Properties

Label 128.5.d.c
Level 128
Weight 5
Character orbit 128.d
Analytic conductor 13.231
Analytic rank 0
Dimension 4
CM No
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: No
Analytic conductor: \(13.2313552747\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{11} \)
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.

\(f(q)\) \(=\) \( q\) \( -\beta_{2} q^{3} \) \( -\beta_{1} q^{5} \) \( + \beta_{3} q^{7} \) \( + 15 q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{2} q^{3} \) \( -\beta_{1} q^{5} \) \( + \beta_{3} q^{7} \) \( + 15 q^{9} \) \( + 11 \beta_{2} q^{11} \) \( -27 \beta_{1} q^{13} \) \( + \beta_{3} q^{15} \) \( -162 q^{17} \) \( -45 \beta_{2} q^{19} \) \( -96 \beta_{1} q^{21} \) \( -9 \beta_{3} q^{23} \) \( + 561 q^{25} \) \( + 66 \beta_{2} q^{27} \) \( -163 \beta_{1} q^{29} \) \( -8 \beta_{3} q^{31} \) \( -1056 q^{33} \) \( + 64 \beta_{2} q^{35} \) \( -189 \beta_{1} q^{37} \) \( + 27 \beta_{3} q^{39} \) \( + 1890 q^{41} \) \( -297 \beta_{2} q^{43} \) \( -15 \beta_{1} q^{45} \) \( + 18 \beta_{3} q^{47} \) \( -3743 q^{49} \) \( + 162 \beta_{2} q^{51} \) \( + 247 \beta_{1} q^{53} \) \( -11 \beta_{3} q^{55} \) \( + 4320 q^{57} \) \( + 231 \beta_{2} q^{59} \) \( + 297 \beta_{1} q^{61} \) \( + 15 \beta_{3} q^{63} \) \( -1728 q^{65} \) \( + 171 \beta_{2} q^{67} \) \( + 864 \beta_{1} q^{69} \) \( -99 \beta_{3} q^{71} \) \( + 2750 q^{73} \) \( -561 \beta_{2} q^{75} \) \( + 1056 \beta_{1} q^{77} \) \( -102 \beta_{3} q^{79} \) \( -7551 q^{81} \) \( -953 \beta_{2} q^{83} \) \( + 162 \beta_{1} q^{85} \) \( + 163 \beta_{3} q^{87} \) \( + 2430 q^{89} \) \( + 1728 \beta_{2} q^{91} \) \( + 768 \beta_{1} q^{93} \) \( + 45 \beta_{3} q^{95} \) \( + 7454 q^{97} \) \( + 165 \beta_{2} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 60q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 60q^{9} \) \(\mathstrut -\mathstrut 648q^{17} \) \(\mathstrut +\mathstrut 2244q^{25} \) \(\mathstrut -\mathstrut 4224q^{33} \) \(\mathstrut +\mathstrut 7560q^{41} \) \(\mathstrut -\mathstrut 14972q^{49} \) \(\mathstrut +\mathstrut 17280q^{57} \) \(\mathstrut -\mathstrut 6912q^{65} \) \(\mathstrut +\mathstrut 11000q^{73} \) \(\mathstrut -\mathstrut 30204q^{81} \) \(\mathstrut +\mathstrut 9720q^{89} \) \(\mathstrut +\mathstrut 29816q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 8 \nu^{2} \)\(/3\)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{3} + 12 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 32 \nu^{3} + 96 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\)\()/64\)
\(\nu^{2}\)\(=\)\(3\) \(\beta_{1}\)\(/8\)
\(\nu^{3}\)\(=\)\((\)\(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(24\) \(\beta_{2}\)\()/64\)

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
1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 1.22474i
−1.22474 + 1.22474i
0 −9.79796 0 8.00000i 0 78.3837i 0 15.0000 0
63.2 0 −9.79796 0 8.00000i 0 78.3837i 0 15.0000 0
63.3 0 9.79796 0 8.00000i 0 78.3837i 0 15.0000 0
63.4 0 9.79796 0 8.00000i 0 78.3837i 0 15.0000 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 kernel of the linear operator \(T_{3}^{2} \) \(\mathstrut -\mathstrut 96 \) acting on \(S_{5}^{\mathrm{new}}(128, [\chi])\).