Properties

Label 20.9.d.b
Level 20
Weight 9
Character orbit 20.d
Self dual yes
Analytic conductor 8.148
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: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 20.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(8.14757220122\)
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

\(f(q)\) \(=\) \( q + 16q^{2} - 158q^{3} + 256q^{4} + 625q^{5} - 2528q^{6} + 1922q^{7} + 4096q^{8} + 18403q^{9} + O(q^{10}) \) \( q + 16q^{2} - 158q^{3} + 256q^{4} + 625q^{5} - 2528q^{6} + 1922q^{7} + 4096q^{8} + 18403q^{9} + 10000q^{10} - 40448q^{12} + 30752q^{14} - 98750q^{15} + 65536q^{16} + 294448q^{18} + 160000q^{20} - 303676q^{21} + 211202q^{23} - 647168q^{24} + 390625q^{25} - 1871036q^{27} + 492032q^{28} + 20642q^{29} - 1580000q^{30} + 1048576q^{32} + 1201250q^{35} + 4711168q^{36} + 2560000q^{40} - 5419198q^{41} - 4858816q^{42} - 2519518q^{43} + 11501875q^{45} + 3379232q^{46} + 9618242q^{47} - 10354688q^{48} - 2070717q^{49} + 6250000q^{50} - 29936576q^{54} + 7872512q^{56} + 330272q^{58} - 25280000q^{60} - 11061598q^{61} + 35370566q^{63} + 16777216q^{64} - 20249758q^{67} - 33369916q^{69} + 19220000q^{70} + 75378688q^{72} - 61718750q^{75} + 40960000q^{80} + 174881605q^{81} - 86707168q^{82} - 30884638q^{83} - 77741056q^{84} - 40312288q^{86} - 3261436q^{87} - 106804798q^{89} + 184030000q^{90} + 54067712q^{92} + 153891872q^{94} - 165675008q^{96} - 33131472q^{98} + O(q^{100}) \)

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
16.0000 −158.000 256.000 625.000 −2528.00 1922.00 4096.00 18403.0 10000.0
\(n\): 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 20.9.d.b yes 1
4.b odd 2 1 20.9.d.a 1
5.b even 2 1 20.9.d.a 1
5.c odd 4 2 100.9.b.b 2
8.b even 2 1 320.9.h.b 1
8.d odd 2 1 320.9.h.a 1
20.d odd 2 1 CM 20.9.d.b yes 1
20.e even 4 2 100.9.b.b 2
40.e odd 2 1 320.9.h.b 1
40.f even 2 1 320.9.h.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.9.d.a 1 4.b odd 2 1
20.9.d.a 1 5.b even 2 1
20.9.d.b yes 1 1.a even 1 1 trivial
20.9.d.b yes 1 20.d odd 2 1 CM
100.9.b.b 2 5.c odd 4 2
100.9.b.b 2 20.e even 4 2
320.9.h.a 1 8.d odd 2 1
320.9.h.a 1 40.f even 2 1
320.9.h.b 1 8.b even 2 1
320.9.h.b 1 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 158 \) acting on \(S_{9}^{\mathrm{new}}(20, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 16 T \)
$3$ \( 1 + 158 T + 6561 T^{2} \)
$5$ \( 1 - 625 T \)
$7$ \( 1 - 1922 T + 5764801 T^{2} \)
$11$ \( ( 1 - 14641 T )( 1 + 14641 T ) \)
$13$ \( ( 1 - 28561 T )( 1 + 28561 T ) \)
$17$ \( ( 1 - 83521 T )( 1 + 83521 T ) \)
$19$ \( ( 1 - 130321 T )( 1 + 130321 T ) \)
$23$ \( 1 - 211202 T + 78310985281 T^{2} \)
$29$ \( 1 - 20642 T + 500246412961 T^{2} \)
$31$ \( ( 1 - 923521 T )( 1 + 923521 T ) \)
$37$ \( ( 1 - 1874161 T )( 1 + 1874161 T ) \)
$41$ \( 1 + 5419198 T + 7984925229121 T^{2} \)
$43$ \( 1 + 2519518 T + 11688200277601 T^{2} \)
$47$ \( 1 - 9618242 T + 23811286661761 T^{2} \)
$53$ \( ( 1 - 7890481 T )( 1 + 7890481 T ) \)
$59$ \( ( 1 - 12117361 T )( 1 + 12117361 T ) \)
$61$ \( 1 + 11061598 T + 191707312997281 T^{2} \)
$67$ \( 1 + 20249758 T + 406067677556641 T^{2} \)
$71$ \( ( 1 - 25411681 T )( 1 + 25411681 T ) \)
$73$ \( ( 1 - 28398241 T )( 1 + 28398241 T ) \)
$79$ \( ( 1 - 38950081 T )( 1 + 38950081 T ) \)
$83$ \( 1 + 30884638 T + 2252292232139041 T^{2} \)
$89$ \( 1 + 106804798 T + 3936588805702081 T^{2} \)
$97$ \( ( 1 - 88529281 T )( 1 + 88529281 T ) \)
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