Properties

Label 320.2.a.b
Level $320$
Weight $2$
Character orbit 320.a
Self dual yes
Analytic conductor $2.555$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(2.55521286468\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q - 2 q^{3} + q^{5} + 2 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{3} + q^{5} + 2 q^{7} + q^{9} - 4 q^{11} + 6 q^{13} - 2 q^{15} + 2 q^{17} + 8 q^{19} - 4 q^{21} + 6 q^{23} + q^{25} + 4 q^{27} + 2 q^{29} - 4 q^{31} + 8 q^{33} + 2 q^{35} - 2 q^{37} - 12 q^{39} - 10 q^{41} - 2 q^{43} + q^{45} + 2 q^{47} - 3 q^{49} - 4 q^{51} - 2 q^{53} - 4 q^{55} - 16 q^{57} - 2 q^{61} + 2 q^{63} + 6 q^{65} - 6 q^{67} - 12 q^{69} + 12 q^{71} + 10 q^{73} - 2 q^{75} - 8 q^{77} + 8 q^{79} - 11 q^{81} - 10 q^{83} + 2 q^{85} - 4 q^{87} - 6 q^{89} + 12 q^{91} + 8 q^{93} + 8 q^{95} + 10 q^{97} - 4 q^{99}+O(q^{100}) \) 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 1.00000 0 2.00000 0 1.00000 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 320.2.a.b 1
3.b odd 2 1 2880.2.a.o 1
4.b odd 2 1 320.2.a.e 1
5.b even 2 1 1600.2.a.t 1
5.c odd 4 2 1600.2.c.c 2
8.b even 2 1 160.2.a.b yes 1
8.d odd 2 1 160.2.a.a 1
12.b even 2 1 2880.2.a.d 1
16.e even 4 2 1280.2.d.b 2
16.f odd 4 2 1280.2.d.h 2
20.d odd 2 1 1600.2.a.e 1
20.e even 4 2 1600.2.c.f 2
24.f even 2 1 1440.2.a.i 1
24.h odd 2 1 1440.2.a.l 1
40.e odd 2 1 800.2.a.i 1
40.f even 2 1 800.2.a.a 1
40.i odd 4 2 800.2.c.b 2
40.k even 4 2 800.2.c.a 2
56.e even 2 1 7840.2.a.w 1
56.h odd 2 1 7840.2.a.e 1
120.i odd 2 1 7200.2.a.l 1
120.m even 2 1 7200.2.a.bp 1
120.q odd 4 2 7200.2.f.w 2
120.w even 4 2 7200.2.f.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.a.a 1 8.d odd 2 1
160.2.a.b yes 1 8.b even 2 1
320.2.a.b 1 1.a even 1 1 trivial
320.2.a.e 1 4.b odd 2 1
800.2.a.a 1 40.f even 2 1
800.2.a.i 1 40.e odd 2 1
800.2.c.a 2 40.k even 4 2
800.2.c.b 2 40.i odd 4 2
1280.2.d.b 2 16.e even 4 2
1280.2.d.h 2 16.f odd 4 2
1440.2.a.i 1 24.f even 2 1
1440.2.a.l 1 24.h odd 2 1
1600.2.a.e 1 20.d odd 2 1
1600.2.a.t 1 5.b even 2 1
1600.2.c.c 2 5.c odd 4 2
1600.2.c.f 2 20.e even 4 2
2880.2.a.d 1 12.b even 2 1
2880.2.a.o 1 3.b odd 2 1
7200.2.a.l 1 120.i odd 2 1
7200.2.a.bp 1 120.m even 2 1
7200.2.f.g 2 120.w even 4 2
7200.2.f.w 2 120.q odd 4 2
7840.2.a.e 1 56.h odd 2 1
7840.2.a.w 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(320))\):

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{7} - 2 \) 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
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T + 4 \) Copy content Toggle raw display
$13$ \( T - 6 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T - 8 \) Copy content Toggle raw display
$23$ \( T - 6 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T + 10 \) Copy content Toggle raw display
$43$ \( T + 2 \) Copy content Toggle raw display
$47$ \( T - 2 \) Copy content Toggle raw display
$53$ \( T + 2 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T + 6 \) Copy content Toggle raw display
$71$ \( T - 12 \) Copy content Toggle raw display
$73$ \( T - 10 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T + 10 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
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