Properties

Label 20.12.a.a
Level 20
Weight 12
Character orbit 20.a
Self dual Yes
Analytic conductor 15.367
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 20.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(15.3668636112\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 306q^{3} \) \(\mathstrut -\mathstrut 3125q^{5} \) \(\mathstrut -\mathstrut 32074q^{7} \) \(\mathstrut -\mathstrut 83511q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 306q^{3} \) \(\mathstrut -\mathstrut 3125q^{5} \) \(\mathstrut -\mathstrut 32074q^{7} \) \(\mathstrut -\mathstrut 83511q^{9} \) \(\mathstrut -\mathstrut 5280q^{11} \) \(\mathstrut +\mathstrut 227834q^{13} \) \(\mathstrut -\mathstrut 956250q^{15} \) \(\mathstrut -\mathstrut 5097318q^{17} \) \(\mathstrut -\mathstrut 16279036q^{19} \) \(\mathstrut -\mathstrut 9814644q^{21} \) \(\mathstrut -\mathstrut 33055038q^{23} \) \(\mathstrut +\mathstrut 9765625q^{25} \) \(\mathstrut -\mathstrut 79761348q^{27} \) \(\mathstrut -\mathstrut 2112786q^{29} \) \(\mathstrut +\mathstrut 91337396q^{31} \) \(\mathstrut -\mathstrut 1615680q^{33} \) \(\mathstrut +\mathstrut 100231250q^{35} \) \(\mathstrut -\mathstrut 109132054q^{37} \) \(\mathstrut +\mathstrut 69717204q^{39} \) \(\mathstrut +\mathstrut 1202079126q^{41} \) \(\mathstrut +\mathstrut 1112512490q^{43} \) \(\mathstrut +\mathstrut 260971875q^{45} \) \(\mathstrut +\mathstrut 507908142q^{47} \) \(\mathstrut -\mathstrut 948585267q^{49} \) \(\mathstrut -\mathstrut 1559779308q^{51} \) \(\mathstrut -\mathstrut 1900361502q^{53} \) \(\mathstrut +\mathstrut 16500000q^{55} \) \(\mathstrut -\mathstrut 4981385016q^{57} \) \(\mathstrut +\mathstrut 2802066708q^{59} \) \(\mathstrut -\mathstrut 9660996838q^{61} \) \(\mathstrut +\mathstrut 2678531814q^{63} \) \(\mathstrut -\mathstrut 711981250q^{65} \) \(\mathstrut +\mathstrut 8370234446q^{67} \) \(\mathstrut -\mathstrut 10114841628q^{69} \) \(\mathstrut -\mathstrut 12173973252q^{71} \) \(\mathstrut +\mathstrut 18053518034q^{73} \) \(\mathstrut +\mathstrut 2988281250q^{75} \) \(\mathstrut +\mathstrut 169350720q^{77} \) \(\mathstrut +\mathstrut 22759013912q^{79} \) \(\mathstrut -\mathstrut 9613249371q^{81} \) \(\mathstrut +\mathstrut 65367228042q^{83} \) \(\mathstrut +\mathstrut 15929118750q^{85} \) \(\mathstrut -\mathstrut 646512516q^{87} \) \(\mathstrut -\mathstrut 13526251734q^{89} \) \(\mathstrut -\mathstrut 7307547716q^{91} \) \(\mathstrut +\mathstrut 27949243176q^{93} \) \(\mathstrut +\mathstrut 50871987500q^{95} \) \(\mathstrut -\mathstrut 155553294334q^{97} \) \(\mathstrut +\mathstrut 440938080q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 306.000 0 −3125.00 0 −32074.0 0 −83511.0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3} \) \(\mathstrut -\mathstrut 306 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(20))\).