# Properties

 Label 5184.2.c.e Level $5184$ Weight $2$ Character orbit 5184.c Analytic conductor $41.394$ Analytic rank $0$ Dimension $4$ CM discriminant -4 Inner twists $4$

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## 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: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ Defining polynomial: $$x^{4} + 4x^{2} + 1$$ x^4 + 4*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 324) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{5}+O(q^{10})$$ q - b2 * q^5 $$q - \beta_{2} q^{5} + ( - \beta_{3} - 2) q^{13} + (\beta_{2} + 3 \beta_1) q^{17} + (\beta_{3} - 9) q^{25} + ( - \beta_{2} + 3 \beta_1) q^{29} + (2 \beta_{3} + 1) q^{37} + \beta_1 q^{41} + 7 q^{49} + 5 \beta_1 q^{53} + (2 \beta_{3} - 5) q^{61} + (\beta_{2} + 13 \beta_1) q^{65} + (\beta_{3} + 8) q^{73} + (2 \beta_{3} + 11) q^{85} + (\beta_{2} - 6 \beta_1) q^{89} + 8 q^{97}+O(q^{100})$$ q - b2 * q^5 + (-b3 - 2) * q^13 + (b2 + 3*b1) * q^17 + (b3 - 9) * q^25 + (-b2 + 3*b1) * q^29 + (2*b3 + 1) * q^37 + b1 * q^41 + 7 * q^49 + 5*b1 * q^53 + (2*b3 - 5) * q^61 + (b2 + 13*b1) * q^65 + (b3 + 8) * q^73 + (2*b3 + 11) * q^85 + (b2 - 6*b1) * q^89 + 8 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 8 q^{13} - 36 q^{25} + 4 q^{37} + 28 q^{49} - 20 q^{61} + 32 q^{73} + 44 q^{85} + 32 q^{97}+O(q^{100})$$ 4 * q - 8 * q^13 - 36 * q^25 + 4 * q^37 + 28 * q^49 - 20 * q^61 + 32 * q^73 + 44 * q^85 + 32 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + 3\nu$$ v^3 + 3*v $$\beta_{2}$$ $$=$$ $$\nu^{3} + 6\nu$$ v^3 + 6*v $$\beta_{3}$$ $$=$$ $$3\nu^{2} + 6$$ 3*v^2 + 6
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 3$$ (b2 - b1) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 6 ) / 3$$ (b3 - 6) / 3 $$\nu^{3}$$ $$=$$ $$-\beta_{2} + 2\beta_1$$ -b2 + 2*b1

## 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
 1.93185i 0.517638i − 0.517638i − 1.93185i
0 0 0 4.38134i 0 0 0 0 0
5183.2 0 0 0 2.96713i 0 0 0 0 0
5183.3 0 0 0 2.96713i 0 0 0 0 0
5183.4 0 0 0 4.38134i 0 0 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
3.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.e 4
3.b odd 2 1 inner 5184.2.c.e 4
4.b odd 2 1 CM 5184.2.c.e 4
8.b even 2 1 324.2.b.a 4
8.d odd 2 1 324.2.b.a 4
12.b even 2 1 inner 5184.2.c.e 4
24.f even 2 1 324.2.b.a 4
24.h odd 2 1 324.2.b.a 4
72.j odd 6 2 324.2.h.e 8
72.l even 6 2 324.2.h.e 8
72.n even 6 2 324.2.h.e 8
72.p odd 6 2 324.2.h.e 8

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

 $$T_{5}^{4} + 28T_{5}^{2} + 169$$ T5^4 + 28*T5^2 + 169 $$T_{7}$$ T7 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 28T^{2} + 169$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$(T^{2} + 4 T - 23)^{2}$$
$17$ $$T^{4} + 52T^{2} + 1$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4} + 76T^{2} + 121$$
$31$ $$T^{4}$$
$37$ $$(T^{2} - 2 T - 107)^{2}$$
$41$ $$(T^{2} + 2)^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$(T^{2} + 50)^{2}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} + 10 T - 83)^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$(T^{2} - 16 T + 37)^{2}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4} + 196T^{2} + 5041$$
$97$ $$(T - 8)^{4}$$
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