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

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

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 −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 \(T_{3} \) \(\mathstrut +\mathstrut 252 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(16))\).