Properties

Label 320.6.a.d
Level $320$
Weight $6$
Character orbit 320.a
Self dual yes
Analytic conductor $51.323$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.3228223402\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 18 q^{3} + 25 q^{5} - 242 q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 18 q^{3} + 25 q^{5} - 242 q^{7} + 81 q^{9} + 656 q^{11} + 206 q^{13} - 450 q^{15} + 1690 q^{17} - 1364 q^{19} + 4356 q^{21} - 2198 q^{23} + 625 q^{25} + 2916 q^{27} + 2218 q^{29} + 1700 q^{31} - 11808 q^{33} - 6050 q^{35} + 846 q^{37} - 3708 q^{39} - 1818 q^{41} + 10534 q^{43} + 2025 q^{45} - 12074 q^{47} + 41757 q^{49} - 30420 q^{51} - 32586 q^{53} + 16400 q^{55} + 24552 q^{57} + 8668 q^{59} + 34670 q^{61} - 19602 q^{63} + 5150 q^{65} - 47566 q^{67} + 39564 q^{69} - 948 q^{71} - 63102 q^{73} - 11250 q^{75} - 158752 q^{77} - 46536 q^{79} - 72171 q^{81} - 88778 q^{83} + 42250 q^{85} - 39924 q^{87} - 104934 q^{89} - 49852 q^{91} - 30600 q^{93} - 34100 q^{95} - 36254 q^{97} + 53136 q^{99}+O(q^{100}) \) 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 −18.0000 0 25.0000 0 −242.000 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)

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 320.6.a.d 1
4.b odd 2 1 320.6.a.m 1
8.b even 2 1 80.6.a.g 1
8.d odd 2 1 40.6.a.a 1
24.f even 2 1 360.6.a.i 1
24.h odd 2 1 720.6.a.k 1
40.e odd 2 1 200.6.a.d 1
40.f even 2 1 400.6.a.b 1
40.i odd 4 2 400.6.c.e 2
40.k even 4 2 200.6.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.a.a 1 8.d odd 2 1
80.6.a.g 1 8.b even 2 1
200.6.a.d 1 40.e odd 2 1
200.6.c.b 2 40.k even 4 2
320.6.a.d 1 1.a even 1 1 trivial
320.6.a.m 1 4.b odd 2 1
360.6.a.i 1 24.f even 2 1
400.6.a.b 1 40.f even 2 1
400.6.c.e 2 40.i odd 4 2
720.6.a.k 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 18 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(320))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 18 \) Copy content Toggle raw display
$5$ \( T - 25 \) Copy content Toggle raw display
$7$ \( T + 242 \) Copy content Toggle raw display
$11$ \( T - 656 \) Copy content Toggle raw display
$13$ \( T - 206 \) Copy content Toggle raw display
$17$ \( T - 1690 \) Copy content Toggle raw display
$19$ \( T + 1364 \) Copy content Toggle raw display
$23$ \( T + 2198 \) Copy content Toggle raw display
$29$ \( T - 2218 \) Copy content Toggle raw display
$31$ \( T - 1700 \) Copy content Toggle raw display
$37$ \( T - 846 \) Copy content Toggle raw display
$41$ \( T + 1818 \) Copy content Toggle raw display
$43$ \( T - 10534 \) Copy content Toggle raw display
$47$ \( T + 12074 \) Copy content Toggle raw display
$53$ \( T + 32586 \) Copy content Toggle raw display
$59$ \( T - 8668 \) Copy content Toggle raw display
$61$ \( T - 34670 \) Copy content Toggle raw display
$67$ \( T + 47566 \) Copy content Toggle raw display
$71$ \( T + 948 \) Copy content Toggle raw display
$73$ \( T + 63102 \) Copy content Toggle raw display
$79$ \( T + 46536 \) Copy content Toggle raw display
$83$ \( T + 88778 \) Copy content Toggle raw display
$89$ \( T + 104934 \) Copy content Toggle raw display
$97$ \( T + 36254 \) Copy content Toggle raw display
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