Properties

Label 20.9.d.a
Level 20
Weight 9
Character orbit 20.d
Self dual Yes
Analytic conductor 8.148
Analytic rank 0
Dimension 1
CM disc. -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 \(2\) and degree \(1\))

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: \( 1 \)
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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
20.d Odd 1 CM by \(\Q(\sqrt{-5}) \) yes

Hecke kernels

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