Properties

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

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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1815) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} -2 q^{4} + \beta q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} -2 q^{4} + \beta q^{7} + q^{9} -2 q^{12} + 4 q^{16} -2 \beta q^{17} + 3 \beta q^{19} + \beta q^{21} -6 q^{23} + q^{27} -2 \beta q^{28} -4 \beta q^{29} + q^{31} -2 q^{36} + 5 q^{37} -2 \beta q^{41} + 6 \beta q^{43} + 12 q^{47} + 4 q^{48} -4 q^{49} -2 \beta q^{51} -6 q^{53} + 3 \beta q^{57} + 7 \beta q^{61} + \beta q^{63} -8 q^{64} + 5 q^{67} + 4 \beta q^{68} -6 q^{69} -6 q^{71} -\beta q^{73} -6 \beta q^{76} + 9 \beta q^{79} + q^{81} -4 \beta q^{83} -2 \beta q^{84} -4 \beta q^{87} -6 q^{89} + 12 q^{92} + q^{93} + 13 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 4q^{4} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 4q^{4} + 2q^{9} - 4q^{12} + 8q^{16} - 12q^{23} + 2q^{27} + 2q^{31} - 4q^{36} + 10q^{37} + 24q^{47} + 8q^{48} - 8q^{49} - 12q^{53} - 16q^{64} + 10q^{67} - 12q^{69} - 12q^{71} + 2q^{81} - 12q^{89} + 24q^{92} + 2q^{93} + 26q^{97} + O(q^{100})$$

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
 −1.73205 1.73205
0 1.00000 −2.00000 0 0 −1.73205 0 1.00000 0
1.2 0 1.00000 −2.00000 0 0 1.73205 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9075.2.a.bl 2
5.b even 2 1 1815.2.a.g 2
11.b odd 2 1 inner 9075.2.a.bl 2
15.d odd 2 1 5445.2.a.q 2
55.d odd 2 1 1815.2.a.g 2
165.d even 2 1 5445.2.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.g 2 5.b even 2 1
1815.2.a.g 2 55.d odd 2 1
5445.2.a.q 2 15.d odd 2 1
5445.2.a.q 2 165.d even 2 1
9075.2.a.bl 2 1.a even 1 1 trivial
9075.2.a.bl 2 11.b odd 2 1 inner

Hecke kernels

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

 $$T_{2}$$ $$T_{7}^{2} - 3$$ $$T_{13}$$ $$T_{17}^{2} - 12$$ $$T_{19}^{2} - 27$$ $$T_{23} + 6$$ $$T_{37} - 5$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$-3 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$-12 + T^{2}$$
$19$ $$-27 + T^{2}$$
$23$ $$( 6 + T )^{2}$$
$29$ $$-48 + T^{2}$$
$31$ $$( -1 + T )^{2}$$
$37$ $$( -5 + T )^{2}$$
$41$ $$-12 + T^{2}$$
$43$ $$-108 + T^{2}$$
$47$ $$( -12 + T )^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$T^{2}$$
$61$ $$-147 + T^{2}$$
$67$ $$( -5 + T )^{2}$$
$71$ $$( 6 + T )^{2}$$
$73$ $$-3 + T^{2}$$
$79$ $$-243 + T^{2}$$
$83$ $$-48 + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$( -13 + T )^{2}$$