# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$20.6972242039$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{13})$$ Defining polynomial: $$x^{4} - x^{3} + 4x^{2} + 3x + 9$$ x^4 - x^3 + 4*x^2 + 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 864) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{5} + ( - \beta_{3} - \beta_{2}) q^{7}+O(q^{10})$$ q - b2 * q^5 + (-b3 - b2) * q^7 $$q - \beta_{2} q^{5} + ( - \beta_{3} - \beta_{2}) q^{7} + ( - \beta_1 + 1) q^{11} - 4 \beta_1 q^{13} - 2 \beta_{3} q^{19} - 6 \beta_1 q^{23} + (8 \beta_1 - 8) q^{25} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{29} - \beta_{2} q^{31} - 13 q^{35} + 10 q^{37} + 2 \beta_{2} q^{41} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{43} + (10 \beta_1 - 10) q^{47} - 6 \beta_1 q^{49} + \beta_{3} q^{53} + \beta_{3} q^{55} - 4 \beta_1 q^{59} + (4 \beta_{3} + 4 \beta_{2}) q^{65} + 2 \beta_{2} q^{67} - 8 q^{71} - 3 q^{73} - \beta_{2} q^{77} + (4 \beta_{3} + 4 \beta_{2}) q^{79} + ( - 9 \beta_1 + 9) q^{83} - 2 \beta_{3} q^{89} + 4 \beta_{3} q^{91} - 26 \beta_1 q^{95} + (7 \beta_1 - 7) q^{97}+O(q^{100})$$ q - b2 * q^5 + (-b3 - b2) * q^7 + (-b1 + 1) * q^11 - 4*b1 * q^13 - 2*b3 * q^19 - 6*b1 * q^23 + (8*b1 - 8) * q^25 + (-2*b3 - 2*b2) * q^29 - b2 * q^31 - 13 * q^35 + 10 * q^37 + 2*b2 * q^41 + (-2*b3 - 2*b2) * q^43 + (10*b1 - 10) * q^47 - 6*b1 * q^49 + b3 * q^53 + b3 * q^55 - 4*b1 * q^59 + (4*b3 + 4*b2) * q^65 + 2*b2 * q^67 - 8 * q^71 - 3 * q^73 - b2 * q^77 + (4*b3 + 4*b2) * q^79 + (-9*b1 + 9) * q^83 - 2*b3 * q^89 + 4*b3 * q^91 - 26*b1 * q^95 + (7*b1 - 7) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 2 q^{11} - 8 q^{13} - 12 q^{23} - 16 q^{25} - 52 q^{35} + 40 q^{37} - 20 q^{47} - 12 q^{49} - 8 q^{59} - 32 q^{71} - 12 q^{73} + 18 q^{83} - 52 q^{95} - 14 q^{97}+O(q^{100})$$ 4 * q + 2 * q^11 - 8 * q^13 - 12 * q^23 - 16 * q^25 - 52 * q^35 + 40 * q^37 - 20 * q^47 - 12 * q^49 - 8 * q^59 - 32 * q^71 - 12 * q^73 + 18 * q^83 - 52 * q^95 - 14 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 4x^{2} + 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 4\nu^{2} - 4\nu + 9 ) / 12$$ (-v^3 + 4*v^2 - 4*v + 9) / 12 $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 4\nu^{2} + 28\nu - 9 ) / 12$$ (v^3 - 4*v^2 + 28*v - 9) / 12 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 5 ) / 2$$ (v^3 + 5) / 2
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 7\beta _1 - 7 ) / 2$$ (b3 + b2 + 7*b1 - 7) / 2 $$\nu^{3}$$ $$=$$ $$2\beta_{3} - 5$$ 2*b3 - 5

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
865.1
 1.15139 + 1.99426i −0.651388 − 1.12824i 1.15139 − 1.99426i −0.651388 + 1.12824i
0 0 0 −1.80278 3.12250i 0 1.80278 3.12250i 0 0 0
865.2 0 0 0 1.80278 + 3.12250i 0 −1.80278 + 3.12250i 0 0 0
1729.1 0 0 0 −1.80278 + 3.12250i 0 1.80278 + 3.12250i 0 0 0
1729.2 0 0 0 1.80278 3.12250i 0 −1.80278 3.12250i 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
9.c even 3 1 inner
12.b even 2 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.i.bc 4
3.b odd 2 1 2592.2.i.bb 4
4.b odd 2 1 2592.2.i.bb 4
9.c even 3 1 864.2.a.m 2
9.c even 3 1 inner 2592.2.i.bc 4
9.d odd 6 1 864.2.a.n yes 2
9.d odd 6 1 2592.2.i.bb 4
12.b even 2 1 inner 2592.2.i.bc 4
36.f odd 6 1 864.2.a.n yes 2
36.f odd 6 1 2592.2.i.bb 4
36.h even 6 1 864.2.a.m 2
36.h even 6 1 inner 2592.2.i.bc 4
72.j odd 6 1 1728.2.a.bc 2
72.l even 6 1 1728.2.a.bd 2
72.n even 6 1 1728.2.a.bd 2
72.p odd 6 1 1728.2.a.bc 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.a.m 2 9.c even 3 1
864.2.a.m 2 36.h even 6 1
864.2.a.n yes 2 9.d odd 6 1
864.2.a.n yes 2 36.f odd 6 1
1728.2.a.bc 2 72.j odd 6 1
1728.2.a.bc 2 72.p odd 6 1
1728.2.a.bd 2 72.l even 6 1
1728.2.a.bd 2 72.n even 6 1
2592.2.i.bb 4 3.b odd 2 1
2592.2.i.bb 4 4.b odd 2 1
2592.2.i.bb 4 9.d odd 6 1
2592.2.i.bb 4 36.f odd 6 1
2592.2.i.bc 4 1.a even 1 1 trivial
2592.2.i.bc 4 9.c even 3 1 inner
2592.2.i.bc 4 12.b even 2 1 inner
2592.2.i.bc 4 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 $$S_{2}^{\mathrm{new}}(2592, [\chi])$$:

 $$T_{5}^{4} + 13T_{5}^{2} + 169$$ T5^4 + 13*T5^2 + 169 $$T_{7}^{4} + 13T_{7}^{2} + 169$$ T7^4 + 13*T7^2 + 169 $$T_{11}^{2} - T_{11} + 1$$ T11^2 - T11 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 13T^{2} + 169$$
$7$ $$T^{4} + 13T^{2} + 169$$
$11$ $$(T^{2} - T + 1)^{2}$$
$13$ $$(T^{2} + 4 T + 16)^{2}$$
$17$ $$T^{4}$$
$19$ $$(T^{2} - 52)^{2}$$
$23$ $$(T^{2} + 6 T + 36)^{2}$$
$29$ $$T^{4} + 52T^{2} + 2704$$
$31$ $$T^{4} + 13T^{2} + 169$$
$37$ $$(T - 10)^{4}$$
$41$ $$T^{4} + 52T^{2} + 2704$$
$43$ $$T^{4} + 52T^{2} + 2704$$
$47$ $$(T^{2} + 10 T + 100)^{2}$$
$53$ $$(T^{2} - 13)^{2}$$
$59$ $$(T^{2} + 4 T + 16)^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4} + 52T^{2} + 2704$$
$71$ $$(T + 8)^{4}$$
$73$ $$(T + 3)^{4}$$
$79$ $$T^{4} + 208 T^{2} + 43264$$
$83$ $$(T^{2} - 9 T + 81)^{2}$$
$89$ $$(T^{2} - 52)^{2}$$
$97$ $$(T^{2} + 7 T + 49)^{2}$$