Properties

Label 320.4.a.g
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 5)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{3} + 5 q^{5} + 6 q^{7} - 23 q^{9} + O(q^{10}) \) \( q - 2 q^{3} + 5 q^{5} + 6 q^{7} - 23 q^{9} - 32 q^{11} + 38 q^{13} - 10 q^{15} + 26 q^{17} - 100 q^{19} - 12 q^{21} - 78 q^{23} + 25 q^{25} + 100 q^{27} + 50 q^{29} - 108 q^{31} + 64 q^{33} + 30 q^{35} - 266 q^{37} - 76 q^{39} + 22 q^{41} - 442 q^{43} - 115 q^{45} - 514 q^{47} - 307 q^{49} - 52 q^{51} - 2 q^{53} - 160 q^{55} + 200 q^{57} - 500 q^{59} + 518 q^{61} - 138 q^{63} + 190 q^{65} - 126 q^{67} + 156 q^{69} + 412 q^{71} - 878 q^{73} - 50 q^{75} - 192 q^{77} + 600 q^{79} + 421 q^{81} - 282 q^{83} + 130 q^{85} - 100 q^{87} - 150 q^{89} + 228 q^{91} + 216 q^{93} - 500 q^{95} + 386 q^{97} + 736 q^{99} + O(q^{100}) \)

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.g 1
4.b odd 2 1 320.4.a.h 1
5.b even 2 1 1600.4.a.bi 1
8.b even 2 1 5.4.a.a 1
8.d odd 2 1 80.4.a.d 1
16.e even 4 2 1280.4.d.e 2
16.f odd 4 2 1280.4.d.l 2
20.d odd 2 1 1600.4.a.s 1
24.f even 2 1 720.4.a.u 1
24.h odd 2 1 45.4.a.d 1
40.e odd 2 1 400.4.a.m 1
40.f even 2 1 25.4.a.c 1
40.i odd 4 2 25.4.b.a 2
40.k even 4 2 400.4.c.k 2
56.h odd 2 1 245.4.a.a 1
56.j odd 6 2 245.4.e.g 2
56.p even 6 2 245.4.e.f 2
72.j odd 6 2 405.4.e.c 2
72.n even 6 2 405.4.e.l 2
88.b odd 2 1 605.4.a.d 1
104.e even 2 1 845.4.a.b 1
120.i odd 2 1 225.4.a.b 1
120.w even 4 2 225.4.b.c 2
136.h even 2 1 1445.4.a.a 1
152.g odd 2 1 1805.4.a.h 1
168.i even 2 1 2205.4.a.q 1
280.c odd 2 1 1225.4.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 8.b even 2 1
25.4.a.c 1 40.f even 2 1
25.4.b.a 2 40.i odd 4 2
45.4.a.d 1 24.h odd 2 1
80.4.a.d 1 8.d odd 2 1
225.4.a.b 1 120.i odd 2 1
225.4.b.c 2 120.w even 4 2
245.4.a.a 1 56.h odd 2 1
245.4.e.f 2 56.p even 6 2
245.4.e.g 2 56.j odd 6 2
320.4.a.g 1 1.a even 1 1 trivial
320.4.a.h 1 4.b odd 2 1
400.4.a.m 1 40.e odd 2 1
400.4.c.k 2 40.k even 4 2
405.4.e.c 2 72.j odd 6 2
405.4.e.l 2 72.n even 6 2
605.4.a.d 1 88.b odd 2 1
720.4.a.u 1 24.f even 2 1
845.4.a.b 1 104.e even 2 1
1225.4.a.k 1 280.c odd 2 1
1280.4.d.e 2 16.e even 4 2
1280.4.d.l 2 16.f odd 4 2
1445.4.a.a 1 136.h even 2 1
1600.4.a.s 1 20.d odd 2 1
1600.4.a.bi 1 5.b even 2 1
1805.4.a.h 1 152.g odd 2 1
2205.4.a.q 1 168.i even 2 1

Hecke kernels

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

\( T_{3} + 2 \)
\( T_{7} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 2 + T \)
$5$ \( -5 + T \)
$7$ \( -6 + T \)
$11$ \( 32 + T \)
$13$ \( -38 + T \)
$17$ \( -26 + T \)
$19$ \( 100 + T \)
$23$ \( 78 + T \)
$29$ \( -50 + T \)
$31$ \( 108 + T \)
$37$ \( 266 + T \)
$41$ \( -22 + T \)
$43$ \( 442 + T \)
$47$ \( 514 + T \)
$53$ \( 2 + T \)
$59$ \( 500 + T \)
$61$ \( -518 + T \)
$67$ \( 126 + T \)
$71$ \( -412 + T \)
$73$ \( 878 + T \)
$79$ \( -600 + T \)
$83$ \( 282 + T \)
$89$ \( 150 + T \)
$97$ \( -386 + T \)
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