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

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

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

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 2 T + 27 T^{2} \)
$5$ \( 1 - 5 T \)
$7$ \( 1 - 6 T + 343 T^{2} \)
$11$ \( 1 + 32 T + 1331 T^{2} \)
$13$ \( 1 - 38 T + 2197 T^{2} \)
$17$ \( 1 - 26 T + 4913 T^{2} \)
$19$ \( 1 + 100 T + 6859 T^{2} \)
$23$ \( 1 + 78 T + 12167 T^{2} \)
$29$ \( 1 - 50 T + 24389 T^{2} \)
$31$ \( 1 + 108 T + 29791 T^{2} \)
$37$ \( 1 + 266 T + 50653 T^{2} \)
$41$ \( 1 - 22 T + 68921 T^{2} \)
$43$ \( 1 + 442 T + 79507 T^{2} \)
$47$ \( 1 + 514 T + 103823 T^{2} \)
$53$ \( 1 + 2 T + 148877 T^{2} \)
$59$ \( 1 + 500 T + 205379 T^{2} \)
$61$ \( 1 - 518 T + 226981 T^{2} \)
$67$ \( 1 + 126 T + 300763 T^{2} \)
$71$ \( 1 - 412 T + 357911 T^{2} \)
$73$ \( 1 + 878 T + 389017 T^{2} \)
$79$ \( 1 - 600 T + 493039 T^{2} \)
$83$ \( 1 + 282 T + 571787 T^{2} \)
$89$ \( 1 + 150 T + 704969 T^{2} \)
$97$ \( 1 - 386 T + 912673 T^{2} \)
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