Properties

Label 14.16.a.a
Level $14$
Weight $16$
Character orbit 14.a
Self dual yes
Analytic conductor $19.977$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 14.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.9770907140\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 128q^{2} + 1350q^{3} + 16384q^{4} - 81060q^{5} - 172800q^{6} + 823543q^{7} - 2097152q^{8} - 12526407q^{9} + O(q^{10}) \) \( q - 128q^{2} + 1350q^{3} + 16384q^{4} - 81060q^{5} - 172800q^{6} + 823543q^{7} - 2097152q^{8} - 12526407q^{9} + 10375680q^{10} + 70121184q^{11} + 22118400q^{12} + 151469552q^{13} - 105413504q^{14} - 109431000q^{15} + 268435456q^{16} - 249756546q^{17} + 1603380096q^{18} - 6476856550q^{19} - 1328087040q^{20} + 1111783050q^{21} - 8975511552q^{22} - 21129196200q^{23} - 2831155200q^{24} - 23946854525q^{25} - 19388102656q^{26} - 36281673900q^{27} + 13492928512q^{28} + 7794825354q^{29} + 14007168000q^{30} - 95032053412q^{31} - 34359738368q^{32} + 94663598400q^{33} + 31968837888q^{34} - 66756395580q^{35} - 205232652288q^{36} - 870082295470q^{37} + 829037638400q^{38} + 204483895200q^{39} + 169995141120q^{40} + 1007666657262q^{41} - 142308230400q^{42} + 155007585272q^{43} + 1148865478656q^{44} + 1015390551420q^{45} + 2704537113600q^{46} - 2551970135004q^{47} + 362387865600q^{48} + 678223072849q^{49} + 3065197379200q^{50} - 337171337100q^{51} + 2481677139968q^{52} + 4047645687774q^{53} + 4644054259200q^{54} - 5684023175040q^{55} - 1727094849536q^{56} - 8743756342500q^{57} - 997737645312q^{58} - 12599248786302q^{59} - 1792917504000q^{60} - 39925031318044q^{61} + 12164102836736q^{62} - 10316034800001q^{63} + 4398046511104q^{64} - 12278121885120q^{65} - 12116940595200q^{66} - 48423780261124q^{67} - 4092011249664q^{68} - 28524414870000q^{69} + 8544818634240q^{70} + 37693101366144q^{71} + 26269779492864q^{72} + 141416194574306q^{73} + 111370533820160q^{74} - 32328253608750q^{75} - 106116817715200q^{76} + 57747810234912q^{77} - 26173938585600q^{78} + 247020521013128q^{79} - 21759378063360q^{80} + 130759989322149q^{81} - 128981332129536q^{82} + 2788789610034q^{83} + 18215453491200q^{84} + 20245265618760q^{85} - 19840970914816q^{86} + 10523014227900q^{87} - 147054781267968q^{88} - 5839634731110q^{89} - 129969990581760q^{90} + 124741689262736q^{91} - 346180750540800q^{92} - 128293272106200q^{93} + 326652177280512q^{94} + 525013991943000q^{95} - 46385646796800q^{96} + 278027158065374q^{97} - 86812553324672q^{98} - 878366490105888q^{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
−128.000 1350.00 16384.0 −81060.0 −172800. 823543. −2.09715e6 −1.25264e7 1.03757e7
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.16.a.a 1
3.b odd 2 1 126.16.a.e 1
4.b odd 2 1 112.16.a.a 1
7.b odd 2 1 98.16.a.b 1
7.c even 3 2 98.16.c.b 2
7.d odd 6 2 98.16.c.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.16.a.a 1 1.a even 1 1 trivial
98.16.a.b 1 7.b odd 2 1
98.16.c.b 2 7.c even 3 2
98.16.c.c 2 7.d odd 6 2
112.16.a.a 1 4.b odd 2 1
126.16.a.e 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1350 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(14))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 128 + T \)
$3$ \( -1350 + T \)
$5$ \( 81060 + T \)
$7$ \( -823543 + T \)
$11$ \( -70121184 + T \)
$13$ \( -151469552 + T \)
$17$ \( 249756546 + T \)
$19$ \( 6476856550 + T \)
$23$ \( 21129196200 + T \)
$29$ \( -7794825354 + T \)
$31$ \( 95032053412 + T \)
$37$ \( 870082295470 + T \)
$41$ \( -1007666657262 + T \)
$43$ \( -155007585272 + T \)
$47$ \( 2551970135004 + T \)
$53$ \( -4047645687774 + T \)
$59$ \( 12599248786302 + T \)
$61$ \( 39925031318044 + T \)
$67$ \( 48423780261124 + T \)
$71$ \( -37693101366144 + T \)
$73$ \( -141416194574306 + T \)
$79$ \( -247020521013128 + T \)
$83$ \( -2788789610034 + T \)
$89$ \( 5839634731110 + T \)
$97$ \( -278027158065374 + T \)
show more
show less