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Properties

Label 927.1.v.a
Level $927$
Weight $1$
Character orbit 927.v
Analytic conductor $0.463$
Analytic rank $0$
Dimension $16$
Projective image $D_{34}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [927,1,Mod(10,927)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(927, base_ring=CyclotomicField(34))
 
chi = DirichletCharacter(H, H._module([0, 15]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("927.10");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 927 = 3^{2} \cdot 103 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 927.v (of order \(34\), 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: \(0.462633266711\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{34})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{34}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{34} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{34}^{11} q^{4} + (\zeta_{34}^{16} + \zeta_{34}^{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{34}^{11} q^{4} + (\zeta_{34}^{16} + \zeta_{34}^{2}) q^{7} + (\zeta_{34}^{15} + \zeta_{34}^{3}) q^{13} - \zeta_{34}^{5} q^{16} + (\zeta_{34}^{12} + \zeta_{34}^{8}) q^{19} - \zeta_{34}^{13} q^{25} + (\zeta_{34}^{13} - \zeta_{34}^{10}) q^{28} + (\zeta_{34}^{6} - \zeta_{34}^{4}) q^{31} + (\zeta_{34}^{10} + \zeta_{34}^{9}) q^{37} + ( - \zeta_{34}^{12} - \zeta_{34}^{3}) q^{43} + ( - \zeta_{34}^{15} + \cdots - \zeta_{34}) q^{49} + \cdots + ( - \zeta_{34}^{8} - 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{4} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + q^{4} - 2 q^{7} + 2 q^{13} - q^{16} - 2 q^{19} - q^{25} + 2 q^{28} - 3 q^{49} - 2 q^{52} + 2 q^{61} + q^{64} + 2 q^{76} + 2 q^{79} - 13 q^{91} - 15 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/927\mathbb{Z}\right)^\times\).

\(n\) \(722\) \(829\)
\(\chi(n)\) \(1\) \(\zeta_{34}^{13}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
−0.0922684 + 0.995734i
0.602635 + 0.798017i
0.273663 − 0.961826i
0.273663 + 0.961826i
0.850217 − 0.526432i
0.982973 − 0.183750i
0.982973 + 0.183750i
0.602635 − 0.798017i
−0.932472 + 0.361242i
−0.739009 + 0.673696i
−0.739009 − 0.673696i
−0.0922684 − 0.995734i
−0.932472 − 0.361242i
0.850217 + 0.526432i
−0.445738 + 0.895163i
−0.445738 − 0.895163i
0 0 0.850217 − 0.526432i 0 0 −0.890705 + 0.811985i 0 0 0
37.1 0 0 −0.739009 − 0.673696i 0 0 −0.876298 + 1.75984i 0 0 0
73.1 0 0 −0.0922684 − 0.995734i 0 0 −1.12388 − 1.48826i 0 0 0
127.1 0 0 −0.0922684 + 0.995734i 0 0 −1.12388 + 1.48826i 0 0 0
145.1 0 0 0.982973 + 0.183750i 0 0 −0.404479 − 1.42160i 0 0 0
172.1 0 0 −0.445738 − 0.895163i 0 0 −0.0505009 − 0.544991i 0 0 0
415.1 0 0 −0.445738 + 0.895163i 0 0 −0.0505009 + 0.544991i 0 0 0
451.1 0 0 −0.739009 + 0.673696i 0 0 −0.876298 − 1.75984i 0 0 0
595.1 0 0 0.602635 − 0.798017i 0 0 1.67148 − 0.312454i 0 0 0
604.1 0 0 0.273663 + 0.961826i 0 0 0.831277 − 0.322039i 0 0 0
640.1 0 0 0.273663 − 0.961826i 0 0 0.831277 + 0.322039i 0 0 0
649.1 0 0 0.850217 + 0.526432i 0 0 −0.890705 − 0.811985i 0 0 0
712.1 0 0 0.602635 + 0.798017i 0 0 1.67148 + 0.312454i 0 0 0
748.1 0 0 0.982973 − 0.183750i 0 0 −0.404479 + 1.42160i 0 0 0
811.1 0 0 −0.932472 − 0.361242i 0 0 −0.156896 + 0.0971461i 0 0 0
919.1 0 0 −0.932472 + 0.361242i 0 0 −0.156896 − 0.0971461i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
103.f odd 34 1 inner
309.k even 34 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 927.1.v.a ✓ 16
3.b odd 2 1 CM 927.1.v.a ✓ 16
103.f odd 34 1 inner 927.1.v.a ✓ 16
309.k even 34 1 inner 927.1.v.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
927.1.v.a ✓ 16 1.a even 1 1 trivial
927.1.v.a ✓ 16 3.b odd 2 1 CM
927.1.v.a ✓ 16 103.f odd 34 1 inner
927.1.v.a ✓ 16 309.k even 34 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(927, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} - 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} + 17 T^{10} + \cdots + 17 \) Copy content Toggle raw display
$37$ \( T^{16} - 51 T^{9} + \cdots + 17 \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( T^{16} - 17 T^{11} + \cdots + 17 \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} - 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{16} - 17 T^{7} + \cdots + 17 \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} - 17 T^{7} + \cdots + 17 \) Copy content Toggle raw display
$79$ \( T^{16} - 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( T^{16} + 15 T^{15} + \cdots + 1 \) Copy content Toggle raw display
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