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Properties

Label 4864.2.a.bp
Level $4864$
Weight $2$
Character orbit 4864.a
Self dual yes
Analytic conductor $38.839$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 13x^{6} + 24x^{5} + 48x^{4} - 68x^{3} - 62x^{2} + 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 152)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + ( - \beta_{2} + 1) q^{5} - \beta_{4} q^{7} + ( - \beta_{7} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} + ( - \beta_{2} + 1) q^{5} - \beta_{4} q^{7} + ( - \beta_{7} + 1) q^{9} + ( - \beta_{7} - 1) q^{11} + ( - \beta_{3} + 1) q^{13} + ( - \beta_{7} + 2 \beta_{5} + \beta_{4} + \cdots - 1) q^{15}+ \cdots + ( - 3 \beta_{7} + 2 \beta_{4} + \cdots + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5} - 4 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{5} - 4 q^{7} + 12 q^{9} - 4 q^{11} + 8 q^{13} - 4 q^{17} - 8 q^{19} + 16 q^{21} + 12 q^{25} + 28 q^{29} - 8 q^{31} + 12 q^{35} + 4 q^{37} + 4 q^{39} - 8 q^{41} + 4 q^{43} + 24 q^{45} - 12 q^{47} + 12 q^{49} - 12 q^{51} + 32 q^{53} + 8 q^{55} - 12 q^{59} + 8 q^{61} + 16 q^{63} + 8 q^{65} + 4 q^{67} + 28 q^{69} + 24 q^{71} + 24 q^{77} + 24 q^{79} - 8 q^{81} - 40 q^{83} + 24 q^{85} - 24 q^{87} + 8 q^{89} + 4 q^{91} + 32 q^{93} - 8 q^{95} + 16 q^{97} + 76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 13x^{6} + 24x^{5} + 48x^{4} - 68x^{3} - 62x^{2} + 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 13\nu^{5} - 2\nu^{4} + 44\nu^{3} + 16\nu^{2} - 22\nu - 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + \nu^{6} - 13\nu^{5} - 13\nu^{4} + 44\nu^{3} + 46\nu^{2} - 22\nu - 18 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{7} + 61\nu^{5} + 6\nu^{4} - 184\nu^{3} - 68\nu^{2} + 78\nu + 40 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} - 2\nu^{6} + 61\nu^{5} + 28\nu^{4} - 184\nu^{3} - 120\nu^{2} + 62\nu + 44 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} + \nu^{6} - 37\nu^{5} - 15\nu^{4} + 114\nu^{3} + 68\nu^{2} - 46\nu - 22 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} + \nu^{6} - 37\nu^{5} - 15\nu^{4} + 116\nu^{3} + 70\nu^{2} - 58\nu - 30 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{7} + \nu^{6} - 50\nu^{5} - 15\nu^{4} + 160\nu^{3} + 72\nu^{2} - 84\nu - 26 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} - \beta_{4} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{6} - 6\beta_{5} - 5\beta_{4} + \beta_{3} - \beta_{2} + 6\beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} - 16\beta_{5} - 9\beta_{4} - 7\beta_{3} + 7\beta_{2} - 2\beta _1 + 50 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{7} + 20\beta_{6} - 36\beta_{5} - 31\beta_{4} + 11\beta_{3} - 13\beta_{2} + 38\beta _1 + 30 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 22\beta_{7} - 22\beta_{6} - 116\beta_{5} - 69\beta_{4} - 47\beta_{3} + 51\beta_{2} - 30\beta _1 + 338 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -22\beta_{7} + 168\beta_{6} - 226\beta_{5} - 207\beta_{4} + 101\beta_{3} - 127\beta_{2} + 256\beta _1 + 128 ) / 2 \) 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
2.28103
2.67705
−2.81542
−0.365796
−0.932340
2.25259
−1.77833
0.681212
0 −3.13611 0 0.594041 0 −3.48756 0 6.83520 0
1.2 0 −2.09554 0 3.36827 0 4.47116 0 1.39129 0
1.3 0 −1.70663 0 −1.66222 0 −1.99556 0 −0.0874066 0
1.4 0 −0.840428 0 4.04855 0 −3.59283 0 −2.29368 0
1.5 0 0.579017 0 −2.10882 0 −2.73436 0 −2.66474 0
1.6 0 1.91886 0 −1.51356 0 0.580162 0 0.682013 0
1.7 0 2.32921 0 3.13887 0 −0.535658 0 2.42523 0
1.8 0 2.95163 0 2.13486 0 3.29464 0 5.71210 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

