Properties

Label 20.4.a.a
Level 20
Weight 4
Character orbit 20.a
Self dual yes
Analytic conductor 1.180
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 20.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.18003820011\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{3} + 5q^{5} - 16q^{7} - 11q^{9} + O(q^{10}) \) \( q + 4q^{3} + 5q^{5} - 16q^{7} - 11q^{9} - 60q^{11} + 86q^{13} + 20q^{15} + 18q^{17} + 44q^{19} - 64q^{21} + 48q^{23} + 25q^{25} - 152q^{27} - 186q^{29} + 176q^{31} - 240q^{33} - 80q^{35} + 254q^{37} + 344q^{39} + 186q^{41} - 100q^{43} - 55q^{45} + 168q^{47} - 87q^{49} + 72q^{51} - 498q^{53} - 300q^{55} + 176q^{57} - 252q^{59} - 58q^{61} + 176q^{63} + 430q^{65} - 1036q^{67} + 192q^{69} + 168q^{71} + 506q^{73} + 100q^{75} + 960q^{77} + 272q^{79} - 311q^{81} + 948q^{83} + 90q^{85} - 744q^{87} - 1014q^{89} - 1376q^{91} + 704q^{93} + 220q^{95} - 766q^{97} + 660q^{99} + O(q^{100}) \)

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 4.00000 0 5.00000 0 −16.0000 0 −11.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 20.4.a.a 1
3.b odd 2 1 180.4.a.a 1
4.b odd 2 1 80.4.a.c 1
5.b even 2 1 100.4.a.a 1
5.c odd 4 2 100.4.c.a 2
7.b odd 2 1 980.4.a.c 1
7.c even 3 2 980.4.i.e 2
7.d odd 6 2 980.4.i.n 2
8.b even 2 1 320.4.a.d 1
8.d odd 2 1 320.4.a.k 1
9.c even 3 2 1620.4.i.d 2
9.d odd 6 2 1620.4.i.j 2
11.b odd 2 1 2420.4.a.d 1
12.b even 2 1 720.4.a.k 1
15.d odd 2 1 900.4.a.m 1
15.e even 4 2 900.4.d.k 2
16.e even 4 2 1280.4.d.n 2
16.f odd 4 2 1280.4.d.c 2
20.d odd 2 1 400.4.a.o 1
20.e even 4 2 400.4.c.j 2
40.e odd 2 1 1600.4.a.p 1
40.f even 2 1 1600.4.a.bl 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.a.a 1 1.a even 1 1 trivial
80.4.a.c 1 4.b odd 2 1
100.4.a.a 1 5.b even 2 1
100.4.c.a 2 5.c odd 4 2
180.4.a.a 1 3.b odd 2 1
320.4.a.d 1 8.b even 2 1
320.4.a.k 1 8.d odd 2 1
400.4.a.o 1 20.d odd 2 1
400.4.c.j 2 20.e even 4 2
720.4.a.k 1 12.b even 2 1
900.4.a.m 1 15.d odd 2 1
900.4.d.k 2 15.e even 4 2
980.4.a.c 1 7.b odd 2 1
980.4.i.e 2 7.c even 3 2
980.4.i.n 2 7.d odd 6 2
1280.4.d.c 2 16.f odd 4 2
1280.4.d.n 2 16.e even 4 2
1600.4.a.p 1 40.e odd 2 1
1600.4.a.bl 1 40.f even 2 1
1620.4.i.d 2 9.c even 3 2
1620.4.i.j 2 9.d odd 6 2
2420.4.a.d 1 11.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(\Gamma_0(20))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 4 T + 27 T^{2} \)
$5$ \( 1 - 5 T \)
$7$ \( 1 + 16 T + 343 T^{2} \)
$11$ \( 1 + 60 T + 1331 T^{2} \)
$13$ \( 1 - 86 T + 2197 T^{2} \)
$17$ \( 1 - 18 T + 4913 T^{2} \)
$19$ \( 1 - 44 T + 6859 T^{2} \)
$23$ \( 1 - 48 T + 12167 T^{2} \)
$29$ \( 1 + 186 T + 24389 T^{2} \)
$31$ \( 1 - 176 T + 29791 T^{2} \)
$37$ \( 1 - 254 T + 50653 T^{2} \)
$41$ \( 1 - 186 T + 68921 T^{2} \)
$43$ \( 1 + 100 T + 79507 T^{2} \)
$47$ \( 1 - 168 T + 103823 T^{2} \)
$53$ \( 1 + 498 T + 148877 T^{2} \)
$59$ \( 1 + 252 T + 205379 T^{2} \)
$61$ \( 1 + 58 T + 226981 T^{2} \)
$67$ \( 1 + 1036 T + 300763 T^{2} \)
$71$ \( 1 - 168 T + 357911 T^{2} \)
$73$ \( 1 - 506 T + 389017 T^{2} \)
$79$ \( 1 - 272 T + 493039 T^{2} \)
$83$ \( 1 - 948 T + 571787 T^{2} \)
$89$ \( 1 + 1014 T + 704969 T^{2} \)
$97$ \( 1 + 766 T + 912673 T^{2} \)
show more
show less