Properties

Label 4.20.a.a
Level 4
Weight 20
Character orbit 4.a
Self dual Yes
Analytic conductor 9.153
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 20 \)
Character orbit: \([\chi]\) = 4.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(9.15266786226\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 36q^{3} \) \(\mathstrut -\mathstrut 196290q^{5} \) \(\mathstrut -\mathstrut 35905576q^{7} \) \(\mathstrut -\mathstrut 1162260171q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 36q^{3} \) \(\mathstrut -\mathstrut 196290q^{5} \) \(\mathstrut -\mathstrut 35905576q^{7} \) \(\mathstrut -\mathstrut 1162260171q^{9} \) \(\mathstrut -\mathstrut 12016099980q^{11} \) \(\mathstrut -\mathstrut 45529656874q^{13} \) \(\mathstrut +\mathstrut 7066440q^{15} \) \(\mathstrut +\mathstrut 496563248178q^{17} \) \(\mathstrut +\mathstrut 1410273986444q^{19} \) \(\mathstrut +\mathstrut 1292600736q^{21} \) \(\mathstrut -\mathstrut 7039745388792q^{23} \) \(\mathstrut -\mathstrut 19034956564025q^{25} \) \(\mathstrut +\mathstrut 83682778968q^{27} \) \(\mathstrut +\mathstrut 38996890912134q^{29} \) \(\mathstrut +\mathstrut 173641323230816q^{31} \) \(\mathstrut +\mathstrut 432579599280q^{33} \) \(\mathstrut +\mathstrut 7047905513040q^{35} \) \(\mathstrut -\mathstrut 1108106825662306q^{37} \) \(\mathstrut +\mathstrut 1639067647464q^{39} \) \(\mathstrut -\mathstrut 1444509198124614q^{41} \) \(\mathstrut +\mathstrut 4646075748354260q^{43} \) \(\mathstrut +\mathstrut 228140048965590q^{45} \) \(\mathstrut +\mathstrut 8950457686524048q^{47} \) \(\mathstrut -\mathstrut 10109684797481367q^{49} \) \(\mathstrut -\mathstrut 17876276934408q^{51} \) \(\mathstrut -\mathstrut 32948524384463538q^{53} \) \(\mathstrut +\mathstrut 2358640265074200q^{55} \) \(\mathstrut -\mathstrut 50769863511984q^{57} \) \(\mathstrut +\mathstrut 36999205673523588q^{59} \) \(\mathstrut +\mathstrut 82929105285760742q^{61} \) \(\mathstrut +\mathstrut 41731620901613496q^{63} \) \(\mathstrut +\mathstrut 8937016347797460q^{65} \) \(\mathstrut -\mathstrut 186668590860047716q^{67} \) \(\mathstrut +\mathstrut 253430833996512q^{69} \) \(\mathstrut -\mathstrut 596514630027659112q^{71} \) \(\mathstrut +\mathstrut 310786775495585306q^{73} \) \(\mathstrut +\mathstrut 685258436304900q^{75} \) \(\mathstrut +\mathstrut 431444991055488480q^{77} \) \(\mathstrut +\mathstrut 700397513485701872q^{79} \) \(\mathstrut +\mathstrut 1350847198802088009q^{81} \) \(\mathstrut -\mathstrut 1357882121724855732q^{83} \) \(\mathstrut -\mathstrut 97470399984859620q^{85} \) \(\mathstrut -\mathstrut 1403888072836824q^{87} \) \(\mathstrut -\mathstrut 5991411253779123894q^{89} \) \(\mathstrut +\mathstrut 1634768555143329424q^{91} \) \(\mathstrut -\mathstrut 6251087636309376q^{93} \) \(\mathstrut -\mathstrut 276822680799092760q^{95} \) \(\mathstrut +\mathstrut 4531118407744664354q^{97} \) \(\mathstrut +\mathstrut 13965834417507896580q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −36.0000 0 −196290. 0 −3.59056e7 0 −1.16226e9 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
\(2\) \(-1\)

Hecke kernels

There are no other newforms in \(S_{20}^{\mathrm{new}}(\Gamma_0(4))\).