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Properties

Label 2808.1.fi.a
Level $2808$
Weight $1$
Character orbit 2808.fi
Analytic conductor $1.401$
Analytic rank $0$
Dimension $18$
Projective image $D_{27}$
CM discriminant -104
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2808,1,Mod(259,2808)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2808, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 9, 4, 9])) N = Newforms(chi, 1, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2808.259"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2808 = 2^{3} \cdot 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2808.fi (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,0,0,0,0,0,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.40137455547\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{54})\)
Copy content comment:defining polynomial
 
Copy content 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

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{54}^{24} q^{2} - \zeta_{54}^{7} q^{3} - \zeta_{54}^{21} q^{4} + ( - \zeta_{54}^{23} - \zeta_{54}) q^{5} + \zeta_{54}^{4} q^{6} + (\zeta_{54}^{20} - \zeta_{54}^{19}) q^{7} + \zeta_{54}^{18} q^{8} + \cdots + (\zeta_{54}^{10} + \cdots + \zeta_{54}^{8}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 9 q^{8} + 18 q^{21} + 18 q^{26} + 18 q^{30} + 18 q^{31} - 9 q^{54} - 9 q^{64} - 9 q^{70} - 9 q^{75} - 9 q^{85} - 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2808\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(1081\) \(1405\) \(2081\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-\zeta_{54}^{21}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
259.1
0.686242 + 0.727374i
−0.973045 + 0.230616i
0.286803 − 0.957990i
0.993238 − 0.116093i
−0.396080 + 0.918216i
−0.597159 − 0.802123i
0.993238 + 0.116093i
−0.396080 − 0.918216i
−0.597159 + 0.802123i
0.686242 − 0.727374i
−0.973045 − 0.230616i
0.286803 + 0.957990i
−0.893633 − 0.448799i
0.0581448 + 0.998308i
0.835488 − 0.549509i
−0.893633 + 0.448799i
0.0581448 − 0.998308i
0.835488 + 0.549509i
0.766044 + 0.642788i −0.835488 + 0.549509i 0.173648 + 0.984808i −1.67948 − 0.611281i −0.993238 − 0.116093i 0.137557 − 0.780125i −0.500000 + 0.866025i 0.396080 − 0.918216i −0.893633 − 1.54782i
259.2 0.766044 + 0.642788i −0.0581448 − 0.998308i 0.173648 + 0.984808i 1.57020 + 0.571507i 0.597159 − 0.802123i −0.344948 + 1.95630i −0.500000 + 0.866025i −0.993238 + 0.116093i 0.835488 + 1.44711i
259.3 0.766044 + 0.642788i 0.893633 + 0.448799i 0.173648 + 0.984808i 0.109277 + 0.0397734i 0.396080 + 0.918216i 0.207391 − 1.17617i −0.500000 + 0.866025i 0.597159 + 0.802123i 0.0581448 + 0.100710i
571.1 −0.939693 − 0.342020i −0.686242 + 0.727374i 0.766044 + 0.642788i −0.0996057 + 0.564892i 0.893633 − 0.448799i −0.0890830 + 0.0747496i −0.500000 − 0.866025i −0.0581448 − 0.998308i 0.286803 − 0.496758i
571.2 −0.939693 − 0.342020i −0.286803 − 0.957990i 0.766044 + 0.642788i 0.337935 − 1.91652i −0.0581448 + 0.998308i −1.28004 + 1.07408i −0.500000 − 0.866025i −0.835488 + 0.549509i −0.973045 + 1.68536i
571.3 −0.939693 − 0.342020i 0.973045 + 0.230616i 0.766044 + 0.642788i −0.238329 + 1.35163i −0.835488 − 0.549509i 1.36912 − 1.14883i −0.500000 − 0.866025i 0.893633 + 0.448799i 0.686242 − 1.18861i
1195.1 −0.939693 + 0.342020i −0.686242 − 0.727374i 0.766044 − 0.642788i −0.0996057 − 0.564892i 0.893633 + 0.448799i −0.0890830 − 0.0747496i −0.500000 + 0.866025i −0.0581448 + 0.998308i 0.286803 + 0.496758i
1195.2 −0.939693 + 0.342020i −0.286803 + 0.957990i 0.766044 − 0.642788i 0.337935 + 1.91652i −0.0581448 − 0.998308i −1.28004 − 1.07408i −0.500000 + 0.866025i −0.835488 − 0.549509i −0.973045 − 1.68536i
1195.3 −0.939693 + 0.342020i 0.973045 − 0.230616i 0.766044 − 0.642788i −0.238329 − 1.35163i −0.835488 + 0.549509i 1.36912 + 1.14883i −0.500000 + 0.866025i 0.893633 − 0.448799i 0.686242 + 1.18861i
1507.1 0.766044 − 0.642788i −0.835488 − 0.549509i 0.173648 − 0.984808i −1.67948 + 0.611281i −0.993238 + 0.116093i 0.137557 + 0.780125i −0.500000 − 0.866025i 0.396080 + 0.918216i −0.893633 + 1.54782i
