# 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 $8$

# Related objects

## 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: $$x^{4} + 1$$ x^4 + 1 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.

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{1}^{\mathrm{new}}(3584, [\chi])$$:

 $$T_{3}$$ T3 $$T_{191}$$ T191

## Hecke characteristic polynomials

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