# Properties

 Label 1521.4.a.n Level $1521$ Weight $4$ Character orbit 1521.a Self dual yes Analytic conductor $89.742$ Analytic rank $1$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1521.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$89.7419051187$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 117) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 18\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 8 q^{4} + \beta q^{7}+O(q^{10})$$ q - 8 * q^4 + b * q^7 $$q - 8 q^{4} + \beta q^{7} + 64 q^{16} - 5 \beta q^{19} - 125 q^{25} - 8 \beta q^{28} + 5 \beta q^{31} - 14 \beta q^{37} + 520 q^{43} + 629 q^{49} - 182 q^{61} - 512 q^{64} + 21 \beta q^{67} - 12 \beta q^{73} + 40 \beta q^{76} - 884 q^{79} + 44 \beta q^{97} +O(q^{100})$$ q - 8 * q^4 + b * q^7 + 64 * q^16 - 5*b * q^19 - 125 * q^25 - 8*b * q^28 + 5*b * q^31 - 14*b * q^37 + 520 * q^43 + 629 * q^49 - 182 * q^61 - 512 * q^64 + 21*b * q^67 - 12*b * q^73 + 40*b * q^76 - 884 * q^79 + 44*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 16 q^{4}+O(q^{10})$$ 2 * q - 16 * q^4 $$2 q - 16 q^{4} + 128 q^{16} - 250 q^{25} + 1040 q^{43} + 1258 q^{49} - 364 q^{61} - 1024 q^{64} - 1768 q^{79}+O(q^{100})$$ 2 * q - 16 * q^4 + 128 * q^16 - 250 * q^25 + 1040 * q^43 + 1258 * q^49 - 364 * q^61 - 1024 * q^64 - 1768 * q^79

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
0 0 −8.00000 0 0 −31.1769 0 0 0
1.2 0 0 −8.00000 0 0 31.1769 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$13$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
13.b even 2 1 inner
39.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.4.a.n 2
3.b odd 2 1 CM 1521.4.a.n 2
13.b even 2 1 inner 1521.4.a.n 2
13.d odd 4 2 117.4.b.b 2
39.d odd 2 1 inner 1521.4.a.n 2
39.f even 4 2 117.4.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.4.b.b 2 13.d odd 4 2
117.4.b.b 2 39.f even 4 2
1521.4.a.n 2 1.a even 1 1 trivial
1521.4.a.n 2 3.b odd 2 1 CM
1521.4.a.n 2 13.b even 2 1 inner
1521.4.a.n 2 39.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1521))$$:

 $$T_{2}$$ T2 $$T_{5}$$ T5 $$T_{7}^{2} - 972$$ T7^2 - 972

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 972$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 24300$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} - 24300$$
$37$ $$T^{2} - 190512$$
$41$ $$T^{2}$$
$43$ $$(T - 520)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 182)^{2}$$
$67$ $$T^{2} - 428652$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 139968$$
$79$ $$(T + 884)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 1881792$$