Properties

Label 20.5.d.b
Level 20
Weight 5
Character orbit 20.d
Self dual Yes
Analytic conductor 2.067
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: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 20.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(2.06739926168\)
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

\(f(q)\) \(=\) \( q + 4q^{2} - 2q^{3} + 16q^{4} + 25q^{5} - 8q^{6} - 82q^{7} + 64q^{8} - 77q^{9} + O(q^{10}) \) \( q + 4q^{2} - 2q^{3} + 16q^{4} + 25q^{5} - 8q^{6} - 82q^{7} + 64q^{8} - 77q^{9} + 100q^{10} - 32q^{12} - 328q^{14} - 50q^{15} + 256q^{16} - 308q^{18} + 400q^{20} + 164q^{21} + 878q^{23} - 128q^{24} + 625q^{25} + 316q^{27} - 1312q^{28} - 1198q^{29} - 200q^{30} + 1024q^{32} - 2050q^{35} - 1232q^{36} + 1600q^{40} + 482q^{41} + 656q^{42} + 2078q^{43} - 1925q^{45} + 3512q^{46} - 4402q^{47} - 512q^{48} + 4323q^{49} + 2500q^{50} + 1264q^{54} - 5248q^{56} - 4792q^{58} - 800q^{60} - 4078q^{61} + 6314q^{63} + 4096q^{64} + 4478q^{67} - 1756q^{69} - 8200q^{70} - 4928q^{72} - 1250q^{75} + 6400q^{80} + 5605q^{81} + 1928q^{82} - 8002q^{83} + 2624q^{84} + 8312q^{86} + 2396q^{87} + 4322q^{89} - 7700q^{90} + 14048q^{92} - 17608q^{94} - 2048q^{96} + 17292q^{98} + O(q^{100}) \)

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
4.00000 −2.00000 16.0000 25.0000 −8.00000 −82.0000 64.0000 −77.0000 100.000
\(n\): 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 \( T_{3} + 2 \) acting on \(S_{5}^{\mathrm{new}}(20, [\chi])\).