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Properties

Label 2289.1.en.a
Level $2289$
Weight $1$
Character orbit 2289.en
Analytic conductor $1.142$
Analytic rank $0$
Dimension $18$
Projective image $D_{27}$
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] = [2289,1,Mod(158,2289)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2289, base_ring=CyclotomicField(54))
 
chi = DirichletCharacter(H, H._module([27, 36, 40]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2289.158");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2289 = 3 \cdot 7 \cdot 109 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2289.en (of order \(54\), degree \(18\), 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: \(1.14235981392\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\Q(\zeta_{54})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{9} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{27}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{27} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{54}^{13} q^{3} + \zeta_{54}^{24} q^{4} + \zeta_{54}^{24} q^{7} + \zeta_{54}^{26} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{54}^{13} q^{3} + \zeta_{54}^{24} q^{4} + \zeta_{54}^{24} q^{7} + \zeta_{54}^{26} q^{9} + \zeta_{54}^{10} q^{12} + ( - \zeta_{54}^{25} + \zeta_{54}^{4}) q^{13} - \zeta_{54}^{21} q^{16} + ( - \zeta_{54}^{17} + \zeta_{54}^{16}) q^{19} + \zeta_{54}^{10} q^{21} + \zeta_{54}^{2} q^{25} + \zeta_{54}^{12} q^{27} - \zeta_{54}^{21} q^{28} + ( - \zeta_{54}^{11} - \zeta_{54}^{5}) q^{31} - \zeta_{54}^{23} q^{36} + ( - \zeta_{54}^{9} - \zeta_{54}^{7}) q^{37} + ( - \zeta_{54}^{17} - \zeta_{54}^{11}) q^{39} + ( - \zeta_{54}^{23} - \zeta_{54}^{19}) q^{43} - \zeta_{54}^{7} q^{48} - \zeta_{54}^{21} q^{49} + (\zeta_{54}^{22} - \zeta_{54}) q^{52} + ( - \zeta_{54}^{3} + \zeta_{54}^{2}) q^{57} + ( - \zeta_{54}^{17} + \zeta_{54}^{6}) q^{61} - \zeta_{54}^{23} q^{63} + \zeta_{54}^{18} q^{64} - \zeta_{54}^{8} q^{67} + ( - \zeta_{54}^{25} + \zeta_{54}^{12}) q^{73} - \zeta_{54}^{15} q^{75} + (\zeta_{54}^{14} - \zeta_{54}^{13}) q^{76} + ( - \zeta_{54}^{19} + \zeta_{54}^{6}) q^{79} - \zeta_{54}^{25} q^{81} - \zeta_{54}^{7} q^{84} + (\zeta_{54}^{22} - \zeta_{54}) q^{91} + (\zeta_{54}^{24} + \zeta_{54}^{18}) q^{93} + (\zeta_{54}^{22} + \zeta_{54}^{16}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 9 q^{37} - 9 q^{64} - 9 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2289\mathbb{Z}\right)^\times\).

