# Properties

 Label 2592.2.s.h Level $2592$ Weight $2$ Character orbit 2592.s Analytic conductor $20.697$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2592 = 2^{5} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2592.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$20.6972242039$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 864) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} - \beta_{2} + 2) q^{5} + ( - \beta_{7} - \beta_{4} + \beta_1) q^{7}+O(q^{10})$$ q + (-b5 - b2 + 2) * q^5 + (-b7 - b4 + b1) * q^7 $$q + ( - \beta_{5} - \beta_{2} + 2) q^{5} + ( - \beta_{7} - \beta_{4} + \beta_1) q^{7} + ( - 2 \beta_{7} + 2 \beta_{4} - 2 \beta_{3} + \beta_1) q^{11} + ( - 4 \beta_{6} + 2 \beta_{5} + 2 \beta_{2}) q^{13} + ( - 4 \beta_{2} + 2) q^{17} + ( - 2 \beta_{7} + 4 \beta_{4}) q^{19} + 2 \beta_{4} q^{23} + (2 \beta_{6} - 4 \beta_{5}) q^{25} + ( - 4 \beta_{6} - 2 \beta_{2} - 2) q^{29} + (2 \beta_{7} - \beta_{4} - 5 \beta_{3} + 5 \beta_1) q^{31} + ( - 4 \beta_{7} - 3 \beta_{3} + 6 \beta_1) q^{35} + (2 \beta_{6} + 2 \beta_{5}) q^{37} + ( - 2 \beta_{5} + 4 \beta_{2} - 8) q^{41} + ( - 2 \beta_{7} - 2 \beta_{4} - 2 \beta_1) q^{43} + ( - 4 \beta_{7} + 4 \beta_{4} + 4 \beta_{3} - 2 \beta_1) q^{47} + ( - 4 \beta_{6} + 2 \beta_{5}) q^{49} + ( - 3 \beta_{6} + 3 \beta_{5} + 10 \beta_{2} - 5) q^{53} + ( - 3 \beta_{7} + 6 \beta_{4} - 7 \beta_{3}) q^{55} + (4 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{59} + (4 \beta_{2} - 4) q^{61} + ( - 8 \beta_{6} + 6 \beta_{2} + 6) q^{65} + (4 \beta_{7} - 2 \beta_{4} + 10 \beta_{3} - 10 \beta_1) q^{67} + (2 \beta_{7} + 6 \beta_{3} - 12 \beta_1) q^{71} + (2 \beta_{6} + 2 \beta_{5} - 3) q^{73} + ( - 5 \beta_{5} - 5 \beta_{2} + 10) q^{77} + 2 \beta_1 q^{79} + ( - 10 \beta_{7} + 10 \beta_{4} + 2 \beta_{3} - \beta_1) q^{83} + (4 \beta_{6} - 2 \beta_{5} - 6 \beta_{2}) q^{85} + ( - 6 \beta_{6} + 6 \beta_{5} - 4 \beta_{2} + 2) q^{89} + (4 \beta_{7} - 8 \beta_{4} + 14 \beta_{3}) q^{91} + (6 \beta_{4} - 4 \beta_{3} - 4 \beta_1) q^{95} + ( - 5 \beta_{2} + 5) q^{97}+O(q^{100})$$ q + (-b5 - b2 + 2) * q^5 + (-b7 - b4 + b1) * q^7 + (-2*b7 + 2*b4 - 2*b3 + b1) * q^11 + (-4*b6 + 2*b5 + 2*b2) * q^13 + (-4*b2 + 2) * q^17 + (-2*b7 + 4*b4) * q^19 + 2*b4 * q^23 + (2*b6 - 4*b5) * q^25 + (-4*b6 - 2*b2 - 2) * q^29 + (2*b7 - b4 - 5*b3 + 5*b1) * q^31 + (-4*b7 - 3*b3 + 6*b1) * q^35 + (2*b6 + 2*b5) * q^37 + (-2*b5 + 4*b2 - 8) * q^41 + (-2*b7 - 2*b4 - 2*b1) * q^43 + (-4*b7 + 4*b4 + 4*b3 - 2*b1) * q^47 + (-4*b6 + 2*b5) * q^49 + (-3*b6 + 3*b5 + 10*b2 - 5) * q^53 + (-3*b7 + 6*b4 - 7*b3) * q^55 + (4*b4 + 2*b3 + 2*b1) * q^59 + (4*b2 - 4) * q^61 + (-8*b6 + 6*b2 + 6) * q^65 + (4*b7 - 2*b4 + 10*b3 - 10*b1) * q^67 + (2*b7 + 6*b3 - 12*b1) * q^71 + (2*b6 + 2*b5 - 3) * q^73 + (-5*b5 - 5*b2 + 10) * q^77 + 2*b1 * q^79 + (-10*b7 + 10*b4 + 2*b3 - b1) * q^83 + (4*b6 - 2*b5 - 6*b2) * q^85 + (-6*b6 + 6*b5 - 4*b2 + 2) * q^89 + (4*b7 - 8*b4 + 14*b3) * q^91 + (6*b4 - 4*b3 - 4*b1) * q^95 + (-5*b2 + 5) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 12 q^{5}+O(q^{10})$$ 8 * q + 12 * q^5 $$8 q + 12 q^{5} + 8 q^{13} - 24 q^{29} - 48 q^{41} - 16 q^{61} + 72 q^{65} - 24 q^{73} + 60 q^{77} - 24 q^{85} + 20 q^{97}+O(q^{100})$$ 8 * q + 12 * q^5 + 8 * q^13 - 24 * q^29 - 48 * q^41 - 16 * q^61 + 72 * q^65 - 24 * q^73 + 60 * q^77 - 24 * q^85 + 20 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{3}$$ $$=$$ $$\zeta_{24}^{6}$$ v^6 $$\beta_{4}$$ $$=$$ $$\zeta_{24}^{7} + \zeta_{24}$$ v^7 + v $$\beta_{5}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}$$ -v^7 + v $$\beta_{6}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3}$$ -v^7 + v^5 + v^3 $$\beta_{7}$$ $$=$$ $$-\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ -v^5 + v^3 + v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{5} + \beta_{4} ) / 2$$ (b5 + b4) / 2 $$\zeta_{24}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{5} ) / 2$$ (b7 + b6 - b5) / 2 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2$$ (-b7 + b6 + b4) / 2 $$\zeta_{24}^{6}$$ $$=$$ $$\beta_{3}$$ b3 $$\zeta_{24}^{7}$$ $$=$$ $$( -\beta_{5} + \beta_{4} ) / 2$$ (-b5 + b4) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1217$$ $$2431$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
863.1
 0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i −0.965926 + 0.258819i 0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i −0.965926 − 0.258819i
0 0 0 0.275255 + 0.158919i 0 −1.25529 + 0.724745i 0 0 0
863.2 0 0 0 0.275255 + 0.158919i 0 1.25529 0.724745i 0 0 0
863.3 0 0 0 2.72474 + 1.57313i 0 −2.98735 + 1.72474i 0 0 0
863.4 0 0 0 2.72474 + 1.57313i 0 2.98735 1.72474i 0 0 0
1727.1 0 0 0 0.275255 0.158919i 0 −1.25529 0.724745i 0 0 0
1727.2 0 0 0 0.275255 0.158919i 0 1.25529 + 0.724745i 0 0 0
1727.3 0 0 0 2.72474 1.57313i 0 −2.98735 1.72474i 0 0 0
