Properties

Label 16.12.a.a
Level 16
Weight 12
Character orbit 16.a
Self dual Yes
Analytic conductor 12.293
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 16 = 2^{4} \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 16.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(12.293490889\)
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

\(f(q)\) \(=\) \( q - 252q^{3} + 4830q^{5} + 16744q^{7} - 113643q^{9} + O(q^{10}) \) \( q - 252q^{3} + 4830q^{5} + 16744q^{7} - 113643q^{9} - 534612q^{11} - 577738q^{13} - 1217160q^{15} - 6905934q^{17} - 10661420q^{19} - 4219488q^{21} - 18643272q^{23} - 25499225q^{25} + 73279080q^{27} + 128406630q^{29} + 52843168q^{31} + 134722224q^{33} + 80873520q^{35} - 182213314q^{37} + 145589976q^{39} + 308120442q^{41} + 17125708q^{43} - 548895690q^{45} - 2687348496q^{47} - 1696965207q^{49} + 1740295368q^{51} - 1596055698q^{53} - 2582175960q^{55} + 2686677840q^{57} + 5189203740q^{59} + 6956478662q^{61} - 1902838392q^{63} - 2790474540q^{65} + 15481826884q^{67} + 4698104544q^{69} - 9791485272q^{71} + 1463791322q^{73} + 6425804700q^{75} - 8951543328q^{77} - 38116845680q^{79} + 1665188361q^{81} + 29335099668q^{83} - 33355661220q^{85} - 32358470760q^{87} - 24992917110q^{89} - 9673645072q^{91} - 13316478336q^{93} - 51494658600q^{95} + 75013568546q^{97} + 60754911516q^{99} + O(q^{100}) \)

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 −252.000 0 4830.00 0 16744.0 0 −113643. 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

This newform can be constructed as the kernel of the linear operator \( T_{3} + 252 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(16))\).