Properties

Label 48.12.a.i
Level $48$
Weight $12$
Character orbit 48.a
Self dual yes
Analytic conductor $36.880$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,12,Mod(1,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 48.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: \(36.8804726669\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
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)
 
\(f(q)\) \(=\) \( q + 243 q^{3} + 9990 q^{5} + 86128 q^{7} + 59049 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 243 q^{3} + 9990 q^{5} + 86128 q^{7} + 59049 q^{9} + 806004 q^{11} - 960250 q^{13} + 2427570 q^{15} - 4306878 q^{17} - 401300 q^{19} + 20929104 q^{21} - 17751528 q^{23} + 50971975 q^{25} + 14348907 q^{27} - 84704994 q^{29} - 140930504 q^{31} + 195858972 q^{33} + 860418720 q^{35} - 413506594 q^{37} - 233340750 q^{39} + 150094890 q^{41} - 706702028 q^{43} + 589899510 q^{45} + 2475725472 q^{47} + 5440705641 q^{49} - 1046571354 q^{51} + 1600124166 q^{53} + 8051979960 q^{55} - 97515900 q^{57} - 3945492396 q^{59} - 885973498 q^{61} + 5085772272 q^{63} - 9592897500 q^{65} + 4881597772 q^{67} - 4313621304 q^{69} - 12631469400 q^{71} + 1423335194 q^{73} + 12386189925 q^{75} + 69419512512 q^{77} - 667407512 q^{79} + 3486784401 q^{81} - 5716071828 q^{83} - 43025711220 q^{85} - 20583313542 q^{87} - 85738736790 q^{89} - 82704412000 q^{91} - 34246112472 q^{93} - 4008987000 q^{95} - 52302647806 q^{97} + 47593730196 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
0 243.000 0 9990.00 0 86128.0 0 59049.0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

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 48.12.a.i 1
3.b odd 2 1 144.12.a.a 1
4.b odd 2 1 12.12.a.a 1
8.b even 2 1 192.12.a.a 1
8.d odd 2 1 192.12.a.k 1
12.b even 2 1 36.12.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.12.a.a 1 4.b odd 2 1
36.12.a.a 1 12.b even 2 1
48.12.a.i 1 1.a even 1 1 trivial
144.12.a.a 1 3.b odd 2 1
192.12.a.a 1 8.b even 2 1
192.12.a.k 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 9990 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(48))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 243 \) Copy content Toggle raw display
$5$ \( T - 9990 \) Copy content Toggle raw display
$7$ \( T - 86128 \) Copy content Toggle raw display
$11$ \( T - 806004 \) Copy content Toggle raw display
$13$ \( T + 960250 \) Copy content Toggle raw display
$17$ \( T + 4306878 \) Copy content Toggle raw display
$19$ \( T + 401300 \) Copy content Toggle raw display
$23$ \( T + 17751528 \) Copy content Toggle raw display
$29$ \( T + 84704994 \) Copy content Toggle raw display
$31$ \( T + 140930504 \) Copy content Toggle raw display
$37$ \( T + 413506594 \) Copy content Toggle raw display
$41$ \( T - 150094890 \) Copy content Toggle raw display
$43$ \( T + 706702028 \) Copy content Toggle raw display
$47$ \( T - 2475725472 \) Copy content Toggle raw display
$53$ \( T - 1600124166 \) Copy content Toggle raw display
$59$ \( T + 3945492396 \) Copy content Toggle raw display
$61$ \( T + 885973498 \) Copy content Toggle raw display
$67$ \( T - 4881597772 \) Copy content Toggle raw display
$71$ \( T + 12631469400 \) Copy content Toggle raw display
$73$ \( T - 1423335194 \) Copy content Toggle raw display
$79$ \( T + 667407512 \) Copy content Toggle raw display
$83$ \( T + 5716071828 \) Copy content Toggle raw display
$89$ \( T + 85738736790 \) Copy content Toggle raw display
$97$ \( T + 52302647806 \) Copy content Toggle raw display
show more
show less