# Properties

 Label 4608.2.c.n Level $4608$ Weight $2$ Character orbit 4608.c Analytic conductor $36.795$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4608 = 2^{9} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4608.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.7950652514$$ 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: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{8}^{2} q^{5} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{7} +O(q^{10})$$ $$q + 2 \zeta_{8}^{2} q^{5} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{7} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{11} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{17} + 6 q^{23} + q^{25} + 2 \zeta_{8}^{2} q^{29} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{31} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{35} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{37} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{41} -12 \zeta_{8}^{2} q^{43} -6 q^{47} + 5 q^{49} + 2 \zeta_{8}^{2} q^{53} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{55} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{61} + 12 \zeta_{8}^{2} q^{67} -6 q^{71} + 6 q^{73} + 4 \zeta_{8}^{2} q^{77} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{79} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{83} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{85} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{89} -8 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + O(q^{10})$$ $$4 q + 24 q^{23} + 4 q^{25} - 24 q^{47} + 20 q^{49} - 24 q^{71} + 24 q^{73} - 32 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times$$.

 $$n$$ $$2053$$ $$3583$$ $$4097$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4607.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
0 0 0 2.00000i 0 1.41421i 0 0 0
4607.2 0 0 0 2.00000i 0 1.41421i 0 0 0
4607.3 0 0 0 2.00000i 0 1.41421i 0 0 0
4607.4 0 0 0 2.00000i 0 1.41421i 0 0 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
8.b even 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4608.2.c.n yes 4
3.b odd 2 1 4608.2.c.k 4
4.b odd 2 1 4608.2.c.k 4
8.b even 2 1 inner 4608.2.c.n yes 4
8.d odd 2 1 4608.2.c.k 4
12.b even 2 1 inner 4608.2.c.n yes 4
16.e even 4 1 4608.2.f.b 2
16.e even 4 1 4608.2.f.f 2
16.f odd 4 1 4608.2.f.c 2
16.f odd 4 1 4608.2.f.g 2
24.f even 2 1 inner 4608.2.c.n yes 4
24.h odd 2 1 4608.2.c.k 4
48.i odd 4 1 4608.2.f.c 2
48.i odd 4 1 4608.2.f.g 2
48.k even 4 1 4608.2.f.b 2
48.k even 4 1 4608.2.f.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4608.2.c.k 4 3.b odd 2 1
4608.2.c.k 4 4.b odd 2 1
4608.2.c.k 4 8.d odd 2 1
4608.2.c.k 4 24.h odd 2 1
4608.2.c.n yes 4 1.a even 1 1 trivial
4608.2.c.n yes 4 8.b even 2 1 inner
4608.2.c.n yes 4 12.b even 2 1 inner
4608.2.c.n yes 4 24.f even 2 1 inner
4608.2.f.b 2 16.e even 4 1
4608.2.f.b 2 48.k even 4 1
4608.2.f.c 2 16.f odd 4 1
4608.2.f.c 2 48.i odd 4 1
4608.2.f.f 2 16.e even 4 1
4608.2.f.f 2 48.k even 4 1
4608.2.f.g 2 16.f odd 4 1
4608.2.f.g 2 48.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4608, [\chi])$$:

 $$T_{5}^{2} + 4$$ $$T_{7}^{2} + 2$$ $$T_{11}^{2} - 8$$ $$T_{13}$$ $$T_{23} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 4 + T^{2} )^{2}$$
$7$ $$( 2 + T^{2} )^{2}$$
$11$ $$( -8 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( 18 + T^{2} )^{2}$$
$19$ $$T^{4}$$
$23$ $$( -6 + T )^{4}$$
$29$ $$( 4 + T^{2} )^{2}$$
$31$ $$( 2 + T^{2} )^{2}$$
$37$ $$( -72 + T^{2} )^{2}$$
$41$ $$( 18 + T^{2} )^{2}$$
$43$ $$( 144 + T^{2} )^{2}$$
$47$ $$( 6 + T )^{4}$$
$53$ $$( 4 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$( -72 + T^{2} )^{2}$$
$67$ $$( 144 + T^{2} )^{2}$$
$71$ $$( 6 + T )^{4}$$
$73$ $$( -6 + T )^{4}$$
$79$ $$( 50 + T^{2} )^{2}$$
$83$ $$( -8 + T^{2} )^{2}$$
$89$ $$( 18 + T^{2} )^{2}$$
$97$ $$( 8 + T )^{4}$$