Properties

Label 128.4.b.b
Level 128
Weight 4
Character orbit 128.b
Analytic conductor 7.552
Analytic rank 0
Dimension 2
CM discriminant -8
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 128.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.55224448073\)
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.

\(f(q)\) \(=\) \( q + \beta q^{3} + 19 q^{9} +O(q^{10})\) \( q + \beta q^{3} + 19 q^{9} + 25 \beta q^{11} -90 q^{17} + 45 \beta q^{19} + 125 q^{25} + 46 \beta q^{27} -200 q^{33} + 522 q^{41} -171 \beta q^{43} -343 q^{49} -90 \beta q^{51} -360 q^{57} -115 \beta q^{59} -387 \beta q^{67} + 430 q^{73} + 125 \beta q^{75} + 145 q^{81} + 241 \beta q^{83} -1026 q^{89} + 1910 q^{97} + 475 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 38q^{9} + O(q^{10}) \) \( 2q + 38q^{9} - 180q^{17} + 250q^{25} - 400q^{33} + 1044q^{41} - 686q^{49} - 720q^{57} + 860q^{73} + 290q^{81} - 2052q^{89} + 3820q^{97} + O(q^{100}) \)

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 19.0000 0
65.2 0 2.82843i 0 0 0 0 0 19.0000 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 128.4.b.b 2
3.b odd 2 1 1152.4.d.e 2
4.b odd 2 1 inner 128.4.b.b 2
8.b even 2 1 inner 128.4.b.b 2
8.d odd 2 1 CM 128.4.b.b 2
12.b even 2 1 1152.4.d.e 2
16.e even 4 2 256.4.a.k 2
16.f odd 4 2 256.4.a.k 2
24.f even 2 1 1152.4.d.e 2
24.h odd 2 1 1152.4.d.e 2
48.i odd 4 2 2304.4.a.bf 2
48.k even 4 2 2304.4.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.4.b.b 2 1.a even 1 1 trivial
128.4.b.b 2 4.b odd 2 1 inner
128.4.b.b 2 8.b even 2 1 inner
128.4.b.b 2 8.d odd 2 1 CM
256.4.a.k 2 16.e even 4 2
256.4.a.k 2 16.f odd 4 2
1152.4.d.e 2 3.b odd 2 1
1152.4.d.e 2 12.b even 2 1
1152.4.d.e 2 24.f even 2 1
1152.4.d.e 2 24.h odd 2 1
2304.4.a.bf 2 48.i odd 4 2
2304.4.a.bf 2 48.k even 4 2

Hecke kernels

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

\( T_{3}^{2} + 8 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 10 T + 27 T^{2} )( 1 + 10 T + 27 T^{2} ) \)
$5$ \( ( 1 - 125 T^{2} )^{2} \)
$7$ \( ( 1 + 343 T^{2} )^{2} \)
$11$ \( ( 1 - 18 T + 1331 T^{2} )( 1 + 18 T + 1331 T^{2} ) \)
$13$ \( ( 1 - 2197 T^{2} )^{2} \)
$17$ \( ( 1 + 90 T + 4913 T^{2} )^{2} \)
$19$ \( ( 1 - 106 T + 6859 T^{2} )( 1 + 106 T + 6859 T^{2} ) \)
$23$ \( ( 1 + 12167 T^{2} )^{2} \)
$29$ \( ( 1 - 24389 T^{2} )^{2} \)
$31$ \( ( 1 + 29791 T^{2} )^{2} \)
$37$ \( ( 1 - 50653 T^{2} )^{2} \)
$41$ \( ( 1 - 522 T + 68921 T^{2} )^{2} \)
$43$ \( ( 1 - 290 T + 79507 T^{2} )( 1 + 290 T + 79507 T^{2} ) \)
$47$ \( ( 1 + 103823 T^{2} )^{2} \)
$53$ \( ( 1 - 148877 T^{2} )^{2} \)
$59$ \( ( 1 - 846 T + 205379 T^{2} )( 1 + 846 T + 205379 T^{2} ) \)
$61$ \( ( 1 - 226981 T^{2} )^{2} \)
$67$ \( ( 1 - 70 T + 300763 T^{2} )( 1 + 70 T + 300763 T^{2} ) \)
$71$ \( ( 1 + 357911 T^{2} )^{2} \)
$73$ \( ( 1 - 430 T + 389017 T^{2} )^{2} \)
$79$ \( ( 1 + 493039 T^{2} )^{2} \)
$83$ \( ( 1 - 1350 T + 571787 T^{2} )( 1 + 1350 T + 571787 T^{2} ) \)
$89$ \( ( 1 + 1026 T + 704969 T^{2} )^{2} \)
$97$ \( ( 1 - 1910 T + 912673 T^{2} )^{2} \)
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