Properties

Label 128.5.d.b
Level 128
Weight 5
Character orbit 128.d
Self dual yes
Analytic conductor 13.231
Analytic rank 0
Dimension 2
CM discriminant -8
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 128.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(13.2313552747\)
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: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 8\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 \beta q^{3} + 47 q^{9} +O(q^{10})\) \( q + 8 \beta q^{3} + 47 q^{9} + 168 \beta q^{11} + 574 q^{17} -408 \beta q^{19} + 625 q^{25} -272 \beta q^{27} + 2688 q^{33} -1246 q^{41} + 840 \beta q^{43} + 2401 q^{49} + 4592 \beta q^{51} -6528 q^{57} -4920 \beta q^{59} -5208 \beta q^{67} -9506 q^{73} + 5000 \beta q^{75} -8159 q^{81} -5688 \beta q^{83} -5474 q^{89} + 9982 q^{97} + 7896 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 94q^{9} + O(q^{10}) \) \( 2q + 94q^{9} + 1148q^{17} + 1250q^{25} + 5376q^{33} - 2492q^{41} + 4802q^{49} - 13056q^{57} - 19012q^{73} - 16318q^{81} - 10948q^{89} + 19964q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
−1.41421
1.41421
0 −11.3137 0 0 0 0 0 47.0000 0
63.2 0 11.3137 0 0 0 0 0 47.0000 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 128.5.d.b 2
3.b odd 2 1 1152.5.b.a 2
4.b odd 2 1 inner 128.5.d.b 2
8.b even 2 1 inner 128.5.d.b 2
8.d odd 2 1 CM 128.5.d.b 2
12.b even 2 1 1152.5.b.a 2
16.e even 4 2 256.5.c.e 2
16.f odd 4 2 256.5.c.e 2
24.f even 2 1 1152.5.b.a 2
24.h odd 2 1 1152.5.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.5.d.b 2 1.a even 1 1 trivial
128.5.d.b 2 4.b odd 2 1 inner
128.5.d.b 2 8.b even 2 1 inner
128.5.d.b 2 8.d odd 2 1 CM
256.5.c.e 2 16.e even 4 2
256.5.c.e 2 16.f odd 4 2
1152.5.b.a 2 3.b odd 2 1
1152.5.b.a 2 12.b even 2 1
1152.5.b.a 2 24.f even 2 1
1152.5.b.a 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 128 \) acting on \(S_{5}^{\mathrm{new}}(128, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 34 T^{2} + 6561 T^{4} \)
$5$ \( ( 1 - 25 T )^{2}( 1 + 25 T )^{2} \)
$7$ \( ( 1 - 49 T )^{2}( 1 + 49 T )^{2} \)
$11$ \( 1 - 27166 T^{2} + 214358881 T^{4} \)
$13$ \( ( 1 - 169 T )^{2}( 1 + 169 T )^{2} \)
$17$ \( ( 1 - 574 T + 83521 T^{2} )^{2} \)
$19$ \( 1 - 72286 T^{2} + 16983563041 T^{4} \)
$23$ \( ( 1 - 529 T )^{2}( 1 + 529 T )^{2} \)
$29$ \( ( 1 - 841 T )^{2}( 1 + 841 T )^{2} \)
$31$ \( ( 1 - 961 T )^{2}( 1 + 961 T )^{2} \)
$37$ \( ( 1 - 1369 T )^{2}( 1 + 1369 T )^{2} \)
$41$ \( ( 1 + 1246 T + 2825761 T^{2} )^{2} \)
$43$ \( 1 + 5426402 T^{2} + 11688200277601 T^{4} \)
$47$ \( ( 1 - 2209 T )^{2}( 1 + 2209 T )^{2} \)
$53$ \( ( 1 - 2809 T )^{2}( 1 + 2809 T )^{2} \)
$59$ \( 1 - 24178078 T^{2} + 146830437604321 T^{4} \)
$61$ \( ( 1 - 3721 T )^{2}( 1 + 3721 T )^{2} \)
$67$ \( 1 - 13944286 T^{2} + 406067677556641 T^{4} \)
$71$ \( ( 1 - 5041 T )^{2}( 1 + 5041 T )^{2} \)
$73$ \( ( 1 + 9506 T + 28398241 T^{2} )^{2} \)
$79$ \( ( 1 - 6241 T )^{2}( 1 + 6241 T )^{2} \)
$83$ \( 1 + 30209954 T^{2} + 2252292232139041 T^{4} \)
$89$ \( ( 1 + 5474 T + 62742241 T^{2} )^{2} \)
$97$ \( ( 1 - 9982 T + 88529281 T^{2} )^{2} \)
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