Properties

Label 36.3.d.b
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 $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 36.d (of order \(2\), degree \(1\), minimal)

Newform invariants

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

\(f(q)\) \(=\) \( q + 2q^{2} + 4q^{4} - 8q^{5} + 8q^{8} + O(q^{10}) \) \( q + 2q^{2} + 4q^{4} - 8q^{5} + 8q^{8} - 16q^{10} - 10q^{13} + 16q^{16} + 16q^{17} - 32q^{20} + 39q^{25} - 20q^{26} + 40q^{29} + 32q^{32} + 32q^{34} - 70q^{37} - 64q^{40} - 80q^{41} + 49q^{49} + 78q^{50} - 40q^{52} - 56q^{53} + 80q^{58} - 22q^{61} + 64q^{64} + 80q^{65} + 64q^{68} + 110q^{73} - 140q^{74} - 128q^{80} - 160q^{82} - 128q^{85} + 160q^{89} - 130q^{97} + 98q^{98} + O(q^{100}) \)

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.

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

Hecke kernels

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

Hecke characteristic polynomials

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