Properties

Label 128.2.b.a
Level 128
Weight 2
Character orbit 128.b
Analytic conductor 1.022
Analytic rank 0
Dimension 2
CM discriminant -8
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 128.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.02208514587\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} -5 q^{9} +O(q^{10})\) \( q + \beta q^{3} -5 q^{9} + \beta q^{11} + 6 q^{17} -3 \beta q^{19} + 5 q^{25} -2 \beta q^{27} -8 q^{33} -6 q^{41} -3 \beta q^{43} -7 q^{49} + 6 \beta q^{51} + 24 q^{57} + 5 \beta q^{59} -3 \beta q^{67} -2 q^{73} + 5 \beta q^{75} + q^{81} + \beta q^{83} -18 q^{89} -10 q^{97} -5 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 10q^{9} + O(q^{10}) \) \( 2q - 10q^{9} + 12q^{17} + 10q^{25} - 16q^{33} - 12q^{41} - 14q^{49} + 48q^{57} - 4q^{73} + 2q^{81} - 36q^{89} - 20q^{97} + O(q^{100}) \)

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
1.41421i
1.41421i
0 2.82843i 0 0 0 0 0 −5.00000 0
65.2 0 2.82843i 0 0 0 0 0 −5.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.2.b.a 2
3.b odd 2 1 1152.2.d.c 2
4.b odd 2 1 inner 128.2.b.a 2
5.b even 2 1 3200.2.d.c 2
5.c odd 4 2 3200.2.f.o 4
8.b even 2 1 inner 128.2.b.a 2
8.d odd 2 1 CM 128.2.b.a 2
12.b even 2 1 1152.2.d.c 2
16.e even 4 2 256.2.a.e 2
16.f odd 4 2 256.2.a.e 2
20.d odd 2 1 3200.2.d.c 2
20.e even 4 2 3200.2.f.o 4
24.f even 2 1 1152.2.d.c 2
24.h odd 2 1 1152.2.d.c 2
32.g even 8 2 1024.2.e.a 2
32.g even 8 2 1024.2.e.f 2
32.h odd 8 2 1024.2.e.a 2
32.h odd 8 2 1024.2.e.f 2
40.e odd 2 1 3200.2.d.c 2
40.f even 2 1 3200.2.d.c 2
40.i odd 4 2 3200.2.f.o 4
40.k even 4 2 3200.2.f.o 4
48.i odd 4 2 2304.2.a.t 2
48.k even 4 2 2304.2.a.t 2
80.k odd 4 2 6400.2.a.by 2
80.q even 4 2 6400.2.a.by 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.b.a 2 1.a even 1 1 trivial
128.2.b.a 2 4.b odd 2 1 inner
128.2.b.a 2 8.b even 2 1 inner
128.2.b.a 2 8.d odd 2 1 CM
256.2.a.e 2 16.e even 4 2
256.2.a.e 2 16.f odd 4 2
1024.2.e.a 2 32.g even 8 2
1024.2.e.a 2 32.h odd 8 2
1024.2.e.f 2 32.g even 8 2
1024.2.e.f 2 32.h odd 8 2
1152.2.d.c 2 3.b odd 2 1
1152.2.d.c 2 12.b even 2 1
1152.2.d.c 2 24.f even 2 1
1152.2.d.c 2 24.h odd 2 1
2304.2.a.t 2 48.i odd 4 2
2304.2.a.t 2 48.k even 4 2
3200.2.d.c 2 5.b even 2 1
3200.2.d.c 2 20.d odd 2 1
3200.2.d.c 2 40.e odd 2 1
3200.2.d.c 2 40.f even 2 1
3200.2.f.o 4 5.c odd 4 2
3200.2.f.o 4 20.e even 4 2
3200.2.f.o 4 40.i odd 4 2
3200.2.f.o 4 40.k even 4 2
6400.2.a.by 2 80.k odd 4 2
6400.2.a.by 2 80.q even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(128, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 2 T + 3 T^{2} )( 1 + 2 T + 3 T^{2} ) \)
$5$ \( ( 1 - 5 T^{2} )^{2} \)
$7$ \( ( 1 + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 6 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} ) \)
$13$ \( ( 1 - 13 T^{2} )^{2} \)
$17$ \( ( 1 - 6 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 - 2 T + 19 T^{2} )( 1 + 2 T + 19 T^{2} ) \)
$23$ \( ( 1 + 23 T^{2} )^{2} \)
$29$ \( ( 1 - 29 T^{2} )^{2} \)
$31$ \( ( 1 + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 37 T^{2} )^{2} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 10 T + 43 T^{2} )( 1 + 10 T + 43 T^{2} ) \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( ( 1 - 53 T^{2} )^{2} \)
$59$ \( ( 1 - 6 T + 59 T^{2} )( 1 + 6 T + 59 T^{2} ) \)
$61$ \( ( 1 - 61 T^{2} )^{2} \)
$67$ \( ( 1 - 14 T + 67 T^{2} )( 1 + 14 T + 67 T^{2} ) \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 2 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 + 79 T^{2} )^{2} \)
$83$ \( ( 1 - 18 T + 83 T^{2} )( 1 + 18 T + 83 T^{2} ) \)
$89$ \( ( 1 + 18 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 + 10 T + 97 T^{2} )^{2} \)
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