Properties

Label 320.4.a.c
Level $320$
Weight $4$
Character orbit 320.a
Self dual yes
Analytic conductor $18.881$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 320.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.8806112018\)
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

\(f(q)\) \(=\) \( q - 6 q^{3} + 5 q^{5} + 34 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 6 q^{3} + 5 q^{5} + 34 q^{7} + 9 q^{9} + 16 q^{11} - 58 q^{13} - 30 q^{15} - 70 q^{17} + 4 q^{19} - 204 q^{21} + 134 q^{23} + 25 q^{25} + 108 q^{27} + 242 q^{29} - 100 q^{31} - 96 q^{33} + 170 q^{35} + 438 q^{37} + 348 q^{39} - 138 q^{41} + 178 q^{43} + 45 q^{45} - 22 q^{47} + 813 q^{49} + 420 q^{51} - 162 q^{53} + 80 q^{55} - 24 q^{57} - 268 q^{59} - 250 q^{61} + 306 q^{63} - 290 q^{65} + 422 q^{67} - 804 q^{69} + 852 q^{71} + 306 q^{73} - 150 q^{75} + 544 q^{77} + 456 q^{79} - 891 q^{81} + 434 q^{83} - 350 q^{85} - 1452 q^{87} - 726 q^{89} - 1972 q^{91} + 600 q^{93} + 20 q^{95} + 1378 q^{97} + 144 q^{99}+O(q^{100}) \) 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 −6.00000 0 5.00000 0 34.0000 0 9.00000 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 320.4.a.c 1
4.b odd 2 1 320.4.a.l 1
5.b even 2 1 1600.4.a.br 1
8.b even 2 1 80.4.a.e 1
8.d odd 2 1 40.4.a.a 1
16.e even 4 2 1280.4.d.a 2
16.f odd 4 2 1280.4.d.p 2
20.d odd 2 1 1600.4.a.j 1
24.f even 2 1 360.4.a.h 1
24.h odd 2 1 720.4.a.bd 1
40.e odd 2 1 200.4.a.i 1
40.f even 2 1 400.4.a.e 1
40.i odd 4 2 400.4.c.f 2
40.k even 4 2 200.4.c.c 2
56.e even 2 1 1960.4.a.h 1
120.m even 2 1 1800.4.a.bi 1
120.q odd 4 2 1800.4.f.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.a.a 1 8.d odd 2 1
80.4.a.e 1 8.b even 2 1
200.4.a.i 1 40.e odd 2 1
200.4.c.c 2 40.k even 4 2
320.4.a.c 1 1.a even 1 1 trivial
320.4.a.l 1 4.b odd 2 1
360.4.a.h 1 24.f even 2 1
400.4.a.e 1 40.f even 2 1
400.4.c.f 2 40.i odd 4 2
720.4.a.bd 1 24.h odd 2 1
1280.4.d.a 2 16.e even 4 2
1280.4.d.p 2 16.f odd 4 2
1600.4.a.j 1 20.d odd 2 1
1600.4.a.br 1 5.b even 2 1
1800.4.a.bi 1 120.m even 2 1
1800.4.f.j 2 120.q odd 4 2
1960.4.a.h 1 56.e even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(320))\):

\( T_{3} + 6 \) Copy content Toggle raw display
\( T_{7} - 34 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 6 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T - 34 \) Copy content Toggle raw display
$11$ \( T - 16 \) Copy content Toggle raw display
$13$ \( T + 58 \) Copy content Toggle raw display
$17$ \( T + 70 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T - 134 \) Copy content Toggle raw display
$29$ \( T - 242 \) Copy content Toggle raw display
$31$ \( T + 100 \) Copy content Toggle raw display
$37$ \( T - 438 \) Copy content Toggle raw display
$41$ \( T + 138 \) Copy content Toggle raw display
$43$ \( T - 178 \) Copy content Toggle raw display
$47$ \( T + 22 \) Copy content Toggle raw display
$53$ \( T + 162 \) Copy content Toggle raw display
$59$ \( T + 268 \) Copy content Toggle raw display
$61$ \( T + 250 \) Copy content Toggle raw display
$67$ \( T - 422 \) Copy content Toggle raw display
$71$ \( T - 852 \) Copy content Toggle raw display
$73$ \( T - 306 \) Copy content Toggle raw display
$79$ \( T - 456 \) Copy content Toggle raw display
$83$ \( T - 434 \) Copy content Toggle raw display
$89$ \( T + 726 \) Copy content Toggle raw display
$97$ \( T - 1378 \) Copy content Toggle raw display
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