Properties

Label 128.4.b.a
Level 128
Weight 4
Character orbit 128.b
Analytic conductor 7.552
Analytic rank 1
Dimension 2
CM No
Inner twists 2

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + 2 \beta q^{3} \) \( -3 \beta q^{5} \) \( -32 q^{7} \) \( -37 q^{9} \) \(+O(q^{10})\) \( q\) \( + 2 \beta q^{3} \) \( -3 \beta q^{5} \) \( -32 q^{7} \) \( -37 q^{9} \) \( + 2 \beta q^{11} \) \( -5 \beta q^{13} \) \( + 96 q^{15} \) \( -98 q^{17} \) \( -22 \beta q^{19} \) \( -64 \beta q^{21} \) \( + 32 q^{23} \) \( -19 q^{25} \) \( -20 \beta q^{27} \) \( + 43 \beta q^{29} \) \( -256 q^{31} \) \( -64 q^{33} \) \( + 96 \beta q^{35} \) \( -23 \beta q^{37} \) \( + 160 q^{39} \) \( -102 q^{41} \) \( + 74 \beta q^{43} \) \( + 111 \beta q^{45} \) \( -320 q^{47} \) \( + 681 q^{49} \) \( -196 \beta q^{51} \) \( -19 \beta q^{53} \) \( + 96 q^{55} \) \( + 704 q^{57} \) \( -102 \beta q^{59} \) \( + 159 \beta q^{61} \) \( + 1184 q^{63} \) \( -240 q^{65} \) \( + 138 \beta q^{67} \) \( + 64 \beta q^{69} \) \( -416 q^{71} \) \( -138 q^{73} \) \( -38 \beta q^{75} \) \( -64 \beta q^{77} \) \( -64 q^{79} \) \( -359 q^{81} \) \( + 98 \beta q^{83} \) \( + 294 \beta q^{85} \) \( -1376 q^{87} \) \( + 582 q^{89} \) \( + 160 \beta q^{91} \) \( -512 \beta q^{93} \) \( -1056 q^{95} \) \( + 238 q^{97} \) \( -74 \beta q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 64q^{7} \) \(\mathstrut -\mathstrut 74q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 64q^{7} \) \(\mathstrut -\mathstrut 74q^{9} \) \(\mathstrut +\mathstrut 192q^{15} \) \(\mathstrut -\mathstrut 196q^{17} \) \(\mathstrut +\mathstrut 64q^{23} \) \(\mathstrut -\mathstrut 38q^{25} \) \(\mathstrut -\mathstrut 512q^{31} \) \(\mathstrut -\mathstrut 128q^{33} \) \(\mathstrut +\mathstrut 320q^{39} \) \(\mathstrut -\mathstrut 204q^{41} \) \(\mathstrut -\mathstrut 640q^{47} \) \(\mathstrut +\mathstrut 1362q^{49} \) \(\mathstrut +\mathstrut 192q^{55} \) \(\mathstrut +\mathstrut 1408q^{57} \) \(\mathstrut +\mathstrut 2368q^{63} \) \(\mathstrut -\mathstrut 480q^{65} \) \(\mathstrut -\mathstrut 832q^{71} \) \(\mathstrut -\mathstrut 276q^{73} \) \(\mathstrut -\mathstrut 128q^{79} \) \(\mathstrut -\mathstrut 718q^{81} \) \(\mathstrut -\mathstrut 2752q^{87} \) \(\mathstrut +\mathstrut 1164q^{89} \) \(\mathstrut -\mathstrut 2112q^{95} \) \(\mathstrut +\mathstrut 476q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
1.00000i
1.00000i
0 8.00000i 0 12.0000i 0 −32.0000 0 −37.0000 0
65.2 0 8.00000i 0 12.0000i 0 −32.0000 0 −37.0000 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
8.b Even 1 yes

Hecke kernels

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

\(T_{3}^{2} \) \(\mathstrut +\mathstrut 64 \)
\(T_{7} \) \(\mathstrut +\mathstrut 32 \)