Properties

Label 36.2.a.a
Level $36$
Weight $2$
Character orbit 36.a
Self dual yes
Analytic conductor $0.287$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 36.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.287461447277\)
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: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q - 4 q^{7} + O(q^{10}) \) \( q - 4 q^{7} + 2 q^{13} + 8 q^{19} - 5 q^{25} - 4 q^{31} - 10 q^{37} + 8 q^{43} + 9 q^{49} + 14 q^{61} - 16 q^{67} - 10 q^{73} - 4 q^{79} - 8 q^{91} + 14 q^{97} + O(q^{100}) \)

Expression as an eta quotient

\(f(z) = \eta(6z)^{4}=q\prod_{n=1}^\infty(1 - q^{6n})^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 0 0 −4.00000 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.2.a.a 1
3.b odd 2 1 CM 36.2.a.a 1
4.b odd 2 1 144.2.a.a 1
5.b even 2 1 900.2.a.g 1
5.c odd 4 2 900.2.d.b 2
7.b odd 2 1 1764.2.a.e 1
7.c even 3 2 1764.2.k.h 2
7.d odd 6 2 1764.2.k.g 2
8.b even 2 1 576.2.a.e 1
8.d odd 2 1 576.2.a.f 1
9.c even 3 2 324.2.e.c 2
9.d odd 6 2 324.2.e.c 2
11.b odd 2 1 4356.2.a.g 1
12.b even 2 1 144.2.a.a 1
13.b even 2 1 6084.2.a.i 1
13.d odd 4 2 6084.2.b.f 2
15.d odd 2 1 900.2.a.g 1
15.e even 4 2 900.2.d.b 2
16.e even 4 2 2304.2.d.q 2
16.f odd 4 2 2304.2.d.a 2
20.d odd 2 1 3600.2.a.e 1
20.e even 4 2 3600.2.f.m 2
21.c even 2 1 1764.2.a.e 1
21.g even 6 2 1764.2.k.g 2
21.h odd 6 2 1764.2.k.h 2
24.f even 2 1 576.2.a.f 1
24.h odd 2 1 576.2.a.e 1
28.d even 2 1 7056.2.a.bb 1
33.d even 2 1 4356.2.a.g 1
36.f odd 6 2 1296.2.i.h 2
36.h even 6 2 1296.2.i.h 2
39.d odd 2 1 6084.2.a.i 1
39.f even 4 2 6084.2.b.f 2
48.i odd 4 2 2304.2.d.q 2
48.k even 4 2 2304.2.d.a 2
60.h even 2 1 3600.2.a.e 1
60.l odd 4 2 3600.2.f.m 2
84.h odd 2 1 7056.2.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.a.a 1 1.a even 1 1 trivial
36.2.a.a 1 3.b odd 2 1 CM
144.2.a.a 1 4.b odd 2 1
144.2.a.a 1 12.b even 2 1
324.2.e.c 2 9.c even 3 2
324.2.e.c 2 9.d odd 6 2
576.2.a.e 1 8.b even 2 1
576.2.a.e 1 24.h odd 2 1
576.2.a.f 1 8.d odd 2 1
576.2.a.f 1 24.f even 2 1
900.2.a.g 1 5.b even 2 1
900.2.a.g 1 15.d odd 2 1
900.2.d.b 2 5.c odd 4 2
900.2.d.b 2 15.e even 4 2
1296.2.i.h 2 36.f odd 6 2
1296.2.i.h 2 36.h even 6 2
1764.2.a.e 1 7.b odd 2 1
1764.2.a.e 1 21.c even 2 1
1764.2.k.g 2 7.d odd 6 2
1764.2.k.g 2 21.g even 6 2
1764.2.k.h 2 7.c even 3 2
1764.2.k.h 2 21.h odd 6 2
2304.2.d.a 2 16.f odd 4 2
2304.2.d.a 2 48.k even 4 2
2304.2.d.q 2 16.e even 4 2
2304.2.d.q 2 48.i odd 4 2
3600.2.a.e 1 20.d odd 2 1
3600.2.a.e 1 60.h even 2 1
3600.2.f.m 2 20.e even 4 2
3600.2.f.m 2 60.l odd 4 2
4356.2.a.g 1 11.b odd 2 1
4356.2.a.g 1 33.d even 2 1
6084.2.a.i 1 13.b even 2 1
6084.2.a.i 1 39.d odd 2 1
6084.2.b.f 2 13.d odd 4 2
6084.2.b.f 2 39.f even 4 2
7056.2.a.bb 1 28.d even 2 1
7056.2.a.bb 1 84.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(\Gamma_0(36))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( 4 + T \)
$11$ \( T \)
$13$ \( -2 + T \)
$17$ \( T \)
$19$ \( -8 + T \)
$23$ \( T \)
$29$ \( T \)
$31$ \( 4 + T \)
$37$ \( 10 + T \)
$41$ \( T \)
$43$ \( -8 + T \)
$47$ \( T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( -14 + T \)
$67$ \( 16 + T \)
$71$ \( T \)
$73$ \( 10 + T \)
$79$ \( 4 + T \)
$83$ \( T \)
$89$ \( T \)
$97$ \( -14 + T \)
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