Properties

Label 320.6.a.c
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 20)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 22 q^{3} + 25 q^{5} + 218 q^{7} + 241 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 22 q^{3} + 25 q^{5} + 218 q^{7} + 241 q^{9} + 480 q^{11} + 622 q^{13} - 550 q^{15} + 186 q^{17} + 1204 q^{19} - 4796 q^{21} - 3186 q^{23} + 625 q^{25} + 44 q^{27} - 5526 q^{29} + 9356 q^{31} - 10560 q^{33} + 5450 q^{35} - 5618 q^{37} - 13684 q^{39} - 14394 q^{41} + 370 q^{43} + 6025 q^{45} + 16146 q^{47} + 30717 q^{49} - 4092 q^{51} + 4374 q^{53} + 12000 q^{55} - 26488 q^{57} + 11748 q^{59} - 13202 q^{61} + 52538 q^{63} + 15550 q^{65} + 11542 q^{67} + 70092 q^{69} - 29532 q^{71} + 33698 q^{73} - 13750 q^{75} + 104640 q^{77} + 31208 q^{79} - 59531 q^{81} + 38466 q^{83} + 4650 q^{85} + 121572 q^{87} + 119514 q^{89} + 135596 q^{91} - 205832 q^{93} + 30100 q^{95} + 94658 q^{97} + 115680 q^{99}+O(q^{100}) \) 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 −22.0000 0 25.0000 0 218.000 0 241.000 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.c 1
4.b odd 2 1 320.6.a.n 1
8.b even 2 1 20.6.a.a 1
8.d odd 2 1 80.6.a.b 1
24.f even 2 1 720.6.a.l 1
24.h odd 2 1 180.6.a.e 1
40.e odd 2 1 400.6.a.m 1
40.f even 2 1 100.6.a.a 1
40.i odd 4 2 100.6.c.a 2
40.k even 4 2 400.6.c.c 2
56.h odd 2 1 980.6.a.b 1
120.i odd 2 1 900.6.a.b 1
120.w even 4 2 900.6.d.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.6.a.a 1 8.b even 2 1
80.6.a.b 1 8.d odd 2 1
100.6.a.a 1 40.f even 2 1
100.6.c.a 2 40.i odd 4 2
180.6.a.e 1 24.h odd 2 1
320.6.a.c 1 1.a even 1 1 trivial
320.6.a.n 1 4.b odd 2 1
400.6.a.m 1 40.e odd 2 1
400.6.c.c 2 40.k even 4 2
720.6.a.l 1 24.f even 2 1
900.6.a.b 1 120.i odd 2 1
900.6.d.h 2 120.w even 4 2
980.6.a.b 1 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 22 \) 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
$3$ \( T + 22 \) Copy content Toggle raw display
$5$ \( T - 25 \) Copy content Toggle raw display
$7$ \( T - 218 \) Copy content Toggle raw display
$11$ \( T - 480 \) Copy content Toggle raw display
$13$ \( T - 622 \) Copy content Toggle raw display
$17$ \( T - 186 \) Copy content Toggle raw display
$19$ \( T - 1204 \) Copy content Toggle raw display
$23$ \( T + 3186 \) Copy content Toggle raw display
$29$ \( T + 5526 \) Copy content Toggle raw display
$31$ \( T - 9356 \) Copy content Toggle raw display
$37$ \( T + 5618 \) Copy content Toggle raw display
$41$ \( T + 14394 \) Copy content Toggle raw display
$43$ \( T - 370 \) Copy content Toggle raw display
$47$ \( T - 16146 \) Copy content Toggle raw display
$53$ \( T - 4374 \) Copy content Toggle raw display
$59$ \( T - 11748 \) Copy content Toggle raw display
$61$ \( T + 13202 \) Copy content Toggle raw display
$67$ \( T - 11542 \) Copy content Toggle raw display
$71$ \( T + 29532 \) Copy content Toggle raw display
$73$ \( T - 33698 \) Copy content Toggle raw display
$79$ \( T - 31208 \) Copy content Toggle raw display
$83$ \( T - 38466 \) Copy content Toggle raw display
$89$ \( T - 119514 \) Copy content Toggle raw display
$97$ \( T - 94658 \) Copy content Toggle raw display
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