Properties

Label 16.12.a.b
Level 16
Weight 12
Character orbit 16.a
Self dual Yes
Analytic conductor 12.293
Analytic rank 0
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: \(0\)
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

\(f(q)\) \(=\) \( q + 36q^{3} - 3490q^{5} + 55464q^{7} - 175851q^{9} + O(q^{10}) \) \( q + 36q^{3} - 3490q^{5} + 55464q^{7} - 175851q^{9} + 597004q^{11} + 1373878q^{13} - 125640q^{15} + 10140850q^{17} + 7297396q^{19} + 1996704q^{21} + 32057464q^{23} - 36648025q^{25} - 12707928q^{27} - 13605402q^{29} - 233160800q^{31} + 21492144q^{33} - 193569360q^{35} - 257786178q^{37} + 49459608q^{39} - 221438598q^{41} + 1697758892q^{43} + 613719990q^{45} - 527509392q^{47} + 1098928553q^{49} + 365070600q^{51} + 3277379822q^{53} - 2083543960q^{55} + 262706256q^{57} + 3001908988q^{59} - 11630023610q^{61} - 9753399864q^{63} - 4794834220q^{65} + 17189000548q^{67} + 1154068704q^{69} - 26169539608q^{71} - 7039021094q^{73} - 1319328900q^{75} + 33112229856q^{77} + 4199910416q^{79} + 30693991689q^{81} + 39739936436q^{83} - 35391566500q^{85} - 489794472q^{87} + 10565331594q^{89} + 76200769392q^{91} - 8393788800q^{93} - 25467912040q^{95} - 69851645662q^{97} - 104983750404q^{99} + O(q^{100}) \)

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 36.0000 0 −3490.00 0 55464.0 0 −175851. 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 \( T_{3} - 36 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(16))\).