# Properties

 Label 5184.2.c.i Level $5184$ Weight $2$ Character orbit 5184.c Analytic conductor $41.394$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5184 = 2^{6} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5184.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$41.3944484078$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 2592) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - \beta_1) q^{5} + \beta_{5} q^{7}+O(q^{10})$$ q + (-b3 - b1) * q^5 + b5 * q^7 $$q + ( - \beta_{3} - \beta_1) q^{5} + \beta_{5} q^{7} + (\beta_{7} + 2 \beta_{6}) q^{11} + ( - 2 \beta_{2} - 1) q^{13} + (\beta_{3} - \beta_1) q^{17} + (\beta_{5} - \beta_{4}) q^{19} - \beta_{6} q^{23} + ( - \beta_{2} + 3) q^{25} + ( - 5 \beta_{3} - 2 \beta_1) q^{29} + ( - 2 \beta_{5} - 3 \beta_{4}) q^{31} - \beta_{6} q^{35} + (\beta_{2} + 6) q^{37} + (2 \beta_{3} - \beta_1) q^{41} + (\beta_{5} + 4 \beta_{4}) q^{43} + ( - \beta_{7} + 5 \beta_{6}) q^{47} + (2 \beta_{2} + 3) q^{49} + ( - 2 \beta_{3} - \beta_1) q^{53} + (3 \beta_{5} + 4 \beta_{4}) q^{55} + (5 \beta_{7} - \beta_{6}) q^{59} + (3 \beta_{2} - 4) q^{61} + (3 \beta_{3} + 5 \beta_1) q^{65} + (7 \beta_{5} + 4 \beta_{4}) q^{67} + ( - 6 \beta_{7} + \beta_{6}) q^{71} + ( - \beta_{2} - 12) q^{73} + ( - 2 \beta_{3} - 2 \beta_1) q^{77} + (9 \beta_{5} + 5 \beta_{4}) q^{79} + (2 \beta_{7} + 4 \beta_{6}) q^{83} - \beta_{2} q^{85} + (7 \beta_{3} + 8 \beta_1) q^{89} + (\beta_{5} - 2 \beta_{4}) q^{91} + \beta_{7} q^{95} + ( - 6 \beta_{2} + 4) q^{97}+O(q^{100})$$ q + (-b3 - b1) * q^5 + b5 * q^7 + (b7 + 2*b6) * q^11 + (-2*b2 - 1) * q^13 + (b3 - b1) * q^17 + (b5 - b4) * q^19 - b6 * q^23 + (-b2 + 3) * q^25 + (-5*b3 - 2*b1) * q^29 + (-2*b5 - 3*b4) * q^31 - b6 * q^35 + (b2 + 6) * q^37 + (2*b3 - b1) * q^41 + (b5 + 4*b4) * q^43 + (-b7 + 5*b6) * q^47 + (2*b2 + 3) * q^49 + (-2*b3 - b1) * q^53 + (3*b5 + 4*b4) * q^55 + (5*b7 - b6) * q^59 + (3*b2 - 4) * q^61 + (3*b3 + 5*b1) * q^65 + (7*b5 + 4*b4) * q^67 + (-6*b7 + b6) * q^71 + (-b2 - 12) * q^73 + (-2*b3 - 2*b1) * q^77 + (9*b5 + 5*b4) * q^79 + (2*b7 + 4*b6) * q^83 - b2 * q^85 + (7*b3 + 8*b1) * q^89 + (b5 - 2*b4) * q^91 + b7 * q^95 + (-6*b2 + 4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 8 q^{13} + 24 q^{25} + 48 q^{37} + 24 q^{49} - 32 q^{61} - 96 q^{73} + 32 q^{97}+O(q^{100})$$ 8 * q - 8 * q^13 + 24 * q^25 + 48 * q^37 + 24 * q^49 - 32 * q^61 - 96 * q^73 + 32 * q^97

