# Properties

 Label 9075.2.a.n Level $9075$ Weight $2$ Character orbit 9075.a Self dual yes Analytic conductor $72.464$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} - q^{4} - q^{6} + 2 q^{7} - 3 q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 - q^4 - q^6 + 2 * q^7 - 3 * q^8 + q^9 $$q + q^{2} - q^{3} - q^{4} - q^{6} + 2 q^{7} - 3 q^{8} + q^{9} + q^{12} - 4 q^{13} + 2 q^{14} - q^{16} + 6 q^{17} + q^{18} - 6 q^{19} - 2 q^{21} - 4 q^{23} + 3 q^{24} - 4 q^{26} - q^{27} - 2 q^{28} + 6 q^{29} + 8 q^{31} + 5 q^{32} + 6 q^{34} - q^{36} + 6 q^{37} - 6 q^{38} + 4 q^{39} - 6 q^{41} - 2 q^{42} + 6 q^{43} - 4 q^{46} - 8 q^{47} + q^{48} - 3 q^{49} - 6 q^{51} + 4 q^{52} - 6 q^{53} - q^{54} - 6 q^{56} + 6 q^{57} + 6 q^{58} + 4 q^{61} + 8 q^{62} + 2 q^{63} + 7 q^{64} - 12 q^{67} - 6 q^{68} + 4 q^{69} + 8 q^{71} - 3 q^{72} - 16 q^{73} + 6 q^{74} + 6 q^{76} + 4 q^{78} + 2 q^{79} + q^{81} - 6 q^{82} + 2 q^{84} + 6 q^{86} - 6 q^{87} + 10 q^{89} - 8 q^{91} + 4 q^{92} - 8 q^{93} - 8 q^{94} - 5 q^{96} + 6 q^{97} - 3 q^{98}+O(q^{100})$$ q + q^2 - q^3 - q^4 - q^6 + 2 * q^7 - 3 * q^8 + q^9 + q^12 - 4 * q^13 + 2 * q^14 - q^16 + 6 * q^17 + q^18 - 6 * q^19 - 2 * q^21 - 4 * q^23 + 3 * q^24 - 4 * q^26 - q^27 - 2 * q^28 + 6 * q^29 + 8 * q^31 + 5 * q^32 + 6 * q^34 - q^36 + 6 * q^37 - 6 * q^38 + 4 * q^39 - 6 * q^41 - 2 * q^42 + 6 * q^43 - 4 * q^46 - 8 * q^47 + q^48 - 3 * q^49 - 6 * q^51 + 4 * q^52 - 6 * q^53 - q^54 - 6 * q^56 + 6 * q^57 + 6 * q^58 + 4 * q^61 + 8 * q^62 + 2 * q^63 + 7 * q^64 - 12 * q^67 - 6 * q^68 + 4 * q^69 + 8 * q^71 - 3 * q^72 - 16 * q^73 + 6 * q^74 + 6 * q^76 + 4 * q^78 + 2 * q^79 + q^81 - 6 * q^82 + 2 * q^84 + 6 * q^86 - 6 * q^87 + 10 * q^89 - 8 * q^91 + 4 * q^92 - 8 * q^93 - 8 * q^94 - 5 * q^96 + 6 * q^97 - 3 * q^98

## 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
1.00000 −1.00000 −1.00000 0 −1.00000 2.00000 −3.00000 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.n 1
5.b even 2 1 1815.2.a.a 1
11.b odd 2 1 9075.2.a.d 1
15.d odd 2 1 5445.2.a.j 1
55.d odd 2 1 1815.2.a.e yes 1
165.d even 2 1 5445.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.a 1 5.b even 2 1
1815.2.a.e yes 1 55.d odd 2 1
5445.2.a.e 1 165.d even 2 1
5445.2.a.j 1 15.d odd 2 1
9075.2.a.d 1 11.b odd 2 1
9075.2.a.n 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9075))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{7} - 2$$ T7 - 2 $$T_{13} + 4$$ T13 + 4 $$T_{17} - 6$$ T17 - 6 $$T_{19} + 6$$ T19 + 6 $$T_{23} + 4$$ T23 + 4 $$T_{37} - 6$$ T37 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T - 2$$
$11$ $$T$$
$13$ $$T + 4$$
$17$ $$T - 6$$
$19$ $$T + 6$$
$23$ $$T + 4$$
$29$ $$T - 6$$
$31$ $$T - 8$$
$37$ $$T - 6$$
$41$ $$T + 6$$
$43$ $$T - 6$$
$47$ $$T + 8$$
$53$ $$T + 6$$
$59$ $$T$$
$61$ $$T - 4$$
$67$ $$T + 12$$
$71$ $$T - 8$$
$73$ $$T + 16$$
$79$ $$T - 2$$
$83$ $$T$$
$89$ $$T - 10$$
$97$ $$T - 6$$