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Properties

Label 1011.1.y.a
Level $1011$
Weight $1$
Character orbit 1011.y
Analytic conductor $0.505$
Analytic rank $0$
Dimension $12$
Projective image $D_{28}$
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] = [1011,1,Mod(164,1011)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1011, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([14, 13]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1011.164");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1011 = 3 \cdot 337 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1011.y (of order \(28\), degree \(12\), 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: \(0.504554727772\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{28})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{28}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{28} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{28}^{3} q^{3} - \zeta_{28}^{10} q^{4} + ( - \zeta_{28}^{6} - \zeta_{28}^{4}) q^{7} + \zeta_{28}^{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{28}^{3} q^{3} - \zeta_{28}^{10} q^{4} + ( - \zeta_{28}^{6} - \zeta_{28}^{4}) q^{7} + \zeta_{28}^{6} q^{9} + \zeta_{28}^{13} q^{12} - \zeta_{28}^{6} q^{16} + (\zeta_{28}^{8} - \zeta_{28}^{3}) q^{19} + (\zeta_{28}^{9} + \zeta_{28}^{7}) q^{21} + \zeta_{28}^{5} q^{25} - \zeta_{28}^{9} q^{27} + ( - \zeta_{28}^{2} - 1) q^{28} + (\zeta_{28}^{2} - \zeta_{28}) q^{31} + \zeta_{28}^{2} q^{36} + ( - \zeta_{28}^{8} - 1) q^{37} + ( - \zeta_{28}^{13} - \zeta_{28}^{9}) q^{43} + \zeta_{28}^{9} q^{48} + (\zeta_{28}^{12} + \cdots + \zeta_{28}^{8}) q^{49} + \cdots + ( - \zeta_{28}^{8} + \zeta_{28}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{4} + 2 q^{9} - 2 q^{16} - 2 q^{19} - 14 q^{28} + 2 q^{31} + 2 q^{36} - 10 q^{37} - 2 q^{49} + 2 q^{57} + 2 q^{61} - 2 q^{64} + 12 q^{67} + 2 q^{73} + 2 q^{75} - 2 q^{76} - 2 q^{81} - 2 q^{93} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1011\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(338\)
\(\chi(n)\) \(\zeta_{28}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
164.1
−0.974928 + 0.222521i
0.974928 − 0.222521i
0.781831 − 0.623490i
−0.781831 − 0.623490i
−0.433884 + 0.900969i
−0.974928 − 0.222521i
0.433884 + 0.900969i
−0.433884 − 0.900969i
0.974928 + 0.222521i
0.433884 − 0.900969i
0.781831 + 0.623490i
−0.781831 + 0.623490i
0 0.781831 − 0.623490i 0.623490 + 0.781831i 0 0 −0.846011 + 1.75676i 0 0.222521 − 0.974928i 0
173.1 0 −0.781831 + 0.623490i 0.623490 + 0.781831i 0 0 −0.846011 + 1.75676i 0 0.222521 − 0.974928i 0
386.1 0 0.433884 + 0.900969i −0.900969 + 0.433884i 0 0 1.52446 − 0.347948i 0 −0.623490 + 0.781831i 0
392.1 0 −0.433884 + 0.900969i −0.900969 − 0.433884i 0 0 1.52446 + 0.347948i 0 −0.623490 − 0.781831i 0
440.1 0 −0.974928 + 0.222521i −0.222521 − 0.974928i 0 0 −0.678448 − 0.541044i 0 0.900969 − 0.433884i 0
524.1 0 0.781831 + 0.623490i 0.623490 − 0.781831i 0 0 −0.846011 − 1.75676i 0 0.222521 + 0.974928i 0
638.1 0 0.974928 + 0.222521i −0.222521 + 0.974928i 0 0 −0.678448 + 0.541044i 0 0.900969 + 0.433884i 0
710.1 0 −0.974928 − 0.222521i −0.222521 + 0.974928i 0 0 −0.678448 + 0.541044i 0 0.900969 + 0.433884i 0
824.1 0 −0.781831 − 0.623490i 0.623490 − 0.781831i 0 0 −0.846011 − 1.75676i 0 0.222521 + 0.974928i 0
908.1 0 0.974928 − 0.222521i −0.222521 − 0.974928i 0 0 −0.678448 − 0.541044i 0 0.900969 − 0.433884i 0
956.1 0 0.433884 − 0.900969i −0.900969 − 0.433884i 0 0 1.52446 + 0.347948i 0 −0.623490 − 0.781831i 0
962.1 0 −0.433884 − 0.900969i −0.900969 + 0.433884i 0 0 1.52446 − 0.347948i 0 −0.623490 + 0.781831i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 164.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
337.m even 28 1 inner
1011.y odd 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1011.1.y.a ✓ 12
3.b odd 2 1 CM 1011.1.y.a ✓ 12
337.m even 28 1 inner 1011.1.y.a ✓ 12
1011.y odd 28 1 inner 1011.1.y.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1011.1.y.a ✓ 12 1.a even 1 1 trivial
1011.1.y.a ✓ 12 3.b odd 2 1 CM
1011.1.y.a ✓ 12 337.m even 28 1 inner
1011.1.y.a ✓ 12 1011.y odd 28 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1011, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - T^{10} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} - 7 T^{3} + 7 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( (T^{6} + 5 T^{5} + 11 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} - 4 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{12} - 12 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{12} + 7 T^{10} + \cdots + 49 \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} - 2 T^{11} + \cdots + 64 \) Copy content Toggle raw display
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