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

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

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
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 \(T_{3} \) \(\mathstrut +\mathstrut 2 \) acting on \(S_{5}^{\mathrm{new}}(20, [\chi])\).