Properties

Label 128.3.c.b
Level 128
Weight 3
Character orbit 128.c
Analytic conductor 3.488
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 128 = 2^{7} \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 128.c (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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{3} \) \( + ( 2 + \beta_{2} ) q^{5} \) \( + ( \beta_{1} + \beta_{3} ) q^{7} \) \( + ( -3 - 2 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \( + ( 2 + \beta_{2} ) q^{5} \) \( + ( \beta_{1} + \beta_{3} ) q^{7} \) \( + ( -3 - 2 \beta_{2} ) q^{9} \) \( + ( \beta_{1} + 2 \beta_{3} ) q^{11} \) \( + ( -6 + \beta_{2} ) q^{13} \) \( + ( 7 \beta_{1} - \beta_{3} ) q^{15} \) \( + ( -2 - 2 \beta_{2} ) q^{17} \) \( + ( 3 \beta_{1} - 2 \beta_{3} ) q^{19} \) \( -8 q^{21} \) \( + ( -5 \beta_{1} + 3 \beta_{3} ) q^{23} \) \( + ( 11 + 4 \beta_{2} ) q^{25} \) \( + ( -4 \beta_{1} + 2 \beta_{3} ) q^{27} \) \( + ( 34 + \beta_{2} ) q^{29} \) \( + ( -12 \beta_{1} - 4 \beta_{3} ) q^{31} \) \( + ( -4 + 2 \beta_{2} ) q^{33} \) \( -4 \beta_{3} q^{35} \) \( + ( 10 - 7 \beta_{2} ) q^{37} \) \( + ( -\beta_{1} - \beta_{3} ) q^{39} \) \( + ( 2 + 4 \beta_{2} ) q^{41} \) \( + ( -13 \beta_{1} - 4 \beta_{3} ) q^{43} \) \( + ( -70 - 7 \beta_{2} ) q^{45} \) \( + ( 6 \beta_{1} - 2 \beta_{3} ) q^{47} \) \( + ( 1 + 8 \beta_{2} ) q^{49} \) \( + ( -12 \beta_{1} + 2 \beta_{3} ) q^{51} \) \( + ( -22 + \beta_{2} ) q^{53} \) \( + ( -7 \beta_{1} - 7 \beta_{3} ) q^{55} \) \( + ( -44 - 10 \beta_{2} ) q^{57} \) \( + 11 \beta_{1} q^{59} \) \( + ( 74 - 7 \beta_{2} ) q^{61} \) \( + ( \beta_{1} + 9 \beta_{3} ) q^{63} \) \( + ( 20 - 4 \beta_{2} ) q^{65} \) \( + ( -9 \beta_{1} + 6 \beta_{3} ) q^{67} \) \( + ( 72 + 16 \beta_{2} ) q^{69} \) \( + ( 9 \beta_{1} + \beta_{3} ) q^{71} \) \( + ( 22 + 6 \beta_{2} ) q^{73} \) \( + ( 31 \beta_{1} - 4 \beta_{3} ) q^{75} \) \( + ( -88 + 16 \beta_{2} ) q^{77} \) \( + ( -22 \beta_{1} + 10 \beta_{3} ) q^{79} \) \( + ( 29 - 6 \beta_{2} ) q^{81} \) \( + ( 29 \beta_{1} + 8 \beta_{3} ) q^{83} \) \( + ( -68 - 6 \beta_{2} ) q^{85} \) \( + ( 39 \beta_{1} - \beta_{3} ) q^{87} \) \( + ( 54 - 10 \beta_{2} ) q^{89} \) \( + ( -8 \beta_{1} - 12 \beta_{3} ) q^{91} \) \( + ( 128 + 16 \beta_{2} ) q^{93} \) \( + ( 35 \beta_{1} + 3 \beta_{3} ) q^{95} \) \( + ( -82 - 10 \beta_{2} ) q^{97} \) \( + ( 15 \beta_{1} + 16 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 8q^{5} \) \(\mathstrut -\mathstrut 12q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 8q^{5} \) \(\mathstrut -\mathstrut 12q^{9} \) \(\mathstrut -\mathstrut 24q^{13} \) \(\mathstrut -\mathstrut 8q^{17} \) \(\mathstrut -\mathstrut 32q^{21} \) \(\mathstrut +\mathstrut 44q^{25} \) \(\mathstrut +\mathstrut 136q^{29} \) \(\mathstrut -\mathstrut 16q^{33} \) \(\mathstrut +\mathstrut 40q^{37} \) \(\mathstrut +\mathstrut 8q^{41} \) \(\mathstrut -\mathstrut 280q^{45} \) \(\mathstrut +\mathstrut 4q^{49} \) \(\mathstrut -\mathstrut 88q^{53} \) \(\mathstrut -\mathstrut 176q^{57} \) \(\mathstrut +\mathstrut 296q^{61} \) \(\mathstrut +\mathstrut 80q^{65} \) \(\mathstrut +\mathstrut 288q^{69} \) \(\mathstrut +\mathstrut 88q^{73} \) \(\mathstrut -\mathstrut 352q^{77} \) \(\mathstrut +\mathstrut 116q^{81} \) \(\mathstrut -\mathstrut 272q^{85} \) \(\mathstrut +\mathstrut 216q^{89} \) \(\mathstrut +\mathstrut 512q^{93} \) \(\mathstrut -\mathstrut 328q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring:

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 2 \zeta_{8} \)
\(\beta_{2}\)\(=\)\( -4 \zeta_{8}^{3} + 4 \zeta_{8} \)
\(\beta_{3}\)\(=\)\( 2 \zeta_{8}^{3} - 6 \zeta_{8}^{2} + 2 \zeta_{8} \)
\(1\)\(=\)\(\beta_0\)
\(\zeta_{8}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\)\()/16\)
\(\zeta_{8}^{2}\)\(=\)\((\)\(-\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\)\()/8\)
\(\zeta_{8}^{3}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\)\()/16\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0 4.82843i 0 7.65685 0 1.65685i 0 −14.3137 0
127.2 0 0.828427i 0 −3.65685 0 9.65685i 0 8.31371 0
127.3 0 0.828427i 0 −3.65685 0 9.65685i 0 8.31371 0
127.4 0 4.82843i 0 7.65685 0 1.65685i 0 −14.3137 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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{5}^{2} \) \(\mathstrut -\mathstrut 4 T_{5} \) \(\mathstrut -\mathstrut 28 \) acting on \(S_{3}^{\mathrm{new}}(128, [\chi])\).