# Properties

 Label 9025.2.a.p Level $9025$ Weight $2$ Character orbit 9025.a Self dual yes Analytic conductor $72.065$ Analytic rank $0$ Dimension $2$ CM discriminant -19 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9025 = 5^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9025.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.0649878242$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{19})$$ Defining polynomial: $$x^{2} - 19$$ x^2 - 19 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1805) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{19}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 q^{4} - \beta q^{7} - 3 q^{9} +O(q^{10})$$ q - 2 * q^4 - b * q^7 - 3 * q^9 $$q - 2 q^{4} - \beta q^{7} - 3 q^{9} + 5 q^{11} + 4 q^{16} - \beta q^{17} - 2 \beta q^{23} + 2 \beta q^{28} + 6 q^{36} - 3 \beta q^{43} - 10 q^{44} - \beta q^{47} + 12 q^{49} - 15 q^{61} + 3 \beta q^{63} - 8 q^{64} + 2 \beta q^{68} - 3 \beta q^{73} - 5 \beta q^{77} + 9 q^{81} + 2 \beta q^{83} + 4 \beta q^{92} - 15 q^{99} +O(q^{100})$$ q - 2 * q^4 - b * q^7 - 3 * q^9 + 5 * q^11 + 4 * q^16 - b * q^17 - 2*b * q^23 + 2*b * q^28 + 6 * q^36 - 3*b * q^43 - 10 * q^44 - b * q^47 + 12 * q^49 - 15 * q^61 + 3*b * q^63 - 8 * q^64 + 2*b * q^68 - 3*b * q^73 - 5*b * q^77 + 9 * q^81 + 2*b * q^83 + 4*b * q^92 - 15 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} - 6 q^{9}+O(q^{10})$$ 2 * q - 4 * q^4 - 6 * q^9 $$2 q - 4 q^{4} - 6 q^{9} + 10 q^{11} + 8 q^{16} + 12 q^{36} - 20 q^{44} + 24 q^{49} - 30 q^{61} - 16 q^{64} + 18 q^{81} - 30 q^{99}+O(q^{100})$$ 2 * q - 4 * q^4 - 6 * q^9 + 10 * q^11 + 8 * q^16 + 12 * q^36 - 20 * q^44 + 24 * q^49 - 30 * q^61 - 16 * q^64 + 18 * q^81 - 30 * q^99

## 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
 4.35890 −4.35890
0 0 −2.00000 0 0 −4.35890 0 −3.00000 0
1.2 0 0 −2.00000 0 0 4.35890 0 −3.00000 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
$$5$$ $$-1$$
$$19$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
5.b even 2 1 inner
95.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9025.2.a.p 2
5.b even 2 1 inner 9025.2.a.p 2
5.c odd 4 2 1805.2.b.d 2
19.b odd 2 1 CM 9025.2.a.p 2
95.d odd 2 1 inner 9025.2.a.p 2
95.g even 4 2 1805.2.b.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.2.b.d 2 5.c odd 4 2
1805.2.b.d 2 95.g even 4 2
9025.2.a.p 2 1.a even 1 1 trivial
9025.2.a.p 2 5.b even 2 1 inner
9025.2.a.p 2 19.b odd 2 1 CM
9025.2.a.p 2 95.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9025))$$:

 $$T_{2}$$ T2 $$T_{3}$$ T3 $$T_{7}^{2} - 19$$ T7^2 - 19 $$T_{11} - 5$$ T11 - 5 $$T_{29}$$ T29

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 19$$
$11$ $$(T - 5)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 19$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 76$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 171$$
$47$ $$T^{2} - 19$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 15)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 171$$
$79$ $$T^{2}$$
$83$ $$T^{2} - 76$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$