# Properties

 Label 9075.2.a.u 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 33) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{2} + q^{3} + 3 \beta q^{4} + ( - \beta - 1) q^{6} - q^{7} + ( - 4 \beta - 1) q^{8} + q^{9}+O(q^{10})$$ q + (-b - 1) * q^2 + q^3 + 3*b * q^4 + (-b - 1) * q^6 - q^7 + (-4*b - 1) * q^8 + q^9 $$q + ( - \beta - 1) q^{2} + q^{3} + 3 \beta q^{4} + ( - \beta - 1) q^{6} - q^{7} + ( - 4 \beta - 1) q^{8} + q^{9} + 3 \beta q^{12} + (2 \beta - 3) q^{13} + (\beta + 1) q^{14} + (3 \beta + 5) q^{16} + (3 \beta - 6) q^{17} + ( - \beta - 1) q^{18} + ( - 3 \beta - 1) q^{19} - q^{21} + ( - 2 \beta + 3) q^{23} + ( - 4 \beta - 1) q^{24} + ( - \beta + 1) q^{26} + q^{27} - 3 \beta q^{28} + 6 q^{29} + ( - 5 \beta + 2) q^{31} + ( - 3 \beta - 6) q^{32} + 3 q^{34} + 3 \beta q^{36} + (2 \beta + 3) q^{37} + (7 \beta + 4) q^{38} + (2 \beta - 3) q^{39} + ( - 2 \beta + 3) q^{41} + (\beta + 1) q^{42} + ( - 6 \beta + 3) q^{43} + (\beta - 1) q^{46} + (5 \beta + 2) q^{47} + (3 \beta + 5) q^{48} - 6 q^{49} + (3 \beta - 6) q^{51} + ( - 3 \beta + 6) q^{52} + ( - \beta + 2) q^{53} + ( - \beta - 1) q^{54} + (4 \beta + 1) q^{56} + ( - 3 \beta - 1) q^{57} + ( - 6 \beta - 6) q^{58} + ( - \beta + 9) q^{59} + (9 \beta - 3) q^{61} + (8 \beta + 3) q^{62} - q^{63} + (6 \beta - 1) q^{64} + ( - 3 \beta + 3) q^{67} + ( - 9 \beta + 9) q^{68} + ( - 2 \beta + 3) q^{69} + (7 \beta - 1) q^{71} + ( - 4 \beta - 1) q^{72} + ( - 6 \beta + 4) q^{73} + ( - 7 \beta - 5) q^{74} + ( - 12 \beta - 9) q^{76} + ( - \beta + 1) q^{78} - 11 q^{79} + q^{81} + (\beta - 1) q^{82} + (4 \beta - 5) q^{83} - 3 \beta q^{84} + (9 \beta + 3) q^{86} + 6 q^{87} + ( - 2 \beta - 5) q^{89} + ( - 2 \beta + 3) q^{91} + (3 \beta - 6) q^{92} + ( - 5 \beta + 2) q^{93} + ( - 12 \beta - 7) q^{94} + ( - 3 \beta - 6) q^{96} + ( - 3 \beta - 3) q^{97} + (6 \beta + 6) q^{98} +O(q^{100})$$ q + (-b - 1) * q^2 + q^3 + 3*b * q^4 + (-b - 1) * q^6 - q^7 + (-4*b - 1) * q^8 + q^9 + 3*b * q^12 + (2*b - 3) * q^13 + (b + 1) * q^14 + (3*b + 5) * q^16 + (3*b - 6) * q^17 + (-b - 1) * q^18 + (-3*b - 1) * q^19 - q^21 + (-2*b + 3) * q^23 + (-4*b - 1) * q^24 + (-b + 1) * q^26 + q^27 - 3*b * q^28 + 6 * q^29 + (-5*b + 2) * q^31 + (-3*b - 6) * q^32 + 3 * q^34 + 3*b * q^36 + (2*b + 3) * q^37 + (7*b + 4) * q^38 + (2*b - 3) * q^39 + (-2*b + 3) * q^41 + (b + 1) * q^42 + (-6*b + 3) * q^43 + (b - 1) * q^46 + (5*b + 2) * q^47 + (3*b + 5) * q^48 - 6 * q^49 + (3*b - 6) * q^51 + (-3*b + 6) * q^52 + (-b + 2) * q^53 + (-b - 1) * q^54 + (4*b + 1) * q^56 + (-3*b - 1) * q^57 + (-6*b - 6) * q^58 + (-b + 9) * q^59 + (9*b - 3) * q^61 + (8*b + 3) * q^62 - q^63 + (6*b - 1) * q^64 + (-3*b + 3) * q^67 + (-9*b + 9) * q^68 + (-2*b + 3) * q^69 + (7*b - 1) * q^71 + (-4*b - 1) * q^72 + (-6*b + 4) * q^73 + (-7*b - 5) * q^74 + (-12*b - 9) * q^76 + (-b + 1) * q^78 - 11 * q^79 + q^81 + (b - 1) * q^82 + (4*b - 5) * q^83 - 3*b * q^84 + (9*b + 3) * q^86 + 6 * q^87 + (-2*b - 5) * q^89 + (-2*b + 3) * q^91 + (3*b - 6) * q^92 + (-5*b + 2) * q^93 + (-12*b - 7) * q^94 + (-3*b - 6) * q^96 + (-3*b - 3) * q^97 + (6*b + 6) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 3 q^{6} - 2 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 - 3 * q^6 - 2 * q^7 - 6 * q^8 + 2 * q^9 $$2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 3 q^{6} - 2 q^{7} - 6 q^{8} + 2 q^{9} + 3 q^{12} - 4 q^{13} + 3 q^{14} + 13 q^{16} - 9 q^{17} - 3 q^{18} - 5 q^{19} - 2 q^{21} + 4 q^{23} - 6 q^{24} + q^{26} + 2 q^{27} - 3 q^{28} + 12 q^{29} - q^{31} - 15 q^{32} + 6 q^{34} + 3 q^{36} + 8 q^{37} + 15 q^{38} - 4 q^{39} + 4 q^{41} + 3 q^{42} - q^{46} + 9 q^{47} + 13 q^{48} - 12 q^{49} - 9 q^{51} + 9 q^{52} + 3 q^{53} - 3 q^{54} + 6 q^{56} - 5 q^{57} - 18 q^{58} + 17 q^{59} + 3 q^{61} + 14 q^{62} - 2 q^{63} + 4 q^{64} + 3 q^{67} + 9 q^{68} + 4 q^{69} + 5 q^{71} - 6 q^{72} + 2 q^{73} - 17 q^{74} - 30 q^{76} + q^{78} - 22 q^{79} + 2 q^{81} - q^{82} - 6 q^{83} - 3 q^{84} + 15 q^{86} + 12 q^{87} - 12 q^{89} + 4 q^{91} - 9 q^{92} - q^{93} - 26 q^{94} - 15 q^{96} - 9 q^{97} + 18 q^{98}+O(q^{100})$$ 2 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 - 3 * q^6 - 2 * q^7 - 6 * q^8 + 2 * q^9 + 3 * q^12 - 4 * q^13 + 3 * q^14 + 13 * q^16 - 9 * q^17 - 3 * q^18 - 5 * q^19 - 2 * q^21 + 4 * q^23 - 6 * q^24 + q^26 + 2 * q^27 - 3 * q^28 + 12 * q^29 - q^31 - 15 * q^32 + 6 * q^34 + 3 * q^36 + 8 * q^37 + 15 * q^38 - 4 * q^39 + 4 * q^41 + 3 * q^42 - q^46 + 9 * q^47 + 13 * q^48 - 12 * q^49 - 9 * q^51 + 9 * q^52 + 3 * q^53 - 3 * q^54 + 6 * q^56 - 5 * q^57 - 18 * q^58 + 17 * q^59 + 3 * q^61 + 14 * q^62 - 2 * q^63 + 4 * q^64 + 3 * q^67 + 9 * q^68 + 4 * q^69 + 5 * q^71 - 6 * q^72 + 2 * q^73 - 17 * q^74 - 30 * q^76 + q^78 - 22 * q^79 + 2 * q^81 - q^82 - 6 * q^83 - 3 * q^84 + 15 * q^86 + 12 * q^87 - 12 * q^89 + 4 * q^91 - 9 * q^92 - q^93 - 26 * q^94 - 15 * q^96 - 9 * q^97 + 18 * 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
