Properties

Label 14.12.a.a
Level 14
Weight 12
Character orbit 14.a
Self dual Yes
Analytic conductor 10.757
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 14 = 2 \cdot 7 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 14.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(10.7568045278\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 32q^{2} \) \(\mathstrut -\mathstrut 396q^{3} \) \(\mathstrut +\mathstrut 1024q^{4} \) \(\mathstrut +\mathstrut 7350q^{5} \) \(\mathstrut +\mathstrut 12672q^{6} \) \(\mathstrut +\mathstrut 16807q^{7} \) \(\mathstrut -\mathstrut 32768q^{8} \) \(\mathstrut -\mathstrut 20331q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 32q^{2} \) \(\mathstrut -\mathstrut 396q^{3} \) \(\mathstrut +\mathstrut 1024q^{4} \) \(\mathstrut +\mathstrut 7350q^{5} \) \(\mathstrut +\mathstrut 12672q^{6} \) \(\mathstrut +\mathstrut 16807q^{7} \) \(\mathstrut -\mathstrut 32768q^{8} \) \(\mathstrut -\mathstrut 20331q^{9} \) \(\mathstrut -\mathstrut 235200q^{10} \) \(\mathstrut -\mathstrut 108780q^{11} \) \(\mathstrut -\mathstrut 405504q^{12} \) \(\mathstrut -\mathstrut 635842q^{13} \) \(\mathstrut -\mathstrut 537824q^{14} \) \(\mathstrut -\mathstrut 2910600q^{15} \) \(\mathstrut +\mathstrut 1048576q^{16} \) \(\mathstrut -\mathstrut 9225918q^{17} \) \(\mathstrut +\mathstrut 650592q^{18} \) \(\mathstrut -\mathstrut 7555372q^{19} \) \(\mathstrut +\mathstrut 7526400q^{20} \) \(\mathstrut -\mathstrut 6655572q^{21} \) \(\mathstrut +\mathstrut 3480960q^{22} \) \(\mathstrut +\mathstrut 26489400q^{23} \) \(\mathstrut +\mathstrut 12976128q^{24} \) \(\mathstrut +\mathstrut 5194375q^{25} \) \(\mathstrut +\mathstrut 20346944q^{26} \) \(\mathstrut +\mathstrut 78201288q^{27} \) \(\mathstrut +\mathstrut 17210368q^{28} \) \(\mathstrut -\mathstrut 169827594q^{29} \) \(\mathstrut +\mathstrut 93139200q^{30} \) \(\mathstrut -\mathstrut 51362704q^{31} \) \(\mathstrut -\mathstrut 33554432q^{32} \) \(\mathstrut +\mathstrut 43076880q^{33} \) \(\mathstrut +\mathstrut 295229376q^{34} \) \(\mathstrut +\mathstrut 123531450q^{35} \) \(\mathstrut -\mathstrut 20818944q^{36} \) \(\mathstrut -\mathstrut 251605906q^{37} \) \(\mathstrut +\mathstrut 241771904q^{38} \) \(\mathstrut +\mathstrut 251793432q^{39} \) \(\mathstrut -\mathstrut 240844800q^{40} \) \(\mathstrut -\mathstrut 928817814q^{41} \) \(\mathstrut +\mathstrut 212978304q^{42} \) \(\mathstrut -\mathstrut 1818895756q^{43} \) \(\mathstrut -\mathstrut 111390720q^{44} \) \(\mathstrut -\mathstrut 149432850q^{45} \) \(\mathstrut -\mathstrut 847660800q^{46} \) \(\mathstrut +\mathstrut 523343136q^{47} \) \(\mathstrut -\mathstrut 415236096q^{48} \) \(\mathstrut +\mathstrut 282475249q^{49} \) \(\mathstrut -\mathstrut 166220000q^{50} \) \(\mathstrut +\mathstrut 3653463528q^{51} \) \(\mathstrut -\mathstrut 651102208q^{52} \) \(\mathstrut +\mathstrut 4199520078q^{53} \) \(\mathstrut -\mathstrut 2502441216q^{54} \) \(\mathstrut -\mathstrut 799533000q^{55} \) \(\mathstrut -\mathstrut 550731776q^{56} \) \(\mathstrut +\mathstrut 2991927312q^{57} \) \(\mathstrut +\mathstrut 5434483008q^{58} \) \(\mathstrut +\mathstrut 9140129196q^{59} \) \(\mathstrut -\mathstrut 2980454400q^{60} \) \(\mathstrut -\mathstrut 6639312802q^{61} \) \(\mathstrut +\mathstrut 1643606528q^{62} \) \(\mathstrut -\mathstrut 341703117q^{63} \) \(\mathstrut +\mathstrut 1073741824q^{64} \) \(\mathstrut -\mathstrut 4673438700q^{65} \) \(\mathstrut -\mathstrut 1378460160q^{66} \) \(\mathstrut -\mathstrut 2878139188q^{67} \) \(\mathstrut -\mathstrut 9447340032q^{68} \) \(\mathstrut -\mathstrut 10489802400q^{69} \) \(\mathstrut -\mathstrut 3953006400q^{70} \) \(\mathstrut -\mathstrut 4345596360q^{71} \) \(\mathstrut +\mathstrut 666206208q^{72} \) \(\mathstrut +\mathstrut 23450332826q^{73} \) \(\mathstrut +\mathstrut 8051388992q^{74} \) \(\mathstrut -\mathstrut 2056972500q^{75} \) \(\mathstrut -\mathstrut 7736700928q^{76} \) \(\mathstrut -\mathstrut 1828265460q^{77} \) \(\mathstrut -\mathstrut 8057389824q^{78} \) \(\mathstrut -\mathstrut 28761853648q^{79} \) \(\mathstrut +\mathstrut 7707033600q^{80} \) \(\mathstrut -\mathstrut 27366134391q^{81} \) \(\mathstrut +\mathstrut 29722170048q^{82} \) \(\mathstrut -\mathstrut 5577757548q^{83} \) \(\mathstrut -\mathstrut 6815305728q^{84} \) \(\mathstrut -\mathstrut 67810497300q^{85} \) \(\mathstrut +\mathstrut 58204664192q^{86} \) \(\mathstrut +\mathstrut 67251727224q^{87} \) \(\mathstrut +\mathstrut 3564503040q^{88} \) \(\mathstrut +\mathstrut 78002173386q^{89} \) \(\mathstrut +\mathstrut 4781851200q^{90} \) \(\mathstrut -\mathstrut 10686596494q^{91} \) \(\mathstrut +\mathstrut 27125145600q^{92} \) \(\mathstrut +\mathstrut 20339630784q^{93} \) \(\mathstrut -\mathstrut 16746980352q^{94} \) \(\mathstrut -\mathstrut 55531984200q^{95} \) \(\mathstrut +\mathstrut 13287555072q^{96} \) \(\mathstrut -\mathstrut 26685859630q^{97} \) \(\mathstrut -\mathstrut 9039207968q^{98} \) \(\mathstrut +\mathstrut 2211606180q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−32.0000 −396.000 1024.00 7350.00 12672.0 16807.0 −32768.0 −20331.0 −235200.
\(n\): 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
\(2\) \(1\)
\(7\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3} \) \(\mathstrut +\mathstrut 396 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(14))\).