# Properties

 Label 9075.2.a.bu 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: no (minimal twist has level 825) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - q^{3} + (\beta - 1) q^{4} - \beta q^{6} + 3 q^{7} + ( - 2 \beta + 1) q^{8} + q^{9} +O(q^{10})$$ q + b * q^2 - q^3 + (b - 1) * q^4 - b * q^6 + 3 * q^7 + (-2*b + 1) * q^8 + q^9 $$q + \beta q^{2} - q^{3} + (\beta - 1) q^{4} - \beta q^{6} + 3 q^{7} + ( - 2 \beta + 1) q^{8} + q^{9} + ( - \beta + 1) q^{12} + ( - 3 \beta + 3) q^{13} + 3 \beta q^{14} - 3 \beta q^{16} + (4 \beta + 1) q^{17} + \beta q^{18} + ( - 2 \beta + 6) q^{19} - 3 q^{21} + ( - \beta - 6) q^{23} + (2 \beta - 1) q^{24} - 3 q^{26} - q^{27} + (3 \beta - 3) q^{28} + ( - \beta - 2) q^{29} + ( - 3 \beta - 4) q^{31} + (\beta - 5) q^{32} + (5 \beta + 4) q^{34} + (\beta - 1) q^{36} + ( - 3 \beta - 4) q^{37} + (4 \beta - 2) q^{38} + (3 \beta - 3) q^{39} + (2 \beta - 3) q^{41} - 3 \beta q^{42} + ( - 2 \beta + 5) q^{43} + ( - 7 \beta - 1) q^{46} + (\beta - 11) q^{47} + 3 \beta q^{48} + 2 q^{49} + ( - 4 \beta - 1) q^{51} + (3 \beta - 6) q^{52} + (4 \beta - 6) q^{53} - \beta q^{54} + ( - 6 \beta + 3) q^{56} + (2 \beta - 6) q^{57} + ( - 3 \beta - 1) q^{58} + (3 \beta - 9) q^{59} + ( - 2 \beta + 9) q^{61} + ( - 7 \beta - 3) q^{62} + 3 q^{63} + (2 \beta + 1) q^{64} + (6 \beta - 1) q^{67} + (\beta + 3) q^{68} + (\beta + 6) q^{69} + (4 \beta - 5) q^{71} + ( - 2 \beta + 1) q^{72} + ( - 3 \beta - 2) q^{73} + ( - 7 \beta - 3) q^{74} + (6 \beta - 8) q^{76} + 3 q^{78} + (9 \beta - 7) q^{79} + q^{81} + ( - \beta + 2) q^{82} + ( - \beta + 12) q^{83} + ( - 3 \beta + 3) q^{84} + (3 \beta - 2) q^{86} + (\beta + 2) q^{87} + (9 \beta - 12) q^{89} + ( - 9 \beta + 9) q^{91} + ( - 6 \beta + 5) q^{92} + (3 \beta + 4) q^{93} + ( - 10 \beta + 1) q^{94} + ( - \beta + 5) q^{96} + (12 \beta - 9) q^{97} + 2 \beta q^{98} +O(q^{100})$$ q + b * q^2 - q^3 + (b - 1) * q^4 - b * q^6 + 3 * q^7 + (-2*b + 1) * q^8 + q^9 + (-b + 1) * q^12 + (-3*b + 3) * q^13 + 3*b * q^14 - 3*b * q^16 + (4*b + 1) * q^17 + b * q^18 + (-2*b + 6) * q^19 - 3 * q^21 + (-b - 6) * q^23 + (2*b - 1) * q^24 - 3 * q^26 - q^27 + (3*b - 3) * q^28 + (-b - 2) * q^29 + (-3*b - 4) * q^31 + (b - 5) * q^32 + (5*b + 4) * q^34 + (b - 1) * q^36 + (-3*b - 4) * q^37 + (4*b - 2) * q^38 + (3*b - 3) * q^39 + (2*b - 3) * q^41 - 3*b * q^42 + (-2*b + 5) * q^43 + (-7*b - 1) * q^46 + (b - 11) * q^47 + 3*b * q^48 + 2 * q^49 + (-4*b - 1) * q^51 + (3*b - 6) * q^52 + (4*b - 6) * q^53 - b * q^54 + (-6*b + 3) * q^56 + (2*b - 6) * q^57 + (-3*b - 1) * q^58 + (3*b - 9) * q^59 + (-2*b + 9) * q^61 + (-7*b - 3) * q^62 + 3 * q^63 + (2*b + 1) * q^64 + (6*b - 1) * q^67 + (b + 3) * q^68 + (b + 6) * q^69 + (4*b - 5) * q^71 + (-2*b + 1) * q^72 + (-3*b - 2) * q^73 + (-7*b - 3) * q^74 + (6*b - 8) * q^76 + 3 * q^78 + (9*b - 7) * q^79 + q^81 + (-b + 2) * q^82 + (-b + 12) * q^83 + (-3*b + 3) * q^84 + (3*b - 2) * q^86 + (b + 2) * q^87 + (9*b - 12) * q^89 + (-9*b + 9) * q^91 + (-6*b + 5) * q^92 + (3*b + 4) * q^93 + (-10*b + 1) * q^94 + (-b + 5) * q^96 + (12*b - 9) * q^97 + 2*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 6 