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4864.2.a.bp 8
4.b odd 2 1 4864.2.a.bq 8
8.b even 2 1 4864.2.a.bn 8
8.d odd 2 1 4864.2.a.bo 8
16.e even 4 2 608.2.c.b 16
16.f odd 4 2 152.2.c.b ✓ 16
48.i odd 4 2 5472.2.g.b 16
48.k even 4 2 1368.2.g.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.c.b ✓ 16 16.f odd 4 2
608.2.c.b 16 16.e even 4 2
1368.2.g.b 16 48.k even 4 2
4864.2.a.bn 8 8.b even 2 1
4864.2.a.bo 8 8.d odd 2 1
4864.2.a.bp 8 1.a even 1 1 trivial
4864.2.a.bq 8 4.b odd 2 1
5472.2.g.b 16 48.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4864))\):

\( T_{3}^{8} - 18T_{3}^{6} + 101T_{3}^{4} + 8T_{3}^{3} - 196T_{3}^{2} - 32T_{3} + 72 \) Copy content Toggle raw display
\( T_{5}^{8} - 8T_{5}^{7} + 6T_{5}^{6} + 80T_{5}^{5} - 119T_{5}^{4} - 280T_{5}^{3} + 388T_{5}^{2} + 368T_{5} - 288 \) Copy content Toggle raw display
\( T_{7}^{8} + 4T_{7}^{7} - 26T_{7}^{6} - 128T_{7}^{5} + 80T_{7}^{4} + 948T_{7}^{3} + 946T_{7}^{2} - 328T_{7} - 313 \) Copy content Toggle raw display
\( T_{11}^{8} + 4T_{11}^{7} - 34T_{11}^{6} - 148T_{11}^{5} + 201T_{11}^{4} + 1272T_{11}^{3} + 1152T_{11}^{2} - 128T_{11} - 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 18 T^{6} + \cdots + 72 \) Copy content Toggle raw display
$5$ \( T^{8} - 8 T^{7} + \cdots - 288 \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} + \cdots - 313 \) Copy content Toggle raw display
$11$ \( T^{8} + 4 T^{7} + \cdots - 256 \) Copy content Toggle raw display
$13$ \( T^{8} - 8 T^{7} + \cdots - 512 \) Copy content Toggle raw display
$17$ \( T^{8} + 4 T^{7} + \cdots + 9409 \) Copy content Toggle raw display
$19$ \( (T + 1)^{8} \) Copy content Toggle raw display
$23$ \( T^{8} - 124 T^{6} + \cdots + 155824 \) Copy content Toggle raw display
$29$ \( T^{8} - 28 T^{7} + \cdots + 2936 \) Copy content Toggle raw display
$31$ \( T^{8} + 8 T^{7} + \cdots + 7552 \) Copy content Toggle raw display
$37$ \( T^{8} - 4 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{7} + \cdots - 3072 \) Copy content Toggle raw display
$43$ \( T^{8} - 4 T^{7} + \cdots + 1184 \) Copy content Toggle raw display
$47$ \( T^{8} + 12 T^{7} + \cdots + 34432 \) Copy content Toggle raw display
$53$ \( T^{8} - 32 T^{7} + \cdots - 363592 \) Copy content Toggle raw display
$59$ \( T^{8} + 12 T^{7} + \cdots + 52176 \) Copy content Toggle raw display
$61$ \( T^{8} - 8 T^{7} + \cdots - 3258128 \) Copy content Toggle raw display
$67$ \( T^{8} - 4 T^{7} + \cdots - 69752 \) Copy content Toggle raw display
$71$ \( T^{8} - 24 T^{7} + \cdots - 9153024 \) Copy content Toggle raw display
$73$ \( T^{8} - 316 T^{6} + \cdots + 6637913 \) Copy content Toggle raw display
$79$ \( T^{8} - 24 T^{7} + \cdots - 70912 \) Copy content Toggle raw display
$83$ \( T^{8} + 40 T^{7} + \cdots + 151552 \) Copy content Toggle raw display
$89$ \( T^{8} - 8 T^{7} + \cdots - 400128 \) Copy content Toggle raw display
$97$ \( T^{8} - 16 T^{7} + \cdots + 8192 \) Copy content Toggle raw display
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