# Properties

 Label 9075.2.a.ba Level $9075$ Weight $2$ Character orbit 9075.a Self dual yes Analytic conductor $72.464$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} - q^{3} + (\beta - 1) q^{4} + \beta q^{6} + ( - \beta + 1) q^{7} + (2 \beta - 1) q^{8} + q^{9} +O(q^{10})$$ q - b * q^2 - q^3 + (b - 1) * q^4 + b * q^6 + (-b + 1) * q^7 + (2*b - 1) * q^8 + q^9 $$q - \beta q^{2} - q^{3} + (\beta - 1) q^{4} + \beta q^{6} + ( - \beta + 1) q^{7} + (2 \beta - 1) q^{8} + q^{9} + ( - \beta + 1) q^{12} + ( - \beta + 2) q^{13} + q^{14} - 3 \beta q^{16} + (2 \beta - 3) q^{17} - \beta q^{18} + ( - 2 \beta - 1) q^{19} + (\beta - 1) q^{21} + ( - 3 \beta + 2) q^{23} + ( - 2 \beta + 1) q^{24} + ( - \beta + 1) q^{26} - q^{27} + (\beta - 2) q^{28} + ( - 2 \beta + 3) q^{29} + (3 \beta - 6) q^{31} + ( - \beta + 5) q^{32} + (\beta - 2) q^{34} + (\beta - 1) q^{36} + (8 \beta - 4) q^{37} + (3 \beta + 2) q^{38} + (\beta - 2) q^{39} + (5 \beta - 3) q^{41} - q^{42} + (4 \beta - 2) q^{43} + (\beta + 3) q^{46} + (6 \beta + 2) q^{47} + 3 \beta q^{48} + ( - \beta - 5) q^{49} + ( - 2 \beta + 3) q^{51} + (2 \beta - 3) q^{52} + ( - 5 \beta + 8) q^{53} + \beta q^{54} + (\beta - 3) q^{56} + (2 \beta + 1) q^{57} + ( - \beta + 2) q^{58} + ( - 6 \beta + 5) q^{59} + (2 \beta + 2) q^{61} + (3 \beta - 3) q^{62} + ( - \beta + 1) q^{63} + (2 \beta + 1) q^{64} + ( - 2 \beta - 2) q^{67} + ( - 3 \beta + 5) q^{68} + (3 \beta - 2) q^{69} + ( - 2 \beta - 3) q^{71} + (2 \beta - 1) q^{72} + ( - \beta - 4) q^{73} + ( - 4 \beta - 8) q^{74} + ( - \beta - 1) q^{76} + (\beta - 1) q^{78} + ( - 6 \beta + 6) q^{79} + q^{81} + ( - 2 \beta - 5) q^{82} + ( - 3 \beta + 4) q^{83} + ( - \beta + 2) q^{84} + ( - 2 \beta - 4) q^{86} + (2 \beta - 3) q^{87} + ( - 7 \beta - 3) q^{89} + ( - 2 \beta + 3) q^{91} + (2 \beta - 5) q^{92} + ( - 3 \beta + 6) q^{93} + ( - 8 \beta - 6) q^{94} + (\beta - 5) q^{96} + ( - 10 \beta + 2) q^{97} + (6 \beta + 1) q^{98} +O(q^{100})$$ q - b * q^2 - q^3 + (b - 1) * q^4 + b * q^6 + (-b + 1) * q^7 + (2*b - 1) * q^8 + q^9 + (-b + 1) * q^12 + (-b + 2) * q^13 + q^14 - 3*b * q^16 + (2*b - 3) * q^17 - b * q^18 + (-2*b - 1) * q^19 + (b - 1) * q^21 + (-3*b + 2) * q^23 + (-2*b + 1) * q^24 + (-b + 1) * q^26 - q^27 + (b - 2) * q^28 + (-2*b + 3) * q^29 + (3*b - 6) * q^31 + (-b + 5) * q^32 + (b - 2) * q^34 + (b - 1) * q^36 + (8*b - 4) * q^37 + (3*b + 2) * q^38 + (b - 2) * q^39 + (5*b - 3) * q^41 - q^42 + (4*b - 2) * q^43 + (b + 3) * q^46 + (6*b + 2) * q^47 + 3*b * q^48 + (-b - 5) * q^49 + (-2*b + 3) * q^51 + (2*b - 3) * q^52 + (-5*b + 8) * q^53 + b * q^54 + (b - 3) * q^56 + (2*b + 1) * q^57 + (-b + 2) * q^58 + (-6*b + 5) * q^59 + (2*b + 2) * q^61 + (3*b - 3) * q^62 + (-b + 1) * q^63 + (2*b + 1) * q^64 + (-2*b - 2) * q^67 + (-3*b + 5) * q^68 + (3*b - 2) * q^69 + (-2*b - 3) * q^71 + (2*b - 1) * q^72 + (-b - 4) * q^73 + (-4*b - 8) * q^74 + (-b - 1) * q^76 + (b - 1) * q^78 + (-6*b + 6) * q^79 + q^81 + (-2*b - 5) * q^82 + (-3*b + 4) * q^83 + (-b + 2) * q^84 + (-2*b - 4) * q^86 + (2*b - 3) * q^87 + (-7*b - 3) * q^89 + (-2*b + 3) * q^91 + (2*b - 5) * q^92 + (-3*b + 6) * q^93 + (-8*b - 6) * q^94 + (b - 5) * q^96 + (-10*b + 2) * q^97 + (6*b + 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} + q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - q^2 - 2 * q^3 - q^4 + q^6 + q^7 + 2 * q^9 $$2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} + q^{7} + 2 q^{9} + q^{12} + 3 q^{13} + 2 q^{14} - 3 q^{16} - 4 q^{17} - q^{18} - 4 q^{19} - q^{21} + q^{23} + q^{26} - 2 q^{27} - 3 q^{28} + 4 q^{29} - 9 q^{31} + 9 q^{32} - 3 q^{34} - q^{36} + 7 q^{38} - 3 q^{39} - q^{41} - 2 q^{42} + 7 q^{46} + 10 q^{47} + 3 q^{48} - 11 q^{49} + 4 q^{51} - 4 q^{52} + 11 q^{53} + q^{54} - 5 q^{56} + 4 q^{57} + 3 q^{58} + 4 q^{59} + 6 q^{61} - 3 q^{62} + q^{63} + 4 q^{64} - 6 q^{67} + 7 q^{68} - q^{69} - 8 q^{71} - 9 q^{73} - 20 q^{74} - 3 q^{76} - q^{78} + 6 q^{79} + 2 q^{81} - 12 q^{82} + 5 q^{83} + 3 q^{84} - 10 q^{86} - 4 q^{87} - 13 q^{89} + 4 q^{91} - 8 q^{92} + 9 q^{93} - 20 q^{94} - 9 q^{96} - 6 q^{97} + 8 q^{98}+O(q^{100})$$ 2 * q - q^2 - 2 * q^3 - q^4 + q^6 + q^7 + 2 * q^9 + q^12 + 3 * q^13 + 2 * q^14 - 3 * q^16 - 4 * q^17 - q^18 - 4 * q^19 - q^21 + q^23 + q^26 - 2 * q^27 - 3 * q^28 + 4 * q^29 - 9 * q^31 + 9 * q^32 - 3 * q^34 - q^36 + 7 * q^38 - 3 * q^39 - q^41 - 2 * q^42 + 7 * q^46 + 10 * q^47 + 3 * q^48 - 11 * q^49 + 4 * q^51 - 4 * q^52 + 11 * q^53 + q^54 - 5 * q^56 + 4 * q^57 + 3 * q^58 + 4 * q^59 + 6 * q^61 - 3 * q^62 + q^63 + 4 * q^64 - 6 * q^67 + 7 * q^68 - q^69 - 8 * q^71 - 9 * q^73 - 20 * q^74 - 3 * q^76 - q^78 + 6 * q^79 + 2 * q^81 - 12 * q^82 + 5 * q^83 + 3 * q^84 - 10 * q^86 - 4 * q^87 - 13 * q^89 + 4 * q^91 - 8 * q^92 + 9 * q^93 - 20 * q^94 - 9 * q^96 - 6 * q^97 + 8 * 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
 1.61803 −0.618034
−1.61803 −1.00000 0.618034 0 1.61803 −0.618034 2.23607 1.00000 0
1.2 0.618034 −1.00000 −1.61803 0 −0.618034 1.61803 −2.23607 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.ba 2
5.b even 2 1 9075.2.a.bw yes 2
11.b odd 2 1 9075.2.a.bs yes 2
55.d odd 2 1 9075.2.a.be yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9075.2.a.ba 2 1.a even 1 1 trivial
9075.2.a.be yes 2 55.d odd 2 1
9075.2.a.bs yes 2 11.b odd 2 1
9075.2.a.bw yes 2 5.b even 2 1

