Properties

Label 1.12.a.a
Level $1$
Weight $12$
Character orbit 1.a
Self dual yes
Analytic conductor $0.768$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

This is the discriminant modular form $\Delta=\sum \tau(n)q^n$, where $\tau$ is the Ramanujan tau function [A000594]. It is the minimal weight newform of level $1$.

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1,12,Mod(1,1)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 1.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: \(0.768343180560\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q - 24 q^{2} + 252 q^{3} - 1472 q^{4} + 4830 q^{5} - 6048 q^{6} - 16744 q^{7} + 84480 q^{8} - 113643 q^{9} - 115920 q^{10} + 534612 q^{11} - 370944 q^{12} - 577738 q^{13} + 401856 q^{14} + 1217160 q^{15}+ \cdots - 60754911516 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Expression as an eta quotient

\(f(z) = \eta(z)^{24}=q\prod_{n=1}^\infty(1 - q^{n})^{24}\)

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
−24.0000 252.000 −1472.00 4830.00 −6048.00 −16744.0 84480.0 −113643. −115920.
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 1.12.a.a 1
3.b odd 2 1 9.12.a.b 1
4.b odd 2 1 16.12.a.a 1
5.b even 2 1 25.12.a.b 1
5.c odd 4 2 25.12.b.b 2
7.b odd 2 1 49.12.a.a 1
7.c even 3 2 49.12.c.b 2
7.d odd 6 2 49.12.c.c 2
8.b even 2 1 64.12.a.b 1
8.d odd 2 1 64.12.a.f 1
9.c even 3 2 81.12.c.d 2
9.d odd 6 2 81.12.c.b 2
11.b odd 2 1 121.12.a.b 1
12.b even 2 1 144.12.a.d 1
13.b even 2 1 169.12.a.a 1
15.d odd 2 1 225.12.a.b 1
15.e even 4 2 225.12.b.d 2
16.e even 4 2 256.12.b.e 2
16.f odd 4 2 256.12.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.12.a.a 1 1.a even 1 1 trivial
9.12.a.b 1 3.b odd 2 1
16.12.a.a 1 4.b odd 2 1
25.12.a.b 1 5.b even 2 1
25.12.b.b 2 5.c odd 4 2
49.12.a.a 1 7.b odd 2 1
49.12.c.b 2 7.c even 3 2
49.12.c.c 2 7.d odd 6 2
64.12.a.b 1 8.b even 2 1
64.12.a.f 1 8.d odd 2 1
81.12.c.b 2 9.d odd 6 2
81.12.c.d 2 9.c even 3 2
121.12.a.b 1 11.b odd 2 1
144.12.a.d 1 12.b even 2 1
169.12.a.a 1 13.b even 2 1
225.12.a.b 1 15.d odd 2 1
225.12.b.d 2 15.e even 4 2
256.12.b.c 2 16.f odd 4 2
256.12.b.e 2 16.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 24 \) Copy content Toggle raw display
$3$ \( T - 252 \) Copy content Toggle raw display
$5$ \( T - 4830 \) Copy content Toggle raw display
$7$ \( T + 16744 \) Copy content Toggle raw display
$11$ \( T - 534612 \) Copy content Toggle raw display
$13$ \( T + 577738 \) Copy content Toggle raw display
$17$ \( T + 6905934 \) Copy content Toggle raw display
$19$ \( T - 10661420 \) Copy content Toggle raw display
$23$ \( T - 18643272 \) Copy content Toggle raw display
$29$ \( T - 128406630 \) Copy content Toggle raw display
$31$ \( T + 52843168 \) Copy content Toggle raw display
$37$ \( T + 182213314 \) Copy content Toggle raw display
$41$ \( T - 308120442 \) Copy content Toggle raw display
$43$ \( T + 17125708 \) Copy content Toggle raw display
$47$ \( T - 2687348496 \) Copy content Toggle raw display
$53$ \( T + 1596055698 \) Copy content Toggle raw display
$59$ \( T + 5189203740 \) Copy content Toggle raw display
$61$ \( T - 6956478662 \) Copy content Toggle raw display
$67$ \( T + 15481826884 \) Copy content Toggle raw display
$71$ \( T - 9791485272 \) Copy content Toggle raw display
$73$ \( T - 1463791322 \) Copy content Toggle raw display
$79$ \( T - 38116845680 \) Copy content Toggle raw display
$83$ \( T + 29335099668 \) Copy content Toggle raw display
$89$ \( T + 24992917110 \) Copy content Toggle raw display
$97$ \( T - 75013568546 \) Copy content Toggle raw display
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Additional information

\(\displaystyle q\prod_{n\geq1}(1-q^n)^{24} \)

\(\eta(z)^{24}\)