# Properties

 Label 9075.2.a.p 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: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} - q^{4} + q^{6} - q^{7} - 3 q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 - q^4 + q^6 - q^7 - 3 * q^8 + q^9 $$q + q^{2} + q^{3} - q^{4} + q^{6} - q^{7} - 3 q^{8} + q^{9} - q^{12} + 2 q^{13} - q^{14} - q^{16} - 3 q^{17} + q^{18} - 3 q^{19} - q^{21} + q^{23} - 3 q^{24} + 2 q^{26} + q^{27} + q^{28} + 6 q^{29} + 2 q^{31} + 5 q^{32} - 3 q^{34} - q^{36} - 3 q^{37} - 3 q^{38} + 2 q^{39} + 3 q^{41} - q^{42} + 12 q^{43} + q^{46} - q^{47} - q^{48} - 6 q^{49} - 3 q^{51} - 2 q^{52} - 6 q^{53} + q^{54} + 3 q^{56} - 3 q^{57} + 6 q^{58} - 3 q^{59} - 10 q^{61} + 2 q^{62} - q^{63} + 7 q^{64} - 6 q^{67} + 3 q^{68} + q^{69} - 7 q^{71} - 3 q^{72} + 2 q^{73} - 3 q^{74} + 3 q^{76} + 2 q^{78} + 7 q^{79} + q^{81} + 3 q^{82} - 6 q^{83} + q^{84} + 12 q^{86} + 6 q^{87} - 14 q^{89} - 2 q^{91} - q^{92} + 2 q^{93} - q^{94} + 5 q^{96} + 3 q^{97} - 6 q^{98}+O(q^{100})$$ q + q^2 + q^3 - q^4 + q^6 - q^7 - 3 * q^8 + q^9 - q^12 + 2 * q^13 - q^14 - q^16 - 3 * q^17 + q^18 - 3 * q^19 - q^21 + q^23 - 3 * q^24 + 2 * q^26 + q^27 + q^28 + 6 * q^29 + 2 * q^31 + 5 * q^32 - 3 * q^34 - q^36 - 3 * q^37 - 3 * q^38 + 2 * q^39 + 3 * q^41 - q^42 + 12 * q^43 + q^46 - q^47 - q^48 - 6 * q^49 - 3 * q^51 - 2 * q^52 - 6 * q^53 + q^54 + 3 * q^56 - 3 * q^57 + 6 * q^58 - 3 * q^59 - 10 * q^61 + 2 * q^62 - q^63 + 7 * q^64 - 6 * q^67 + 3 * q^68 + q^69 - 7 * q^71 - 3 * q^72 + 2 * q^73 - 3 * q^74 + 3 * q^76 + 2 * q^78 + 7 * q^79 + q^81 + 3 * q^82 - 6 * q^83 + q^84 + 12 * q^86 + 6 * q^87 - 14 * q^89 - 2 * q^91 - q^92 + 2 * q^93 - q^94 + 5 * q^96 + 3 * q^97 - 6 * 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 −1.00000 −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.p yes 1
5.b even 2 1 9075.2.a.e 1
11.b odd 2 1 9075.2.a.h yes 1
55.d odd 2 1 9075.2.a.m yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9075.2.a.e 1 5.b even 2 1
9075.2.a.h yes 1 11.b odd 2 1
9075.2.a.m yes 1 55.d odd 2 1
9075.2.a.p yes 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 $$T_{7} + 1$$ T7 + 1 $$T_{13} - 2$$ T13 - 2 $$T_{17} + 3$$ T17 + 3 $$T_{19} + 3$$ T19 + 3 $$T_{23} - 1$$ T23 - 1 $$T_{37} + 3$$ T37 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T$$
$13$ $$T - 2$$
$17$ $$T + 3$$
$19$ $$T + 3$$
$23$ $$T - 1$$
$29$ $$T - 6$$
$31$ $$T - 2$$
$37$ $$T + 3$$
$41$ $$T - 3$$
$43$ $$T - 12$$
$47$ $$T + 1$$
$53$ $$T + 6$$
$59$ $$T + 3$$
$61$ $$T + 10$$
$67$ $$T + 6$$
$71$ $$T + 7$$
$73$ $$T - 2$$
$79$ $$T - 7$$
$83$ $$T + 6$$
$89$ $$T + 14$$
$97$ $$T - 3$$