Properties

Label 2880.1.i.b
Level $2880$
Weight $1$
Character orbit 2880.i
Analytic conductor $1.437$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -40
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.i (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.43730723638\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.5400.2

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{8}^{2} q^{5} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8}^{2} q^{5} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{7} + (\zeta_{8}^{3} - \zeta_{8}) q^{11} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{13} + q^{23} - q^{25} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{35} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{37} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{41} - q^{49} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{55} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{59} + (\zeta_{8}^{3} + \zeta_{8}) q^{65} + \zeta_{8}^{2} q^{77} + (\zeta_{8}^{3} + \zeta_{8}) q^{89} - \zeta_{8}^{2} q^{91} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{23} - 4 q^{25} - 4 q^{49}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1889.1
−0.707107 + 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
0 0 0 1.00000i 0 1.41421i 0 0 0
1889.2 0 0 0 1.00000i 0 1.41421i 0 0 0
1889.3 0 0 0 1.00000i 0 1.41421i 0 0 0
1889.4 0 0 0 1.00000i 0 1.41421i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
8.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
24.f even 2 1 inner
120.i odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.1.i.b yes 4
3.b odd 2 1 2880.1.i.a 4
4.b odd 2 1 2880.1.i.a 4
5.b even 2 1 2880.1.i.a 4
8.b even 2 1 inner 2880.1.i.b yes 4
8.d odd 2 1 2880.1.i.a 4
12.b even 2 1 inner 2880.1.i.b yes 4
15.d odd 2 1 inner 2880.1.i.b yes 4
20.d odd 2 1 inner 2880.1.i.b yes 4
24.f even 2 1 inner 2880.1.i.b yes 4
24.h odd 2 1 2880.1.i.a 4
40.e odd 2 1 CM 2880.1.i.b yes 4
40.f even 2 1 2880.1.i.a 4
60.h even 2 1 2880.1.i.a 4
120.i odd 2 1 inner 2880.1.i.b yes 4
120.m even 2 1 2880.1.i.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2880.1.i.a 4 3.b odd 2 1
2880.1.i.a 4 4.b odd 2 1
2880.1.i.a 4 5.b even 2 1
2880.1.i.a 4 8.d odd 2 1
2880.1.i.a 4 24.h odd 2 1
2880.1.i.a 4 40.f even 2 1
2880.1.i.a 4 60.h even 2 1
2880.1.i.a 4 120.m even 2 1
2880.1.i.b yes 4 1.a even 1 1 trivial
2880.1.i.b yes 4 8.b even 2 1 inner
2880.1.i.b yes 4 12.b even 2 1 inner
2880.1.i.b yes 4 15.d odd 2 1 inner
2880.1.i.b yes 4 20.d odd 2 1 inner
2880.1.i.b yes 4 24.f even 2 1 inner
2880.1.i.b yes 4 40.e odd 2 1 CM
2880.1.i.b yes 4 120.i odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23} - 2 \) acting on \(S_{1}^{\mathrm{new}}(2880, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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