# Properties

 Label 9075.2.a.bc Level $9075$ Weight $2$ Character orbit 9075.a Self dual yes Analytic conductor $72.464$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9075 = 3 \cdot 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9075.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$72.4642398343$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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 $$q - \beta q^{2} + q^{3} + (\beta - 1) q^{4} - \beta q^{6} + ( - 2 \beta - 2) q^{7} + (2 \beta - 1) q^{8} + q^{9} + (\beta - 1) q^{12} + 2 \beta q^{13} + (4 \beta + 2) q^{14} - 3 \beta q^{16} + 2 q^{17} - \beta q^{18} - 5 q^{19} + ( - 2 \beta - 2) q^{21} + ( - \beta - 3) q^{23} + (2 \beta - 1) q^{24} + ( - 2 \beta - 2) q^{26} + q^{27} - 2 \beta q^{28} + ( - 3 \beta + 4) q^{29} + 7 q^{31} + ( - \beta + 5) q^{32} - 2 \beta q^{34} + (\beta - 1) q^{36} + ( - 4 \beta + 5) q^{37} + 5 \beta q^{38} + 2 \beta q^{39} + ( - 4 \beta - 5) q^{41} + (4 \beta + 2) q^{42} + (5 \beta + 1) q^{43} + (4 \beta + 1) q^{46} + (5 \beta - 2) q^{47} - 3 \beta q^{48} + (12 \beta + 1) q^{49} + 2 q^{51} + 2 q^{52} + ( - \beta + 7) q^{53} - \beta q^{54} + ( - 6 \beta - 2) q^{56} - 5 q^{57} + ( - \beta + 3) q^{58} + (6 \beta - 3) q^{59} - 7 q^{61} - 7 \beta q^{62} + ( - 2 \beta - 2) q^{63} + (2 \beta + 1) q^{64} + (\beta + 10) q^{67} + (2 \beta - 2) q^{68} + ( - \beta - 3) q^{69} - 8 q^{71} + (2 \beta - 1) q^{72} + (4 \beta - 11) q^{73} + ( - \beta + 4) q^{74} + ( - 5 \beta + 5) q^{76} + ( - 2 \beta - 2) q^{78} + ( - 5 \beta + 5) q^{79} + q^{81} + (9 \beta + 4) q^{82} + (6 \beta - 2) q^{83} - 2 \beta q^{84} + ( - 6 \beta - 5) q^{86} + ( - 3 \beta + 4) q^{87} + ( - 3 \beta + 9) q^{89} + ( - 8 \beta - 4) q^{91} + ( - 3 \beta + 2) q^{92} + 7 q^{93} + ( - 3 \beta - 5) q^{94} + ( - \beta + 5) q^{96} + (4 \beta - 9) q^{97} + ( - 13 \beta - 12) q^{98} +O(q^{100})$$ 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 * q^13 + (4*b + 2) * q^14 - 3*b * q^16 + 2 * q^17 - b * q^18 - 5 * q^19 + (-2*b - 2) * q^21 + (-b - 3) * q^23 + (2*b - 1) * q^24 + (-2*b - 2) * q^26 + q^27 - 2*b * q^28 + (-3*b + 4) * q^29 + 7 * q^31 + (-b + 5) * q^32 - 2*b * q^34 + (b - 1) * q^36 + (-4*b + 5) * q^37 + 5*b * q^38 + 2*b * q^39 + (-4*b - 5) * q^41 + (4*b + 2) * q^42 + (5*b + 1) * q^43 + (4*b + 1) * q^46 + (5*b - 2) * q^47 - 3*b * q^48 + (12*b + 1) * q^49 + 2 * q^51 + 2 * q^52 + (-b + 7) * q^53 - b * q^54 + (-6*b - 2) * q^56 - 5 * q^57 + (-b + 3) * q^58 + (6*b - 3) * q^59 - 7 * q^61 - 7*b * q^62 + (-2*b - 2) * q^63 + (2*b + 1) * q^64 + (b + 10) * q^67 + (2*b - 2) * q^68 + (-b - 3) * q^69 - 8 * q^71 + (2*b - 1) * q^72 + (4*b - 11) * q^73 + (-b + 4) * q^74 + (-5*b + 5) * q^76 + (-2*b - 2) * q^78 + (-5*b + 5) * q^79 + q^81 + (9*b + 4) * q^82 + (6*b - 2) * q^83 - 