Basis of coefficient ring

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1217$$ $$2431$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5183.1
 0.965926 − 0.258819i −0.965926 − 0.258819i −0.258819 − 0.965926i 0.258819 − 0.965926i 0.258819 + 0.965926i −0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 + 0.258819i
0 0 0 1.93185i 0 0.732051i 0 0 0
5183.2 0 0 0 1.93185i 0 0.732051i 0 0 0
5183.3 0 0 0 0.517638i 0 2.73205i 0 0 0
5183.4 0 0 0 0.517638i 0 2.73205i 0 0 0
5183.5 0 0 0 0.517638i 0 2.73205i 0 0 0
5183.6 0 0 0 0.517638i 0 2.73205i 0 0 0
5183.7 0 0 0 1.93185i 0 0.732051i 0 0 0
5183.8 0 0 0 1.93185i 0 0.732051i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5183.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
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 5184.2.c.i 8
3.b odd 2 1 inner 5184.2.c.i 8
4.b odd 2 1 inner 5184.2.c.i 8
8.b even 2 1 2592.2.c.a 8
8.d odd 2 1 2592.2.c.a 8
12.b even 2 1 inner 5184.2.c.i 8
24.f even 2 1 2592.2.c.a 8
24.h odd 2 1 2592.2.c.a 8
72.j odd 6 1 2592.2.s.b 8
72.j odd 6 1 2592.2.s.f 8
72.l even 6 1 2592.2.s.b 8
72.l even 6 1 2592.2.s.f 8
72.n even 6 1 2592.2.s.b 8
72.n even 6 1 2592.2.s.f 8
72.p odd 6 1 2592.2.s.b 8
72.p odd 6 1 2592.2.s.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2592.2.c.a 8 8.b even 2 1
2592.2.c.a 8 8.d odd 2 1
2592.2.c.a 8 24.f even 2 1
2592.2.c.a 8 24.h odd 2 1
2592.2.s.b 8 72.j odd 6 1
2592.2.s.b 8 72.l even 6 1
2592.2.s.b 8 72.n even 6 1
2592.2.s.b 8 72.p odd 6 1
2592.2.s.f 8 72.j odd 6 1
2592.2.s.f 8 72.l even 6 1
2592.2.s.f 8 72.n even 6 1
2592.2.s.f 8 72.p odd 6 1
5184.2.c.i 8 1.a even 1 1 trivial
5184.2.c.i 8 3.b odd 2 1 inner
5184.2.c.i 8 4.b odd 2 1 inner
5184.2.c.i 8 12.b 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 $$S_{2}^{\mathrm{new}}(5184, [\chi])$$:

 $$T_{5}^{4} + 4T_{5}^{2} + 1$$ T5^4 + 4*T5^2 + 1 $$T_{7}^{4} + 8T_{7}^{2} + 4$$ T7^4 + 8*T7^2 + 4 $$T_{11}^{4} - 28T_{11}^{2} + 4$$ T11^4 - 28*T11^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + 4 T^{2} + 1)^{2}$$
$7$ $$(T^{4} + 8 T^{2} + 4)^{2}$$
$11$ $$(T^{4} - 28 T^{2} + 4)^{2}$$
$13$ $$(T^{2} + 2 T - 11)^{4}$$
$17$ $$(T^{4} + 12 T^{2} + 9)^{2}$$
$19$ $$(T^{4} + 24 T^{2} + 36)^{2}$$
$23$ $$(T^{2} - 2)^{4}$$
$29$ $$(T^{4} + 76 T^{2} + 1369)^{2}$$
$31$ $$(T^{4} + 56 T^{2} + 16)^{2}$$
$37$ $$(T^{2} - 12 T + 33)^{4}$$
$41$ $$(T^{4} + 28 T^{2} + 4)^{2}$$
$43$ $$(T^{4} + 104 T^{2} + 2116)^{2}$$
$47$ $$(T^{4} - 112 T^{2} + 1936)^{2}$$
$53$ $$(T^{2} + 6)^{4}$$
$59$ $$(T^{4} - 304 T^{2} + 21904)^{2}$$
$61$ $$(T^{2} + 8 T - 11)^{4}$$
$67$ $$(T^{4} + 296 T^{2} + 21316)^{2}$$
$71$ $$(T^{4} - 436 T^{2} + 45796)^{2}$$
$73$ $$(T^{2} + 24 T + 141)^{4}$$
$79$ $$(T^{4} + 488 T^{2} + 58564)^{2}$$
$83$ $$(T^{4} - 112 T^{2} + 64)^{2}$$
$89$ $$(T^{4} + 228 T^{2} + 1089)^{2}$$
$97$ $$(T^{2} - 8 T - 92)^{4}$$