# Properties

 Label 4608.2.c.g Level $4608$ Weight $2$ Character orbit 4608.c Analytic conductor $36.795$ Analytic rank $0$ 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.7950652514$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 3 \beta q^{7} +O(q^{10})$$ $$q + 3 \beta q^{7} + 4 q^{11} -6 q^{13} -3 \beta q^{17} -2 \beta q^{19} -6 q^{23} + 5 q^{25} -6 \beta q^{29} -3 \beta q^{31} -6 q^{37} + \beta q^{41} -2 \beta q^{43} + 6 q^{47} -11 q^{49} -6 \beta q^{53} -4 q^{59} + 6 q^{61} -8 \beta q^{67} + 6 q^{71} + 6 q^{73} + 12 \beta q^{77} -3 \beta q^{79} + 16 q^{83} + 9 \beta q^{89} -18 \beta q^{91} -12 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + O(q^{10})$$ $$2 q + 8 q^{11} - 12 q^{13} - 12 q^{23} + 10 q^{25} - 12 q^{37} + 12 q^{47} - 22 q^{49} - 8 q^{59} + 12 q^{61} + 12 q^{71} + 12 q^{73} + 32 q^{83} - 24 q^{97} + O(q^{100})$$

## 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
 − 1.41421i 1.41421i
0 0 0 0 0 4.24264i 0 0 0
4607.2 0 0 0 0 0 4.24264i 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
12.b 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.g yes 2
3.b odd 2 1 4608.2.c.a 2
4.b odd 2 1 4608.2.c.a 2
8.b even 2 1 4608.2.c.b yes 2
8.d odd 2 1 4608.2.c.h yes 2
12.b even 2 1 inner 4608.2.c.g yes 2
16.e even 4 2 4608.2.f.k 4
16.f odd 4 2 4608.2.f.i 4
24.f even 2 1 4608.2.c.b yes 2
24.h odd 2 1 4608.2.c.h yes 2
48.i odd 4 2 4608.2.f.i 4
48.k even 4 2 4608.2.f.k 4

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

## Hecke kernels

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

 $$T_{5}$$ $$T_{7}^{2} + 18$$ $$T_{11} - 4$$ $$T_{13} + 6$$ $$T_{23} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$18 + T^{2}$$
$11$ $$( -4 + T )^{2}$$
$13$ $$( 6 + T )^{2}$$
$17$ $$18 + T^{2}$$
$19$ $$8 + T^{2}$$
$23$ $$( 6 + T )^{2}$$
$29$ $$72 + T^{2}$$
$31$ $$18 + T^{2}$$
$37$ $$( 6 + T )^{2}$$
$41$ $$2 + T^{2}$$
$43$ $$8 + T^{2}$$
$47$ $$( -6 + T )^{2}$$
$53$ $$72 + T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$( -6 + T )^{2}$$
$67$ $$128 + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$( -6 + T )^{2}$$
$79$ $$18 + T^{2}$$
$83$ $$( -16 + T )^{2}$$
$89$ $$162 + T^{2}$$
$97$ $$( 12 + T )^{2}$$