# Properties

 Label 4608.2.d.q Level $4608$ Weight $2$ Character orbit 4608.d Analytic conductor $36.795$ Analytic rank $0$ Dimension $8$ CM discriminant -24 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4608 = 2^{9} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4608.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.7950652514$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{18}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{5} - \beta_{3} q^{7}+O(q^{10})$$ q + b5 * q^5 - b3 * q^7 $$q + \beta_{5} q^{5} - \beta_{3} q^{7} + \beta_1 q^{11} + (\beta_{6} - 5) q^{25} + ( - \beta_{7} + \beta_{5}) q^{29} + (\beta_{4} - 2 \beta_{3}) q^{31} + ( - \beta_{2} - 3 \beta_1) q^{35} + ( - \beta_{6} + 7) q^{49} + (\beta_{7} + 3 \beta_{5}) q^{53} + (\beta_{4} - 3 \beta_{3}) q^{55} - \beta_{2} q^{59} - \beta_{6} q^{73} + ( - \beta_{7} - 4 \beta_{5}) q^{77} + ( - 2 \beta_{4} - \beta_{3}) q^{79} + 5 \beta_1 q^{83} - 2 q^{97}+O(q^{100})$$ q + b5 * q^5 - b3 * q^7 + b1 * q^11 + (b6 - 5) * q^25 + (-b7 + b5) * q^29 + (b4 - 2*b3) * q^31 + (-b2 - 3*b1) * q^35 + (-b6 + 7) * q^49 + (b7 + 3*b5) * q^53 + (b4 - 3*b3) * q^55 - b2 * q^59 - b6 * q^73 + (-b7 - 4*b5) * q^77 + (-2*b4 - b3) * q^79 + 5*b1 * q^83 - 2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 40 q^{25} + 56 q^{49} - 16 q^{97}+O(q^{100})$$ 8 * q - 40 * q^25 + 56 * q^49 - 16 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$4\zeta_{24}^{4} - 2$$ 4*v^4 - 2 $$\beta_{2}$$ $$=$$ $$-8\zeta_{24}^{5} - 8\zeta_{24}^{3} + 8\zeta_{24}$$ -8*v^5 - 8*v^3 + 8*v $$\beta_{3}$$ $$=$$ $$-2\zeta_{24}^{6} + \zeta_{24}^{5} - \zeta_{24}^{3} + 4\zeta_{24}^{2} - \zeta_{24}$$ -2*v^6 + v^5 - v^3 + 4*v^2 - v $$\beta_{4}$$ $$=$$ $$-2\zeta_{24}^{6} - 3\zeta_{24}^{5} + 3\zeta_{24}^{3} + 4\zeta_{24}^{2} + 3\zeta_{24}$$ -2*v^6 - 3*v^5 + 3*v^3 + 4*v^2 + 3*v $$\beta_{5}$$ $$=$$ $$-2\zeta_{24}^{7} + 2\zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24}$$ -2*v^7 + 2*v^6 - v^5 + v^3 - v $$\beta_{6}$$ $$=$$ $$-8\zeta_{24}^{7} + 4\zeta_{24}^{5} + 4\zeta_{24}^{3} + 4\zeta_{24}$$ -8*v^7 + 4*v^5 + 4*v^3 + 4*v $$\beta_{7}$$ $$=$$ $$4\zeta_{24}^{7} + 4\zeta_{24}^{6} + 2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24}$$ 4*v^7 + 4*v^6 + 2*v^5 - 2*v^3 + 2*v
 $$\zeta_{24}$$ $$=$$ $$( 2\beta_{7} + 2\beta_{6} - 4\beta_{5} + 2\beta_{4} - 2\beta_{3} + \beta_{2} ) / 32$$ (2*b7 + 2*b6 - 4*b5 + 2*b4 - 2*b3 + b2) / 32 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{7} + 2\beta_{5} + \beta_{4} + 3\beta_{3} ) / 16$$ (b7 + 2*b5 + b4 + 3*b3) / 16 $$\zeta_{24}^{3}$$ $$=$$ $$( 2\beta_{4} - 2\beta_{3} - \beta_{2} ) / 16$$ (2*b4 - 2*b3 - b2) / 16 $$\zeta_{24}^{4}$$ $$=$$ $$( \beta _1 + 2 ) / 4$$ (b1 + 2) / 4 $$\zeta_{24}^{5}$$ $$=$$ $$( 2\beta_{7} + 2\beta_{6} - 4\beta_{5} - 2\beta_{4} + 2\beta_{3} - \beta_{2} ) / 32$$ (2*b7 + 2*b6 - 4*b5 - 2*b4 + 2*b3 - b2) / 32 $$\zeta_{24}^{6}$$ $$=$$ $$( \beta_{7} + 2\beta_{5} ) / 8$$ (b7 + 2*b5) / 8 $$\zeta_{24}^{7}$$ $$=$$ $$( 2\beta_{7} - 2\beta_{6} - 4\beta_{5} + 2\beta_{4} - 2\beta_{3} - \beta_{2} ) / 32$$ (2*b7 - 2*b6 - 4*b5 + 2*b4 - 2*b3 - b2) / 32

