Properties

Label 1440.1.cm.a
Level $1440$
Weight $1$
Character orbit 1440.cm
Analytic conductor $0.719$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -15
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1440.cm (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.718653618192\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.36238786560000.11

$q$-expansion

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(-1\) \(1\) \(\zeta_{16}^{2}\) \(-1\)

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
−0.923880 + 0.382683i
0.923880 0.382683i
−0.923880 0.382683i
0.923880 + 0.382683i
0.382683 + 0.923880i
−0.382683 0.923880i
0.382683 0.923880i
−0.382683 + 0.923880i
−0.382683 + 0.923880i 0 −0.707107 0.707107i −0.382683 0.923880i 0 0 0.923880 0.382683i 0 1.00000
19.2 0.382683 0.923880i 0 −0.707107 0.707107i 0.382683 + 0.923880i 0 0 −0.923880 + 0.382683i 0 1.00000
379.1 −0.382683 0.923880i 0 −0.707107 + 0.707107i −0.382683 + 0.923880i 0 0 0.923880 + 0.382683i 0 1.00000
379.2 0.382683 + 0.923880i 0 −0.707107 + 0.707107i 0.382683 0.923880i 0 0 −0.923880 0.382683i 0 1.00000
739.1 −0.923880 0.382683i 0 0.707107 + 0.707107i −0.923880 + 0.382683i 0 0 −0.382683 0.923880i 0 1.00000
739.2 0.923880 + 0.382683i 0 0.707107 + 0.707107i 0.923880 0.382683i 0 0 0.382683 + 0.923880i 0 1.00000
1099.1 −0.923880 + 0.382683i 0 0.707107 0.707107i −0.923880 0.382683i 0 0 −0.382683 + 0.923880i 0 1.00000
1099.2 0.923880 0.382683i 0 0.707107 0.707107i 0.923880 + 0.382683i 0 0 0.382683 0.923880i 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1099.2
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
32.h odd 8 1 inner
96.o even 8 1 inner
160.y odd 8 1 inner
480.bs even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.1.cm.a 8
3.b odd 2 1 inner 1440.1.cm.a 8
5.b even 2 1 inner 1440.1.cm.a 8
15.d odd 2 1 CM 1440.1.cm.a 8
32.h odd 8 1 inner 1440.1.cm.a 8
96.o even 8 1 inner 1440.1.cm.a 8
160.y odd 8 1 inner 1440.1.cm.a 8
480.bs even 8 1 inner 1440.1.cm.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.1.cm.a 8 1.a even 1 1 trivial
1440.1.cm.a 8 3.b odd 2 1 inner
1440.1.cm.a 8 5.b even 2 1 inner
1440.1.cm.a 8 15.d odd 2 1 CM
1440.1.cm.a 8 32.h odd 8 1 inner
1440.1.cm.a 8 96.o even 8 1 inner
1440.1.cm.a 8 160.y odd 8 1 inner
1440.1.cm.a 8 480.bs even 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1440, [\chi])\).

Hecke characteristic polynomials

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