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 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

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

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 \(T_{3} \) \(\mathstrut +\mathstrut 158 \) acting on \(S_{9}^{\mathrm{new}}(20, [\chi])\).