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Properties

Label 388.1.r.a
Level $388$
Weight $1$
Character orbit 388.r
Analytic conductor $0.194$
Analytic rank $0$
Dimension $8$
Projective image $D_{24}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [388,1,Mod(43,388)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(388, base_ring=CyclotomicField(24))
 
chi = DirichletCharacter(H, H._module([12, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("388.43");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 388 = 2^{2} \cdot 97 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 388.r (of order \(24\), degree \(8\), 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: \(0.193637224902\)
Analytic rank: \(0\)
Dimension: \(8\)
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q - \zeta_{24}^{11} q^{2} - \zeta_{24}^{10} q^{4} + (\zeta_{24}^{4} - \zeta_{24}^{3}) q^{5} - \zeta_{24}^{9} q^{8} + \zeta_{24}^{10} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24}^{11} q^{2} - \zeta_{24}^{10} q^{4} + (\zeta_{24}^{4} - \zeta_{24}^{3}) q^{5} - \zeta_{24}^{9} q^{8} + \zeta_{24}^{10} q^{9} + (\zeta_{24}^{3} - \zeta_{24}^{2}) q^{10} + (\zeta_{24}^{7} - 1) q^{13} - \zeta_{24}^{8} q^{16} + ( - \zeta_{24}^{6} - \zeta_{24}^{5}) q^{17} + \zeta_{24}^{9} q^{18} + (\zeta_{24}^{2} - \zeta_{24}) q^{20} + (\zeta_{24}^{8} + \cdots + \zeta_{24}^{6}) q^{25} + \cdots - q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{5} - 8 q^{13} + 4 q^{16} - 4 q^{25} - 4 q^{34} - 4 q^{36} - 4 q^{37} - 8 q^{40} + 4 q^{41} + 4 q^{58} - 4 q^{65} - 4 q^{68} - 4 q^{74} + 8 q^{80} + 4 q^{81} - 4 q^{85} + 8 q^{90} - 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/388\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(195\)
\(\chi(n)\) \(-\zeta_{24}^{7}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
0.258819 − 0.965926i
−0.258819 + 0.965926i
−0.258819 − 0.965926i
−0.965926 − 0.258819i
0.965926 − 0.258819i
−0.965926 + 0.258819i
0.965926 + 0.258819i
0.258819 + 0.965926i
0.258819 + 0.965926i 0 −0.866025 + 0.500000i 1.20711 + 0.158919i 0 0 −0.707107 − 0.707107i 0.866025 − 0.500000i 0.158919 + 1.20711i
151.1 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i −0.207107 + 1.57313i 0 0 0.707107 + 0.707107i 0.866025 − 0.500000i 1.57313 − 0.207107i
203.1 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i −0.207107 − 1.57313i 0 0 0.707107 − 0.707107i 0.866025 + 0.500000i 1.57313 + 0.207107i
267.1 −0.965926 + 0.258819i 0 0.866025 − 0.500000i 1.20711 + 1.57313i 0 0 −0.707107 + 0.707107i −0.866025 + 0.500000i −1.57313 − 1.20711i
287.1 0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.207107 − 0.158919i 0 0 0.707107 + 0.707107i −0.866025 − 0.500000i −0.158919 − 0.207107i
295.1 −0.965926 − 0.258819i 0 0.866025 + 0.500000i 1.20711 − 1.57313i 0 0 −0.707107 − 0.707107i −0.866025 − 0.500000i −1.57313 + 1.20711i
315.1 0.965926 − 0.258819i 0 0.866025 − 0.500000i −0.207107 + 0.158919i 0 0 0.707107 − 0.707107i −0.866025 + 0.500000i −0.158919 + 0.207107i
379.1 0.258819 − 0.965926i 0 −0.866025 − 0.500000i 1.20711 − 0.158919i 0 0 −0.707107 + 0.707107i 0.866025 + 0.500000i 0.158919 − 1.20711i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
97.i even 24 1 inner
388.r odd 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 388.1.r.a ✓ 8
3.b odd 2 1 3492.1.da.a 8
4.b odd 2 1 CM 388.1.r.a ✓ 8
12.b even 2 1 3492.1.da.a 8
97.i even 24 1 inner 388.1.r.a ✓ 8
291.r odd 24 1 3492.1.da.a 8
388.r odd 24 1 inner 388.1.r.a ✓ 8
1164.bg even 24 1 3492.1.da.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
388.1.r.a ✓ 8 1.a even 1 1 trivial
388.1.r.a ✓ 8 4.b odd 2 1 CM
388.1.r.a ✓ 8 97.i even 24 1 inner
388.1.r.a ✓ 8 388.r odd 24 1 inner
3492.1.da.a 8 3.b odd 2 1
3492.1.da.a 8 12.b even 2 1
3492.1.da.a 8 291.r odd 24 1
3492.1.da.a 8 1164.bg even 24 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(388, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{4} + 1 \) 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
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} - 2 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + 4 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{8} - 4 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} - 9T^{4} + 81 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
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