Properties

Label 8.14.a.a
Level $8$
Weight $14$
Character orbit 8.a
Self dual yes
Analytic conductor $8.578$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8,14,Mod(1,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 8.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: \(8.57847431615\)
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)
 
\(f(q)\) \(=\) \( q - 12 q^{3} - 4330 q^{5} - 139992 q^{7} - 1594179 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 12 q^{3} - 4330 q^{5} - 139992 q^{7} - 1594179 q^{9} - 6484324 q^{11} - 22588034 q^{13} + 51960 q^{15} - 23732270 q^{17} + 325344836 q^{19} + 1679904 q^{21} + 921600632 q^{23} - 1201954225 q^{25} + 38262024 q^{27} - 3865879218 q^{29} - 2253401440 q^{31} + 77811888 q^{33} + 606165360 q^{35} + 18250384566 q^{37} + 271056408 q^{39} + 34422845322 q^{41} - 17192501444 q^{43} + 6902795070 q^{45} - 67371749904 q^{47} - 77291250343 q^{49} + 284787240 q^{51} - 87281218426 q^{53} + 28077122920 q^{55} - 3904138032 q^{57} + 540214518668 q^{59} - 51276568850 q^{61} + 223172306568 q^{63} + 97806187220 q^{65} + 25519930676 q^{67} - 11059207584 q^{69} - 1387500699032 q^{71} - 819049441238 q^{73} + 14423450700 q^{75} + 907753485408 q^{77} - 4030935615344 q^{79} + 2541177101529 q^{81} + 4180823831428 q^{83} + 102760729100 q^{85} + 46390550616 q^{87} + 2677027798266 q^{89} + 3162144055728 q^{91} + 27040817280 q^{93} - 1408743139880 q^{95} - 14039464316446 q^{97} + 10337173149996 q^{99}+O(q^{100}) \) 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
0 −12.0000 0 −4330.00 0 −139992. 0 −1.59418e6 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)

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 8.14.a.a 1
3.b odd 2 1 72.14.a.a 1
4.b odd 2 1 16.14.a.c 1
5.b even 2 1 200.14.a.a 1
5.c odd 4 2 200.14.c.a 2
8.b even 2 1 64.14.a.f 1
8.d odd 2 1 64.14.a.d 1
12.b even 2 1 144.14.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.14.a.a 1 1.a even 1 1 trivial
16.14.a.c 1 4.b odd 2 1
64.14.a.d 1 8.d odd 2 1
64.14.a.f 1 8.b even 2 1
72.14.a.a 1 3.b odd 2 1
144.14.a.f 1 12.b even 2 1
200.14.a.a 1 5.b even 2 1
200.14.c.a 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 12 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(8))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 12 \) Copy content Toggle raw display
$5$ \( T + 4330 \) Copy content Toggle raw display
$7$ \( T + 139992 \) Copy content Toggle raw display
$11$ \( T + 6484324 \) Copy content Toggle raw display
$13$ \( T + 22588034 \) Copy content Toggle raw display
$17$ \( T + 23732270 \) Copy content Toggle raw display
$19$ \( T - 325344836 \) Copy content Toggle raw display
$23$ \( T - 921600632 \) Copy content Toggle raw display
$29$ \( T + 3865879218 \) Copy content Toggle raw display
$31$ \( T + 2253401440 \) Copy content Toggle raw display
$37$ \( T - 18250384566 \) Copy content Toggle raw display
$41$ \( T - 34422845322 \) Copy content Toggle raw display
$43$ \( T + 17192501444 \) Copy content Toggle raw display
$47$ \( T + 67371749904 \) Copy content Toggle raw display
$53$ \( T + 87281218426 \) Copy content Toggle raw display
$59$ \( T - 540214518668 \) Copy content Toggle raw display
$61$ \( T + 51276568850 \) Copy content Toggle raw display
$67$ \( T - 25519930676 \) Copy content Toggle raw display
$71$ \( T + 1387500699032 \) Copy content Toggle raw display
$73$ \( T + 819049441238 \) Copy content Toggle raw display
$79$ \( T + 4030935615344 \) Copy content Toggle raw display
$83$ \( T - 4180823831428 \) Copy content Toggle raw display
$89$ \( T - 2677027798266 \) Copy content Toggle raw display
$97$ \( T + 14039464316446 \) Copy content Toggle raw display
show more
show less