Properties

Label 20.8.a.a
Level 20
Weight 8
Character orbit 20.a
Self dual yes
Analytic conductor 6.248
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 20.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.24770050968\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - 6q^{3} - 125q^{5} - 706q^{7} - 2151q^{9} + O(q^{10}) \) \( q - 6q^{3} - 125q^{5} - 706q^{7} - 2151q^{9} - 3840q^{11} - 4054q^{13} + 750q^{15} + 858q^{17} + 21044q^{19} + 4236q^{21} + 85338q^{23} + 15625q^{25} + 26028q^{27} - 83106q^{29} - 145564q^{31} + 23040q^{33} + 88250q^{35} - 498886q^{37} + 24324q^{39} - 689514q^{41} + 867890q^{43} + 268875q^{45} + 235638q^{47} - 325107q^{49} - 5148q^{51} + 1835442q^{53} + 480000q^{55} - 126264q^{57} + 629508q^{59} - 2667958q^{61} + 1518606q^{63} + 506750q^{65} - 3373306q^{67} - 512028q^{69} - 2600052q^{71} - 1628494q^{73} - 93750q^{75} + 2711040q^{77} - 4243528q^{79} + 4548069q^{81} + 1251378q^{83} - 107250q^{85} + 498636q^{87} + 6299466q^{89} + 2862124q^{91} + 873384q^{93} - 2630500q^{95} + 3976514q^{97} + 8259840q^{99} + O(q^{100}) \)

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 −6.00000 0 −125.000 0 −706.000 0 −2151.00 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 20.8.a.a 1
3.b odd 2 1 180.8.a.c 1
4.b odd 2 1 80.8.a.b 1
5.b even 2 1 100.8.a.a 1
5.c odd 4 2 100.8.c.a 2
8.b even 2 1 320.8.a.e 1
8.d odd 2 1 320.8.a.d 1
20.d odd 2 1 400.8.a.j 1
20.e even 4 2 400.8.c.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.8.a.a 1 1.a even 1 1 trivial
80.8.a.b 1 4.b odd 2 1
100.8.a.a 1 5.b even 2 1
100.8.c.a 2 5.c odd 4 2
180.8.a.c 1 3.b odd 2 1
320.8.a.d 1 8.d odd 2 1
320.8.a.e 1 8.b even 2 1
400.8.a.j 1 20.d odd 2 1
400.8.c.l 2 20.e even 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 6 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(20))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + 6 T + 2187 T^{2} \)
$5$ \( 1 + 125 T \)
$7$ \( 1 + 706 T + 823543 T^{2} \)
$11$ \( 1 + 3840 T + 19487171 T^{2} \)
$13$ \( 1 + 4054 T + 62748517 T^{2} \)
$17$ \( 1 - 858 T + 410338673 T^{2} \)
$19$ \( 1 - 21044 T + 893871739 T^{2} \)
$23$ \( 1 - 85338 T + 3404825447 T^{2} \)
$29$ \( 1 + 83106 T + 17249876309 T^{2} \)
$31$ \( 1 + 145564 T + 27512614111 T^{2} \)
$37$ \( 1 + 498886 T + 94931877133 T^{2} \)
$41$ \( 1 + 689514 T + 194754273881 T^{2} \)
$43$ \( 1 - 867890 T + 271818611107 T^{2} \)
$47$ \( 1 - 235638 T + 506623120463 T^{2} \)
$53$ \( 1 - 1835442 T + 1174711139837 T^{2} \)
$59$ \( 1 - 629508 T + 2488651484819 T^{2} \)
$61$ \( 1 + 2667958 T + 3142742836021 T^{2} \)
$67$ \( 1 + 3373306 T + 6060711605323 T^{2} \)
$71$ \( 1 + 2600052 T + 9095120158391 T^{2} \)
$73$ \( 1 + 1628494 T + 11047398519097 T^{2} \)
$79$ \( 1 + 4243528 T + 19203908986159 T^{2} \)
$83$ \( 1 - 1251378 T + 27136050989627 T^{2} \)
$89$ \( 1 - 6299466 T + 44231334895529 T^{2} \)
$97$ \( 1 - 3976514 T + 80798284478113 T^{2} \)
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