# Properties

 Label 4608.2.f.l Level $4608$ Weight $2$ Character orbit 4608.f 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.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.7950652514$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} - \beta_{2} q^{7}+O(q^{10})$$ q + b3 * q^5 - b2 * q^7 $$q + \beta_{3} q^{5} - \beta_{2} q^{7} - \beta_1 q^{13} - \beta_{2} q^{17} + \beta_{3} q^{19} + 6 q^{23} + 3 q^{25} + 2 \beta_{3} q^{29} + 7 \beta_{2} q^{31} - 2 \beta_1 q^{35} - \beta_1 q^{37} - 3 \beta_{2} q^{41} - 3 \beta_{3} q^{43} - 10 q^{47} + 5 q^{49} + 2 \beta_{3} q^{53} + 6 \beta_1 q^{59} - 5 \beta_1 q^{61} - 4 \beta_{2} q^{65} + 2 \beta_{3} q^{67} + 2 q^{71} + 10 q^{73} - 9 \beta_{2} q^{79} - 2 \beta_1 q^{83} - 2 \beta_1 q^{85} - 7 \beta_{2} q^{89} - \beta_{3} q^{91} + 8 q^{95} + 12 q^{97}+O(q^{100})$$ q + b3 * q^5 - b2 * q^7 - b1 * q^13 - b2 * q^17 + b3 * q^19 + 6 * q^23 + 3 * q^25 + 2*b3 * q^29 + 7*b2 * q^31 - 2*b1 * q^35 - b1 * q^37 - 3*b2 * q^41 - 3*b3 * q^43 - 10 * q^47 + 5 * q^49 + 2*b3 * q^53 + 6*b1 * q^59 - 5*b1 * q^61 - 4*b2 * q^65 + 2*b3 * q^67 + 2 * q^71 + 10 * q^73 - 9*b2 * q^79 - 2*b1 * q^83 - 2*b1 * q^85 - 7*b2 * q^89 - b3 * q^91 + 8 * q^95 + 12 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 24 q^{23} + 12 q^{25} - 40 q^{47} + 20 q^{49} + 8 q^{71} + 40 q^{73} + 32 q^{95} + 48 q^{97}+O(q^{100})$$ 4 * q + 24 * q^23 + 12 * q^25 - 40 * q^47 + 20 * q^49 + 8 * q^71 + 40 * q^73 + 32 * q^95 + 48 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{8}^{2}$$ 2*v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-2\zeta_{8}^{3} + 2\zeta_{8}$$ -2*v^3 + 2*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + 2\beta_{2} ) / 4$$ (b3 + 2*b2) / 4 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} ) / 4$$ (-b3 + 2*b2) / 4

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2303.1
 −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i
0 0 0 −2.82843 0 1.41421i 0 0 0
2303.2 0 0 0 −2.82843 0 1.41421i 0 0 0
2303.3 0 0 0 2.82843 0 1.41421i 0 0 0
2303.4 0 0 0 2.82843 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.f.l 4
3.b odd 2 1 4608.2.f.j 4
4.b odd 2 1 4608.2.f.j 4
8.b even 2 1 inner 4608.2.f.l 4
8.d odd 2 1 4608.2.f.j 4
12.b even 2 1 inner 4608.2.f.l 4
16.e even 4 1 4608.2.c.c 2
16.e even 4 1 4608.2.c.e yes 2
16.f odd 4 1 4608.2.c.d yes 2
16.f odd 4 1 4608.2.c.f yes 2
24.f even 2 1 inner 4608.2.f.l 4
24.h odd 2 1 4608.2.f.j 4
48.i odd 4 1 4608.2.c.d yes 2
48.i odd 4 1 4608.2.c.f yes 2
48.k even 4 1 4608.2.c.c 2
48.k even 4 1 4608.2.c.e yes 2

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

## Hecke kernels

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

 $$T_{5}^{2} - 8$$ T5^2 - 8 $$T_{7}^{2} + 2$$ T7^2 + 2 $$T_{19}^{2} - 8$$ T19^2 - 8 $$T_{23} - 6$$ T23 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 8)^{2}$$
$7$ $$(T^{2} + 2)^{2}$$
$11$ $$T^{4}$$
$13$ $$(T^{2} + 4)^{2}$$
$17$ $$(T^{2} + 2)^{2}$$
$19$ $$(T^{2} - 8)^{2}$$
$23$ $$(T - 6)^{4}$$
$29$ $$(T^{2} - 32)^{2}$$
$31$ $$(T^{2} + 98)^{2}$$
$37$ $$(T^{2} + 4)^{2}$$
$41$ $$(T^{2} + 18)^{2}$$
$43$ $$(T^{2} - 72)^{2}$$
$47$ $$(T + 10)^{4}$$
$53$ $$(T^{2} - 32)^{2}$$
$59$ $$(T^{2} + 144)^{2}$$
$61$ $$(T^{2} + 100)^{2}$$
$67$ $$(T^{2} - 32)^{2}$$
$71$ $$(T - 2)^{4}$$
$73$ $$(T - 10)^{4}$$
$79$ $$(T^{2} + 162)^{2}$$
$83$ $$(T^{2} + 16)^{2}$$
$89$ $$(T^{2} + 98)^{2}$$
$97$ $$(T - 12)^{4}$$