# Properties

 Label 9.10.a.b Level 9 Weight 10 Character orbit 9.a Self dual yes Analytic conductor 4.635 Analytic rank 1 Dimension 1 CM discriminant -3 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$9 = 3^{2}$$ Weight: $$k$$ = $$10$$ Character orbit: $$[\chi]$$ = 9.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.63532252547$$ 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: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 512q^{4} - 12580q^{7} + O(q^{10})$$ $$q - 512q^{4} - 12580q^{7} + 118370q^{13} + 262144q^{16} - 976696q^{19} - 1953125q^{25} + 6440960q^{28} + 1691228q^{31} - 15384490q^{37} - 16577080q^{43} + 117902793q^{49} - 60605440q^{52} - 117903058q^{61} - 134217728q^{64} + 112542320q^{67} + 296368310q^{73} + 500068352q^{76} - 616732324q^{79} - 1489094600q^{91} + 1288928270q^{97} + O(q^{100})$$

## 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 0 −512.000 0 0 −12580.0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.10.a.b 1
3.b odd 2 1 CM 9.10.a.b 1
4.b odd 2 1 144.10.a.h 1
5.b even 2 1 225.10.a.d 1
5.c odd 4 2 225.10.b.f 2
9.c even 3 2 81.10.c.c 2
9.d odd 6 2 81.10.c.c 2
12.b even 2 1 144.10.a.h 1
15.d odd 2 1 225.10.a.d 1
15.e even 4 2 225.10.b.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.10.a.b 1 1.a even 1 1 trivial
9.10.a.b 1 3.b odd 2 1 CM
81.10.c.c 2 9.c even 3 2
81.10.c.c 2 9.d odd 6 2
144.10.a.h 1 4.b odd 2 1
144.10.a.h 1 12.b even 2 1
225.10.a.d 1 5.b even 2 1
225.10.a.d 1 15.d odd 2 1
225.10.b.f 2 5.c odd 4 2
225.10.b.f 2 15.e even 4 2

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(9))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 512 T^{2}$$
$3$ 
$5$ $$1 + 1953125 T^{2}$$
$7$ $$1 + 12580 T + 40353607 T^{2}$$
$11$ $$1 + 2357947691 T^{2}$$
$13$ $$1 - 118370 T + 10604499373 T^{2}$$
$17$ $$1 + 118587876497 T^{2}$$
$19$ $$1 + 976696 T + 322687697779 T^{2}$$
$23$ $$1 + 1801152661463 T^{2}$$
$29$ $$1 + 14507145975869 T^{2}$$
$31$ $$1 - 1691228 T + 26439622160671 T^{2}$$
$37$ $$1 + 15384490 T + 129961739795077 T^{2}$$
$41$ $$1 + 327381934393961 T^{2}$$
$43$ $$1 + 16577080 T + 502592611936843 T^{2}$$
$47$ $$1 + 1119130473102767 T^{2}$$
$53$ $$1 + 3299763591802133 T^{2}$$
$59$ $$1 + 8662995818654939 T^{2}$$
$61$ $$1 + 117903058 T + 11694146092834141 T^{2}$$
$67$ $$1 - 112542320 T + 27206534396294947 T^{2}$$
$71$ $$1 + 45848500718449031 T^{2}$$
$73$ $$1 - 296368310 T + 58871586708267913 T^{2}$$
$79$ $$1 + 616732324 T + 119851595982618319 T^{2}$$
$83$ $$1 + 186940255267540403 T^{2}$$
$89$ $$1 + 350356403707485209 T^{2}$$
$97$ $$1 - 1288928270 T + 760231058654565217 T^{2}$$