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Properties

Label 1028.1.l.a
Level $1028$
Weight $1$
Character orbit 1028.l
Analytic conductor $0.513$
Analytic rank $0$
Dimension $16$
Projective image $D_{32}$
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] = [1028,1,Mod(15,1028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1028, base_ring=CyclotomicField(32))
 
chi = DirichletCharacter(H, H._module([16, 7]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1028.15");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1028 = 2^{2} \cdot 257 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1028.l (of order \(32\), degree \(16\), 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: \(0.513038832987\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{32})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{32}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{32} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{32}^{8} q^{2} - q^{4} + (\zeta_{32}^{13} + \zeta_{32}^{2}) q^{5} + \zeta_{32}^{8} q^{8} + \zeta_{32}^{9} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{32}^{8} q^{2} - q^{4} + (\zeta_{32}^{13} + \zeta_{32}^{2}) q^{5} + \zeta_{32}^{8} q^{8} + \zeta_{32}^{9} q^{9} + ( - \zeta_{32}^{10} + \zeta_{32}^{5}) q^{10} + (\zeta_{32}^{14} + \zeta_{32}^{12}) q^{13} + q^{16} + (\zeta_{32}^{15} + \zeta_{32}^{9}) q^{17} + \zeta_{32} q^{18} + ( - \zeta_{32}^{13} - \zeta_{32}^{2}) q^{20} + (\zeta_{32}^{15} + \cdots + \zeta_{32}^{4}) q^{25} + \cdots - \zeta_{32}^{5} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 16 q^{16} - 16 q^{64} - 16 q^{65}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1028\mathbb{Z}\right)^\times\).

\(n\) \(515\) \(517\)
\(\chi(n)\) \(-1\) \(\zeta_{32}^{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
−0.980785 + 0.195090i
0.831470 − 0.555570i
−0.555570 − 0.831470i
0.555570 + 0.831470i
−0.831470 + 0.555570i
0.980785 − 0.195090i
−0.195090 − 0.980785i
−0.195090 + 0.980785i
0.980785 + 0.195090i
−0.831470 − 0.555570i
−0.555570 + 0.831470i
0.555570 − 0.831470i
0.831470 + 0.555570i
−0.980785 − 0.195090i
0.195090 − 0.980785i
0.195090 + 0.980785i
1.00000i 0 −1.00000 1.75535 + 0.172887i 0 0 − 1.00000i 0.195090 + 0.980785i −0.172887 + 1.75535i
223.1 − 1.00000i 0 −1.00000 0.577774 − 1.90466i 0 0 1.00000i 0.555570 + 0.831470i −1.90466 − 0.577774i
227.1 − 1.00000i 0 −1.00000 −1.36347 + 0.728789i 0 0 1.00000i 0.831470 − 0.555570i 0.728789 + 1.36347i
287.1 − 1.00000i 0 −1.00000 0.598102 + 1.11897i 0 0 1.00000i −0.831470 + 0.555570i 1.11897 − 0.598102i
291.1 − 1.00000i 0 −1.00000 0.187593 + 0.0569057i 0 0 1.00000i −0.555570 − 0.831470i 0.0569057 − 0.187593i
499.1 1.00000i 0 −1.00000 0.0924099 − 0.938254i 0 0 − 1.00000i −0.195090 − 0.980785i 0.938254 + 0.0924099i
531.1 1.00000i 0 −1.00000 −1.47945 + 1.21415i 0 0 − 1.00000i −0.980785 + 0.195090i −1.21415 − 1.47945i
635.1 − 1.00000i 0 −1.00000 −1.47945 − 1.21415i 0 0 1.00000i −0.980785 − 0.195090i −1.21415 + 1.47945i
651.1 − 1.00000i 0 −1.00000 0.0924099 + 0.938254i 0 0 1.00000i −0.195090 + 0.980785i 0.938254 − 0.0924099i
703.1 1.00000i 0 −1.00000 0.187593 − 0.0569057i 0 0 − 1.00000i −0.555570 + 0.831470i 0.0569057 + 0.187593i
711.1 1.00000i 0 −1.00000 −1.36347 − 0.728789i 0 0 − 1.00000i 0.831470 + 0.555570i 0.728789 − 1.36347i
831.1 1.00000i 0 −1.00000 0.598102 − 1.11897i 0 0 − 1.00000i −0.831470 − 0.555570i 1.11897 + 0.598102i
839.1 1.00000i 0 −1.00000 0.577774 + 1.90466i 0 0 − 1.00000i 0.555570 − 0.831470i −1.90466 + 0.577774i
891.1 − 1.00000i 0 −1.00000 1.75535 − 0.172887i 0 0 1.00000i 0.195090 − 0.980785i −0.172887 − 1.75535i
907.1 − 1.00000i 0 −1.00000 −0.368309 + 0.448786i 0 0 1.00000i 0.980785 + 0.195090i 0.448786 + 0.368309i
1011.1 1.00000i 0 −1.00000 −0.368309 − 0.448786i 0 0 − 1.00000i 0.980785 − 0.195090i 0.448786 − 0.368309i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
257.f even 32 1 inner
1028.l odd 32 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1028.1.l.a ✓ 16
4.b odd 2 1 CM 1028.1.l.a ✓ 16
257.f even 32 1 inner 1028.1.l.a ✓ 16
1028.l odd 32 1 inner 1028.1.l.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1028.1.l.a ✓ 16 1.a even 1 1 trivial
1028.1.l.a ✓ 16 4.b odd 2 1 CM
1028.1.l.a ✓ 16 257.f even 32 1 inner
1028.1.l.a ✓ 16 1028.l odd 32 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1028, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 32 T^{11} + \cdots + 2 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{8} + 2 T^{4} - 16 T^{3} + \cdots + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 24 T^{12} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} + 256 \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} + 4 T^{12} + \cdots + 2 \) Copy content Toggle raw display
$41$ \( T^{16} + 16 T^{13} + \cdots + 2 \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} - 32 T^{11} + \cdots + 2 \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} - 16 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} - 16 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( (T^{8} + 2 T^{4} - 16 T^{3} + \cdots + 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + 4 T^{12} + \cdots + 2 \) Copy content Toggle raw display
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