Properties

Label 128.2.g.b
Level 128
Weight 2
Character orbit 128.g
Analytic conductor 1.022
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 128 = 2^{7} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 128.g (of order \(8\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(1.02208514587\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: 8.0.18939904.2
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( \beta_{2} + \beta_{6} - \beta_{7} ) q^{3} \) \( + ( -\beta_{6} - \beta_{7} ) q^{5} \) \( + ( 1 - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{7} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( \beta_{2} + \beta_{6} - \beta_{7} ) q^{3} \) \( + ( -\beta_{6} - \beta_{7} ) q^{5} \) \( + ( 1 - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{7} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{9} \) \( + ( -1 + \beta_{1} - \beta_{6} + \beta_{7} ) q^{11} \) \( + ( -2 - 2 \beta_{3} + \beta_{4} - \beta_{7} ) q^{13} \) \( + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{15} \) \( + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{17} \) \( + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{19} \) \( + ( 1 - 2 \beta_{1} - \beta_{4} - 3 \beta_{6} + 3 \beta_{7} ) q^{21} \) \( + ( 1 - \beta_{1} - \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{23} \) \( + ( -1 + 3 \beta_{4} + \beta_{7} ) q^{25} \) \( + ( -\beta_{2} + \beta_{3} + 2 \beta_{6} + 3 \beta_{7} ) q^{27} \) \( + ( 2 \beta_{1} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{29} \) \( + ( -4 + 2 \beta_{4} + 2 \beta_{6} ) q^{31} \) \( + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{33} \) \( + ( -2 - \beta_{1} - \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{35} \) \( + ( -2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{37} \) \( + ( -3 + \beta_{1} - \beta_{3} - 4 \beta_{4} + \beta_{7} ) q^{39} \) \( + ( 1 - 2 \beta_{1} - 2 \beta_{3} - 4 \beta_{6} + \beta_{7} ) q^{41} \) \( + ( 1 + \beta_{1} + 2 \beta_{4} + 3 \beta_{6} - 3 \beta_{7} ) q^{43} \) \( + ( 1 + 2 \beta_{3} + \beta_{6} ) q^{45} \) \( + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{47} \) \( + ( -2 - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{49} \) \( + ( 2 + 4 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} ) q^{51} \) \( + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 5 \beta_{6} - 3 \beta_{7} ) q^{53} \) \( + ( 1 + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{55} \) \( + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{57} \) \( + ( 3 - \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{59} \) \( + ( 2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{61} \) \( + ( 5 - \beta_{1} + \beta_{2} - 4 \beta_{4} - 4 \beta_{6} ) q^{63} \) \( + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{6} ) q^{65} \) \( + ( 6 - 3 \beta_{2} + 3 \beta_{6} + 3 \beta_{7} ) q^{67} \) \( + ( 3 + 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{69} \) \( + ( 3 + 3 \beta_{2} + 5 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{71} \) \( + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{73} \) \( + ( 3 - 3 \beta_{1} - \beta_{2} - \beta_{3} - 5 \beta_{6} + 4 \beta_{7} ) q^{75} \) \( + ( 1 