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2305.1
 −0.965926 + 0.258819i −0.258819 + 0.965926i 0.965926 + 0.258819i 0.258819 + 0.965926i 0.965926 − 0.258819i 0.258819 − 0.965926i −0.965926 − 0.258819i −0.258819 − 0.965926i
0 0 0 4.44949i 0 −4.87832 0 0 0
2305.2 0 0 0 4.44949i 0 4.87832 0 0 0
2305.3 0 0 0 0.449490i 0 −2.04989 0 0 0
2305.4 0 0 0 0.449490i 0 2.04989 0 0 0
2305.5 0 0 0 0.449490i 0 −2.04989 0 0 0
2305.6 0 0 0 0.449490i 0 2.04989 0 0 0
2305.7 0 0 0 4.44949i 0 −4.87832 0 0 0
2305.8 0 0 0 4.44949i 0 4.87832 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2305.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 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.d.q 8
3.b odd 2 1 inner 4608.2.d.q 8
4.b odd 2 1 inner 4608.2.d.q 8
8.b even 2 1 inner 4608.2.d.q 8
8.d odd 2 1 inner 4608.2.d.q 8
12.b even 2 1 inner 4608.2.d.q 8
16.e even 4 1 4608.2.a.s 4
16.e even 4 1 4608.2.a.bc yes 4
16.f odd 4 1 4608.2.a.s 4
16.f odd 4 1 4608.2.a.bc yes 4
24.f even 2 1 inner 4608.2.d.q 8
24.h odd 2 1 CM 4608.2.d.q 8
48.i odd 4 1 4608.2.a.s 4
48.i odd 4 1 4608.2.a.bc yes 4
48.k even 4 1 4608.2.a.s 4
48.k even 4 1 4608.2.a.bc yes 4

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

## Hecke kernels

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

 $$T_{5}^{4} + 20T_{5}^{2} + 4$$ T5^4 + 20*T5^2 + 4 $$T_{7}^{4} - 28T_{7}^{2} + 100$$ T7^4 - 28*T7^2 + 100 $$T_{17}$$ T17 $$T_{23}$$ T23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + 20 T^{2} + 4)^{2}$$
$7$ $$(T^{4} - 28 T^{2} + 100)^{2}$$
$11$ $$(T^{2} + 12)^{4}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$(T^{4} + 116 T^{2} + 2500)^{2}$$
$31$ $$(T^{4} - 124 T^{2} + 1444)^{2}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$(T^{4} + 212 T^{2} + 8836)^{2}$$
$59$ $$(T^{2} + 128)^{4}$$
$61$ $$T^{8}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$(T^{2} - 96)^{4}$$
$79$ $$(T^{4} - 316 T^{2} + 3364)^{2}$$
$83$ $$(T^{2} + 300)^{4}$$
$89$ $$T^{8}$$
$97$ $$(T + 2)^{8}$$