2*b * q^84 + (-6*b - 5) * q^86 + (-3*b + 4) * q^87 + (-3*b + 9) * q^89 + (-8*b - 4) * q^91 + (-3*b + 2) * q^92 + 7 * q^93 + (-3*b - 5) * q^94 + (-b + 5) * q^96 + (4*b - 9) * q^97 + (-13*b - 12) * 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 $$2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} - 6 q^{7} + 2 q^{9} - q^{12} + 2 q^{13} + 8 q^{14} - 3 q^{16} + 4 q^{17} - q^{18} - 10 q^{19} - 6 q^{21} - 7 q^{23} - 6 q^{26} + 2 q^{27} - 2 q^{28} + 5 q^{29} + 14 q^{31} + 9 q^{32} - 2 q^{34} - q^{36} + 6 q^{37} + 5 q^{38} + 2 q^{39} - 14 q^{41} + 8 q^{42} + 7 q^{43} + 6 q^{46} + q^{47} - 3 q^{48} + 14 q^{49} + 4 q^{51} + 4 q^{52} + 13 q^{53} - q^{54} - 10 q^{56} - 10 q^{57} + 5 q^{58} - 14 q^{61} - 7 q^{62} - 6 q^{63} + 4 q^{64} + 21 q^{67} - 2 q^{68} - 7 q^{69} - 16 q^{71} - 18 q^{73} + 7 q^{74} + 5 q^{76} - 6 q^{78} + 5 q^{79} + 2 q^{81} + 17 q^{82} + 2 q^{83} - 2 q^{84} - 16 q^{86} + 5 q^{87} + 15 q^{89} - 16 q^{91} + q^{92} + 14 q^{93} - 13 q^{94} + 9 q^{96} - 14 q^{97} - 37 q^{98}+O(q^{100})$$ 2 * q - q^2 + 2 * q^3 - q^4 - q^6 - 6 * q^7 + 2 * q^9 - q^12 + 2 * q^13 + 8 * q^14 - 3 * q^16 + 4 * q^17 - q^18 - 10 * q^19 - 6 * q^21 - 7 * q^23 - 6 * q^26 + 2 * q^27 - 2 * q^28 + 5 * q^29 + 14 * q^31 + 9 * q^32 - 2 * q^34 - q^36 + 6 * q^37 + 5 * q^38 + 2 * q^39 - 14 * q^41 + 8 * q^42 + 7 * q^43 + 6 * q^46 + q^47 - 3 * q^48 + 14 * q^49 + 4 * q^51 + 4 * q^52 + 13 * q^53 - q^54 - 10 * q^56 - 10 * q^57 + 5 * q^58 - 14 * q^61 - 7 * q^62 - 6 * q^63 + 4 * q^64 + 21 * q^67 - 2 * q^68 - 7 * q^69 - 16 * q^71 - 18 * q^73 + 7 * q^74 + 5 * q^76 - 6 * q^78 + 5 * q^79 + 2 * q^81 + 17 * q^82 + 2 * q^83 - 2 * q^84 - 16 * q^86 + 5 * q^87 + 15 * q^89 - 16 * q^91 + q^92 + 14 * q^93 - 13 * q^94 + 9 * q^96 - 14 * q^97 - 37 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 −5.23607 2.23607 1.00000 0
1.2 0.618034 1.00000 −1.61803 0 0.618034 −0.763932 −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.bc 2
5.b even 2 1 9075.2.a.bt 2
11.b odd 2 1 9075.2.a.by 2
11.d odd 10 2 825.2.n.d yes 4
55.d odd 2 1 9075.2.a.z 2
55.h odd 10 2 825.2.n.b 4
55.l even 20 4 825.2.bx.c 8

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 1$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 6T + 4$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 2T - 4$$
$17$ $$(T - 2)^{2}$$
$19$ $$(T + 5)^{2}$$
$23$ $$T^{2} + 7T + 11$$
$29$ $$T^{2} - 5T - 5$$
$31$ $$(T - 7)^{2}$$
$37$ $$T^{2} - 6T - 11$$
$41$ $$T^{2} + 14T + 29$$
$43$ $$T^{2} - 7T - 19$$
$47$ $$T^{2} - T - 31$$
$53$ $$T^{2} - 13T + 41$$
$59$ $$T^{2} - 45$$
$61$ $$(T + 7)^{2}$$
$67$ $$T^{2} - 21T + 109$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} + 18T + 61$$
$79$ $$T^{2} - 5T - 25$$
$83$ $$T^{2} - 2T - 44$$
$89$ $$T^{2} - 15T + 45$$
$97$ $$T^{2} + 14T + 29$$
show more
show less