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Properties

Label 2880.1.ec.a
Level $2880$
Weight $1$
Character orbit 2880.ec
Analytic conductor $1.437$
Analytic rank $0$
Dimension $16$
Projective image $D_{16}$
CM discriminant -15
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2880,1,Mod(19,2880)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2880, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([8, 7, 0, 8])) 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("2880.19"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2880.ec (of order \(16\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.43730723638\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{16})\)
Coefficient field: \(\Q(\zeta_{32})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{16}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{16} + \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{32}^{7} q^{2} + \zeta_{32}^{14} q^{4} - \zeta_{32} q^{5} + \zeta_{32}^{5} q^{8} + \zeta_{32}^{8} q^{10} - \zeta_{32}^{12} q^{16} + (\zeta_{32}^{13} - \zeta_{32}^{11}) q^{17} + (\zeta_{32}^{12} + \zeta_{32}^{2}) q^{19} + \cdots - \zeta_{32}^{11} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{76} + 16 q^{79} - 16 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(-1\) \(1\) \(-\zeta_{32}^{2}\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.980785 − 0.195090i
−0.980785 + 0.195090i
−0.195090 + 0.980785i
0.195090 − 0.980785i
−0.831470 − 0.555570i
0.831470 + 0.555570i
−0.831470 + 0.555570i
0.831470 − 0.555570i
−0.195090 − 0.980785i
0.195090 + 0.980785i
0.980785 + 0.195090i
−0.980785 − 0.195090i
0.555570 − 0.831470i
−0.555570 + 0.831470i
0.555570 + 0.831470i
−0.555570 − 0.831470i
−0.195090 + 0.980785i 0 −0.923880 − 0.382683i −0.980785 + 0.195090i 0 0 0.555570 − 0.831470i 0 − 1.00000i
19.2 0.195090 − 0.980785i 0 −0.923880 − 0.382683i 0.980785 − 0.195090i 0 0 −0.555570 + 0.831470i 0 − 1.00000i
379.1 −0.980785 + 0.195090i 0 0.923880 − 0.382683i 0.195090 − 0.980785i 0 0 −0.831470 + 0.555570i 0 1.00000i
379.2 0.980785 − 0.195090i 0 0.923880 − 0.382683i −0.195090 + 0.980785i 0 0 0.831470 − 0.555570i 0 1.00000i
739.1 −0.555570 − 0.831470i 0 −0.382683 + 0.923880i 0.831470 + 0.555570i 0 0 0.980785 − 0.195090i 0 − 1.00000i
739.2 0.555570 + 0.831470i 0 −0.382683 + 0.923880i −0.831470 − 0.555570i 0 0 −0.980785 + 0.195090i 0 − 1.00000i
1099.1 −0.555570 + 0.831470i 0 −0.382683 − 0.923880i 0.831470 − 0.555570i 0 0 0.980785 + 0.195090i 0 1.00000i
1099.2 0.555570 − 0.831470i 0 −0.382683 − 0.923880i −0.831470 + 0.555570i 0 0 −0.980785 − 0.195090i 0 1.00000i
1459.1 −0.980785 − 0.195090i 0 0.923880 + 0.382683i 0.195090 + 0.980785i 0 0 −0.831470 − 0.555570i 0 − 1.00000i
1459.2 0.980785 + 0.195090i 0 0.923880 + 0.382683i −0.195090 − 0.980785i 0 0 0.831470 + 0.555570i 0 − 1.00000i
1819.1 −0.195090 − 0.980785i 0 −0.923880 + 0.382683i −0.980785 − 0.195090i 0 0 0.555570 + 0.831470i 0 1.00000i
1819.2 0.195090 + 0.980785i 0 −0.923880 + 0.382683i 0.980785 + 0.195090i 0 0 −0.555570 − 0.831470i 0 1.00000i
2179.1 −0.831470 + 0.555570i 0 0.382683 − 0.923880i −0.555570 + 0.831470i 0 0 0.195090 + 0.980785i 0 − 1.00000i
2179.2 0.831470 − 0.555570i 0 0.382683 − 0.923880i 0.555570 − 0.831470i 0 0 −0.195090 − 0.980785i 0 − 1.00000i
2539.1 −0.831470 − 0.555570i 0 0.382683 + 0.923880i −0.555570 − 0.831470i 0 0 0.195090 − 0.980785i 0 1.00000i
2539.2 0.831470 + 0.555570i 0 0.382683 + 0.923880i 0.555570 + 0.831470i 0 0 −0.195090 + 0.980785i 0 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
64.j odd 16 1 inner
192.s even 16 1 inner
320.bh odd 16 1 inner
960.cp even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.1.ec.a ✓ 16
3.b odd 2 1 inner 2880.1.ec.a ✓ 16
5.b even 2 1 inner 2880.1.ec.a ✓ 16
15.d odd 2 1 CM 2880.1.ec.a ✓ 16
64.j odd 16 1 inner 2880.1.ec.a ✓ 16
192.s even 16 1 inner 2880.1.ec.a ✓ 16
320.bh odd 16 1 inner 2880.1.ec.a ✓ 16
960.cp even 16 1 inner 2880.1.ec.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2880.1.ec.a ✓ 16 1.a even 1 1 trivial
2880.1.ec.a ✓ 16 3.b odd 2 1 inner
2880.1.ec.a ✓ 16 5.b even 2 1 inner
2880.1.ec.a ✓ 16 15.d odd 2 1 CM
2880.1.ec.a ✓ 16 64.j odd 16 1 inner
2880.1.ec.a ✓ 16 192.s even 16 1 inner
2880.1.ec.a ✓ 16 320.bh odd 16 1 inner
2880.1.ec.a ✓ 16 960.cp even 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2880, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 1 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} + 24 T^{12} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( (T^{8} + 8 T^{5} + 2 T^{4} + \cdots + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 16 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{2} + 2)^{4} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} + 24 T^{12} + \cdots + 4 \) Copy content Toggle raw display
$53$ \( T^{16} - 16 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( (T^{8} + 4 T^{6} + 6 T^{4} + \cdots + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( (T^{2} - 2 T + 2)^{8} \) Copy content Toggle raw display
$83$ \( T^{16} - 16 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( T^{16} \) Copy content Toggle raw display
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