1507.2 0.766044 − 0.642788i −0.0581448 + 0.998308i 0.173648 − 0.984808i 1.57020 − 0.571507i 0.597159 + 0.802123i −0.344948 − 1.95630i −0.500000 − 0.866025i −0.993238 − 0.116093i 0.835488 − 1.44711i
1507.3 0.766044 − 0.642788i 0.893633 − 0.448799i 0.173648 − 0.984808i 0.109277 − 0.0397734i 0.396080 − 0.918216i 0.207391 + 1.17617i −0.500000 − 0.866025i 0.597159 − 0.802123i 0.0581448 − 0.100710i
2131.1 0.173648 − 0.984808i −0.993238 − 0.116093i −0.939693 − 0.342020i 0.606829 − 0.509190i −0.286803 + 0.957990i −1.82873 + 0.665602i −0.500000 + 0.866025i 0.973045 + 0.230616i −0.396080 − 0.686030i
2131.2 0.173648 − 0.984808i 0.396080 + 0.918216i −0.939693 − 0.342020i 0.914900 − 0.767692i 0.973045 − 0.230616i 1.28971 − 0.469417i −0.500000 + 0.866025i −0.686242 + 0.727374i −0.597159 − 1.03431i
2131.3 0.173648 − 0.984808i 0.597159 − 0.802123i −0.939693 − 0.342020i −1.52173 + 1.27688i −0.686242 − 0.727374i 0.539014 − 0.196185i −0.500000 + 0.866025i −0.286803 − 0.957990i 0.993238 + 1.72034i
2443.1 0.173648 + 0.984808i −0.993238 + 0.116093i −0.939693 + 0.342020i 0.606829 + 0.509190i −0.286803 − 0.957990i −1.82873 − 0.665602i −0.500000 − 0.866025i 0.973045 − 0.230616i −0.396080 + 0.686030i
2443.2 0.173648 + 0.984808i 0.396080 − 0.918216i −0.939693 + 0.342020i 0.914900 + 0.767692i 0.973045 + 0.230616i 1.28971 + 0.469417i −0.500000 − 0.866025i −0.686242 − 0.727374i −0.597159 + 1.03431i
2443.3 0.173648 + 0.984808i 0.597159 + 0.802123i −0.939693 + 0.342020i −1.52173 − 1.27688i −0.686242 + 0.727374i 0.539014 + 0.196185i −0.500000 − 0.866025i −0.286803 + 0.957990i 0.993238 − 1.72034i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 259.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
104.h odd 2 1 CM by \(\Q(\sqrt{-26}) \)
27.e even 9 1 inner
2808.fi odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2808.1.fi.a ✓ 18
8.d odd 2 1 2808.1.fi.b yes 18
13.b even 2 1 2808.1.fi.b yes 18
27.e even 9 1 inner 2808.1.fi.a ✓ 18
104.h odd 2 1 CM 2808.1.fi.a ✓ 18
216.r odd 18 1 2808.1.fi.b yes 18
351.bl even 18 1 2808.1.fi.b yes 18
2808.fi odd 18 1 inner 2808.1.fi.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2808.1.fi.a ✓ 18 1.a even 1 1 trivial
2808.1.fi.a ✓ 18 27.e even 9 1 inner
2808.1.fi.a ✓ 18 104.h odd 2 1 CM
2808.1.fi.a ✓ 18 2808.fi odd 18 1 inner
2808.1.fi.b yes 18 8.d odd 2 1
2808.1.fi.b yes 18 13.b even 2 1
2808.1.fi.b yes 18 216.r odd 18 1
2808.1.fi.b yes 18 351.bl even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} - 18 T_{5}^{13} + 30 T_{5}^{12} + 27 T_{5}^{11} - 18 T_{5}^{10} + 2 T_{5}^{9} + 81 T_{5}^{8} + \cdots + 1 \) acting on \(S_{1}^{\mathrm{new}}(2808, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{3} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{18} + T^{9} + 1 \) Copy content Toggle raw display
$5$ \( T^{18} - 18 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{18} + 9 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{18} \) Copy content Toggle raw display
$13$ \( (T^{6} + T^{3} + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{18} + 9 T^{16} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{18} \) Copy content Toggle raw display
$23$ \( T^{18} \) Copy content Toggle raw display
$29$ \( T^{18} \) Copy content Toggle raw display
$31$ \( (T^{6} - 6 T^{5} + 15 T^{4} + \cdots + 1)^{3} \) Copy content Toggle raw display
$37$ \( T^{18} + 9 T^{16} + \cdots + 1 \) 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
$47$ \( T^{18} - 18 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{18} \) Copy content Toggle raw display
$59$ \( T^{18} \) Copy content Toggle raw display
$61$ \( T^{18} \) Copy content Toggle raw display
$67$ \( T^{18} \) Copy content Toggle raw display
$71$ \( T^{18} + 9 T^{16} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{18} \) Copy content Toggle raw display
$79$ \( T^{18} \) Copy content Toggle raw display
$83$ \( T^{18} \) Copy content Toggle raw display
$89$ \( T^{18} \) Copy content Toggle raw display
$97$ \( T^{18} \) Copy content Toggle raw display
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