## Hecke kernels

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

 $$T_{2}^{2} + T_{2} - 1$$ T2^2 + T2 - 1 $$T_{7}^{2} - T_{7} - 1$$ T7^2 - T7 - 1 $$T_{13}^{2} - 3T_{13} + 1$$ T13^2 - 3*T13 + 1 $$T_{17}^{2} + 4T_{17} - 1$$ T17^2 + 4*T17 - 1 $$T_{19}^{2} + 4T_{19} - 1$$ T19^2 + 4*T19 - 1 $$T_{23}^{2} - T_{23} - 11$$ T23^2 - T23 - 11 $$T_{37}^{2} - 80$$ T37^2 - 80

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 1$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - T - 1$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 3T + 1$$
$17$ $$T^{2} + 4T - 1$$
$19$ $$T^{2} + 4T - 1$$
$23$ $$T^{2} - T - 11$$
$29$ $$T^{2} - 4T - 1$$
$31$ $$T^{2} + 9T + 9$$
$37$ $$T^{2} - 80$$
$41$ $$T^{2} + T - 31$$
$43$ $$T^{2} - 20$$
$47$ $$T^{2} - 10T - 20$$
$53$ $$T^{2} - 11T - 1$$
$59$ $$T^{2} - 4T - 41$$
$61$ $$T^{2} - 6T + 4$$
$67$ $$T^{2} + 6T + 4$$
$71$ $$T^{2} + 8T + 11$$
$73$ $$T^{2} + 9T + 19$$
$79$ $$T^{2} - 6T - 36$$
$83$ $$T^{2} - 5T - 5$$
$89$ $$T^{2} + 13T - 19$$
$97$ $$T^{2} + 6T - 116$$