Properties

Label 128.3.d.b
Level 128
Weight 3
Character orbit 128.d
Self dual Yes
Analytic conductor 3.488
Analytic rank 0
Dimension 2
CM disc. -8
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 128 = 2^{7} \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 128.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(3.48774738381\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{3} \) \( + 23 q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta q^{3} \) \( + 23 q^{9} \) \( -3 \beta q^{11} \) \( -2 q^{17} \) \( -3 \beta q^{19} \) \( + 25 q^{25} \) \( + 14 \beta q^{27} \) \( -96 q^{33} \) \( -46 q^{41} \) \( -15 \beta q^{43} \) \( + 49 q^{49} \) \( -2 \beta q^{51} \) \( -96 q^{57} \) \( -15 \beta q^{59} \) \( + 21 \beta q^{67} \) \( + 142 q^{73} \) \( + 25 \beta q^{75} \) \( + 241 q^{81} \) \( + 9 \beta q^{83} \) \( -146 q^{89} \) \( + 94 q^{97} \) \( -69 \beta q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 46q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 46q^{9} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 50q^{25} \) \(\mathstrut -\mathstrut 192q^{33} \) \(\mathstrut -\mathstrut 92q^{41} \) \(\mathstrut +\mathstrut 98q^{49} \) \(\mathstrut -\mathstrut 192q^{57} \) \(\mathstrut +\mathstrut 284q^{73} \) \(\mathstrut +\mathstrut 482q^{81} \) \(\mathstrut -\mathstrut 292q^{89} \) \(\mathstrut +\mathstrut 188q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −5.65685 0 0 0 0 0 23.0000 0
63.2 0 5.65685 0 0 0 0 0 23.0000 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 \(T_{3}^{2} \) \(\mathstrut -\mathstrut 32 \) acting on \(S_{3}^{\mathrm{new}}(128, [\chi])\).