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Properties

Label 992.1.y.b
Level $992$
Weight $1$
Character orbit 992.y
Analytic conductor $0.495$
Analytic rank $0$
Dimension $8$
Projective image $D_{24}$
CM discriminant -31
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [992,1,Mod(61,992)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(992, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("992.61");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 992 = 2^{5} \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 992.y (of order \(8\), degree \(4\), minimal)

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: no
Analytic conductor: \(0.495072492532\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{24}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{24}^{10} q^{2} - \zeta_{24}^{8} q^{4} + (\zeta_{24}^{8} - \zeta_{24}) q^{5} + ( - \zeta_{24}^{11} - \zeta_{24}^{7}) q^{7} - \zeta_{24}^{6} q^{8} - \zeta_{24}^{3} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24}^{10} q^{2} - \zeta_{24}^{8} q^{4} + (\zeta_{24}^{8} - \zeta_{24}) q^{5} + ( - \zeta_{24}^{11} - \zeta_{24}^{7}) q^{7} - \zeta_{24}^{6} q^{8} - \zeta_{24}^{3} q^{9} + (\zeta_{24}^{11} + \zeta_{24}^{6}) q^{10} + ( - \zeta_{24}^{9} - \zeta_{24}^{5}) q^{14} - \zeta_{24}^{4} q^{16} - \zeta_{24} q^{18} + ( - \zeta_{24}^{10} + \zeta_{24}^{5}) q^{19} + (\zeta_{24}^{9} + \zeta_{24}^{4}) q^{20} + ( - \zeta_{24}^{9} + \cdots + \zeta_{24}^{2}) q^{25} + \cdots + ( - \zeta_{24}^{8} - \zeta_{24}^{4} - 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} - 4 q^{5} - 4 q^{16} + 4 q^{20} - 4 q^{25} - 8 q^{31} - 12 q^{35} + 4 q^{38} + 4 q^{45} + 8 q^{50} + 4 q^{59} - 8 q^{64} - 4 q^{71} + 8 q^{80} - 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/992\mathbb{Z}\right)^\times\).

\(n\) \(63\) \(65\) \(869\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{24}^{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
0.258819 + 0.965926i
−0.965926 − 0.258819i
0.258819 − 0.965926i
−0.965926 + 0.258819i
−0.258819 − 0.965926i
0.965926 + 0.258819i
−0.258819 + 0.965926i
0.965926 − 0.258819i
−0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.758819 − 1.83195i 0 1.22474 − 1.22474i − 1.00000i 0.707107 + 0.707107i −0.258819 + 1.96593i
61.2 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.465926 + 1.12484i 0 −1.22474 + 1.22474i − 1.00000i 0.707107 + 0.707107i 0.965926 + 0.741181i
309.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.758819 + 1.83195i 0 1.22474 + 1.22474i 1.00000i 0.707107 − 0.707107i −0.258819 − 1.96593i
309.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.465926 − 1.12484i 0 −1.22474 − 1.22474i 1.00000i 0.707107 − 0.707107i 0.965926 − 0.741181i
557.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.241181 + 0.0999004i 0 −1.22474 + 1.22474i − 1.00000i −0.707107 − 0.707107i 0.258819 + 0.0340742i
557.2 0.866025 − 0.500000i 0 0.500000 − 0.866025i −1.46593 + 0.607206i 0 1.22474 − 1.22474i − 1.00000i −0.707107 − 0.707107i −0.965926 + 1.25882i
805.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.241181 − 0.0999004i 0 −1.22474 − 1.22474i 1.00000i −0.707107 + 0.707107i 0.258819 − 0.0340742i
805.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.46593 − 0.607206i 0 1.22474 + 1.22474i 1.00000i −0.707107 + 0.707107i −0.965926 − 1.25882i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 CM by \(\Q(\sqrt{-31}) \)
32.g even 8 1 inner
992.y odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 992.1.y.b ✓ 8
4.b odd 2 1 3968.1.y.b 8
31.b odd 2 1 CM 992.1.y.b ✓ 8
32.g even 8 1 inner 992.1.y.b ✓ 8
32.h odd 8 1 3968.1.y.b 8
124.d even 2 1 3968.1.y.b 8
992.w even 8 1 3968.1.y.b 8
992.y odd 8 1 inner 992.1.y.b ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
992.1.y.b ✓ 8 1.a even 1 1 trivial
992.1.y.b ✓ 8 31.b odd 2 1 CM
992.1.y.b ✓ 8 32.g even 8 1 inner
992.1.y.b ✓ 8 992.y odd 8 1 inner
3968.1.y.b 8 4.b odd 2 1
3968.1.y.b 8 32.h odd 8 1
3968.1.y.b 8 124.d even 2 1
3968.1.y.b 8 992.w even 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 4T_{5}^{7} + 10T_{5}^{6} + 16T_{5}^{5} + 18T_{5}^{4} + 20T_{5}^{3} + 22T_{5}^{2} + 8T_{5} + 1 \) acting on \(S_{1}^{\mathrm{new}}(992, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 4 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} - 2 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T + 1)^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} - 4 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 2 T^{2} - 4 T + 2)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} - 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
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