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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Newspace parameters

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

Self dual: no
Analytic conductor: \(1.43730723638\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{16})\)
Coefficient field: \(\Q(\zeta_{32})\)
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

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

\(f(q)\) \(=\) \( q -\zeta_{32}^{7} q^{2} + \zeta_{32}^{14} q^{4} -\zeta_{32} q^{5} + \zeta_{32}^{5} q^{8} +O(q^{10})\) \( q -\zeta_{32}^{7} q^{2} + \zeta_{32}^{14} q^{4} -\zeta_{32} q^{5} + \zeta_{32}^{5} q^{8} + \zeta_{32}^{8} q^{10} -\zeta_{32}^{12} q^{16} + ( -\zeta_{32}^{11} + \zeta_{32}^{13} ) q^{17} + ( \zeta_{32}^{2} + \zeta_{32}^{12} ) q^{19} -\zeta_{32}^{15} q^{20} + ( \zeta_{32}^{5} - \zeta_{32}^{7} ) q^{23} + \zeta_{32}^{2} q^{25} + ( \zeta_{32}^{6} - \zeta_{32}^{10} ) q^{31} -\zeta_{32}^{3} q^{32} + ( -\zeta_{32}^{2} + \zeta_{32}^{4} ) q^{34} + ( \zeta_{32}^{3} - \zeta_{32}^{9} ) q^{38} -\zeta_{32}^{6} q^{40} + ( -\zeta_{32}^{12} + \zeta_{32}^{14} ) q^{46} + ( -\zeta_{32}^{9} - \zeta_{32}^{15} ) q^{47} + \zeta_{32}^{4} q^{49} -\zeta_{32}^{9} q^{50} + ( -\zeta_{32}^{11} + \zeta_{32}^{15} ) q^{53} + ( -\zeta_{32}^{8} - \zeta_{32}^{14} ) q^{61} + ( -\zeta_{32} - \zeta_{32}^{13} ) q^{62} + \zeta_{32}^{10} q^{64} + ( \zeta_{32}^{9} - \zeta_{32}^{11} ) q^{68} + ( -1 - \zeta_{32}^{10} ) q^{76} + ( 1 - \zeta_{32}^{8} ) q^{79} + \zeta_{32}^{13} q^{80} + ( \zeta_{32} - \zeta_{32}^{13} ) q^{83} + ( \zeta_{32}^{12} - \zeta_{32}^{14} ) q^{85} + ( -\zeta_{32}^{3} + \zeta_{32}^{5} ) q^{92} + ( -1 - \zeta_{32}^{6} ) q^{94} + ( -\zeta_{32}^{3} - \zeta_{32}^{13} ) q^{95} -\zeta_{32}^{11} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 16q^{76} + 16q^{79} - 16q^{94} + O(q^{100}) \)

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.

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 2539.2
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

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