Properties

Label 8.12.a.a
Level $8$
Weight $12$
Character orbit 8.a
Self dual yes
Analytic conductor $6.147$
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] = [8,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 12 \)
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: \(6.14674544448\)
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)
 
\(f(q)\) \(=\) \( q - 36 q^{3} - 3490 q^{5} - 55464 q^{7} - 175851 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 36 q^{3} - 3490 q^{5} - 55464 q^{7} - 175851 q^{9} - 597004 q^{11} + 1373878 q^{13} + 125640 q^{15} + 10140850 q^{17} - 7297396 q^{19} + 1996704 q^{21} - 32057464 q^{23} - 36648025 q^{25} + 12707928 q^{27} - 13605402 q^{29} + 233160800 q^{31} + 21492144 q^{33} + 193569360 q^{35} - 257786178 q^{37} - 49459608 q^{39} - 221438598 q^{41} - 1697758892 q^{43} + 613719990 q^{45} + 527509392 q^{47} + 1098928553 q^{49} - 365070600 q^{51} + 3277379822 q^{53} + 2083543960 q^{55} + 262706256 q^{57} - 3001908988 q^{59} - 11630023610 q^{61} + 9753399864 q^{63} - 4794834220 q^{65} - 17189000548 q^{67} + 1154068704 q^{69} + 26169539608 q^{71} - 7039021094 q^{73} + 1319328900 q^{75} + 33112229856 q^{77} - 4199910416 q^{79} + 30693991689 q^{81} - 39739936436 q^{83} - 35391566500 q^{85} + 489794472 q^{87} + 10565331594 q^{89} - 76200769392 q^{91} - 8393788800 q^{93} + 25467912040 q^{95} - 69851645662 q^{97} + 104983750404 q^{99}+O(q^{100}) \) 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 −36.0000 0 −3490.00 0 −55464.0 0 −175851. 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 8.12.a.a 1
3.b odd 2 1 72.12.a.c 1
4.b odd 2 1 16.12.a.b 1
5.b even 2 1 200.12.a.b 1
5.c odd 4 2 200.12.c.b 2
8.b even 2 1 64.12.a.e 1
8.d odd 2 1 64.12.a.c 1
12.b even 2 1 144.12.a.j 1
16.e even 4 2 256.12.b.g 2
16.f odd 4 2 256.12.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.12.a.a 1 1.a even 1 1 trivial
16.12.a.b 1 4.b odd 2 1
64.12.a.c 1 8.d odd 2 1
64.12.a.e 1 8.b even 2 1
72.12.a.c 1 3.b odd 2 1
144.12.a.j 1 12.b even 2 1
200.12.a.b 1 5.b even 2 1
200.12.c.b 2 5.c odd 4 2
256.12.b.a 2 16.f odd 4 2
256.12.b.g 2 16.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 36 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(8))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 36 \) Copy content Toggle raw display
$5$ \( T + 3490 \) Copy content Toggle raw display
$7$ \( T + 55464 \) Copy content Toggle raw display
$11$ \( T + 597004 \) Copy content Toggle raw display
$13$ \( T - 1373878 \) Copy content Toggle raw display
$17$ \( T - 10140850 \) Copy content Toggle raw display
$19$ \( T + 7297396 \) Copy content Toggle raw display
$23$ \( T + 32057464 \) Copy content Toggle raw display
$29$ \( T + 13605402 \) Copy content Toggle raw display
$31$ \( T - 233160800 \) Copy content Toggle raw display
$37$ \( T + 257786178 \) Copy content Toggle raw display
$41$ \( T + 221438598 \) Copy content Toggle raw display
$43$ \( T + 1697758892 \) Copy content Toggle raw display
$47$ \( T - 527509392 \) Copy content Toggle raw display
$53$ \( T - 3277379822 \) Copy content Toggle raw display
$59$ \( T + 3001908988 \) Copy content Toggle raw display
$61$ \( T + 11630023610 \) Copy content Toggle raw display
$67$ \( T + 17189000548 \) Copy content Toggle raw display
$71$ \( T - 26169539608 \) Copy content Toggle raw display
$73$ \( T + 7039021094 \) Copy content Toggle raw display
$79$ \( T + 4199910416 \) Copy content Toggle raw display
$83$ \( T + 39739936436 \) Copy content Toggle raw display
$89$ \( T - 10565331594 \) Copy content Toggle raw display
$97$ \( T + 69851645662 \) Copy content Toggle raw display
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