Properties

Label 32.3.d.a
Level $32$
Weight $3$
Character orbit 32.d
Self dual yes
Analytic conductor $0.872$
Analytic rank $0$
Dimension $1$
CM discriminant -8
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 32.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.871936845953\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{3} - 5q^{9} + O(q^{10}) \) \( q + 2q^{3} - 5q^{9} - 14q^{11} + 2q^{17} + 34q^{19} + 25q^{25} - 28q^{27} - 28q^{33} - 46q^{41} - 14q^{43} + 49q^{49} + 4q^{51} + 68q^{57} + 82q^{59} - 62q^{67} - 142q^{73} + 50q^{75} - 11q^{81} - 158q^{83} + 146q^{89} - 94q^{97} + 70q^{99} + O(q^{100}) \)

Expression as an eta quotient

\(f(z) = \dfrac{\eta(4z)^{5}\eta(8z)^{5}}{\eta(2z)^{2}\eta(16z)^{2}}=q\prod_{n=1}^\infty(1 - q^{2n})^{-2}(1 - q^{4n})^{5}(1 - q^{8n})^{5}(1 - q^{16n})^{-2}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(31\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
0
0 2.00000 0 0 0 0 0 −5.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.3.d.a 1
3.b odd 2 1 288.3.b.a 1
4.b odd 2 1 8.3.d.a 1
5.b even 2 1 800.3.g.a 1
5.c odd 4 2 800.3.e.a 2
7.b odd 2 1 1568.3.g.a 1
8.b even 2 1 8.3.d.a 1
8.d odd 2 1 CM 32.3.d.a 1
12.b even 2 1 72.3.b.a 1
16.e even 4 2 256.3.c.e 2
16.f odd 4 2 256.3.c.e 2
20.d odd 2 1 200.3.g.a 1
20.e even 4 2 200.3.e.a 2
24.f even 2 1 288.3.b.a 1
24.h odd 2 1 72.3.b.a 1
28.d even 2 1 392.3.g.a 1
28.f even 6 2 392.3.k.b 2
28.g odd 6 2 392.3.k.d 2
40.e odd 2 1 800.3.g.a 1
40.f even 2 1 200.3.g.a 1
40.i odd 4 2 200.3.e.a 2
40.k even 4 2 800.3.e.a 2
48.i odd 4 2 2304.3.g.j 2
48.k even 4 2 2304.3.g.j 2
56.e even 2 1 1568.3.g.a 1
56.h odd 2 1 392.3.g.a 1
56.j odd 6 2 392.3.k.b 2
56.p even 6 2 392.3.k.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.3.d.a 1 4.b odd 2 1
8.3.d.a 1 8.b even 2 1
32.3.d.a 1 1.a even 1 1 trivial
32.3.d.a 1 8.d odd 2 1 CM
72.3.b.a 1 12.b even 2 1
72.3.b.a 1 24.h odd 2 1
200.3.e.a 2 20.e even 4 2
200.3.e.a 2 40.i odd 4 2
200.3.g.a 1 20.d odd 2 1
200.3.g.a 1 40.f even 2 1
256.3.c.e 2 16.e even 4 2
256.3.c.e 2 16.f odd 4 2
288.3.b.a 1 3.b odd 2 1
288.3.b.a 1 24.f even 2 1
392.3.g.a 1 28.d even 2 1
392.3.g.a 1 56.h odd 2 1
392.3.k.b 2 28.f even 6 2
392.3.k.b 2 56.j odd 6 2
392.3.k.d 2 28.g odd 6 2
392.3.k.d 2 56.p even 6 2
800.3.e.a 2 5.c odd 4 2
800.3.e.a 2 40.k even 4 2
800.3.g.a 1 5.b even 2 1
800.3.g.a 1 40.e odd 2 1
1568.3.g.a 1 7.b odd 2 1
1568.3.g.a 1 56.e even 2 1
2304.3.g.j 2 48.i odd 4 2
2304.3.g.j 2 48.k even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(32, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -2 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( 14 + T \)
$13$ \( T \)
$17$ \( -2 + T \)
$19$ \( -34 + T \)
$23$ \( T \)
$29$ \( T \)
$31$ \( T \)
$37$ \( T \)
$41$ \( 46 + T \)
$43$ \( 14 + T \)
$47$ \( T \)
$53$ \( T \)
$59$ \( -82 + T \)
$61$ \( T \)
$67$ \( 62 + T \)
$71$ \( T \)
$73$ \( 142 + T \)
$79$ \( T \)
$83$ \( 158 + T \)
$89$ \( -146 + T \)
$97$ \( 94 + T \)
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