Properties

Label 8.6.a.a
Level $8$
Weight $6$
Character orbit 8.a
Self dual yes
Analytic conductor $1.283$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 8.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.28307055850\)
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

\(f(q)\) \(=\) \( q + 20 q^{3} - 74 q^{5} - 24 q^{7} + 157 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 20 q^{3} - 74 q^{5} - 24 q^{7} + 157 q^{9} + 124 q^{11} + 478 q^{13} - 1480 q^{15} - 1198 q^{17} + 3044 q^{19} - 480 q^{21} + 184 q^{23} + 2351 q^{25} - 1720 q^{27} - 3282 q^{29} - 5728 q^{31} + 2480 q^{33} + 1776 q^{35} + 10326 q^{37} + 9560 q^{39} - 8886 q^{41} - 9188 q^{43} - 11618 q^{45} + 23664 q^{47} - 16231 q^{49} - 23960 q^{51} + 11686 q^{53} - 9176 q^{55} + 60880 q^{57} + 16876 q^{59} - 18482 q^{61} - 3768 q^{63} - 35372 q^{65} - 15532 q^{67} + 3680 q^{69} - 31960 q^{71} - 4886 q^{73} + 47020 q^{75} - 2976 q^{77} + 44560 q^{79} - 72551 q^{81} + 67364 q^{83} + 88652 q^{85} - 65640 q^{87} + 71994 q^{89} - 11472 q^{91} - 114560 q^{93} - 225256 q^{95} + 48866 q^{97} + 19468 q^{99}+O(q^{100}) \) 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 20.0000 0 −74.0000 0 −24.0000 0 157.000 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 8.6.a.a 1
3.b odd 2 1 72.6.a.f 1
4.b odd 2 1 16.6.a.a 1
5.b even 2 1 200.6.a.a 1
5.c odd 4 2 200.6.c.a 2
7.b odd 2 1 392.6.a.b 1
7.c even 3 2 392.6.i.b 2
7.d odd 6 2 392.6.i.e 2
8.b even 2 1 64.6.a.a 1
8.d odd 2 1 64.6.a.g 1
11.b odd 2 1 968.6.a.a 1
12.b even 2 1 144.6.a.k 1
16.e even 4 2 256.6.b.f 2
16.f odd 4 2 256.6.b.d 2
20.d odd 2 1 400.6.a.l 1
20.e even 4 2 400.6.c.d 2
24.f even 2 1 576.6.a.h 1
24.h odd 2 1 576.6.a.g 1
28.d even 2 1 784.6.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.6.a.a 1 1.a even 1 1 trivial
16.6.a.a 1 4.b odd 2 1
64.6.a.a 1 8.b even 2 1
64.6.a.g 1 8.d odd 2 1
72.6.a.f 1 3.b odd 2 1
144.6.a.k 1 12.b even 2 1
200.6.a.a 1 5.b even 2 1
200.6.c.a 2 5.c odd 4 2
256.6.b.d 2 16.f odd 4 2
256.6.b.f 2 16.e even 4 2
392.6.a.b 1 7.b odd 2 1
392.6.i.b 2 7.c even 3 2
392.6.i.e 2 7.d odd 6 2
400.6.a.l 1 20.d odd 2 1
400.6.c.d 2 20.e even 4 2
576.6.a.g 1 24.h odd 2 1
576.6.a.h 1 24.f even 2 1
784.6.a.l 1 28.d even 2 1
968.6.a.a 1 11.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(\Gamma_0(8))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 20 \) Copy content Toggle raw display
$5$ \( T + 74 \) Copy content Toggle raw display
$7$ \( T + 24 \) Copy content Toggle raw display
$11$ \( T - 124 \) Copy content Toggle raw display
$13$ \( T - 478 \) Copy content Toggle raw display
$17$ \( T + 1198 \) Copy content Toggle raw display
$19$ \( T - 3044 \) Copy content Toggle raw display
$23$ \( T - 184 \) Copy content Toggle raw display
$29$ \( T + 3282 \) Copy content Toggle raw display
$31$ \( T + 5728 \) Copy content Toggle raw display
$37$ \( T - 10326 \) Copy content Toggle raw display
$41$ \( T + 8886 \) Copy content Toggle raw display
$43$ \( T + 9188 \) Copy content Toggle raw display
$47$ \( T - 23664 \) Copy content Toggle raw display
$53$ \( T - 11686 \) Copy content Toggle raw display
$59$ \( T - 16876 \) Copy content Toggle raw display
$61$ \( T + 18482 \) Copy content Toggle raw display
$67$ \( T + 15532 \) Copy content Toggle raw display
$71$ \( T + 31960 \) Copy content Toggle raw display
$73$ \( T + 4886 \) Copy content Toggle raw display
$79$ \( T - 44560 \) Copy content Toggle raw display
$83$ \( T - 67364 \) Copy content Toggle raw display
$89$ \( T - 71994 \) Copy content Toggle raw display
$97$ \( T - 48866 \) Copy content Toggle raw display
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