Properties

Label 20.3.d.a
Level 20
Weight 3
Character orbit 20.d
Self dual yes
Analytic conductor 0.545
Analytic rank 0
Dimension 1
CM discriminant -20
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 20.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.544960528721\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} + 4q^{3} + 4q^{4} - 5q^{5} - 8q^{6} - 4q^{7} - 8q^{8} + 7q^{9} + O(q^{10}) \) \( q - 2q^{2} + 4q^{3} + 4q^{4} - 5q^{5} - 8q^{6} - 4q^{7} - 8q^{8} + 7q^{9} + 10q^{10} + 16q^{12} + 8q^{14} - 20q^{15} + 16q^{16} - 14q^{18} - 20q^{20} - 16q^{21} + 44q^{23} - 32q^{24} + 25q^{25} - 8q^{27} - 16q^{28} - 22q^{29} + 40q^{30} - 32q^{32} + 20q^{35} + 28q^{36} + 40q^{40} + 62q^{41} + 32q^{42} - 76q^{43} - 35q^{45} - 88q^{46} - 4q^{47} + 64q^{48} - 33q^{49} - 50q^{50} + 16q^{54} + 32q^{56} + 44q^{58} - 80q^{60} - 58q^{61} - 28q^{63} + 64q^{64} + 116q^{67} + 176q^{69} - 40q^{70} - 56q^{72} + 100q^{75} - 80q^{80} - 95q^{81} - 124q^{82} - 76q^{83} - 64q^{84} + 152q^{86} - 88q^{87} - 142q^{89} + 70q^{90} + 176q^{92} + 8q^{94} - 128q^{96} + 66q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0
−2.00000 4.00000 4.00000 −5.00000 −8.00000 −4.00000 −8.00000 7.00000 10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.3.d.a 1
3.b odd 2 1 180.3.f.b 1
4.b odd 2 1 20.3.d.b yes 1
5.b even 2 1 20.3.d.b yes 1
5.c odd 4 2 100.3.b.c 2
8.b even 2 1 320.3.h.a 1
8.d odd 2 1 320.3.h.b 1
12.b even 2 1 180.3.f.a 1
15.d odd 2 1 180.3.f.a 1
15.e even 4 2 900.3.c.h 2
16.e even 4 2 1280.3.e.c 2
16.f odd 4 2 1280.3.e.b 2
20.d odd 2 1 CM 20.3.d.a 1
20.e even 4 2 100.3.b.c 2
40.e odd 2 1 320.3.h.a 1
40.f even 2 1 320.3.h.b 1
40.i odd 4 2 1600.3.b.f 2
40.k even 4 2 1600.3.b.f 2
60.h even 2 1 180.3.f.b 1
60.l odd 4 2 900.3.c.h 2
80.k odd 4 2 1280.3.e.c 2
80.q even 4 2 1280.3.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.d.a 1 1.a even 1 1 trivial
20.3.d.a 1 20.d odd 2 1 CM
20.3.d.b yes 1 4.b odd 2 1
20.3.d.b yes 1 5.b even 2 1
100.3.b.c 2 5.c odd 4 2
100.3.b.c 2 20.e even 4 2
180.3.f.a 1 12.b even 2 1
180.3.f.a 1 15.d odd 2 1
180.3.f.b 1 3.b odd 2 1
180.3.f.b 1 60.h even 2 1
320.3.h.a 1 8.b even 2 1
320.3.h.a 1 40.e odd 2 1
320.3.h.b 1 8.d odd 2 1
320.3.h.b 1 40.f even 2 1
900.3.c.h 2 15.e even 4 2
900.3.c.h 2 60.l odd 4 2
1280.3.e.b 2 16.f odd 4 2
1280.3.e.b 2 80.q even 4 2
1280.3.e.c 2 16.e even 4 2
1280.3.e.c 2 80.k odd 4 2
1600.3.b.f 2 40.i odd 4 2
1600.3.b.f 2 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 4 \) acting on \(S_{3}^{\mathrm{new}}(20, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T \)
$3$ \( 1 - 4 T + 9 T^{2} \)
$5$ \( 1 + 5 T \)
$7$ \( 1 + 4 T + 49 T^{2} \)
$11$ \( ( 1 - 11 T )( 1 + 11 T ) \)
$13$ \( ( 1 - 13 T )( 1 + 13 T ) \)
$17$ \( ( 1 - 17 T )( 1 + 17 T ) \)
$19$ \( ( 1 - 19 T )( 1 + 19 T ) \)
$23$ \( 1 - 44 T + 529 T^{2} \)
$29$ \( 1 + 22 T + 841 T^{2} \)
$31$ \( ( 1 - 31 T )( 1 + 31 T ) \)
$37$ \( ( 1 - 37 T )( 1 + 37 T ) \)
$41$ \( 1 - 62 T + 1681 T^{2} \)
$43$ \( 1 + 76 T + 1849 T^{2} \)
$47$ \( 1 + 4 T + 2209 T^{2} \)
$53$ \( ( 1 - 53 T )( 1 + 53 T ) \)
$59$ \( ( 1 - 59 T )( 1 + 59 T ) \)
$61$ \( 1 + 58 T + 3721 T^{2} \)
$67$ \( 1 - 116 T + 4489 T^{2} \)
$71$ \( ( 1 - 71 T )( 1 + 71 T ) \)
$73$ \( ( 1 - 73 T )( 1 + 73 T ) \)
$79$ \( ( 1 - 79 T )( 1 + 79 T ) \)
$83$ \( 1 + 76 T + 6889 T^{2} \)
$89$ \( 1 + 142 T + 7921 T^{2} \)
$97$ \( ( 1 - 97 T )( 1 + 97 T ) \)
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