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 disc. -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 \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.063880321617\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\zeta_{8})\)
Artin image size \(8\)
Artin image $D_4$
Artin field Galois closure of 4.0.512.1

$q$-expansion

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 CM by \(\Q(\sqrt{-1}) \) yes
8.d Odd 1 CM by \(\Q(\sqrt{-2}) \) yes
8.b Even 1 RM by \(\Q(\sqrt{2}) \) yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(128, [\chi])\).