# Properties

 Label 2592.2.s.b Level $2592$ Weight $2$ Character orbit 2592.s Analytic conductor $20.697$ Analytic rank $1$ 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: $$1$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}) q^{5} + (\zeta_{24}^{6} + \zeta_{24}^{4} - \zeta_{24}^{2} - 2) q^{7}+O(q^{10})$$ q + (z^7 - z^5 + z) * q^5 + (z^6 + z^4 - z^2 - 2) * q^7 $$q + (\zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}) q^{5} + (\zeta_{24}^{6} + \zeta_{24}^{4} - \zeta_{24}^{2} - 2) q^{7} + ( - 3 \zeta_{24}^{7} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{3} - 3 \zeta_{24}) q^{11} + ( - 4 \zeta_{24}^{6} + \zeta_{24}^{4} + 2 \zeta_{24}^{2} - 1) q^{13} + ( - \zeta_{24}^{7} - 2 \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24}) q^{17} + (3 \zeta_{24}^{6} + 2 \zeta_{24}^{4} - 1) q^{19} + (\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3}) q^{23} + ( - \zeta_{24}^{6} - 3 \zeta_{24}^{4} - \zeta_{24}^{2}) q^{25} + (3 \zeta_{24}^{7} + 5 \zeta_{24}^{5} - 5 \zeta_{24}^{3} - 3 \zeta_{24}) q^{29} + ( - 2 \zeta_{24}^{4} + 4 \zeta_{24}^{2} - 2) q^{31} + (\zeta_{24}^{5} - \zeta_{24}^{3} - \zeta_{24}) q^{35} + ( - \zeta_{24}^{6} + 2 \zeta_{24}^{2} - 6) q^{37} + ( - \zeta_{24}^{7} + \zeta_{24}^{5} + 3 \zeta_{24}^{3} + 2 \zeta_{24}) q^{41} + (7 \zeta_{24}^{6} - \zeta_{24}^{4} - 7 \zeta_{24}^{2} + 2) q^{43} + (4 \zeta_{24}^{7} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{3} + 4 \zeta_{24}) q^{47} + ( - 4 \zeta_{24}^{6} + 3 \zeta_{24}^{4} + 2 \zeta_{24}^{2} - 3) q^{49} + ( - 2 \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24}) q^{53} + (5 \zeta_{24}^{6} - 6 \zeta_{24}^{4} + 3) q^{55} + ( - 6 \zeta_{24}^{7} - 6 \zeta_{24}^{5} - 4 \zeta_{24}^{3} + 10 \zeta_{24}) q^{59} + ( - 3 \zeta_{24}^{6} - 4 \zeta_{24}^{4} - 3 \zeta_{24}^{2}) q^{61} + ( - 2 \zeta_{24}^{7} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{3} + 2 \zeta_{24}) q^{65} + ( - 7 \zeta_{24}^{4} + \zeta_{24}^{2} - 7) q^{67} + (12 \zeta_{24}^{7} - 5 \zeta_{24}^{5} - 7 \zeta_{24}^{3} - 7 \zeta_{24}) q^{71} + ( - \zeta_{24}^{6} + 