6} + 249T_{5}^{4} - 712T_{5}^{2} + 656 \) Copy content Toggle raw display
\( T_{7}^{8} - 32T_{7}^{6} + 326T_{7}^{4} - 1072T_{7}^{2} + 41 \) Copy content Toggle raw display
\( T_{11}^{4} + 10T_{11}^{3} + 25T_{11}^{2} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 2 T^{3} - 5 T^{2} + \cdots + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 30 T^{6} + \cdots + 656 \) Copy content Toggle raw display
$7$ \( T^{8} - 32 T^{6} + \cdots + 41 \) Copy content Toggle raw display
$11$ \( (T^{4} + 10 T^{3} + \cdots - 32)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 74 T^{6} + \cdots + 2624 \) Copy content Toggle raw display
$17$ \( (T^{2} + 2 T - 7)^{4} \) Copy content Toggle raw display
$19$ \( (T - 1)^{8} \) Copy content Toggle raw display
$23$ \( T^{8} - 74 T^{6} + \cdots + 2624 \) Copy content Toggle raw display
$29$ \( T^{8} - 110 T^{6} + \cdots + 164 \) Copy content Toggle raw display
$31$ \( T^{8} - 52 T^{6} + \cdots + 2624 \) Copy content Toggle raw display
$37$ \( T^{8} - 252 T^{6} + \cdots + 4410944 \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} - 30 T^{2} + \cdots + 32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 14 T^{3} + \cdots - 1756)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 230 T^{6} + \cdots + 758336 \) Copy content Toggle raw display
$53$ \( T^{8} - 198 T^{6} + \cdots + 393764 \) Copy content Toggle raw display
$59$ \( (T^{4} + 18 T^{3} + 69 T^{2} + \cdots - 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} - 174 T^{6} + \cdots + 189584 \) Copy content Toggle raw display
$67$ \( (T^{4} + 14 T^{3} + \cdots - 562)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 448 T^{6} + \cdots + 52910336 \) Copy content Toggle raw display
$73$ \( (T^{4} + 4 T^{3} + \cdots - 191)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 468 T^{6} + \cdots + 13983296 \) Copy content Toggle raw display
$83$ \( (T^{4} + 20 T^{3} + \cdots - 64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 4 T^{3} + \cdots - 1864)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 20 T^{3} + \cdots - 32)^{2} \) Copy content Toggle raw display
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