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9 $$2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 6 q^{7} + 2 q^{9} + q^{12} + 3 q^{13} + 3 q^{14} - 3 q^{16} + 6 q^{17} + q^{18} + 10 q^{19} - 6 q^{21} - 13 q^{23} - 6 q^{26} - 2 q^{27} - 3 q^{28} - 5 q^{29} - 11 q^{31} - 9 q^{32} + 13 q^{34} - q^{36} - 11 q^{37} - 3 q^{39} - 4 q^{41} - 3 q^{42} + 8 q^{43} - 9 q^{46} - 21 q^{47} + 3 q^{48} + 4 q^{49} - 6 q^{51} - 9 q^{52} - 8 q^{53} - q^{54} - 10 q^{57} - 5 q^{58} - 15 q^{59} + 16 q^{61} - 13 q^{62} + 6 q^{63} + 4 q^{64} + 4 q^{67} + 7 q^{68} + 13 q^{69} - 6 q^{71} - 7 q^{73} - 13 q^{74} - 10 q^{76} + 6 q^{78} - 5 q^{79} + 2 q^{81} + 3 q^{82} + 23 q^{83} + 3 q^{84} - q^{86} + 5 q^{87} - 15 q^{89} + 9 q^{91} + 4 q^{92} + 11 q^{93} - 8 q^{94} + 9 q^{96} - 6 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9 + q^12 + 3 * q^13 + 3 * q^14 - 3 * q^16 + 6 * q^17 + q^18 + 10 * q^19 - 6 * q^21 - 13 * q^23 - 6 * q^26 - 2 * q^27 - 3 * q^28 - 5 * q^29 - 11 * q^31 - 9 * q^32 + 13 * q^34 - q^36 - 11 * q^37 - 3 * q^39 - 4 * q^41 - 3 * q^42 + 8 * q^43 - 9 * q^46 - 21 * q^47 + 3 * q^48 + 4 * q^49 - 6 * q^51 - 9 * q^52 - 8 * q^53 - q^54 - 10 * q^57 - 5 * q^58 - 15 * q^59 + 16 * q^61 - 13 * q^62 + 6 * q^63 + 4 * q^64 + 4 * q^67 + 7 * q^68 + 13 * q^69 - 6 * q^71 - 7 * q^73 - 13 * q^74 - 10 * q^76 + 6 * q^78 - 5 * q^79 + 2 * q^81 + 3 * q^82 + 23 * q^83 + 3 * q^84 - q^86 + 5 * q^87 - 15 * q^89 + 9 * q^91 + 4 * q^92 + 11 * q^93 - 8 * q^94 + 9 * q^96 - 6 * q^97 + 2 * 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
 −0.618034 1.61803
−0.618034 −1.00000 −1.61803 0 0.618034 3.00000 2.23607 1.00000 0
1.2 1.61803 −1.00000 0.618034 0 −1.61803 3.00000 −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.bu 2
5.b even 2 1 9075.2.a.bb 2
11.b odd 2 1 9075.2.a.y 2
11.d odd 10 2 825.2.n.e yes 4
55.d odd 2 1 9075.2.a.bz 2
55.h odd 10 2 825.2.n.a 4
55.l even 20 4 825.2.bx.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.n.a 4 55.h odd 10 2
825.2.n.e yes 4 11.d odd 10 2
825.2.bx.a 8 55.l even 20 4
9075.2.a.y 2 11.b odd 2 1
9075.2.a.bb 2 5.b even 2 1
9075.2.a.bu 2 1.a even 1 1 trivial
9075.2.a.bz 2 55.d odd 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 $$T_{7} - 3$$ T7 - 3 $$T_{13}^{2} - 3T_{13} - 9$$ T13^2 - 3*T13 - 9 $$T_{17}^{2} - 6T_{17} - 11$$ T17^2 - 6*T17 - 11 $$T_{19}^{2} - 10T_{19} + 20$$ T19^2 - 10*T19 + 20 $$T_{23}^{2} + 13T_{23} + 41$$ T23^2 + 13*T23 + 41 $$T_{37}^{2} + 11T_{37} + 19$$ T37^2 + 11*T37 + 19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 1$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$(T - 3)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 3T - 9$$
$17$ $$T^{2} - 6T - 11$$
$19$ $$T^{2} - 10T + 20$$
$23$ $$T^{2} + 13T + 41$$
$29$ $$T^{2} + 5T + 5$$
$31$ $$T^{2} + 11T + 19$$
$37$ $$T^{2} + 11T + 19$$
$41$ $$T^{2} + 4T - 1$$
$43$ $$T^{2} - 8T + 11$$
$47$ $$T^{2} + 21T + 109$$
$53$ $$T^{2} + 8T - 4$$
$59$ $$T^{2} + 15T + 45$$
$61$ $$T^{2} - 16T + 59$$
$67$ $$T^{2} - 4T - 41$$
$71$ $$T^{2} + 6T - 11$$
$73$ $$T^{2} + 7T + 1$$
$79$ $$T^{2} + 5T - 95$$
$83$ $$T^{2} - 23T + 131$$
$89$ $$T^{2} + 15T - 45$$
$97$ $$T^{2} + 6T - 171$$