Properties

Label 128.3.d.c
Level 128
Weight 3
Character orbit 128.d
Analytic conductor 3.488
Analytic rank 0
Dimension 4
CM No
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: No
Analytic conductor: \(3.48774738381\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7} \)
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{2} q^{3} \) \( + \beta_{1} q^{5} \) \( + \beta_{3} q^{7} \) \(- q^{9}\) \(+O(q^{10})\) \( q\) \( + \beta_{2} q^{3} \) \( + \beta_{1} q^{5} \) \( + \beta_{3} q^{7} \) \(- q^{9}\) \( + 5 \beta_{2} q^{11} \) \( -5 \beta_{1} q^{13} \) \( + \beta_{3} q^{15} \) \( -10 q^{17} \) \( + 5 \beta_{2} q^{19} \) \( + 8 \beta_{1} q^{21} \) \( -\beta_{3} q^{23} \) \( + 9 q^{25} \) \( -10 \beta_{2} q^{27} \) \( -5 \beta_{1} q^{29} \) \( + 40 q^{33} \) \( -16 \beta_{2} q^{35} \) \( + 5 \beta_{1} q^{37} \) \( -5 \beta_{3} q^{39} \) \( -30 q^{41} \) \( + \beta_{2} q^{43} \) \( -\beta_{1} q^{45} \) \( -6 \beta_{3} q^{47} \) \( -79 q^{49} \) \( -10 \beta_{2} q^{51} \) \( -15 \beta_{1} q^{53} \) \( + 5 \beta_{3} q^{55} \) \( + 40 q^{57} \) \( -15 \beta_{2} q^{59} \) \( + 7 \beta_{1} q^{61} \) \( -\beta_{3} q^{63} \) \( + 80 q^{65} \) \( + 29 \beta_{2} q^{67} \) \( -8 \beta_{1} q^{69} \) \( + 5 \beta_{3} q^{71} \) \( -10 q^{73} \) \( + 9 \beta_{2} q^{75} \) \( + 40 \beta_{1} q^{77} \) \( + 10 \beta_{3} q^{79} \) \( -71 q^{81} \) \( + 9 \beta_{2} q^{83} \) \( -10 \beta_{1} q^{85} \) \( -5 \beta_{3} q^{87} \) \( + 22 q^{89} \) \( + 80 \beta_{2} q^{91} \) \( + 5 \beta_{3} q^{95} \) \( + 150 q^{97} \) \( -5 \beta_{2} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut -\mathstrut 40q^{17} \) \(\mathstrut +\mathstrut 36q^{25} \) \(\mathstrut +\mathstrut 160q^{33} \) \(\mathstrut -\mathstrut 120q^{41} \) \(\mathstrut -\mathstrut 316q^{49} \) \(\mathstrut +\mathstrut 160q^{57} \) \(\mathstrut +\mathstrut 320q^{65} \) \(\mathstrut -\mathstrut 40q^{73} \) \(\mathstrut -\mathstrut 284q^{81} \) \(\mathstrut +\mathstrut 88q^{89} \) \(\mathstrut +\mathstrut 600q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring:

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 4 \zeta_{8}^{2} \)
\(\beta_{2}\)\(=\)\( -2 \zeta_{8}^{3} + 2 \zeta_{8} \)
\(\beta_{3}\)\(=\)\( 8 \zeta_{8}^{3} + 8 \zeta_{8} \)
\(1\)\(=\)\(\beta_0\)
\(\zeta_{8}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{2}\)\()/16\)
\(\zeta_{8}^{2}\)\(=\)\(\beta_{1}\)\(/4\)
\(\zeta_{8}^{3}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{2}\)\()/16\)

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
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
0 −2.82843 0 4.00000i 0 11.3137i 0 −1.00000 0
63.2 0 −2.82843 0 4.00000i 0 11.3137i 0 −1.00000 0
63.3 0 2.82843 0 4.00000i 0 11.3137i 0 −1.00000 0
63.4 0 2.82843 0 4.00000i 0 11.3137i 0 −1.00000 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 kernel of the linear operator \(T_{3}^{2} \) \(\mathstrut -\mathstrut 8 \) acting on \(S_{3}^{\mathrm{new}}(128, [\chi])\).