Properties

Label 3584.1.c.b
Level $3584$
Weight $1$
Character orbit 3584.c
Analytic conductor $1.789$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
RM discriminant 56
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3584 = 2^{9} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3584.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.78864900528\)
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 4.0.25088.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( - \zeta_{8}^{3} - \zeta_{8}) q^{5} + \zeta_{8}^{2} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{8}^{3} - \zeta_{8}) q^{5} + \zeta_{8}^{2} q^{7} + q^{9} + (\zeta_{8}^{3} - \zeta_{8}) q^{11} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{13} - q^{25} - \zeta_{8}^{2} q^{31} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{35} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{43} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{45} - \zeta_{8}^{2} q^{47} - q^{49} + \zeta_{8}^{2} q^{55} + (\zeta_{8}^{3} + \zeta_{8}) q^{61} + \zeta_{8}^{2} q^{63} + ( - \zeta_{8}^{2} - 2) q^{65} + (\zeta_{8}^{3} - \zeta_{8}) q^{67} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{77} + q^{81} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{91} + (\zeta_{8}^{3} - \zeta_{8}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{9} - 4 q^{25} - 4 q^{49} - 8 q^{65} + 4 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2561.1
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0 0 0 1.41421i 0 1.00000i 0 1.00000 0
2561.2 0 0 0 1.41421i 0 1.00000i 0 1.00000 0
2561.3 0 0 0 1.41421i 0 1.00000i 0 1.00000 0
2561.4 0 0 0 1.41421i 0 1.00000i 0 1.00000 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
56.e even 2 1 RM by \(\Q(\sqrt{14}) \)
4.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
28.d even 2 1 inner
56.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3584.1.c.b 4
4.b odd 2 1 inner 3584.1.c.b 4
7.b odd 2 1 inner 3584.1.c.b 4
8.b even 2 1 inner 3584.1.c.b 4
8.d odd 2 1 inner 3584.1.c.b 4
16.e even 4 2 3584.1.h.b 4
16.f odd 4 2 3584.1.h.b 4
28.d even 2 1 inner 3584.1.c.b 4
56.e even 2 1 RM 3584.1.c.b 4
56.h odd 2 1 inner 3584.1.c.b 4
112.j even 4 2 3584.1.h.b 4
112.l odd 4 2 3584.1.h.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.1.c.b 4 1.a even 1 1 trivial
3584.1.c.b 4 4.b odd 2 1 inner
3584.1.c.b 4 7.b odd 2 1 inner
3584.1.c.b 4 8.b even 2 1 inner
3584.1.c.b 4 8.d odd 2 1 inner
3584.1.c.b 4 28.d even 2 1 inner
3584.1.c.b 4 56.e even 2 1 RM
3584.1.c.b 4 56.h odd 2 1 inner
3584.1.h.b 4 16.e even 4 2
3584.1.h.b 4 16.f odd 4 2
3584.1.h.b 4 112.j even 4 2
3584.1.h.b 4 112.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3584, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{191} \) 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
$5$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) 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
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 2)^{2} \) 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
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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