# Properties

 Label 5625.2.a.k Level $5625$ Weight $2$ Character orbit 5625.a Self dual yes Analytic conductor $44.916$ Analytic rank $1$ Dimension $4$ CM discriminant -3 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5625 = 3^{2} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5625.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.9158511370$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{15})^+$$ Defining polynomial: $$x^{4} - x^{3} - 4x^{2} + 4x + 1$$ x^4 - x^3 - 4*x^2 + 4*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - 2 q^{4} + (2 \beta_{3} - \beta_{2}) q^{7}+O(q^{10})$$ q - 2 * q^4 + (2*b3 - b2) * q^7 $$q - 2 q^{4} + (2 \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{3} - 3 \beta_1 - 1) q^{13} + 4 q^{16} + ( - 2 \beta_{3} + 5 \beta_1 - 2) q^{19} + ( - 4 \beta_{3} + 2 \beta_{2}) q^{28} + ( - \beta_{3} + 5 \beta_{2}) q^{31} + (7 \beta_{3} - 4 \beta_1 + 7) q^{37} + ( - 6 \beta_{3} + 7 \beta_1 - 6) q^{43} + ( - 3 \beta_{3} - 5 \beta_1 + 4) q^{49} + (2 \beta_{3} + 6 \beta_1 + 2) q^{52} + ( - 9 \beta_{3} - 5 \beta_{2}) q^{61} - 8 q^{64} + (7 \beta_{3} + 9 \beta_{2}) q^{67} + ( - 9 \beta_{3} - 8 \beta_{2}) q^{73} + (4 \beta_{3} - 10 \beta_1 + 4) q^{76} + (3 \beta_{3} + 10 \beta_{2}) q^{79} + ( - 3 \beta_{3} - 5 \beta_{2} + 10 \beta_1 - 9) q^{91} + ( - 3 \beta_{3} - 11 \beta_{2}) q^{97}+O(q^{100})$$ q - 2 * q^4 + (2*b3 - b2) * q^7 + (-b3 - 3*b1 - 1) * q^13 + 4 * q^16 + (-2*b3 + 5*b1 - 2) * q^19 + (-4*b3 + 2*b2) * q^28 + (-b3 + 5*b2) * q^31 + (7*b3 - 4*b1 + 7) * q^37 + (-6*b3 + 7*b1 - 6) * q^43 + (-3*b3 - 5*b1 + 4) * q^49 + (2*b3 + 6*b1 + 2) * q^52 + (-9*b3 - 5*b2) * q^61 - 8 * q^64 + (7*b3 + 9*b2) * q^67 + (-9*b3 - 8*b2) * q^73 + (4*b3 - 10*b1 + 4) * q^76 + (3*b3 + 10*b2) * q^79 + (-3*b3 - 5*b2 + 10*b1 - 9) * q^91 + (-3*b3 - 11*b2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4} - 5 q^{7}+O(q^{10})$$ 4 * q - 8 * q^4 - 5 * q^7 $$4 q - 8 q^{4} - 5 q^{7} - 5 q^{13} + 16 q^{16} + q^{19} + 10 q^{28} + 7 q^{31} + 10 q^{37} - 5 q^{43} + 17 q^{49} + 10 q^{52} + 13 q^{61} - 32 q^{64} - 5 q^{67} + 10 q^{73} - 2 q^{76} + 4 q^{79} - 25 q^{91} - 5 q^{97}+O(q^{100})$$ 4 * q - 8 * q^4 - 5 * q^7 - 5 * q^13 + 16 * q^16 + q^19 + 10 * q^28 + 7 * q^31 + 10 * q^37 - 5 * q^43 + 17 * q^49 + 10 * q^52 + 13 * q^61 - 32 * q^64 - 5 * q^67 + 10 * q^73 - 2 * q^76 + 4 * q^79 - 25 * q^91 - 5 * q^97

Basis of coefficient ring in terms of $$\nu = \zeta_{15} + \zeta_{15}^{-1}$$:

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

## 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
 −1.95630 1.33826 1.82709 −0.209057
0 0 −2.00000 0 0 −5.06316 0 0 0
1.2 0 0 −2.00000 0 0 −3.02701 0 0 0
1.3 0 0 −2.00000 0 0 −0.102193 0 0 0
1.4 0 0 −2.00000 0 0 3.19236 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
$$3$$ $$1$$
$$5$$ $$1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5625.2.a.k 4
3.b odd 2 1 CM 5625.2.a.k 4
5.b even 2 1 5625.2.a.l yes 4
15.d odd 2 1 5625.2.a.l yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5625.2.a.k 4 1.a even 1 1 trivial
5625.2.a.k 4 3.b odd 2 1 CM
5625.2.a.l yes 4 5.b even 2 1
5625.2.a.l yes 4 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5625))$$:

 $$T_{2}$$ T2 $$T_{7}^{4} + 5T_{7}^{3} - 10T_{7}^{2} - 50T_{7} - 5$$ T7^4 + 5*T7^3 - 10*T7^2 - 50*T7 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 5 T^{3} - 10 T^{2} - 50 T - 5$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 5 T^{3} - 40 T^{2} - 200 T - 155$$
$17$ $$T^{4}$$
$19$ $$T^{4} - T^{3} - 94 T^{2} + 94 T + 1711$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4} - 7 T^{3} - 106 T^{2} + \cdots - 389$$
$37$ $$T^{4} - 10 T^{3} - 85 T^{2} + \cdots - 1655$$
$41$ $$T^{4}$$
$43$ $$T^{4} + 5 T^{3} - 190 T^{2} + \cdots + 4495$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4} - 13 T^{3} - 136 T^{2} + \cdots - 4379$$
$67$ $$T^{4} + 5 T^{3} - 310 T^{2} + \cdots + 14695$$
$71$ $$T^{4}$$
$73$ $$T^{4} - 10 T^{3} - 265 T^{2} + \cdots + 145$$
$79$ $$T^{4} - 4 T^{3} - 379 T^{2} + \cdots + 25141$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4} + 5 T^{3} - 460 T^{2} + \cdots + 35545$$