Properties

Label 36.3.d.a
Level $36$
Weight $3$
Character orbit 36.d
Self dual yes
Analytic conductor $0.981$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [36,3,Mod(19,36)]
 
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([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("36.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 36.d (of order \(2\), degree \(1\), minimal)

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: \(0.980928951697\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} + 8 q^{5} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 4 q^{4} + 8 q^{5} - 8 q^{8} - 16 q^{10} - 10 q^{13} + 16 q^{16} - 16 q^{17} + 32 q^{20} + 39 q^{25} + 20 q^{26} - 40 q^{29} - 32 q^{32} + 32 q^{34} - 70 q^{37} - 64 q^{40} + 80 q^{41} + 49 q^{49} - 78 q^{50} - 40 q^{52} + 56 q^{53} + 80 q^{58} - 22 q^{61} + 64 q^{64} - 80 q^{65} - 64 q^{68} + 110 q^{73} + 140 q^{74} + 128 q^{80} - 160 q^{82} - 128 q^{85} - 160 q^{89} - 130 q^{97} - 98 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0
−2.00000 0 4.00000 8.00000 0 0 −8.00000 0 −16.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.3.d.a 1
3.b odd 2 1 36.3.d.b yes 1
4.b odd 2 1 CM 36.3.d.a 1
5.b even 2 1 900.3.c.c 1
5.c odd 4 2 900.3.f.b 2
8.b even 2 1 576.3.g.a 1
8.d odd 2 1 576.3.g.a 1
9.c even 3 2 324.3.f.f 2
9.d odd 6 2 324.3.f.e 2
12.b even 2 1 36.3.d.b yes 1
15.d odd 2 1 900.3.c.b 1
15.e even 4 2 900.3.f.a 2
16.e even 4 2 2304.3.b.d 2
16.f odd 4 2 2304.3.b.d 2
20.d odd 2 1 900.3.c.c 1
20.e even 4 2 900.3.f.b 2
24.f even 2 1 576.3.g.c 1
24.h odd 2 1 576.3.g.c 1
36.f odd 6 2 324.3.f.f 2
36.h even 6 2 324.3.f.e 2
48.i odd 4 2 2304.3.b.e 2
48.k even 4 2 2304.3.b.e 2
60.h even 2 1 900.3.c.b 1
60.l odd 4 2 900.3.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.d.a 1 1.a even 1 1 trivial
36.3.d.a 1 4.b odd 2 1 CM
36.3.d.b yes 1 3.b odd 2 1
36.3.d.b yes 1 12.b even 2 1
324.3.f.e 2 9.d odd 6 2
324.3.f.e 2 36.h even 6 2
324.3.f.f 2 9.c even 3 2
324.3.f.f 2 36.f odd 6 2
576.3.g.a 1 8.b even 2 1
576.3.g.a 1 8.d odd 2 1
576.3.g.c 1 24.f even 2 1
576.3.g.c 1 24.h odd 2 1
900.3.c.b 1 15.d odd 2 1
900.3.c.b 1 60.h even 2 1
900.3.c.c 1 5.b even 2 1
900.3.c.c 1 20.d odd 2 1
900.3.f.a 2 15.e even 4 2
900.3.f.a 2 60.l odd 4 2
900.3.f.b 2 5.c odd 4 2
900.3.f.b 2 20.e even 4 2
2304.3.b.d 2 16.e even 4 2
2304.3.b.d 2 16.f odd 4 2
2304.3.b.e 2 48.i odd 4 2
2304.3.b.e 2 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 8 \) acting on \(S_{3}^{\mathrm{new}}(36, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 8 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 10 \) Copy content Toggle raw display
$17$ \( T + 16 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 40 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T + 70 \) Copy content Toggle raw display
$41$ \( T - 80 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 56 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 22 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 110 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 160 \) Copy content Toggle raw display
$97$ \( T + 130 \) Copy content Toggle raw display
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