2 \zeta_{24}^{2} - 12) q^{73} + (2 \zeta_{24}^{7} - 2 \zeta_{24}^{5} + 2 \zeta_{24}) q^{77} + ( - \zeta_{24}^{6} + 9 \zeta_{24}^{4} + \zeta_{24}^{2} - 18) q^{79} + ( - 6 \zeta_{24}^{7} + 4 \zeta_{24}^{5} + 4 \zeta_{24}^{3} - 6 \zeta_{24}) q^{83} + ( - 2 \zeta_{24}^{6} + \zeta_{24}^{2}) q^{85} + ( - 7 \zeta_{24}^{7} + \zeta_{24}^{5} + 8 \zeta_{24}^{3} - 8 \zeta_{24}) q^{89} + (5 \zeta_{24}^{6} + 2 \zeta_{24}^{4} - 1) q^{91} + (\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} - 2 \zeta_{24}) q^{95} + ( - 6 \zeta_{24}^{6} - 4 \zeta_{24}^{4} - 6 \zeta_{24}^{2}) q^{97}+O(q^{100})$$ q + (z^7 - z^5 + z) * q^5 + (z^6 + z^4 - z^2 - 2) * q^7 + (-3*z^7 + 2*z^5 + 2*z^3 - 3*z) * q^11 + (-4*z^6 + z^4 + 2*z^2 - 1) * q^13 + (-z^7 - 2*z^5 - z^3 + z) * q^17 + (3*z^6 + 2*z^4 - 1) * q^19 + (z^7 + z^5 - z^3) * q^23 + (-z^6 - 3*z^4 - z^2) * q^25 + (3*z^7 + 5*z^5 - 5*z^3 - 3*z) * q^29 + (-2*z^4 + 4*z^2 - 2) * q^31 + (z^5 - z^3 - z) * q^35 + (-z^6 + 2*z^2 - 6) * q^37 + (-z^7 + z^5 + 3*z^3 + 2*z) * q^41 + (7*z^6 - z^4 - 7*z^2 + 2) * q^43 + (4*z^7 + 2*z^5 + 2*z^3 + 4*z) * q^47 + (-4*z^6 + 3*z^4 + 2*z^2 - 3) * q^49 + (-2*z^7 - z^5 + z^3 - z) * q^53 + (5*z^6 - 6*z^4 + 3) * q^55 + (-6*z^7 - 6*z^5 - 4*z^3 + 10*z) * q^59 + (-3*z^6 - 4*z^4 - 3*z^2) * q^61 + (-2*z^7 + 3*z^5 - 3*z^3 + 2*z) * q^65 + (-7*z^4 + z^2 - 7) * q^67 + (12*z^7 - 5*z^5 - 7*z^3 - 7*z) * q^71 + (-z^6 + 2*z^2 - 12) * q^73 + (2*z^7 - 2*z^5 + 2*z) * q^77 + (-z^6 + 9*z^4 + z^2 - 18) * q^79 + (-6*z^7 + 4*z^5 + 4*z^3 - 6*z) * q^83 + (-2*z^6 + z^2) * q^85 + (-7*z^7 + z^5 + 8*z^3 - 8*z) * q^89 + (5*z^6 + 2*z^4 - 1) * q^91 + (z^7 + z^5 + z^3 - 2*z) * q^95 + (-6*z^6 - 4*z^4 - 6*z^2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 12 q^{7}+O(q^{10})$$ 8 * q - 12 * q^7 $$8 q - 12 q^{7} - 4 q^{13} - 12 q^{25} - 24 q^{31} - 48 q^{37} + 12 q^{43} - 12 q^{49} - 16 q^{61} - 84 q^{67} - 96 q^{73} - 108 q^{79} - 16 q^{97}+O(q^{100})$$ 8 * q - 12 * q^7 - 4 * q^13 - 12 * q^25 - 24 * q^31 - 48 * q^37 + 12 * q^43 - 12 * q^49 - 16 * q^61 - 84 * q^67 - 96 * q^73 - 108 * q^79 - 16 * q^97

