Properties

Label 128.1.d.a
Level $128$
Weight $1$
Character orbit 128.d
Self dual yes
Analytic conductor $0.064$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -4, -8, 8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 128.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.0638803216170\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\zeta_{8})\)
Artin image $D_4$
Artin field Galois closure of 4.0.512.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{9} + O(q^{10}) \) \( q - q^{9} - 2q^{17} + q^{25} + 2q^{41} + q^{49} - 2q^{73} + q^{81} - 2q^{89} - 2q^{97} + O(q^{100}) \)

Expression as an eta quotient

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
0
0 0 0 0 0 0 0 −1.00000 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
8.b even 2 1 RM by \(\Q(\sqrt{2}) \)
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 128.1.d.a 1
3.b odd 2 1 1152.1.b.a 1
4.b odd 2 1 CM 128.1.d.a 1
5.b even 2 1 3200.1.g.a 1
5.c odd 4 2 3200.1.e.a 2
8.b even 2 1 RM 128.1.d.a 1
8.d odd 2 1 CM 128.1.d.a 1
12.b even 2 1 1152.1.b.a 1
16.e even 4 2 256.1.c.a 1
16.f odd 4 2 256.1.c.a 1
20.d odd 2 1 3200.1.g.a 1
20.e even 4 2 3200.1.e.a 2
24.f even 2 1 1152.1.b.a 1
24.h odd 2 1 1152.1.b.a 1
32.g even 8 4 1024.1.f.b 2
32.h odd 8 4 1024.1.f.b 2
40.e odd 2 1 3200.1.g.a 1
40.f even 2 1 3200.1.g.a 1
40.i odd 4 2 3200.1.e.a 2
40.k even 4 2 3200.1.e.a 2
48.i odd 4 2 2304.1.g.b 1
48.k even 4 2 2304.1.g.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.1.d.a 1 1.a even 1 1 trivial
128.1.d.a 1 4.b odd 2 1 CM
128.1.d.a 1 8.b even 2 1 RM
128.1.d.a 1 8.d odd 2 1 CM
256.1.c.a 1 16.e even 4 2
256.1.c.a 1 16.f odd 4 2
1024.1.f.b 2 32.g even 8 4
1024.1.f.b 2 32.h odd 8 4
1152.1.b.a 1 3.b odd 2 1
1152.1.b.a 1 12.b even 2 1
1152.1.b.a 1 24.f even 2 1
1152.1.b.a 1 24.h odd 2 1
2304.1.g.b 1 48.i odd 4 2
2304.1.g.b 1 48.k even 4 2
3200.1.e.a 2 5.c odd 4 2
3200.1.e.a 2 20.e even 4 2
3200.1.e.a 2 40.i odd 4 2
3200.1.e.a 2 40.k even 4 2
3200.1.g.a 1 5.b even 2 1
3200.1.g.a 1 20.d odd 2 1
3200.1.g.a 1 40.e odd 2 1
3200.1.g.a 1 40.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(128, [\chi])\).

Hecke characteristic polynomials

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