# Properties

 Label 9075.2.a.bd 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 1815) 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.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + q^{3} + (\beta - 1) q^{4} - \beta q^{6} + (2 \beta - 2) q^{7} + (2 \beta - 1) q^{8} + q^{9} +O(q^{10})$$ q - b * q^2 + q^3 + (b - 1) * q^4 - b * q^6 + (2*b - 2) * q^7 + (2*b - 1) * q^8 + q^9 $$q - \beta q^{2} + q^{3} + (\beta - 1) q^{4} - \beta q^{6} + (2 \beta - 2) q^{7} + (2 \beta - 1) q^{8} + q^{9} + (\beta - 1) q^{12} + (2 \beta - 4) q^{13} - 2 q^{14} - 3 \beta q^{16} + ( - 4 \beta + 3) q^{17} - \beta q^{18} + ( - 4 \beta + 4) q^{19} + (2 \beta - 2) q^{21} + ( - 6 \beta + 1) q^{23} + (2 \beta - 1) q^{24} + (2 \beta - 2) q^{26} + q^{27} + ( - 2 \beta + 4) q^{28} + 2 \beta q^{29} + (6 \beta - 3) q^{31} + ( - \beta + 5) q^{32} + (\beta + 4) q^{34} + (\beta - 1) q^{36} + ( - 2 \beta - 2) q^{37} + 4 q^{38} + (2 \beta - 4) q^{39} + (4 \beta + 6) q^{41} - 2 q^{42} + ( - 2 \beta + 4) q^{43} + (5 \beta + 6) q^{46} + (6 \beta - 5) q^{47} - 3 \beta q^{48} + ( - 4 \beta + 1) q^{49} + ( - 4 \beta + 3) q^{51} + ( - 4 \beta + 6) q^{52} + (8 \beta - 5) q^{53} - \beta q^{54} + ( - 2 \beta + 6) q^{56} + ( - 4 \beta + 4) q^{57} + ( - 2 \beta - 2) q^{58} + ( - 6 \beta + 8) q^{59} + (4 \beta + 1) q^{61} + ( - 3 \beta - 6) q^{62} + (2 \beta - 2) q^{63} + (2 \beta + 1) q^{64} + (2 \beta - 10) q^{67} + (3 \beta - 7) q^{68} + ( - 6 \beta + 1) q^{69} + (4 \beta - 12) q^{71} + (2 \beta - 1) q^{72} + ( - 4 \beta + 8) q^{73} + (4 \beta + 2) q^{74} + (4 \beta - 8) q^{76} + (2 \beta - 2) q^{78} + ( - 6 \beta + 9) q^{79} + q^{81} + ( - 10 \beta - 4) q^{82} + 4 q^{83} + ( - 2 \beta + 4) q^{84} + ( - 2 \beta + 2) q^{86} + 2 \beta q^{87} + (2 \beta + 6) q^{89} + ( - 8 \beta + 12) q^{91} + (\beta - 7) q^{92} + (6 \beta - 3) q^{93} + ( - \beta - 6) q^{94} + ( - \beta + 5) q^{96} + ( - 2 \beta - 14) q^{97} + (3 \beta + 4) q^{98} +O(q^{100})$$ q - b * q^2 + q^3 + (b - 1) * q^4 - b * q^6 + (2*b - 2) * q^7 + (2*b - 1) * q^8 + q^9 + (b - 1) * q^12 + (2*b - 4) * q^13 - 2 * q^14 - 3*b * q^16 + (-4*b + 3) * q^17 - b * q^18 + (-4*b + 4) * q^19 + (2*b - 2) * q^21 + (-6*b + 1) * q^23 + (2*b - 1) * q^24 + (2*b - 2) * q^26 + q^27 + (-2*b + 4) * q^28 + 2*b * q^29 + (6*b - 3) * q^31 + (-b + 5) * q^32 + (b + 4) * q^34 + (b - 1) * q^36 + (-2*b - 2) * q^37 + 4 * q^38 + (2*b - 4) * q^39 + (4*b + 6) * q^41 - 2 * q^42 + (-2*b + 4) * q^43 + (5*b + 6) * q^46 + (6*b - 5) * q^47 - 3*b * q^48 + (-4*b + 1) * q^49 + (-4*b + 3) * q^51 + (-4*b + 6) * q^52 + (8*b - 5) * q^53 - b * q^54 + (-2*b + 6) * q^56 + (-4*b + 4) * q^57 + (-2*b - 2) * q^58 + (-6*b + 8) * q^59 + (4*b + 1) * q^61 + (-3*b - 6) * q^62 + (2*b - 2) * q^63 + (2*b + 1) * q^64 + (2*b - 10) * q^67 + (3*b - 7) * q^68 + (-6*b + 1) * q^69 + (4*b - 12) * q^71 + (2*b - 1) * q^72 + (-4*b + 8) * q^73 + (4*b + 2) * q^74 + (4*b - 8) * q^76 + (2*b - 2) * q^78 + (-6*b + 9) * q^79 + q^81 + (-10*b - 4) * q^82 + 4 * q^83 + (-2*b + 4) * q^84 + (-2*b + 2) * q^86 + 2*b * q^87 + (2*b + 6) * q^89 + (-8*b + 12) * q^91 + (b - 7) * q^92 + (6*b - 3) * q^93 + (-b - 6) * q^94 + (-b + 5) * q^96 + (-2*b - 14) * q^97 + (3*b + 4) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} - 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - q^2 + 2 * q^3 - q^4 - q^6 - 2 * q^7 + 2 * q^9 $$2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} - 2 q^{7} + 2 q^{9} - q^{12} - 6 q^{13} - 4 q^{14} - 3 q^{16} + 2 q^{17} - q^{18} + 4 q^{19} - 2 q^{21} - 4 q^{23} - 2 q^{26} + 2 q^{27} + 6 q^{28} + 2 q^{29} + 9 q^{32} + 9 q^{34} - q^{36} - 6 q^{37} + 8 q^{38} - 6 q^{39} + 16 q^{41} - 4 q^{42} + 6 q^{43} + 17 q^{46} - 4 q^{47} - 3 q^{48} - 2 q^{49} + 2 q^{51} + 8 q^{52} - 2 q^{53} - q^{54} + 10 q^{56} + 4 q^{57} - 6 q^{58} + 10 q^{59} + 6 q^{61} - 15 q^{62} - 2 q^{63} + 4 q^{64} - 18 q^{67} - 11 q^{68} - 4 q^{69} - 20 q^{71} + 12 q^{73} + 8 q^{74} - 12 q^{76} - 2 q^{78} + 12 q^{79} + 2 q^{81} - 18 q^{82} + 8 q^{83} + 6 q^{84} + 2 q^{86} + 2 q^{87} + 14 q^{89} + 16 q^{91} - 13 q^{92} - 13 q^{94} + 9 q^{96} - 30 q^{97} + 11 q^{98}+O(q^{100})$$ 2 * q - q^2 + 2 * q^3 - q^4 - q^6 - 2 * q^7 + 2 * q^9 - q^12 - 6 * q^13 - 4 * q^14 - 3 * q^16 + 2 * q^17 - q^18 + 4 * q^19 - 2 * q^21 - 4 * q^23 - 2 * q^26 + 2 * q^27 + 6 * q^28 + 2 * q^29 + 9 * q^32 + 9 * q^34 - q^36 - 6 * q^37 + 8 * q^38 - 6 * q^39 + 16 * q^41 - 4 * q^42 + 6 * q^43 + 17 * q^46 - 4 * q^47 - 3 * q^48 - 2 * q^49 + 2 * q^51 + 8 * q^52 - 2 * q^53 - q^54 + 10 * q^56 + 4 * q^57 - 6 * q^58 + 10 * q^59 + 6 * q^61 - 15 * q^62 - 2 * q^63 + 4 * q^64 - 18 * q^67 - 11 * q^68 - 4 * q^69 - 20 * q^71 + 12 * q^73 + 8 * q^74 - 12 * q^76 - 2 * q^78 + 12 * q^79 + 2 * q^81 - 18 * q^82 + 8 * q^83 + 6 * q^84 + 2 * q^86 + 2 * q^87 + 14 * q^89 + 16 * q^91 - 13 * q^92 - 13 * q^94 + 9 * q^96 - 30 * q^97 + 11 * 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 1.23607 2.23607 1.00000 0
1.2 0.618034 1.00000 −1.61803 0 0.618034 −3.23607 −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.bd 2
5.b even 2 1 1815.2.a.j yes 2
11.b odd 2 1 9075.2.a.bx 2
15.d odd 2 1 5445.2.a.o 2
55.d odd 2 1 1815.2.a.f 2
165.d even 2 1 5445.2.a.x 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.f 2 55.d odd 2 1
1815.2.a.j yes 2 5.b even 2 1
5445.2.a.o 2 15.d odd 2 1
5445.2.a.x 2 165.d even 2 1
9075.2.a.bd 2 1.a even 1 1 trivial
9075.2.a.bx 2 11.b 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 $$T_{7}^{2} + 2T_{7} - 4$$ T7^2 + 2*T7 - 4 $$T_{13}^{2} + 6T_{13} + 4$$ T13^2 + 6*T13 + 4 $$T_{17}^{2} - 2T_{17} - 19$$ T17^2 - 2*T17 - 19 $$T_{19}^{2} - 4T_{19} - 16$$ T19^2 - 4*T19 - 16 $$T_{23}^{2} + 4T_{23} - 41$$ T23^2 + 4*T23 - 41 $$T_{37}^{2} + 6T_{37} + 4$$ T37^2 + 6*T37 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 1$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2T - 4$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 6T + 4$$
$17$ $$T^{2} - 2T - 19$$
$19$ $$T^{2} - 4T - 16$$
$23$ $$T^{2} + 4T - 41$$
$29$ $$T^{2} - 2T - 4$$
$31$ $$T^{2} - 45$$
$37$ $$T^{2} + 6T + 4$$
$41$ $$T^{2} - 16T + 44$$
$43$ $$T^{2} - 6T + 4$$
$47$ $$T^{2} + 4T - 41$$
$53$ $$T^{2} + 2T - 79$$
$59$ $$T^{2} - 10T - 20$$
$61$ $$T^{2} - 6T - 11$$
$67$ $$T^{2} + 18T + 76$$
$71$ $$T^{2} + 20T + 80$$
$73$ $$T^{2} - 12T + 16$$
$79$ $$T^{2} - 12T - 9$$
$83$ $$(T - 4)^{2}$$
$89$ $$T^{2} - 14T + 44$$
$97$ $$T^{2} + 30T + 220$$