# Properties

 Label 9075.2.a.bk Level $9075$ Weight $2$ Character orbit 9075.a Self dual yes Analytic conductor $72.464$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

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: $$0$$ 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: $$2$$ Twist minimal: yes 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} - q^{3} + 3 q^{4} + \beta q^{6} - \beta q^{7} - \beta q^{8} + q^{9} +O(q^{10})$$ q - b * q^2 - q^3 + 3 * q^4 + b * q^6 - b * q^7 - b * q^8 + q^9 $$q - \beta q^{2} - q^{3} + 3 q^{4} + \beta q^{6} - \beta q^{7} - \beta q^{8} + q^{9} - 3 q^{12} - 2 \beta q^{13} + 5 q^{14} - q^{16} - \beta q^{17} - \beta q^{18} + \beta q^{19} + \beta q^{21} - q^{23} + \beta q^{24} + 10 q^{26} - q^{27} - 3 \beta q^{28} - 2 \beta q^{29} + 10 q^{31} + 3 \beta q^{32} + 5 q^{34} + 3 q^{36} - 7 q^{37} - 5 q^{38} + 2 \beta q^{39} - 3 \beta q^{41} - 5 q^{42} + \beta q^{46} - 3 q^{47} + q^{48} - 2 q^{49} + \beta q^{51} - 6 \beta q^{52} + 14 q^{53} + \beta q^{54} + 5 q^{56} - \beta q^{57} + 10 q^{58} - 5 q^{59} - 2 \beta q^{61} - 10 \beta q^{62} - \beta q^{63} - 13 q^{64} + 2 q^{67} - 3 \beta q^{68} + q^{69} - 13 q^{71} - \beta q^{72} - 6 \beta q^{73} + 7 \beta q^{74} + 3 \beta q^{76} - 10 q^{78} + 7 \beta q^{79} + q^{81} + 15 q^{82} - 2 \beta q^{83} + 3 \beta q^{84} + 2 \beta q^{87} + 6 q^{89} + 10 q^{91} - 3 q^{92} - 10 q^{93} + 3 \beta q^{94} - 3 \beta q^{96} - 17 q^{97} + 2 \beta q^{98} +O(q^{100})$$ q - b * q^2 - q^3 + 3 * q^4 + b * q^6 - b * q^7 - b * q^8 + q^9 - 3 * q^12 - 2*b * q^13 + 5 * q^14 - q^16 - b * q^17 - b * q^18 + b * q^19 + b * q^21 - q^23 + b * q^24 + 10 * q^26 - q^27 - 3*b * q^28 - 2*b * q^29 + 10 * q^31 + 3*b * q^32 + 5 * q^34 + 3 * q^36 - 7 * q^37 - 5 * q^38 + 2*b * q^39 - 3*b * q^41 - 5 * q^42 + b * q^46 - 3 * q^47 + q^48 - 2 * q^49 + b * q^51 - 6*b * q^52 + 14 * q^53 + b * q^54 + 5 * q^56 - b * q^57 + 10 * q^58 - 5 * q^59 - 2*b * q^61 - 10*b * q^62 - b * q^63 - 13 * q^64 + 2 * q^67 - 3*b * q^68 + q^69 - 13 * q^71 - b * q^72 - 6*b * q^73 + 7*b * q^74 + 3*b * q^76 - 10 * q^78 + 7*b * q^79 + q^81 + 15 * q^82 - 2*b * q^83 + 3*b * q^84 + 2*b * q^87 + 6 * q^89 + 10 * q^91 - 3 * q^92 - 10 * q^93 + 3*b * q^94 - 3*b * q^96 - 17 * q^97 + 2*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 $$2 q - 2 q^{3} + 6 q^{4} + 2 q^{9} - 6 q^{12} + 10 q^{14} - 2 q^{16} - 2 q^{23} + 20 q^{26} - 2 q^{27} + 20 q^{31} + 10 q^{34} + 6 q^{36} - 14 q^{37} - 10 q^{38} - 10 q^{42} - 6 q^{47} + 2 q^{48} - 4 q^{49} + 28 q^{53} + 10 q^{56} + 20 q^{58} - 10 q^{59} - 26 q^{64} + 4 q^{67} + 2 q^{69} - 26 q^{71} - 20 q^{78} + 2 q^{81} + 30 q^{82} + 12 q^{89} + 20 q^{91} - 6 q^{92} - 20 q^{93} - 34 q^{97}+O(q^{100})$$ 2 * q - 2 * q^3 + 6 * q^4 + 2 * q^9 - 6 * q^12 + 10 * q^14 - 2 * q^16 - 2 * q^23 + 20 * q^26 - 2 * q^27 + 20 * q^31 + 10 * q^34 + 6 * q^36 - 14 * q^37 - 10 * q^38 - 10 * q^42 - 6 * q^47 + 2 * q^48 - 4 * q^49 + 28 * q^53 + 10 * q^56 + 20 * q^58 - 10 * q^59 - 26 * q^64 + 4 * q^67 + 2 * q^69 - 26 * q^71 - 20 * q^78 + 2 * q^81 + 30 * q^82 + 12 * q^89 + 20 * q^91 - 6 * q^92 - 20 * q^93 - 34 * 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.

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

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.bk 2
5.b even 2 1 9075.2.a.br yes 2
11.b odd 2 1 inner 9075.2.a.bk 2
55.d odd 2 1 9075.2.a.br yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9075.2.a.bk 2 1.a even 1 1 trivial
9075.2.a.bk 2 11.b odd 2 1 inner
9075.2.a.br yes 2 5.b even 2 1
9075.2.a.br yes 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9075))$$:

 $$T_{2}^{2} - 5$$ T2^2 - 5 $$T_{7}^{2} - 5$$ T7^2 - 5 $$T_{13}^{2} - 20$$ T13^2 - 20 $$T_{17}^{2} - 5$$ T17^2 - 5 $$T_{19}^{2} - 5$$ T19^2 - 5 $$T_{23} + 1$$ T23 + 1 $$T_{37} + 7$$ T37 + 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 5$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 5$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 20$$
$17$ $$T^{2} - 5$$
$19$ $$T^{2} - 5$$
$23$ $$(T + 1)^{2}$$
$29$ $$T^{2} - 20$$
$31$ $$(T - 10)^{2}$$
$37$ $$(T + 7)^{2}$$
$41$ $$T^{2} - 45$$
$43$ $$T^{2}$$
$47$ $$(T + 3)^{2}$$
$53$ $$(T - 14)^{2}$$
$59$ $$(T + 5)^{2}$$
$61$ $$T^{2} - 20$$
$67$ $$(T - 2)^{2}$$
$71$ $$(T + 13)^{2}$$
$73$ $$T^{2} - 180$$
$79$ $$T^{2} - 245$$
$83$ $$T^{2} - 20$$
$89$ $$(T - 6)^{2}$$
$97$ $$(T + 17)^{2}$$