Properties

Label 128.2.a.d
Level $128$
Weight $2$
Character orbit 128.a
Self dual yes
Analytic conductor $1.022$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(1.02208514587\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q + 2 q^{3} + 2 q^{5} - 4 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{3} + 2 q^{5} - 4 q^{7} + q^{9} - 2 q^{11} + 2 q^{13} + 4 q^{15} - 2 q^{17} + 2 q^{19} - 8 q^{21} + 4 q^{23} - q^{25} - 4 q^{27} - 6 q^{29} - 4 q^{33} - 8 q^{35} + 10 q^{37} + 4 q^{39} - 6 q^{41} + 6 q^{43} + 2 q^{45} - 8 q^{47} + 9 q^{49} - 4 q^{51} - 6 q^{53} - 4 q^{55} + 4 q^{57} + 14 q^{59} + 2 q^{61} - 4 q^{63} + 4 q^{65} + 10 q^{67} + 8 q^{69} + 12 q^{71} + 14 q^{73} - 2 q^{75} + 8 q^{77} - 8 q^{79} - 11 q^{81} - 6 q^{83} - 4 q^{85} - 12 q^{87} - 2 q^{89} - 8 q^{91} + 4 q^{95} - 2 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 2.00000 0 −4.00000 0 1.00000 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 128.2.a.d yes 1
3.b odd 2 1 1152.2.a.c 1
4.b odd 2 1 128.2.a.b yes 1
5.b even 2 1 3200.2.a.h 1
5.c odd 4 2 3200.2.c.e 2
7.b odd 2 1 6272.2.a.a 1
8.b even 2 1 128.2.a.a 1
8.d odd 2 1 128.2.a.c yes 1
12.b even 2 1 1152.2.a.h 1
16.e even 4 2 256.2.b.c 2
16.f odd 4 2 256.2.b.a 2
20.d odd 2 1 3200.2.a.u 1
20.e even 4 2 3200.2.c.k 2
24.f even 2 1 1152.2.a.r 1
24.h odd 2 1 1152.2.a.m 1
28.d even 2 1 6272.2.a.g 1
32.g even 8 4 1024.2.e.i 4
32.h odd 8 4 1024.2.e.m 4
40.e odd 2 1 3200.2.a.e 1
40.f even 2 1 3200.2.a.x 1
40.i odd 4 2 3200.2.c.l 2
40.k even 4 2 3200.2.c.f 2
48.i odd 4 2 2304.2.d.r 2
48.k even 4 2 2304.2.d.b 2
56.e even 2 1 6272.2.a.b 1
56.h odd 2 1 6272.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.a.a 1 8.b even 2 1
128.2.a.b yes 1 4.b odd 2 1
128.2.a.c yes 1 8.d odd 2 1
128.2.a.d yes 1 1.a even 1 1 trivial
256.2.b.a 2 16.f odd 4 2
256.2.b.c 2 16.e even 4 2
1024.2.e.i 4 32.g even 8 4
1024.2.e.m 4 32.h odd 8 4
1152.2.a.c 1 3.b odd 2 1
1152.2.a.h 1 12.b even 2 1
1152.2.a.m 1 24.h odd 2 1
1152.2.a.r 1 24.f even 2 1
2304.2.d.b 2 48.k even 4 2
2304.2.d.r 2 48.i odd 4 2
3200.2.a.e 1 40.e odd 2 1
3200.2.a.h 1 5.b even 2 1
3200.2.a.u 1 20.d odd 2 1
3200.2.a.x 1 40.f even 2 1
3200.2.c.e 2 5.c odd 4 2
3200.2.c.f 2 40.k even 4 2
3200.2.c.k 2 20.e even 4 2
3200.2.c.l 2 40.i odd 4 2
6272.2.a.a 1 7.b odd 2 1
6272.2.a.b 1 56.e even 2 1
6272.2.a.g 1 28.d even 2 1
6272.2.a.h 1 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(128))\):

\( T_{3} - 2 \) Copy content Toggle raw display
\( T_{5} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 2 \) Copy content Toggle raw display
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T + 2 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 10 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T - 6 \) Copy content Toggle raw display
$47$ \( T + 8 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T - 14 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T - 10 \) Copy content Toggle raw display
$71$ \( T - 12 \) Copy content Toggle raw display
$73$ \( T - 14 \) Copy content Toggle raw display
$79$ \( T + 8 \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T + 2 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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