Properties

Label 128.4.a.a
Level $128$
Weight $4$
Character orbit 128.a
Self dual yes
Analytic conductor $7.552$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 128.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.55224448073\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{3} - 6q^{5} + 20q^{7} - 23q^{9} + O(q^{10}) \) \( q - 2q^{3} - 6q^{5} + 20q^{7} - 23q^{9} - 14q^{11} - 54q^{13} + 12q^{15} - 66q^{17} - 162q^{19} - 40q^{21} + 172q^{23} - 89q^{25} + 100q^{27} + 2q^{29} - 128q^{31} + 28q^{33} - 120q^{35} - 158q^{37} + 108q^{39} + 202q^{41} + 298q^{43} + 138q^{45} - 408q^{47} + 57q^{49} + 132q^{51} + 690q^{53} + 84q^{55} + 324q^{57} + 322q^{59} + 298q^{61} - 460q^{63} + 324q^{65} - 202q^{67} - 344q^{69} - 700q^{71} - 418q^{73} + 178q^{75} - 280q^{77} + 744q^{79} + 421q^{81} + 678q^{83} + 396q^{85} - 4q^{87} - 82q^{89} - 1080q^{91} + 256q^{93} + 972q^{95} - 1122q^{97} + 322q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 −2.00000 0 −6.00000 0 20.0000 0 −23.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.4.a.a 1
3.b odd 2 1 1152.4.a.j 1
4.b odd 2 1 128.4.a.c yes 1
8.b even 2 1 128.4.a.d yes 1
8.d odd 2 1 128.4.a.b yes 1
12.b even 2 1 1152.4.a.i 1
16.e even 4 2 256.4.b.b 2
16.f odd 4 2 256.4.b.f 2
24.f even 2 1 1152.4.a.c 1
24.h odd 2 1 1152.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.4.a.a 1 1.a even 1 1 trivial
128.4.a.b yes 1 8.d odd 2 1
128.4.a.c yes 1 4.b odd 2 1
128.4.a.d yes 1 8.b even 2 1
256.4.b.b 2 16.e even 4 2
256.4.b.f 2 16.f odd 4 2
1152.4.a.c 1 24.f even 2 1
1152.4.a.d 1 24.h odd 2 1
1152.4.a.i 1 12.b even 2 1
1152.4.a.j 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(128))\):

\( T_{3} + 2 \)
\( T_{5} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 2 + T \)
$5$ \( 6 + T \)
$7$ \( -20 + T \)
$11$ \( 14 + T \)
$13$ \( 54 + T \)
$17$ \( 66 + T \)
$19$ \( 162 + T \)
$23$ \( -172 + T \)
$29$ \( -2 + T \)
$31$ \( 128 + T \)
$37$ \( 158 + T \)
$41$ \( -202 + T \)
$43$ \( -298 + T \)
$47$ \( 408 + T \)
$53$ \( -690 + T \)
$59$ \( -322 + T \)
$61$ \( -298 + T \)
$67$ \( 202 + T \)
$71$ \( 700 + T \)
$73$ \( 418 + T \)
$79$ \( -744 + T \)
$83$ \( -678 + T \)
$89$ \( 82 + T \)
$97$ \( 1122 + T \)
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