# Properties

 Label 9025.2.a.b Level $9025$ Weight $2$ Character orbit 9025.a Self dual yes Analytic conductor $72.065$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 3 q^{9} + O(q^{10})$$ $$q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 3 q^{9} - q^{11} + 2 q^{13} - 8 q^{14} - 4 q^{16} + 2 q^{17} + 6 q^{18} + 2 q^{22} - 6 q^{23} - 4 q^{26} + 8 q^{28} + 9 q^{29} - 7 q^{31} + 8 q^{32} - 4 q^{34} - 6 q^{36} - 2 q^{37} + 2 q^{41} - 2 q^{43} - 2 q^{44} + 12 q^{46} - 6 q^{47} + 9 q^{49} + 4 q^{52} - 4 q^{53} - 18 q^{58} + 9 q^{59} - 7 q^{61} + 14 q^{62} - 12 q^{63} - 8 q^{64} + 10 q^{67} + 4 q^{68} + q^{71} - 10 q^{73} + 4 q^{74} - 4 q^{77} + q^{79} + 9 q^{81} - 4 q^{82} + 6 q^{83} + 4 q^{86} - 11 q^{89} + 8 q^{91} - 12 q^{92} + 12 q^{94} + 6 q^{97} - 18 q^{98} + 3 q^{99} + O(q^{100})$$

## 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
 0
−2.00000 0 2.00000 0 0 4.00000 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9025.2.a.b 1
5.b even 2 1 9025.2.a.i 1
5.c odd 4 2 1805.2.b.b 2
19.b odd 2 1 9025.2.a.j 1
19.c even 3 2 475.2.e.c 2
95.d odd 2 1 9025.2.a.a 1
95.g even 4 2 1805.2.b.a 2
95.i even 6 2 475.2.e.a 2
95.m odd 12 4 95.2.i.a 4
285.v even 12 4 855.2.be.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.i.a 4 95.m odd 12 4
475.2.e.a 2 95.i even 6 2
475.2.e.c 2 19.c even 3 2
855.2.be.a 4 285.v even 12 4
1805.2.b.a 2 95.g even 4 2
1805.2.b.b 2 5.c odd 4 2
9025.2.a.a 1 95.d odd 2 1
9025.2.a.b 1 1.a even 1 1 trivial
9025.2.a.i 1 5.b even 2 1
9025.2.a.j 1 19.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9025))$$:

 $$T_{2} + 2$$ $$T_{3}$$ $$T_{7} - 4$$ $$T_{11} + 1$$ $$T_{29} - 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-4 + T$$
$11$ $$1 + T$$
$13$ $$-2 + T$$
$17$ $$-2 + T$$
$19$ $$T$$
$23$ $$6 + T$$
$29$ $$-9 + T$$
$31$ $$7 + T$$
$37$ $$2 + T$$
$41$ $$-2 + T$$
$43$ $$2 + T$$
$47$ $$6 + T$$
$53$ $$4 + T$$
$59$ $$-9 + T$$
$61$ $$7 + T$$
$67$ $$-10 + T$$
$71$ $$-1 + T$$
$73$ $$10 + T$$
$79$ $$-1 + T$$
$83$ $$-6 + T$$
$89$ $$11 + T$$
$97$ $$-6 + T$$