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{77} \) \( + ( -4 + \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 6 \beta_{7} ) q^{79} \) \( + ( \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{81} \) \( + ( -4 + 3 \beta_{1} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} ) q^{83} \) \( + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} ) q^{85} \) \( + ( -7 + 3 \beta_{1} + 3 \beta_{3} + 2 \beta_{6} - 7 \beta_{7} ) q^{87} \) \( + ( -1 - \beta_{1} - 3 \beta_{2} + \beta_{3} - 5 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{89} \) \( + ( -8 + 3 \beta_{2} - 3 \beta_{3} + 8 \beta_{4} - 3 \beta_{7} ) q^{91} \) \( + ( -2 \beta_{1} - 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 6 \beta_{6} + 6 \beta_{7} ) q^{93} \) \( + ( 1 - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{95} \) \( + ( 4 + \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{97} \) \( + ( -4 + 3 \beta_{1} + \beta_{2} + \beta_{4} + 3 \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 8q^{7} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut -\mathstrut 8q^{13} \) \(\mathstrut -\mathstrut 4q^{19} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut -\mathstrut 8q^{25} \) \(\mathstrut -\mathstrut 8q^{27} \) \(\mathstrut -\mathstrut 32q^{31} \) \(\mathstrut -\mathstrut 16q^{33} \) \(\mathstrut -\mathstrut 16q^{35} \) \(\mathstrut -\mathstrut 8q^{37} \) \(\mathstrut -\mathstrut 16q^{39} \) \(\mathstrut +\mathstrut 8q^{41} \) \(\mathstrut +\mathstrut 12q^{43} \) \(\mathstrut +\mathstrut 16q^{51} \) \(\mathstrut +\mathstrut 8q^{53} \) \(\mathstrut +\mathstrut 16q^{55} \) \(\mathstrut +\mathstrut 16q^{57} \) \(\mathstrut +\mathstrut 20q^{59} \) \(\mathstrut +\mathstrut 24q^{61} \) \(\mathstrut +\mathstrut 40q^{63} \) \(\mathstrut +\mathstrut 36q^{67} \) \(\mathstrut +\mathstrut 32q^{69} \) \(\mathstrut +\mathstrut 24q^{71} \) \(\mathstrut -\mathstrut 32q^{73} \) \(\mathstrut +\mathstrut 12q^{75} \) \(\mathstrut +\mathstrut 16q^{77} \) \(\mathstrut -\mathstrut 20q^{83} \) \(\mathstrut +\mathstrut 8q^{85} \) \(\mathstrut -\mathstrut 56q^{87} \) \(\mathstrut -\mathstrut 16q^{89} \) \(\mathstrut -\mathstrut 40q^{91} \) \(\mathstrut -\mathstrut 16q^{93} \) \(\mathstrut +\mathstrut 8q^{95} \) \(\mathstrut +\mathstrut 32q^{97} \) \(\mathstrut -\mathstrut 28q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(4\) \(x^{7}\mathstrut +\mathstrut \) \(14\) \(x^{6}\mathstrut -\mathstrut \) \(28\) \(x^{5}\mathstrut +\mathstrut \) \(43\) \(x^{4}\mathstrut -\mathstrut \) \(44\) \(x^{3}\mathstrut +\mathstrut \) \(30\) \(x^{2}\mathstrut -\mathstrut \) \(12\) \(x\mathstrut +\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{7} - 4 \nu^{6} + 14 \nu^{5} - 27 \nu^{4} + 41 \nu^{3} - 37 \nu^{2} + 24 \nu - 5 \)
\(\beta_{2}\)\(=\)\( \nu^{7} - 4 \nu^{6} + 14 \nu^{5} - 28 \nu^{4} + 43 \nu^{3} - 44 \nu^{2} + 30 \nu - 10 \)
\(\beta_{3}\)\(=\)\( -5 \nu^{7} + 17 \nu^{6} - 59 \nu^{5} + 102 \nu^{4} - 146 \nu^{3} + 121 \nu^{2} - 66 \nu + 15 \)
\(\beta_{4}\)\(=\)\( 5 \nu^{7} - 17 \nu^{6} + 60 \nu^{5} - 105 \nu^{4} + 155 \nu^{3} - 133 \nu^{2} + 77 \nu - 19 \)
\(\beta_{5}\)\(=\)\( -5 \nu^{7} + 18 \nu^{6} - 62 \nu^{5} + 113 \nu^{4} - 163 \nu^{3} + 145 \nu^{2} - 82 \nu + 20 \)
