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 $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [14,16,Mod(1,14)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(14, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("14.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 14.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 128 q^{2} + 1350 q^{3} + 16384 q^{4} - 81060 q^{5} - 172800 q^{6} + 823543 q^{7} - 2097152 q^{8} - 12526407 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 128 q^{2} + 1350 q^{3} + 16384 q^{4} - 81060 q^{5} - 172800 q^{6} + 823543 q^{7} - 2097152 q^{8} - 12526407 q^{9} + 10375680 q^{10} + 70121184 q^{11} + 22118400 q^{12} + 151469552 q^{13} - 105413504 q^{14} - 109431000 q^{15} + 268435456 q^{16} - 249756546 q^{17} + 1603380096 q^{18} - 6476856550 q^{19} - 1328087040 q^{20} + 1111783050 q^{21} - 8975511552 q^{22} - 21129196200 q^{23} - 2831155200 q^{24} - 23946854525 q^{25} - 19388102656 q^{26} - 36281673900 q^{27} + 13492928512 q^{28} + 7794825354 q^{29} + 14007168000 q^{30} - 95032053412 q^{31} - 34359738368 q^{32} + 94663598400 q^{33} + 31968837888 q^{34} - 66756395580 q^{35} - 205232652288 q^{36} - 870082295470 q^{37} + 829037638400 q^{38} + 204483895200 q^{39} + 169995141120 q^{40} + 1007666657262 q^{41} - 142308230400 q^{42} + 155007585272 q^{43} + 1148865478656 q^{44} + 1015390551420 q^{45} + 2704537113600 q^{46} - 2551970135004 q^{47} + 362387865600 q^{48} + 678223072849 q^{49} + 3065197379200 q^{50} - 337171337100 q^{51} + 2481677139968 q^{52} + 4047645687774 q^{53} + 4644054259200 q^{54} - 5684023175040 q^{55} - 1727094849536 q^{56} - 8743756342500 q^{57} - 997737645312 q^{58} - 12599248786302 q^{59} - 1792917504000 q^{60} - 39925031318044 q^{61} + 12164102836736 q^{62} - 10316034800001 q^{63} + 4398046511104 q^{64} - 12278121885120 q^{65} - 12116940595200 q^{66} - 48423780261124 q^{67} - 4092011249664 q^{68} - 28524414870000 q^{69} + 8544818634240 q^{70} + 37693101366144 q^{71} + 26269779492864 q^{72} + 141416194574306 q^{73} + 111370533820160 q^{74} - 32328253608750 q^{75} - 106116817715200 q^{76} + 57747810234912 q^{77} - 26173938585600 q^{78} + 247020521013128 q^{79} - 21759378063360 q^{80} + 130759989322149 q^{81} - 128981332129536 q^{82} + 2788789610034 q^{83} + 18215453491200 q^{84} + 20245265618760 q^{85} - 19840970914816 q^{86} + 10523014227900 q^{87} - 147054781267968 q^{88} - 5839634731110 q^{89} - 129969990581760 q^{90} + 124741689262736 q^{91} - 346180750540800 q^{92} - 128293272106200 q^{93} + 326652177280512 q^{94} + 525013991943000 q^{95} - 46385646796800 q^{96} + 278027158065374 q^{97} - 86812553324672 q^{98} - 878366490105888 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 \(S_{16}^{\mathrm{new}}(\Gamma_0(14))\). Copy content Toggle raw display

Hecke characteristic polynomials

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