Properties

Label 128.5.d.b
Level 128
Weight 5
Character orbit 128.d
Self dual Yes
Analytic conductor 13.231
Analytic rank 0
Dimension 2
CM disc. -8
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: Yes
Analytic conductor: \(13.2313552747\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 8\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{3} \) \( + 47 q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta q^{3} \) \( + 47 q^{9} \) \( + 21 \beta q^{11} \) \( + 574 q^{17} \) \( -51 \beta q^{19} \) \( + 625 q^{25} \) \( -34 \beta q^{27} \) \( + 2688 q^{33} \) \( -1246 q^{41} \) \( + 105 \beta q^{43} \) \( + 2401 q^{49} \) \( + 574 \beta q^{51} \) \( -6528 q^{57} \) \( -615 \beta q^{59} \) \( -651 \beta q^{67} \) \( -9506 q^{73} \) \( + 625 \beta q^{75} \) \( -8159 q^{81} \) \( -711 \beta q^{83} \) \( -5474 q^{89} \) \( + 9982 q^{97} \) \( + 987 \beta q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 94q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 94q^{9} \) \(\mathstrut +\mathstrut 1148q^{17} \) \(\mathstrut +\mathstrut 1250q^{25} \) \(\mathstrut +\mathstrut 5376q^{33} \) \(\mathstrut -\mathstrut 2492q^{41} \) \(\mathstrut +\mathstrut 4802q^{49} \) \(\mathstrut -\mathstrut 13056q^{57} \) \(\mathstrut -\mathstrut 19012q^{73} \) \(\mathstrut -\mathstrut 16318q^{81} \) \(\mathstrut -\mathstrut 10948q^{89} \) \(\mathstrut +\mathstrut 19964q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−1.41421
1.41421
0 −11.3137 0 0 0 0 0 47.0000 0
63.2 0 11.3137 0 0 0 0 0 47.0000 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
8.d Odd 1 CM by \(\Q(\sqrt{-2}) \) yes
4.b Odd 1 yes
8.b Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{2} \) \(\mathstrut -\mathstrut 128 \) acting on \(S_{5}^{\mathrm{new}}(128, [\chi])\).