# Properties

 Label 9075.2.a.da Level $9075$ Weight $2$ Character orbit 9075.a Self dual yes Analytic conductor $72.464$ Analytic rank $1$ Dimension $4$ 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: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{7})$$ Defining polynomial: $$x^{4} - 5 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + q^{3} -\beta_{2} q^{4} + \beta_{3} q^{6} + \beta_{1} q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{2} + q^{3} -\beta_{2} q^{4} + \beta_{3} q^{6} + \beta_{1} q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + q^{9} -\beta_{2} q^{12} -\beta_{3} q^{13} - q^{14} + ( 1 + \beta_{2} ) q^{16} + ( -\beta_{1} - \beta_{3} ) q^{17} + \beta_{3} q^{18} + ( -\beta_{1} - \beta_{3} ) q^{19} + \beta_{1} q^{21} + ( -3 + \beta_{2} ) q^{23} + ( \beta_{1} + \beta_{3} ) q^{24} + ( -2 + \beta_{2} ) q^{26} + q^{27} + ( -2 \beta_{1} - \beta_{3} ) q^{28} + ( -\beta_{1} - \beta_{3} ) q^{29} + ( -5 + \beta_{2} ) q^{31} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{32} + ( -1 + \beta_{2} ) q^{34} -\beta_{2} q^{36} + ( -2 + 2 \beta_{2} ) q^{37} + ( -1 + \beta_{2} ) q^{38} -\beta_{3} q^{39} + ( \beta_{1} - 2 \beta_{3} ) q^{41} - q^{42} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{43} + ( -\beta_{1} - 6 \beta_{3} ) q^{46} + ( -2 - 2 \beta_{2} ) q^{47} + ( 1 + \beta_{2} ) q^{48} + ( -4 + \beta_{2} ) q^{49} + ( -\beta_{1} - \beta_{3} ) q^{51} + ( -\beta_{1} - 3 \beta_{3} ) q^{52} + ( 5 + \beta_{2} ) q^{53} + \beta_{3} q^{54} + ( 2 + \beta_{2} ) q^{56} + ( -\beta_{1} - \beta_{3} ) q^{57} + ( -1 + \beta_{2} ) q^{58} + ( -5 - 4 \beta_{2} ) q^{59} + ( -2 \beta_{1} + 6 \beta_{3} ) q^{61} + ( -\beta_{1} - 8 \beta_{3} ) q^{62} + \beta_{1} q^{63} + ( -7 + 2 \beta_{2} ) q^{64} + ( -2 - 4 \beta_{2} ) q^{67} + ( \beta_{1} - 2 \beta_{3} ) q^{68} + ( -3 + \beta_{2} ) q^{69} + ( 5 + 6 \beta_{2} ) q^{71} + ( \beta_{1} + \beta_{3} ) q^{72} + ( -4 \beta_{1} + \beta_{3} ) q^{73} + ( -2 \beta_{1} - 8 \beta_{3} ) q^{74} + ( \beta_{1} - 2 \beta_{3} ) q^{76} + ( -2 + \beta_{2} ) q^{78} + ( 8 \beta_{1} + 6 \beta_{3} ) q^{79} + q^{81} + ( -5 + 2 \beta_{2} ) q^{82} + ( -6 \beta_{1} + 3 \beta_{3} ) q^{83} + ( -2 \beta_{1} - \beta_{3} ) q^{84} + ( -2 + 2 \beta_{2} ) q^{86} + ( -\beta_{1} - \beta_{3} ) q^{87} + ( -8 + 3 \beta_{2} ) q^{89} + q^{91} + ( -5 + 4 \beta_{2} ) q^{92} + ( -5 + \beta_{2} ) q^{93} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{94} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{96} + ( -10 - 2 \beta_{2} ) q^{97} + ( -\beta_{1} - 7 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 2 q^{4} + 4 q^{9} + O(q^{10})$$ $$4 q + 4 q^{3} + 2 q^{4} + 4 q^{9} + 2 q^{12} - 4 q^{14} + 2 q^{16} - 14 q^{23} - 10 q^{26} + 4 q^{27} - 22 q^{31} - 6 q^{34} + 2 q^{36} - 12 q^{37} - 6 q^{38} - 4 q^{42} - 4 q^{47} + 2 q^{48} - 18 q^{49} + 18 q^{53} + 6 q^{56} - 6 q^{58} - 12 q^{59} - 32 q^{64} - 14 q^{69} + 8 q^{71} - 10 q^{78} + 4 q^{81} - 24 q^{82} - 12 q^{86} - 38 q^{89} + 4 q^{91} - 28 q^{92} - 22 q^{93} - 36 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5 \beta_{1}$$

## 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
 0.456850 2.18890 −2.18890 −0.456850
−2.18890 1.00000 2.79129 0 −2.18890 0.456850 −1.73205 1.00000 0
1.2 −0.456850 1.00000 −1.79129 0 −0.456850 2.18890 1.73205 1.00000 0
1.3 0.456850 1.00000 −1.79129 0 0.456850 −2.18890 −1.73205 1.00000 0
1.4 2.18890 1.00000 2.79129 0 2.18890 −0.456850 1.73205 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.da yes 4
5.b even 2 1 9075.2.a.ct 4
11.b odd 2 1 inner 9075.2.a.da yes 4
55.d odd 2 1 9075.2.a.ct 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9075.2.a.ct 4 5.b even 2 1
9075.2.a.ct 4 55.d odd 2 1
9075.2.a.da yes 4 1.a even 1 1 trivial
9075.2.a.da yes 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9075))$$:

 $$T_{2}^{4} - 5 T_{2}^{2} + 1$$ $$T_{7}^{4} - 5 T_{7}^{2} + 1$$ $$T_{13}^{4} - 5 T_{13}^{2} + 1$$ $$T_{17}^{2} - 3$$ $$T_{19}^{2} - 3$$ $$T_{23}^{2} + 7 T_{23} + 7$$ $$T_{37}^{2} + 6 T_{37} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 5 T^{2} + T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$T^{4}$$
$7$ $$1 - 5 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$1 - 5 T^{2} + T^{4}$$
$17$ $$( -3 + T^{2} )^{2}$$
$19$ $$( -3 + T^{2} )^{2}$$
$23$ $$( 7 + 7 T + T^{2} )^{2}$$
$29$ $$( -3 + T^{2} )^{2}$$
$31$ $$( 25 + 11 T + T^{2} )^{2}$$
$37$ $$( -12 + 6 T + T^{2} )^{2}$$
$41$ $$225 - 33 T^{2} + T^{4}$$
$43$ $$( -12 + T^{2} )^{2}$$
$47$ $$( -20 + 2 T + T^{2} )^{2}$$
$53$ $$( 15 - 9 T + T^{2} )^{2}$$
$59$ $$( -75 + 6 T + T^{2} )^{2}$$
$61$ $$10000 - 248 T^{2} + T^{4}$$
$67$ $$( -84 + T^{2} )^{2}$$
$71$ $$( -185 - 4 T + T^{2} )^{2}$$
$73$ $$1369 - 101 T^{2} + T^{4}$$
$79$ $$19600 - 308 T^{2} + T^{4}$$
$83$ $$18225 - 297 T^{2} + T^{4}$$
$89$ $$( 43 + 19 T + T^{2} )^{2}$$
$97$ $$( 60 + 18 T + T^{2} )^{2}$$