## Character values

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

 $$n$$ $$325$$ $$1217$$ $$2431$$ $$\chi(n)$$ $$1$$ $$1 - \zeta_{24}^{4}$$ $$-1$$

## 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
 0.258819 − 0.965926i −0.965926 − 0.258819i 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 0 0 −1.67303 0.965926i 0 −0.633975 + 0.366025i 0 0 0
863.2 0 0 0 −0.448288 0.258819i 0 −2.36603 + 1.36603i 0 0 0
863.3 0 0 0 0.448288 + 0.258819i 0 −2.36603 + 1.36603i 0 0 0
863.4 0 0 0 1.67303 + 0.965926i 0 −0.633975 + 0.366025i 0 0 0
1727.1 0 0 0 −1.67303 + 0.965926i 0 −0.633975 0.366025i 0 0 0
1727.2 0 0 0 −0.448288 + 0.258819i 0 −2.36603 1.36603i 0 0 0
1727.3 0 0 0 0.448288 0.258819i 0 −2.36603 1.36603i 0 0 0
1727.4 0 0 0 1.67303 0.965926i 0 −0.633975 0.366025i 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
3.b odd 2 1 inner
36.f 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.b 8
3.b odd 2 1 inner 2592.2.s.b 8
4.b odd 2 1 2592.2.s.f 8
9.c even 3 1 2592.2.c.a 8
9.c even 3 1 2592.2.s.f 8
9.d odd 6 1 2592.2.c.a 8
9.d odd 6 1 2592.2.s.f 8
12.b even 2 1 2592.2.s.f 8
36.f odd 6 1 2592.2.c.a 8
36.f odd 6 1 inner 2592.2.s.b 8
36.h even 6 1 2592.2.c.a 8
36.h even 6 1 inner 2592.2.s.b 8
72.j odd 6 1 5184.2.c.i 8
72.l even 6 1 5184.2.c.i 8
72.n even 6 1 5184.2.c.i 8
72.p odd 6 1 5184.2.c.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2592.2.c.a 8 9.c even 3 1
2592.2.c.a 8 9.d odd 6 1
2592.2.c.a 8 36.f odd 6 1
2592.2.c.a 8 36.h even 6 1
2592.2.s.b 8 1.a even 1 1 trivial
2592.2.s.b 8 3.b odd 2 1 inner
2592.2.s.b 8 36.f odd 6 1 inner
2592.2.s.b 8 36.h even 6 1 inner
2592.2.s.f 8 4.b odd 2 1
2592.2.s.f 8 9.c even 3 1
2592.2.s.f 8 9.d odd 6 1
2592.2.s.f 8 12.b even 2 1
5184.2.c.i 8 72.j odd 6 1
5184.2.c.i 8 72.l even 6 1
5184.2.c.i 8 72.n even 6 1
5184.2.c.i 8 72.p odd 6 1

## Hecke kernels

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

 $$T_{5}^{8} - 4T_{5}^{6} + 15T_{5}^{4} - 4T_{5}^{2} + 1$$ T5^8 - 4*T5^6 + 15*T5^4 - 4*T5^2 + 1 $$T_{7}^{4} + 6T_{7}^{3} + 14T_{7}^{2} + 12T_{7} + 4$$ T7^4 + 6*T7^3 + 14*T7^2 + 12*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 4 T^{6} + 15 T^{4} - 4 T^{2} + \cdots + 1$$
$7$ $$(T^{4} + 6 T^{3} + 14 T^{2} + 12 T + 4)^{2}$$
$11$ $$T^{8} + 28 T^{6} + 780 T^{4} + \cdots + 16$$
$13$ $$(T^{4} + 2 T^{3} + 15 T^{2} - 22 T + 121)^{2}$$
$17$ $$(T^{4} + 12 T^{2} + 9)^{2}$$
$19$ $$(T^{4} + 24 T^{2} + 36)^{2}$$
$23$ $$(T^{4} + 2 T^{2} + 4)^{2}$$
$29$ $$T^{8} - 76 T^{6} + 4407 T^{4} + \cdots + 1874161$$
$31$ $$(T^{4} + 12 T^{3} + 44 T^{2} - 48 T + 16)^{2}$$
$37$ $$(T^{2} + 12 T + 33)^{4}$$
$41$ $$T^{8} - 28 T^{6} + 780 T^{4} + \cdots + 16$$
$43$ $$(T^{4} - 6 T^{3} - 34 T^{2} + 276 T + 2116)^{2}$$
$47$ $$T^{8} + 112 T^{6} + 10608 T^{4} + \cdots + 3748096$$
$53$ $$(T^{2} + 6)^{4}$$
$59$ $$T^{8} + 304 T^{6} + \cdots + 479785216$$
$61$ $$(T^{4} + 8 T^{3} + 75 T^{2} - 88 T + 121)^{2}$$
$67$ $$(T^{4} + 42 T^{3} + 734 T^{2} + \cdots + 21316)^{2}$$
$71$ $$(T^{4} - 436 T^{2} + 45796)^{2}$$
$73$ $$(T^{2} + 24 T + 141)^{4}$$
$79$ $$(T^{4} + 54 T^{3} + 1214 T^{2} + \cdots + 58564)^{2}$$
$83$ $$T^{8} + 112 T^{6} + 12480 T^{4} + \cdots + 4096$$
$89$ $$(T^{4} + 228 T^{2} + 1089)^{2}$$
$97$ $$(T^{4} + 8 T^{3} + 156 T^{2} - 736 T + 8464)^{2}$$