Properties

Label 128.5.d.a
Level 128
Weight 5
Character orbit 128.d
Analytic conductor 13.231
Analytic rank 0
Dimension 2
CM disc. -4
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 128 = 2^{7} \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 128.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(13.2313552747\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta q^{5} \) \( -81 q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta q^{5} \) \( -81 q^{9} \) \( + 5 \beta q^{13} \) \( -322 q^{17} \) \( -1679 q^{25} \) \( -35 \beta q^{29} \) \( + 35 \beta q^{37} \) \( -3038 q^{41} \) \( + 81 \beta q^{45} \) \( + 2401 q^{49} \) \( -105 \beta q^{53} \) \( -55 \beta q^{61} \) \( + 11520 q^{65} \) \( -1442 q^{73} \) \( + 6561 q^{81} \) \( + 322 \beta q^{85} \) \( + 9758 q^{89} \) \( + 1918 q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 162q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 162q^{9} \) \(\mathstrut -\mathstrut 644q^{17} \) \(\mathstrut -\mathstrut 3358q^{25} \) \(\mathstrut -\mathstrut 6076q^{41} \) \(\mathstrut +\mathstrut 4802q^{49} \) \(\mathstrut +\mathstrut 23040q^{65} \) \(\mathstrut -\mathstrut 2884q^{73} \) \(\mathstrut +\mathstrut 13122q^{81} \) \(\mathstrut +\mathstrut 19516q^{89} \) \(\mathstrut +\mathstrut 3836q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
1.00000i
1.00000i
0 0 0 48.0000i 0 0 0 −81.0000 0
63.2 0 0 0 48.0000i 0 0 0 −81.0000 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.b Even 1 yes
8.d Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3} \) acting on \(S_{5}^{\mathrm{new}}(128, [\chi])\).