\(n\) \(442\) \(764\) \(1963\)
\(\chi(n)\) \(-\zeta_{54}^{23}\) \(-1\) \(\zeta_{54}^{18}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
158.1
−0.396080 + 0.918216i
0.835488 − 0.549509i
0.686242 + 0.727374i
−0.597159 − 0.802123i
−0.973045 − 0.230616i
0.0581448 − 0.998308i
0.286803 − 0.957990i
−0.893633 + 0.448799i
0.0581448 + 0.998308i
−0.893633 − 0.448799i
−0.973045 + 0.230616i
0.993238 − 0.116093i
−0.396080 − 0.918216i
0.686242 − 0.727374i
0.993238 + 0.116093i
0.286803 + 0.957990i
−0.597159 + 0.802123i
0.835488 + 0.549509i
0 −0.835488 − 0.549509i −0.939693 − 0.342020i 0 0 −0.939693 − 0.342020i 0 0.396080 + 0.918216i 0
443.1 0 −0.286803 + 0.957990i 0.173648 − 0.984808i 0 0 0.173648 − 0.984808i 0 −0.835488 − 0.549509i 0
548.1 0 0.396080 + 0.918216i 0.766044 + 0.642788i 0 0 0.766044 + 0.642788i 0 −0.686242 + 0.727374i 0
788.1 0 0.893633 − 0.448799i −0.939693 − 0.342020i 0 0 −0.939693 − 0.342020i 0 0.597159 − 0.802123i 0
893.1 0 −0.993238 + 0.116093i 0.766044 − 0.642788i 0 0 0.766044 − 0.642788i 0 0.973045 − 0.230616i 0
1208.1 0 −0.686242 + 0.727374i 0.173648 + 0.984808i 0 0 0.173648 + 0.984808i 0 −0.0581448 − 0.998308i 0
1388.1 0 0.597159 − 0.802123i 0.766044 + 0.642788i 0 0 0.766044 + 0.642788i 0 −0.286803 − 0.957990i 0
1439.1 0 0.973045 + 0.230616i 0.173648 + 0.984808i 0 0 0.173648 + 0.984808i 0 0.893633 + 0.448799i 0
1514.1 0 −0.686242 − 0.727374i 0.173648 − 0.984808i 0 0 0.173648 − 0.984808i 0 −0.0581448 + 0.998308i 0
1640.1 0 0.973045 − 0.230616i 0.173648 − 0.984808i 0 0 0.173648 − 0.984808i 0 0.893633 − 0.448799i 0
1661.1 0 −0.993238 − 0.116093i 0.766044 + 0.642788i 0 0 0.766044 + 0.642788i 0 0.973045 + 0.230616i 0
1670.1 0 −0.0581448 + 0.998308i −0.939693 − 0.342020i 0 0 −0.939693 − 0.342020i 0 −0.993238 − 0.116093i 0
1724.1 0 −0.835488 + 0.549509i −0.939693 + 0.342020i 0 0 −0.939693 + 0.342020i 0 0.396080 − 0.918216i 0
1817.1 0 0.396080 − 0.918216i 0.766044 − 0.642788i 0 0 0.766044 − 0.642788i 0 −0.686242 − 0.727374i 0
1934.1 0 −0.0581448 − 0.998308i −0.939693 + 0.342020i 0 0 −0.939693 + 0.342020i 0 −0.993238 + 0.116093i 0
2195.1 0 0.597159 + 0.802123i 0.766044 − 0.642788i 0 0 0.766044 − 0.642788i 0 −0.286803 + 0.957990i 0
2228.1 0 0.893633 + 0.448799i −0.939693 + 0.342020i 0 0 −0.939693 + 0.342020i 0 0.597159 + 0.802123i 0
2258.1 0 −0.286803 − 0.957990i 0.173648 + 0.984808i 0 0 0.173648 + 0.984808i 0 −0.835488 + 0.549509i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 158.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
763.br even 27 1 inner
2289.en odd 54 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2289.1.en.a ✓ 18
3.b odd 2 1 CM 2289.1.en.a ✓ 18
7.c even 3 1 2289.1.ep.a yes 18
21.h odd 6 1 2289.1.ep.a yes 18
109.i even 27 1 2289.1.ep.a yes 18
327.v odd 54 1 2289.1.ep.a yes 18
763.br even 27 1 inner 2289.1.en.a ✓ 18
2289.en odd 54 1 inner 2289.1.en.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2289.1.en.a ✓ 18 1.a even 1 1 trivial
2289.1.en.a ✓ 18 3.b odd 2 1 CM
2289.1.en.a ✓ 18 763.br even 27 1 inner
2289.1.en.a ✓ 18 2289.en odd 54 1 inner
2289.1.ep.a yes 18 7.c even 3 1
2289.1.ep.a yes 18 21.h odd 6 1
2289.1.ep.a yes 18 109.i even 27 1
2289.1.ep.a yes 18 327.v odd 54 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2289, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + T^{9} + 1 \) Copy content Toggle raw display
$5$ \( T^{18} \) Copy content Toggle raw display
$7$ \( (T^{6} + T^{3} + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{18} \) Copy content Toggle raw display
$13$ \( T^{18} - 18 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{18} \) Copy content Toggle raw display
$19$ \( T^{18} - 18 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{18} \) Copy content Toggle raw display
$29$ \( T^{18} \) Copy content Toggle raw display
$31$ \( T^{18} - 18 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{18} + 9 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{18} \) Copy content Toggle raw display
$43$ \( T^{18} + 9 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{18} \) Copy content Toggle raw display
$53$ \( T^{18} \) Copy content Toggle raw display
$59$ \( T^{18} \) Copy content Toggle raw display
$61$ \( T^{18} + 3 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{18} - T^{9} + 1 \) Copy content Toggle raw display
$71$ \( T^{18} \) Copy content Toggle raw display
$73$ \( T^{18} + 3 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{18} + 3 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{18} \) Copy content Toggle raw display
$89$ \( T^{18} \) Copy content Toggle raw display
$97$ \( T^{18} - 18 T^{15} + \cdots + 1 \) Copy content Toggle raw display
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