Properties

Label 320.6.a.i
Level $320$
Weight $6$
Character orbit 320.a
Self dual yes
Analytic conductor $51.323$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q + 2 q^{3} + 25 q^{5} - 62 q^{7} - 239 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{3} + 25 q^{5} - 62 q^{7} - 239 q^{9} + 144 q^{11} + 654 q^{13} + 50 q^{15} - 1190 q^{17} - 556 q^{19} - 124 q^{21} + 2182 q^{23} + 625 q^{25} - 964 q^{27} + 1578 q^{29} + 9660 q^{31} + 288 q^{33} - 1550 q^{35} + 3534 q^{37} + 1308 q^{39} + 7462 q^{41} + 7114 q^{43} - 5975 q^{45} - 28294 q^{47} - 12963 q^{49} - 2380 q^{51} + 13046 q^{53} + 3600 q^{55} - 1112 q^{57} + 37092 q^{59} - 39570 q^{61} + 14818 q^{63} + 16350 q^{65} + 56734 q^{67} + 4364 q^{69} + 45588 q^{71} + 11842 q^{73} + 1250 q^{75} - 8928 q^{77} + 94216 q^{79} + 56149 q^{81} + 31482 q^{83} - 29750 q^{85} + 3156 q^{87} - 94054 q^{89} - 40548 q^{91} + 19320 q^{93} - 13900 q^{95} + 23714 q^{97} - 34416 q^{99}+O(q^{100}) \) 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 2.00000 0 25.0000 0 −62.0000 0 −239.000 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.i 1
4.b odd 2 1 320.6.a.h 1
8.b even 2 1 40.6.a.c 1
8.d odd 2 1 80.6.a.d 1
24.f even 2 1 720.6.a.t 1
24.h odd 2 1 360.6.a.f 1
40.e odd 2 1 400.6.a.h 1
40.f even 2 1 200.6.a.b 1
40.i odd 4 2 200.6.c.d 2
40.k even 4 2 400.6.c.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.a.c 1 8.b even 2 1
80.6.a.d 1 8.d odd 2 1
200.6.a.b 1 40.f even 2 1
200.6.c.d 2 40.i odd 4 2
320.6.a.h 1 4.b odd 2 1
320.6.a.i 1 1.a even 1 1 trivial
360.6.a.f 1 24.h odd 2 1
400.6.a.h 1 40.e odd 2 1
400.6.c.k 2 40.k even 4 2
720.6.a.t 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 2 \) 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
$3$ \( T - 2 \) Copy content Toggle raw display
$5$ \( T - 25 \) Copy content Toggle raw display
$7$ \( T + 62 \) Copy content Toggle raw display
$11$ \( T - 144 \) Copy content Toggle raw display
$13$ \( T - 654 \) Copy content Toggle raw display
$17$ \( T + 1190 \) Copy content Toggle raw display
$19$ \( T + 556 \) Copy content Toggle raw display
$23$ \( T - 2182 \) Copy content Toggle raw display
$29$ \( T - 1578 \) Copy content Toggle raw display
$31$ \( T - 9660 \) Copy content Toggle raw display
$37$ \( T - 3534 \) Copy content Toggle raw display
$41$ \( T - 7462 \) Copy content Toggle raw display
$43$ \( T - 7114 \) Copy content Toggle raw display
$47$ \( T + 28294 \) Copy content Toggle raw display
$53$ \( T - 13046 \) Copy content Toggle raw display
$59$ \( T - 37092 \) Copy content Toggle raw display
$61$ \( T + 39570 \) Copy content Toggle raw display
$67$ \( T - 56734 \) Copy content Toggle raw display
$71$ \( T - 45588 \) Copy content Toggle raw display
$73$ \( T - 11842 \) Copy content Toggle raw display
$79$ \( T - 94216 \) Copy content Toggle raw display
$83$ \( T - 31482 \) Copy content Toggle raw display
$89$ \( T + 94054 \) Copy content Toggle raw display
$97$ \( T - 23714 \) Copy content Toggle raw display
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