Properties

Label 36.12.a.b
Level $36$
Weight $12$
Character orbit 36.a
Self dual yes
Analytic conductor $27.660$
Analytic rank $0$
Dimension $1$
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] = [36,12,Mod(1,36)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(36, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("36.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 36.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: \(27.6603545001\)
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)
 
\(f(q)\) \(=\) \( q - 2862 q^{5} + 9128 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 2862 q^{5} + 9128 q^{7} - 668196 q^{11} + 2052950 q^{13} - 1604178 q^{17} - 230500 q^{19} + 43012728 q^{23} - 40637081 q^{25} + 141745194 q^{29} + 233221904 q^{31} - 26124336 q^{35} + 278269694 q^{37} + 1181577510 q^{41} + 856975172 q^{43} + 1664054928 q^{47} - 1894006359 q^{49} + 3851181666 q^{53} + 1912376952 q^{55} - 10339000596 q^{59} + 185948102 q^{61} - 5875542900 q^{65} + 2915010572 q^{67} - 12662314200 q^{71} - 15201270694 q^{73} - 6099293088 q^{77} - 36644027488 q^{79} + 9217637028 q^{83} + 4591157436 q^{85} - 30573828810 q^{89} + 18739327600 q^{91} + 659691000 q^{95} + 145701815906 q^{97}+O(q^{100}) \) 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 0 0 −2862.00 0 9128.00 0 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 36.12.a.b 1
3.b odd 2 1 12.12.a.b 1
4.b odd 2 1 144.12.a.f 1
12.b even 2 1 48.12.a.c 1
24.f even 2 1 192.12.a.n 1
24.h odd 2 1 192.12.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.12.a.b 1 3.b odd 2 1
36.12.a.b 1 1.a even 1 1 trivial
48.12.a.c 1 12.b even 2 1
144.12.a.f 1 4.b odd 2 1
192.12.a.d 1 24.h odd 2 1
192.12.a.n 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 2862 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(36))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 2862 \) Copy content Toggle raw display
$7$ \( T - 9128 \) Copy content Toggle raw display
$11$ \( T + 668196 \) Copy content Toggle raw display
$13$ \( T - 2052950 \) Copy content Toggle raw display
$17$ \( T + 1604178 \) Copy content Toggle raw display
$19$ \( T + 230500 \) Copy content Toggle raw display
$23$ \( T - 43012728 \) Copy content Toggle raw display
$29$ \( T - 141745194 \) Copy content Toggle raw display
$31$ \( T - 233221904 \) Copy content Toggle raw display
$37$ \( T - 278269694 \) Copy content Toggle raw display
$41$ \( T - 1181577510 \) Copy content Toggle raw display
$43$ \( T - 856975172 \) Copy content Toggle raw display
$47$ \( T - 1664054928 \) Copy content Toggle raw display
$53$ \( T - 3851181666 \) Copy content Toggle raw display
$59$ \( T + 10339000596 \) Copy content Toggle raw display
$61$ \( T - 185948102 \) Copy content Toggle raw display
$67$ \( T - 2915010572 \) Copy content Toggle raw display
$71$ \( T + 12662314200 \) Copy content Toggle raw display
$73$ \( T + 15201270694 \) Copy content Toggle raw display
$79$ \( T + 36644027488 \) Copy content Toggle raw display
$83$ \( T - 9217637028 \) Copy content Toggle raw display
$89$ \( T + 30573828810 \) Copy content Toggle raw display
$97$ \( T - 145701815906 \) Copy content Toggle raw display
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