−2.61803 1.00000 4.85410 0 −2.61803 −1.00000 −7.47214 1.00000 0
1.2 −0.381966 1.00000 −1.85410 0 −0.381966 −1.00000 1.47214 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.u 2
5.b even 2 1 363.2.a.i 2
11.b odd 2 1 9075.2.a.cb 2
11.d odd 10 2 825.2.n.c 4
15.d odd 2 1 1089.2.a.l 2
20.d odd 2 1 5808.2.a.ci 2
55.d odd 2 1 363.2.a.d 2
55.h odd 10 2 33.2.e.b 4
55.h odd 10 2 363.2.e.k 4
55.j even 10 2 363.2.e.b 4
55.j even 10 2 363.2.e.f 4
55.l even 20 4 825.2.bx.d 8
165.d even 2 1 1089.2.a.t 2
165.r even 10 2 99.2.f.a 4
220.g even 2 1 5808.2.a.cj 2
220.o even 10 2 528.2.y.b 4
495.bo even 30 4 891.2.n.b 8
495.br odd 30 4 891.2.n.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.e.b 4 55.h odd 10 2
99.2.f.a 4 165.r even 10 2
363.2.a.d 2 55.d odd 2 1
363.2.a.i 2 5.b even 2 1
363.2.e.b 4 55.j even 10 2
363.2.e.f 4 55.j even 10 2
363.2.e.k 4 55.h odd 10 2
528.2.y.b 4 220.o even 10 2
825.2.n.c 4 11.d odd 10 2
825.2.bx.d 8 55.l even 20 4
891.2.n.b 8 495.bo even 30 4
891.2.n.c 8 495.br odd 30 4
1089.2.a.l 2 15.d odd 2 1
1089.2.a.t 2 165.d even 2 1
5808.2.a.ci 2 20.d odd 2 1
5808.2.a.cj 2 220.g even 2 1
9075.2.a.u 2 1.a even 1 1 trivial
9075.2.a.cb 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9075))$$:

 $$T_{2}^{2} + 3T_{2} + 1$$ T2^2 + 3*T2 + 1 $$T_{7} + 1$$ T7 + 1 $$T_{13}^{2} + 4T_{13} - 1$$ T13^2 + 4*T13 - 1 $$T_{17}^{2} + 9T_{17} + 9$$ T17^2 + 9*T17 + 9 $$T_{19}^{2} + 5T_{19} - 5$$ T19^2 + 5*T19 - 5 $$T_{23}^{2} - 4T_{23} - 1$$ T23^2 - 4*T23 - 1 $$T_{37}^{2} - 8T_{37} + 11$$ T37^2 - 8*T37 + 11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T + 1$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 4T - 1$$
$17$ $$T^{2} + 9T + 9$$
$19$ $$T^{2} + 5T - 5$$
$23$ $$T^{2} - 4T - 1$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} + T - 31$$
$37$ $$T^{2} - 8T + 11$$
$41$ $$T^{2} - 4T - 1$$
$43$ $$T^{2} - 45$$
$47$ $$T^{2} - 9T - 11$$
$53$ $$T^{2} - 3T + 1$$
$59$ $$T^{2} - 17T + 71$$
$61$ $$T^{2} - 3T - 99$$
$67$ $$T^{2} - 3T - 9$$
$71$ $$T^{2} - 5T - 55$$
$73$ $$T^{2} - 2T - 44$$
$79$ $$(T + 11)^{2}$$
$83$ $$T^{2} + 6T - 11$$
$89$ $$T^{2} + 12T + 31$$
$97$ $$T^{2} + 9T + 9$$