1727.4 0 0 0 2.72474 1.57313i 0 2.98735 + 1.72474i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1727.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.s.h 8
3.b odd 2 1 2592.2.s.a 8
4.b odd 2 1 inner 2592.2.s.h 8
9.c even 3 1 864.2.c.a 8
9.c even 3 1 2592.2.s.a 8
9.d odd 6 1 864.2.c.a 8
9.d odd 6 1 inner 2592.2.s.h 8
12.b even 2 1 2592.2.s.a 8
36.f odd 6 1 864.2.c.a 8
36.f odd 6 1 2592.2.s.a 8
36.h even 6 1 864.2.c.a 8
36.h even 6 1 inner 2592.2.s.h 8
72.j odd 6 1 1728.2.c.g 8
72.l even 6 1 1728.2.c.g 8
72.n even 6 1 1728.2.c.g 8
72.p odd 6 1 1728.2.c.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.c.a 8 9.c even 3 1
864.2.c.a 8 9.d odd 6 1
864.2.c.a 8 36.f odd 6 1
864.2.c.a 8 36.h even 6 1
1728.2.c.g 8 72.j odd 6 1
1728.2.c.g 8 72.l even 6 1
1728.2.c.g 8 72.n even 6 1
1728.2.c.g 8 72.p odd 6 1
2592.2.s.a 8 3.b odd 2 1
2592.2.s.a 8 9.c even 3 1
2592.2.s.a 8 12.b even 2 1
2592.2.s.a 8 36.f odd 6 1
2592.2.s.h 8 1.a even 1 1 trivial
2592.2.s.h 8 4.b odd 2 1 inner
2592.2.s.h 8 9.d odd 6 1 inner
2592.2.s.h 8 36.h even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2592, [\chi])$$:

 $$T_{5}^{4} - 6T_{5}^{3} + 13T_{5}^{2} - 6T_{5} + 1$$ T5^4 - 6*T5^3 + 13*T5^2 - 6*T5 + 1 $$T_{7}^{8} - 14T_{7}^{6} + 171T_{7}^{4} - 350T_{7}^{2} + 625$$ T7^8 - 14*T7^6 + 171*T7^4 - 350*T7^2 + 625

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - 6 T^{3} + 13 T^{2} - 6 T + 1)^{2}$$
$7$ $$T^{8} - 14 T^{6} + 171 T^{4} + \cdots + 625$$
$11$ $$T^{8} + 22 T^{6} + 459 T^{4} + \cdots + 625$$
$13$ $$(T^{4} - 4 T^{3} + 36 T^{2} + 80 T + 400)^{2}$$
$17$ $$(T^{2} + 12)^{4}$$
$19$ $$(T^{2} + 24)^{4}$$
$23$ $$(T^{4} + 8 T^{2} + 64)^{2}$$
$29$ $$(T^{4} + 12 T^{3} + 28 T^{2} - 240 T + 400)^{2}$$
$31$ $$T^{8} - 62 T^{6} + 3483 T^{4} + \cdots + 130321$$
$37$ $$(T^{2} - 24)^{4}$$
$41$ $$(T^{4} + 24 T^{3} + 232 T^{2} + 960 T + 1600)^{2}$$
$43$ $$T^{8} - 56 T^{6} + 2736 T^{4} + \cdots + 160000$$
$47$ $$T^{8} + 88 T^{6} + 7344 T^{4} + \cdots + 160000$$
$53$ $$(T^{4} + 186 T^{2} + 3249)^{2}$$
$59$ $$T^{8} + 88 T^{6} + 7344 T^{4} + \cdots + 160000$$
$61$ $$(T^{2} + 4 T + 16)^{4}$$
$67$ $$T^{8} - 248 T^{6} + \cdots + 33362176$$
$71$ $$(T^{4} - 232 T^{2} + 10000)^{2}$$
$73$ $$(T^{2} + 6 T - 15)^{4}$$
$79$ $$(T^{4} - 4 T^{2} + 16)^{2}$$
$83$ $$T^{8} + 406 T^{6} + \cdots + 1506138481$$
$89$ $$(T^{4} + 168 T^{2} + 3600)^{2}$$
$97$ $$(T^{2} - 5 T + 25)^{4}$$