Properties

Label 128.4.b.b
Level 128
Weight 4
Character orbit 128.b
Analytic conductor 7.552
Analytic rank 0
Dimension 2
CM disc. -8
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 128 = 2^{7} \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 128.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(7.55224448073\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + \beta q^{3} \) \( + 19 q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta q^{3} \) \( + 19 q^{9} \) \( + 25 \beta q^{11} \) \( -90 q^{17} \) \( + 45 \beta q^{19} \) \( + 125 q^{25} \) \( + 46 \beta q^{27} \) \( -200 q^{33} \) \( + 522 q^{41} \) \( -171 \beta q^{43} \) \( -343 q^{49} \) \( -90 \beta q^{51} \) \( -360 q^{57} \) \( -115 \beta q^{59} \) \( -387 \beta q^{67} \) \( + 430 q^{73} \) \( + 125 \beta q^{75} \) \( + 145 q^{81} \) \( + 241 \beta q^{83} \) \( -1026 q^{89} \) \( + 1910 q^{97} \) \( + 475 \beta q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 38q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 38q^{9} \) \(\mathstrut -\mathstrut 180q^{17} \) \(\mathstrut +\mathstrut 250q^{25} \) \(\mathstrut -\mathstrut 400q^{33} \) \(\mathstrut +\mathstrut 1044q^{41} \) \(\mathstrut -\mathstrut 686q^{49} \) \(\mathstrut -\mathstrut 720q^{57} \) \(\mathstrut +\mathstrut 860q^{73} \) \(\mathstrut +\mathstrut 290q^{81} \) \(\mathstrut -\mathstrut 2052q^{89} \) \(\mathstrut +\mathstrut 3820q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
1.41421i
1.41421i
0 2.82843i 0 0 0 0 0 19.0000 0
65.2 0 2.82843i 0 0 0 0 0 19.0000 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 intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(128, [\chi])\):

\(T_{3}^{2} \) \(\mathstrut +\mathstrut 8 \)
\(T_{7} \)