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Properties

Label 993.1.p.a
Level $993$
Weight $1$
Character orbit 993.p
Analytic conductor $0.496$
Analytic rank $0$
Dimension $10$
Projective image $D_{11}$
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] = [993,1,Mod(74,993)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(993, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 16]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("993.74");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 993 = 3 \cdot 331 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 993.p (of order \(22\), degree \(10\), 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: \(0.495571557545\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{11}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{11} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{22}^{9} q^{3} + q^{4} + ( - \zeta_{22}^{5} - \zeta_{22}) q^{7} - \zeta_{22}^{7} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{22}^{9} q^{3} + q^{4} + ( - \zeta_{22}^{5} - \zeta_{22}) q^{7} - \zeta_{22}^{7} q^{9} - \zeta_{22}^{9} q^{12} + (\zeta_{22}^{8} + \zeta_{22}^{6}) q^{13} + q^{16} + ( - \zeta_{22}^{9} - \zeta_{22}) q^{19} + (\zeta_{22}^{10} - \zeta_{22}^{3}) q^{21} + \zeta_{22}^{2} q^{25} - \zeta_{22}^{5} q^{27} + ( - \zeta_{22}^{5} - \zeta_{22}) q^{28} + (\zeta_{22}^{8} - \zeta_{22}^{3}) q^{31} - \zeta_{22}^{7} q^{36} + \zeta_{22}^{4} q^{37} + (\zeta_{22}^{6} + \zeta_{22}^{4}) q^{39} + (\zeta_{22}^{10} + \zeta_{22}^{4}) q^{43} - \zeta_{22}^{9} q^{48} + (\zeta_{22}^{10} + \cdots + \zeta_{22}^{2}) q^{49} + \cdots - \zeta_{22}^{3} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{3} + 10 q^{4} - 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{3} + 10 q^{4} - 2 q^{7} - q^{9} - q^{12} - 2 q^{13} + 10 q^{16} - 2 q^{19} - 2 q^{21} - q^{25} - q^{27} - 2 q^{28} - 2 q^{31} - q^{36} - 2 q^{37} - 2 q^{39} - 2 q^{43} - q^{48} - 3 q^{49} - 2 q^{52} - 2 q^{57} - 2 q^{61} - 2 q^{63} + 10 q^{64} - 2 q^{67} - 2 q^{73} + 10 q^{75} - 2 q^{76} - 2 q^{79} - q^{81} - 2 q^{84} + 7 q^{91} - 2 q^{93} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/993\mathbb{Z}\right)^\times\).

\(n\) \(332\) \(334\)
\(\chi(n)\) \(-1\) \(-\zeta_{22}^{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1
−0.415415 + 0.909632i
0.959493 − 0.281733i
−0.841254 − 0.540641i
0.142315 − 0.989821i
−0.415415 − 0.909632i
0.654861 − 0.755750i
−0.841254 + 0.540641i
0.959493 + 0.281733i
0.654861 + 0.755750i
0.142315 + 0.989821i
0 −0.654861 + 0.755750i 1.00000 0 0 1.25667 − 0.368991i 0 −0.142315 − 0.989821i 0
80.1 0 0.841254 + 0.540641i 1.00000 0 0 −1.10181 + 1.27155i 0 0.415415 + 0.909632i 0
167.1 0 0.415415 − 0.909632i 1.00000 0 0 −0.118239 + 0.822373i 0 −0.654861 − 0.755750i 0
293.1 0 −0.959493 + 0.281733i 1.00000 0 0 −0.797176 + 1.74557i 0 0.841254 − 0.540641i 0
416.1 0 −0.654861 − 0.755750i 1.00000 0 0 1.25667 + 0.368991i 0 −0.142315 + 0.989821i 0
605.1 0 −0.142315 + 0.989821i 1.00000 0 0 −0.239446 − 0.153882i 0 −0.959493 − 0.281733i 0
773.1 0 0.415415 + 0.909632i 1.00000 0 0 −0.118239 − 0.822373i 0 −0.654861 + 0.755750i 0
782.1 0 0.841254 − 0.540641i 1.00000 0 0 −1.10181 − 1.27155i 0 0.415415 − 0.909632i 0
842.1 0 −0.142315 − 0.989821i 1.00000 0 0 −0.239446 + 0.153882i 0 −0.959493 + 0.281733i 0
932.1 0 −0.959493 − 0.281733i 1.00000 0 0 −0.797176 − 1.74557i 0 0.841254 + 0.540641i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 74.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
331.g even 11 1 inner
993.p odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 993.1.p.a ✓ 10
3.b odd 2 1 CM 993.1.p.a ✓ 10
331.g even 11 1 inner 993.1.p.a ✓ 10
993.p odd 22 1 inner 993.1.p.a ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
993.1.p.a ✓ 10 1.a even 1 1 trivial
993.1.p.a ✓ 10 3.b odd 2 1 CM
993.1.p.a ✓ 10 331.g even 11 1 inner
993.1.p.a ✓ 10 993.p odd 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(993, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + T^{9} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{10} \) Copy content Toggle raw display
$19$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{10} \) Copy content Toggle raw display
$29$ \( T^{10} \) Copy content Toggle raw display
$31$ \( (T^{5} + T^{4} - 4 T^{3} + \cdots + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 2 T^{9} + \cdots + 1024 \) Copy content Toggle raw display
$41$ \( T^{10} \) Copy content Toggle raw display
$43$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{10} \) Copy content Toggle raw display
$53$ \( T^{10} \) Copy content Toggle raw display
$59$ \( T^{10} \) Copy content Toggle raw display
$61$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{10} \) Copy content Toggle raw display
$73$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{10} \) Copy content Toggle raw display
$89$ \( T^{10} \) Copy content Toggle raw display
$97$ \( T^{10} + 2 T^{9} + \cdots + 1024 \) Copy content Toggle raw display
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