# Properties

 Label 4608.2.a.j Level $4608$ Weight $2$ Character orbit 4608.a Self dual yes Analytic conductor $36.795$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4608 = 2^{9} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4608.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.7950652514$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1536) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} + \beta q^{7} +O(q^{10})$$ $$q + \beta q^{5} + \beta q^{7} + 2 q^{11} -2 q^{17} -4 q^{19} -2 \beta q^{23} -3 q^{25} -7 \beta q^{29} -5 \beta q^{31} + 2 q^{35} -6 \beta q^{37} -6 q^{41} -8 q^{43} -2 \beta q^{47} -5 q^{49} + \beta q^{53} + 2 \beta q^{55} + 12 q^{59} + 10 \beta q^{61} -8 q^{67} + 10 \beta q^{71} -8 q^{73} + 2 \beta q^{77} -3 \beta q^{79} + 6 q^{83} -2 \beta q^{85} -2 q^{89} -4 \beta q^{95} -14 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 4q^{11} - 4q^{17} - 8q^{19} - 6q^{25} + 4q^{35} - 12q^{41} - 16q^{43} - 10q^{49} + 24q^{59} - 16q^{67} - 16q^{73} + 12q^{83} - 4q^{89} - 28q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
0 0 0 −1.41421 0 −1.41421 0 0 0
1.2 0 0 0 1.41421 0 1.41421 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4608.2.a.j 2
3.b odd 2 1 1536.2.a.c 2
4.b odd 2 1 4608.2.a.h 2
8.b even 2 1 4608.2.a.h 2
8.d odd 2 1 inner 4608.2.a.j 2
12.b even 2 1 1536.2.a.j yes 2
16.e even 4 2 4608.2.d.g 4
16.f odd 4 2 4608.2.d.g 4
24.f even 2 1 1536.2.a.c 2
24.h odd 2 1 1536.2.a.j yes 2
48.i odd 4 2 1536.2.d.d 4
48.k even 4 2 1536.2.d.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.a.c 2 3.b odd 2 1
1536.2.a.c 2 24.f even 2 1
1536.2.a.j yes 2 12.b even 2 1
1536.2.a.j yes 2 24.h odd 2 1
1536.2.d.d 4 48.i odd 4 2
1536.2.d.d 4 48.k even 4 2
4608.2.a.h 2 4.b odd 2 1
4608.2.a.h 2 8.b even 2 1
4608.2.a.j 2 1.a even 1 1 trivial
4608.2.a.j 2 8.d odd 2 1 inner
4608.2.d.g 4 16.e even 4 2
4608.2.d.g 4 16.f odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4608))$$:

 $$T_{5}^{2} - 2$$ $$T_{7}^{2} - 2$$ $$T_{11} - 2$$ $$T_{17} + 2$$ $$T_{23}^{2} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-2 + T^{2}$$
$7$ $$-2 + T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$-8 + T^{2}$$
$29$ $$-98 + T^{2}$$
$31$ $$-50 + T^{2}$$
$37$ $$-72 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( 8 + T )^{2}$$
$47$ $$-8 + T^{2}$$
$53$ $$-2 + T^{2}$$
$59$ $$( -12 + T )^{2}$$
$61$ $$-200 + T^{2}$$
$67$ $$( 8 + T )^{2}$$
$71$ $$-200 + T^{2}$$
$73$ $$( 8 + T )^{2}$$
$79$ $$-18 + T^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$( 2 + T )^{2}$$
$97$ $$( 14 + T )^{2}$$