Properties

 Label 9075.2.a.bi Level $9075$ Weight $2$ Character orbit 9075.a Self dual yes Analytic conductor $72.464$ Analytic rank $1$ Dimension $2$ 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: $$2$$ 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: $$2$$ Twist minimal: no (minimal twist has level 363) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

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

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
−2.23607 −1.00000 3.00000 0 2.23607 4.47214 −2.23607 1.00000 0
1.2 2.23607 −1.00000 3.00000 0 −2.23607 −4.47214 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

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.bi 2
5.b even 2 1 363.2.a.g 2
11.b odd 2 1 inner 9075.2.a.bi 2
15.d odd 2 1 1089.2.a.p 2
20.d odd 2 1 5808.2.a.bx 2
55.d odd 2 1 363.2.a.g 2
55.h odd 10 2 363.2.e.a 4
55.h odd 10 2 363.2.e.l 4
55.j even 10 2 363.2.e.a 4
55.j even 10 2 363.2.e.l 4
165.d even 2 1 1089.2.a.p 2
220.g even 2 1 5808.2.a.bx 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.a.g 2 5.b even 2 1
363.2.a.g 2 55.d odd 2 1
363.2.e.a 4 55.h odd 10 2
363.2.e.a 4 55.j even 10 2
363.2.e.l 4 55.h odd 10 2
363.2.e.l 4 55.j even 10 2
1089.2.a.p 2 15.d odd 2 1
1089.2.a.p 2 165.d even 2 1
5808.2.a.bx 2 20.d odd 2 1
5808.2.a.bx 2 220.g even 2 1
9075.2.a.bi 2 1.a even 1 1 trivial
9075.2.a.bi 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9075))$$:

 $$T_{2}^{2} - 5$$ T2^2 - 5 $$T_{7}^{2} - 20$$ T7^2 - 20 $$T_{13}$$ T13 $$T_{17}^{2} - 20$$ T17^2 - 20 $$T_{19}^{2} - 20$$ T19^2 - 20 $$T_{23} - 4$$ T23 - 4 $$T_{37} + 2$$ T37 + 2

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 5$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 20$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 20$$
$19$ $$T^{2} - 20$$
$23$ $$(T - 4)^{2}$$
$29$ $$T^{2} - 20$$
$31$ $$T^{2}$$
$37$ $$(T + 2)^{2}$$
$41$ $$T^{2} - 20$$
$43$ $$T^{2} - 20$$
$47$ $$(T + 8)^{2}$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 80$$
$67$ $$(T - 12)^{2}$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} - 80$$
$79$ $$T^{2} - 180$$
$83$ $$T^{2} - 80$$
$89$ $$(T + 14)^{2}$$
$97$ $$(T + 2)^{2}$$