# Properties

 Label 9075.2.a.cz 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: 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + q^{3} -\beta_{2} q^{4} + \beta_{3} q^{6} -2 \beta_{3} q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{2} + q^{3} -\beta_{2} q^{4} + \beta_{3} q^{6} -2 \beta_{3} q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + q^{9} -\beta_{2} q^{12} + 2 \beta_{3} q^{13} + ( -4 + 2 \beta_{2} ) q^{14} + ( 1 + \beta_{2} ) q^{16} + ( 3 \beta_{1} - \beta_{3} ) q^{17} + \beta_{3} q^{18} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{19} -2 \beta_{3} q^{21} + ( -3 + 2 \beta_{2} ) q^{23} + ( \beta_{1} + \beta_{3} ) q^{24} + ( 4 - 2 \beta_{2} ) q^{26} + q^{27} + ( -2 \beta_{1} - 6 \beta_{3} ) q^{28} -2 \beta_{1} q^{29} + ( -1 + 2 \beta_{2} ) q^{31} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{32} + ( -5 + \beta_{2} ) q^{34} -\beta_{2} q^{36} + ( -6 - 2 \beta_{2} ) q^{37} + ( 6 - 2 \beta_{2} ) q^{38} + 2 \beta_{3} q^{39} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{41} + ( -4 + 2 \beta_{2} ) q^{42} + 2 \beta_{1} q^{43} + ( -2 \beta_{1} - 9 \beta_{3} ) q^{46} + ( -3 - 2 \beta_{2} ) q^{47} + ( 1 + \beta_{2} ) q^{48} + ( 1 - 4 \beta_{2} ) q^{49} + ( 3 \beta_{1} - \beta_{3} ) q^{51} + ( 2 \beta_{1} + 6 \beta_{3} ) q^{52} -5 q^{53} + \beta_{3} q^{54} + ( -2 + 2 \beta_{2} ) q^{56} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{57} + 2 q^{58} -2 \beta_{2} q^{59} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{61} + ( -2 \beta_{1} - 7 \beta_{3} ) q^{62} -2 \beta_{3} q^{63} + ( -7 + 2 \beta_{2} ) q^{64} + ( 4 + 6 \beta_{2} ) q^{67} + ( -7 \beta_{1} - 6 \beta_{3} ) q^{68} + ( -3 + 2 \beta_{2} ) q^{69} + ( -2 - 4 \beta_{2} ) q^{71} + ( \beta_{1} + \beta_{3} ) q^{72} + ( 8 \beta_{1} + 4 \beta_{3} ) q^{73} + 2 \beta_{1} q^{74} + ( 6 \beta_{1} + 8 \beta_{3} ) q^{76} + ( 4 - 2 \beta_{2} ) q^{78} + ( \beta_{1} - \beta_{3} ) q^{79} + q^{81} + ( -4 + 4 \beta_{2} ) q^{82} + ( -6 \beta_{1} - 2 \beta_{3} ) q^{83} + ( -2 \beta_{1} - 6 \beta_{3} ) q^{84} -2 q^{86} -2 \beta_{1} q^{87} + ( 6 + 6 \beta_{2} ) q^{89} + ( -8 + 4 \beta_{2} ) q^{91} + ( -10 + 5 \beta_{2} ) q^{92} + ( -1 + 2 \beta_{2} ) q^{93} + ( 2 \beta_{1} + 3 \beta_{3} ) q^{94} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{96} + 2 \beta_{2} q^{97} + ( 4 \beta_{1} + 13 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 2 q^{4} + 4 q^{9} + O(q^{10})$$ $$4 q + 4 q^{3} + 2 q^{4} + 4 q^{9} + 2 q^{12} - 20 q^{14} + 2 q^{16} - 16 q^{23} + 20 q^{26} + 4 q^{27} - 8 q^{31} - 22 q^{34} + 2 q^{36} - 20 q^{37} + 28 q^{38} - 20 q^{42} - 8 q^{47} + 2 q^{48} + 12 q^{49} - 20 q^{53} - 12 q^{56} + 8 q^{58} + 4 q^{59} - 32 q^{64} + 4 q^{67} - 16 q^{69} + 20 q^{78} + 4 q^{81} - 24 q^{82} - 8 q^{86} + 12 q^{89} - 40 q^{91} - 50 q^{92} - 8 q^{93} - 4 q^{97} + O(q^{100})$$

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 4.37780 −1.73205 1.00000 0
1.2 −0.456850 1.00000 −1.79129 0 −0.456850 0.913701 1.73205 1.00000 0
1.3 0.456850 1.00000 −1.79129 0 0.456850 −0.913701 −1.73205 1.00000 0
1.4 2.18890 1.00000 2.79129 0 2.18890 −4.37780 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.cz 4
5.b even 2 1 9075.2.a.cs 4
5.c odd 4 2 1815.2.c.f 8
11.b odd 2 1 inner 9075.2.a.cz 4
55.d odd 2 1 9075.2.a.cs 4
55.e even 4 2 1815.2.c.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.c.f 8 5.c odd 4 2
1815.2.c.f 8 55.e even 4 2
9075.2.a.cs 4 5.b even 2 1
9075.2.a.cs 4 55.d odd 2 1
9075.2.a.cz 4 1.a even 1 1 trivial
9075.2.a.cz 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9075))$$:

 $$T_{2}^{4} - 5 T_{2}^{2} + 1$$ $$T_{7}^{4} - 20 T_{7}^{2} + 16$$ $$T_{13}^{4} - 20 T_{13}^{2} + 16$$ $$T_{17}^{4} - 62 T_{17}^{2} + 625$$ $$T_{19}^{2} - 28$$ $$T_{23}^{2} + 8 T_{23} - 5$$ $$T_{37}^{2} + 10 T_{37} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 5 T^{2} + T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$T^{4}$$
$7$ $$16 - 20 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$16 - 20 T^{2} + T^{4}$$
$17$ $$625 - 62 T^{2} + T^{4}$$
$19$ $$( -28 + T^{2} )^{2}$$
$23$ $$( -5 + 8 T + T^{2} )^{2}$$
$29$ $$16 - 20 T^{2} + T^{4}$$
$31$ $$( -17 + 4 T + T^{2} )^{2}$$
$37$ $$( 4 + 10 T + T^{2} )^{2}$$
$41$ $$( -48 + T^{2} )^{2}$$
$43$ $$16 - 20 T^{2} + T^{4}$$
$47$ $$( -17 + 4 T + T^{2} )^{2}$$
$53$ $$( 5 + T )^{4}$$
$59$ $$( -20 - 2 T + T^{2} )^{2}$$
$61$ $$( -75 + T^{2} )^{2}$$
$67$ $$( -188 - 2 T + T^{2} )^{2}$$
$71$ $$( -84 + T^{2} )^{2}$$
$73$ $$6400 - 272 T^{2} + T^{4}$$
$79$ $$( -7 + T^{2} )^{2}$$
$83$ $$400 - 152 T^{2} + T^{4}$$
$89$ $$( -180 - 6 T + T^{2} )^{2}$$
$97$ $$( -20 + 2 T + T^{2} )^{2}$$