\(\beta_{6}\)\(=\)\( -5 \nu^{7} + 18 \nu^{6} - 63 \nu^{5} + 115 \nu^{4} - 170 \nu^{3} + 152 \nu^{2} - 89 \nu + 23 \)
\(\beta_{7}\)\(=\)\( -8 \nu^{7} + 28 \nu^{6} - 98 \nu^{5} + 175 \nu^{4} - 256 \nu^{3} + 223 \nu^{2} - 126 \nu + 31 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut -\mathstrut \) \(4\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(5\) \(\beta_{7}\mathstrut -\mathstrut \) \(5\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(3\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(11\) \(\beta_{7}\mathstrut -\mathstrut \) \(19\) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(5\) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{2}\mathstrut -\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(12\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(13\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(15\) \(\beta_{5}\mathstrut -\mathstrut \) \(16\) \(\beta_{4}\mathstrut -\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(13\) \(\beta_{2}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(23\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(67\) \(\beta_{7}\mathstrut +\mathstrut \) \(90\) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{5}\mathstrut -\mathstrut \) \(10\) \(\beta_{4}\mathstrut +\mathstrut \) \(16\) \(\beta_{3}\mathstrut +\mathstrut \) \(31\) \(\beta_{2}\mathstrut +\mathstrut \) \(33\) \(\beta_{1}\mathstrut -\mathstrut \) \(28\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(7\) \(\beta_{7}\mathstrut +\mathstrut \) \(87\) \(\beta_{6}\mathstrut -\mathstrut \) \(68\) \(\beta_{5}\mathstrut +\mathstrut \) \(71\) \(\beta_{4}\mathstrut +\mathstrut \) \(65\) \(\beta_{3}\mathstrut -\mathstrut \) \(26\) \(\beta_{2}\mathstrut +\mathstrut \) \(37\) \(\beta_{1}\mathstrut -\mathstrut \) \(125\)\()/2\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(-\beta_{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.500000 + 0.691860i
0.500000 2.10607i
0.500000 + 0.0297061i
0.500000 1.44392i
0.500000 0.0297061i
0.500000 + 1.44392i
0.500000 0.691860i
0.500000 + 2.10607i
0 −0.0794708 + 0.191860i 0 0.707107 0.292893i 0 2.27133 2.27133i 0 2.09083 + 2.09083i 0
17.2 0 1.07947 2.60607i 0 0.707107 0.292893i 0 −1.68554 + 1.68554i 0 −3.50504 3.50504i 0
49.1 0 −1.27882 + 0.529706i 0 −0.707107 + 1.70711i 0 2.74912 + 2.74912i 0 −0.766519 + 0.766519i 0
49.2 0 2.27882 0.943920i 0 −0.707107 + 1.70711i 0 0.665096 + 0.665096i 0 2.18073 2.18073i 0
81.1 0 −1.27882 0.529706i 0 −0.707107 1.70711i 0 2.74912 2.74912i 0 −0.766519 0.766519i 0
81.2 0 2.27882 + 0.943920i 0 −0.707107 1.70711i 0 0.665096 0.665096i 0 2.18073 + 2.18073i 0
113.1 0 −0.0794708 0.191860i 0 0.707107 + 0.292893i 0 2.27133 + 2.27133i 0 2.09083 2.09083i 0
113.2 0 1.07947 + 2.60607i 0 0.707107 + 0.292893i 0 −1.68554 1.68554i 0 −3.50504 + 3.50504i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
32.g Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{8} \) \(\mathstrut -\mathstrut 4 T_{3}^{7} \) \(\mathstrut +\mathstrut 8 T_{3}^{6} \) \(\mathstrut -\mathstrut 32 T_{3}^{4} \) \(\mathstrut +\mathstrut 24 T_{3}^{3} \) \(\mathstrut +\mathstrut 96 T_{3}^{2} \) \(\mathstrut +\mathstrut 16 T_{3} \) \(\mathstrut +\mathstrut 4 \) acting on \(S_{2}^{\mathrm{new}}(128, [\chi])\).