Properties

Label 128.4.b.e
Level 128
Weight 4
Character orbit 128.b
Analytic conductor 7.552
Analytic rank 0
Dimension 4
CM No
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: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-5})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{3} \) \( + \beta_{3} q^{5} \) \( + \beta_{2} q^{7} \) \( -13 q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{3} \) \( + \beta_{3} q^{5} \) \( + \beta_{2} q^{7} \) \( -13 q^{9} \) \( + 7 \beta_{1} q^{11} \) \( -\beta_{3} q^{13} \) \( + 5 \beta_{2} q^{15} \) \( + 70 q^{17} \) \( -13 \beta_{1} q^{19} \) \( -8 \beta_{3} q^{21} \) \( + 7 \beta_{2} q^{23} \) \( -195 q^{25} \) \( -14 \beta_{1} q^{27} \) \( + 7 \beta_{3} q^{29} \) \( + 280 q^{33} \) \( + 64 \beta_{1} q^{35} \) \( + 21 \beta_{3} q^{37} \) \( -5 \beta_{2} q^{39} \) \( -182 q^{41} \) \( -21 \beta_{1} q^{43} \) \( -13 \beta_{3} q^{45} \) \( -14 \beta_{2} q^{47} \) \( + 169 q^{49} \) \( -70 \beta_{1} q^{51} \) \( -7 \beta_{3} q^{53} \) \( -35 \beta_{2} q^{55} \) \( -520 q^{57} \) \( -13 \beta_{1} q^{59} \) \( -13 \beta_{3} q^{61} \) \( -13 \beta_{2} q^{63} \) \( + 320 q^{65} \) \( + 35 \beta_{1} q^{67} \) \( -56 \beta_{3} q^{69} \) \( + 5 \beta_{2} q^{71} \) \( + 910 q^{73} \) \( + 195 \beta_{1} q^{75} \) \( + 56 \beta_{3} q^{77} \) \( -30 \beta_{2} q^{79} \) \( -911 q^{81} \) \( -113 \beta_{1} q^{83} \) \( + 70 \beta_{3} q^{85} \) \( + 35 \beta_{2} q^{87} \) \( -546 q^{89} \) \( -64 \beta_{1} q^{91} \) \( + 65 \beta_{2} q^{95} \) \( -490 q^{97} \) \( -91 \beta_{1} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 52q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 52q^{9} \) \(\mathstrut +\mathstrut 280q^{17} \) \(\mathstrut -\mathstrut 780q^{25} \) \(\mathstrut +\mathstrut 1120q^{33} \) \(\mathstrut -\mathstrut 728q^{41} \) \(\mathstrut +\mathstrut 676q^{49} \) \(\mathstrut -\mathstrut 2080q^{57} \) \(\mathstrut +\mathstrut 1280q^{65} \) \(\mathstrut +\mathstrut 3640q^{73} \) \(\mathstrut -\mathstrut 3644q^{81} \) \(\mathstrut -\mathstrut 2184q^{89} \) \(\mathstrut -\mathstrut 1960q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(4\) \(x^{2}\mathstrut +\mathstrut \) \(9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{3} + 14 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( -16 \nu^{3} - 16 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( 8 \nu^{2} + 16 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\)\()/32\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(16\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(7\) \(\beta_{2}\mathstrut -\mathstrut \) \(8\) \(\beta_{1}\)\()/32\)

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
−0.707107 + 1.58114i
0.707107 + 1.58114i
0.707107 1.58114i
−0.707107 1.58114i
0 6.32456i 0 17.8885i 0 −22.6274 0 −13.0000 0
65.2 0 6.32456i 0 17.8885i 0 22.6274 0 −13.0000 0
65.3 0 6.32456i 0 17.8885i 0 22.6274 0 −13.0000 0
65.4 0 6.32456i 0 17.8885i 0 −22.6274 0 −13.0000 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 yes
8.b Even 1 yes
8.d Odd 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(128, [\chi])\):

\(T_{3}^{2} \) \(\mathstrut +\mathstrut 40 \)
\(T_{7}^{2} \) \(\mathstrut -\mathstrut 512 \)