# Properties

 Label 9075.2.a.j Level $9075$ Weight $2$ Character orbit 9075.a Self dual yes Analytic conductor $72.464$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9075 = 3 \cdot 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9075.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.4642398343$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1815) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} - 2q^{4} - q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} - 2q^{4} - q^{7} + q^{9} + 2q^{12} + 2q^{13} + 4q^{16} + 6q^{17} + 7q^{19} + q^{21} + 6q^{23} - q^{27} + 2q^{28} - q^{31} - 2q^{36} + 7q^{37} - 2q^{39} + 6q^{41} + 8q^{43} - 4q^{48} - 6q^{49} - 6q^{51} - 4q^{52} + 6q^{53} - 7q^{57} - 12q^{59} + q^{61} - q^{63} - 8q^{64} + 7q^{67} - 12q^{68} - 6q^{69} + 6q^{71} - 13q^{73} - 14q^{76} - 11q^{79} + q^{81} - 2q^{84} - 18q^{89} - 2q^{91} - 12q^{92} + q^{93} + q^{97} + O(q^{100})$$

## 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
0 −1.00000 −2.00000 0 0 −1.00000 0 1.00000 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
$$3$$ $$1$$
$$5$$ $$1$$
$$11$$ $$-1$$

## 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 9075.2.a.j 1
5.b even 2 1 1815.2.a.c yes 1
11.b odd 2 1 9075.2.a.k 1
15.d odd 2 1 5445.2.a.g 1
55.d odd 2 1 1815.2.a.b 1
165.d even 2 1 5445.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.b 1 55.d odd 2 1
1815.2.a.c yes 1 5.b even 2 1
5445.2.a.f 1 165.d even 2 1
5445.2.a.g 1 15.d odd 2 1
9075.2.a.j 1 1.a even 1 1 trivial
9075.2.a.k 1 11.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9075))$$:

 $$T_{2}$$ $$T_{7} + 1$$ $$T_{13} - 2$$ $$T_{17} - 6$$ $$T_{19} - 7$$ $$T_{23} - 6$$ $$T_{37} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$1 + T$$
$11$ $$T$$
$13$ $$-2 + T$$
$17$ $$-6 + T$$
$19$ $$-7 + T$$
$23$ $$-6 + T$$
$29$ $$T$$
$31$ $$1 + T$$
$37$ $$-7 + T$$
$41$ $$-6 + T$$
$43$ $$-8 + T$$
$47$ $$T$$
$53$ $$-6 + T$$
$59$ $$12 + T$$
$61$ $$-1 + T$$
$67$ $$-7 + T$$
$71$ $$-6 + T$$
$73$ $$13 + T$$
$79$ $$11 + T$$
$83$ $$T$$
$89$ $$18 + T$$
$97$ $$-1 + T$$