Properties

Label 3584.1.v.b
Level $3584$
Weight $1$
Character orbit 3584.v
Analytic conductor $1.789$
Analytic rank $0$
Dimension $4$
Projective image $D_{8}$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3584 = 2^{9} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3584.v (of order \(8\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.78864900528\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.5156108238848.3

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{8}^{3} q^{7} -\zeta_{8} q^{9} +O(q^{10})\) \( q + \zeta_{8}^{3} q^{7} -\zeta_{8} q^{9} + ( -1 - \zeta_{8}^{3} ) q^{11} + ( 1 + \zeta_{8}^{2} ) q^{23} -\zeta_{8}^{3} q^{25} + ( 1 - \zeta_{8} ) q^{29} + ( 1 - \zeta_{8}^{3} ) q^{37} + ( -\zeta_{8} - \zeta_{8}^{2} ) q^{43} -\zeta_{8}^{2} q^{49} + ( -\zeta_{8} - \zeta_{8}^{2} ) q^{53} + q^{63} + ( -\zeta_{8}^{2} - \zeta_{8}^{3} ) q^{67} + ( \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{77} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{79} + \zeta_{8}^{2} q^{81} + ( -1 + \zeta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{11} + 4q^{23} + 4q^{29} + 4q^{37} + 4q^{63} - 4q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{8}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
0 0 0 0 0 0.707107 0.707107i 0 0.707107 + 0.707107i 0
1217.1 0 0 0 0 0 0.707107 + 0.707107i 0 0.707107 0.707107i 0
2113.1 0 0 0 0 0 −0.707107 + 0.707107i 0 −0.707107 0.707107i 0
3009.1 0 0 0 0 0 −0.707107 0.707107i 0 −0.707107 + 0.707107i 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
32.g even 8 1 inner
224.v odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3584.1.v.b 4
4.b odd 2 1 3584.1.v.c 4
7.b odd 2 1 CM 3584.1.v.b 4
8.b even 2 1 3584.1.v.d 4
8.d odd 2 1 3584.1.v.a 4
16.e even 4 1 224.1.v.a 4
16.e even 4 1 1792.1.v.a 4
16.f odd 4 1 896.1.v.a 4
16.f odd 4 1 1792.1.v.b 4
28.d even 2 1 3584.1.v.c 4
32.g even 8 1 224.1.v.a 4
32.g even 8 1 1792.1.v.a 4
32.g even 8 1 inner 3584.1.v.b 4
32.g even 8 1 3584.1.v.d 4
32.h odd 8 1 896.1.v.a 4
32.h odd 8 1 1792.1.v.b 4
32.h odd 8 1 3584.1.v.a 4
32.h odd 8 1 3584.1.v.c 4
48.i odd 4 1 2016.1.dp.b 4
56.e even 2 1 3584.1.v.a 4
56.h odd 2 1 3584.1.v.d 4
96.p odd 8 1 2016.1.dp.b 4
112.j even 4 1 896.1.v.a 4
112.j even 4 1 1792.1.v.b 4
112.l odd 4 1 224.1.v.a 4
112.l odd 4 1 1792.1.v.a 4
112.w even 12 2 1568.1.bl.a 8
112.x odd 12 2 1568.1.bl.a 8
224.v odd 8 1 224.1.v.a 4
224.v odd 8 1 1792.1.v.a 4
224.v odd 8 1 inner 3584.1.v.b 4
224.v odd 8 1 3584.1.v.d 4
224.x even 8 1 896.1.v.a 4
224.x even 8 1 1792.1.v.b 4
224.x even 8 1 3584.1.v.a 4
224.x even 8 1 3584.1.v.c 4
224.bc odd 24 2 1568.1.bl.a 8
224.bd even 24 2 1568.1.bl.a 8
336.y even 4 1 2016.1.dp.b 4
672.bo even 8 1 2016.1.dp.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.1.v.a 4 16.e even 4 1
224.1.v.a 4 32.g even 8 1
224.1.v.a 4 112.l odd 4 1
224.1.v.a 4 224.v odd 8 1
896.1.v.a 4 16.f odd 4 1
896.1.v.a 4 32.h odd 8 1
896.1.v.a 4 112.j even 4 1
896.1.v.a 4 224.x even 8 1
1568.1.bl.a 8 112.w even 12 2
1568.1.bl.a 8 112.x odd 12 2
1568.1.bl.a 8 224.bc odd 24 2
1568.1.bl.a 8 224.bd even 24 2
1792.1.v.a 4 16.e even 4 1
1792.1.v.a 4 32.g even 8 1
1792.1.v.a 4 112.l odd 4 1
1792.1.v.a 4 224.v odd 8 1
1792.1.v.b 4 16.f odd 4 1
1792.1.v.b 4 32.h odd 8 1
1792.1.v.b 4 112.j even 4 1
1792.1.v.b 4 224.x even 8 1
2016.1.dp.b 4 48.i odd 4 1
2016.1.dp.b 4 96.p odd 8 1
2016.1.dp.b 4 336.y even 4 1
2016.1.dp.b 4 672.bo even 8 1
3584.1.v.a 4 8.d odd 2 1
3584.1.v.a 4 32.h odd 8 1
3584.1.v.a 4 56.e even 2 1
3584.1.v.a 4 224.x even 8 1
3584.1.v.b 4 1.a even 1 1 trivial
3584.1.v.b 4 7.b odd 2 1 CM
3584.1.v.b 4 32.g even 8 1 inner
3584.1.v.b 4 224.v odd 8 1 inner
3584.1.v.c 4 4.b odd 2 1
3584.1.v.c 4 28.d even 2 1
3584.1.v.c 4 32.h odd 8 1
3584.1.v.c 4 224.x even 8 1
3584.1.v.d 4 8.b even 2 1
3584.1.v.d 4 32.g even 8 1
3584.1.v.d 4 56.h odd 2 1
3584.1.v.d 4 224.v odd 8 1

Hecke kernels

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

\( T_{11}^{4} + 4 T_{11}^{3} + 6 T_{11}^{2} + 4 T_{11} + 2 \)
\( T_{23}^{2} - 2 T_{23} + 2 \)

Hecke characteristic polynomials

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