Properties

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

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(18.8806112018\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{3} + 5 q^{5} + 6 q^{7} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{3} + 5 q^{5} + 6 q^{7} - 23 q^{9} - 60 q^{11} - 50 q^{13} + 10 q^{15} - 30 q^{17} - 40 q^{19} + 12 q^{21} + 178 q^{23} + 25 q^{25} - 100 q^{27} - 166 q^{29} + 20 q^{31} - 120 q^{33} + 30 q^{35} - 10 q^{37} - 100 q^{39} - 250 q^{41} - 142 q^{43} - 115 q^{45} + 214 q^{47} - 307 q^{49} - 60 q^{51} - 490 q^{53} - 300 q^{55} - 80 q^{57} + 800 q^{59} - 250 q^{61} - 138 q^{63} - 250 q^{65} + 774 q^{67} + 356 q^{69} + 100 q^{71} - 230 q^{73} + 50 q^{75} - 360 q^{77} - 1320 q^{79} + 421 q^{81} - 982 q^{83} - 150 q^{85} - 332 q^{87} + 874 q^{89} - 300 q^{91} + 40 q^{93} - 200 q^{95} - 310 q^{97} + 1380 q^{99}+O(q^{100}) \) 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 5.00000 0 6.00000 0 −23.0000 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 320.4.a.i 1
4.b odd 2 1 320.4.a.f 1
5.b even 2 1 1600.4.a.r 1
8.b even 2 1 160.4.a.a 1
8.d odd 2 1 160.4.a.b yes 1
16.e even 4 2 1280.4.d.f 2
16.f odd 4 2 1280.4.d.k 2
20.d odd 2 1 1600.4.a.bj 1
24.f even 2 1 1440.4.a.n 1
24.h odd 2 1 1440.4.a.o 1
40.e odd 2 1 800.4.a.d 1
40.f even 2 1 800.4.a.h 1
40.i odd 4 2 800.4.c.f 2
40.k even 4 2 800.4.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.a 1 8.b even 2 1
160.4.a.b yes 1 8.d odd 2 1
320.4.a.f 1 4.b odd 2 1
320.4.a.i 1 1.a even 1 1 trivial
800.4.a.d 1 40.e odd 2 1
800.4.a.h 1 40.f even 2 1
800.4.c.e 2 40.k even 4 2
800.4.c.f 2 40.i odd 4 2
1280.4.d.f 2 16.e even 4 2
1280.4.d.k 2 16.f odd 4 2
1440.4.a.n 1 24.f even 2 1
1440.4.a.o 1 24.h odd 2 1
1600.4.a.r 1 5.b even 2 1
1600.4.a.bj 1 20.d odd 2 1

Hecke kernels

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

\( T_{3} - 2 \) Copy content Toggle raw display
\( T_{7} - 6 \) 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
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T - 6 \) Copy content Toggle raw display
$11$ \( T + 60 \) Copy content Toggle raw display
$13$ \( T + 50 \) Copy content Toggle raw display
$17$ \( T + 30 \) Copy content Toggle raw display
$19$ \( T + 40 \) Copy content Toggle raw display
$23$ \( T - 178 \) Copy content Toggle raw display
$29$ \( T + 166 \) Copy content Toggle raw display
$31$ \( T - 20 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T + 250 \) Copy content Toggle raw display
$43$ \( T + 142 \) Copy content Toggle raw display
$47$ \( T - 214 \) Copy content Toggle raw display
$53$ \( T + 490 \) Copy content Toggle raw display
$59$ \( T - 800 \) Copy content Toggle raw display
$61$ \( T + 250 \) Copy content Toggle raw display
$67$ \( T - 774 \) Copy content Toggle raw display
$71$ \( T - 100 \) Copy content Toggle raw display
$73$ \( T + 230 \) Copy content Toggle raw display
$79$ \( T + 1320 \) Copy content Toggle raw display
$83$ \( T + 982 \) Copy content Toggle raw display
$89$ \( T - 874 \) Copy content Toggle raw display
$97$ \( T + 310 \) Copy content Toggle raw display
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