# Properties

 Label 9075.2.a.i Level $9075$ Weight $2$ Character orbit 9075.a Self dual yes Analytic conductor $72.464$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{3} - 2 q^{4} - q^{7} + q^{9}+O(q^{10})$$ q - q^3 - 2 * q^4 - q^7 + q^9 $$q - q^{3} - 2 q^{4} - q^{7} + q^{9} + 2 q^{12} - q^{13} + 4 q^{16} - 6 q^{17} + 7 q^{19} + q^{21} - 6 q^{23} - q^{27} + 2 q^{28} + 6 q^{29} - 7 q^{31} - 2 q^{36} - 2 q^{37} + q^{39} + 6 q^{41} - q^{43} - 4 q^{48} - 6 q^{49} + 6 q^{51} + 2 q^{52} + 6 q^{53} - 7 q^{57} - 5 q^{61} - q^{63} - 8 q^{64} - 5 q^{67} + 12 q^{68} + 6 q^{69} - 12 q^{71} + 14 q^{73} - 14 q^{76} + 4 q^{79} + q^{81} - 6 q^{83} - 2 q^{84} - 6 q^{87} + 6 q^{89} + q^{91} + 12 q^{92} + 7 q^{93} - 17 q^{97}+O(q^{100})$$ q - q^3 - 2 * q^4 - q^7 + q^9 + 2 * q^12 - q^13 + 4 * q^16 - 6 * q^17 + 7 * q^19 + q^21 - 6 * q^23 - q^27 + 2 * q^28 + 6 * q^29 - 7 * q^31 - 2 * q^36 - 2 * q^37 + q^39 + 6 * q^41 - q^43 - 4 * q^48 - 6 * q^49 + 6 * q^51 + 2 * q^52 + 6 * q^53 - 7 * q^57 - 5 * q^61 - q^63 - 8 * q^64 - 5 * q^67 + 12 * q^68 + 6 * q^69 - 12 * q^71 + 14 * q^73 - 14 * q^76 + 4 * q^79 + q^81 - 6 * q^83 - 2 * q^84 - 6 * q^87 + 6 * q^89 + q^91 + 12 * q^92 + 7 * q^93 - 17 * 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
 0
0 −1.00000 −2.00000 0 0 −1.00000 0 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.i 1
5.b even 2 1 9075.2.a.l 1
11.b odd 2 1 825.2.a.b 1
33.d even 2 1 2475.2.a.f 1
55.d odd 2 1 825.2.a.c yes 1
55.e even 4 2 825.2.c.b 2
165.d even 2 1 2475.2.a.e 1
165.l odd 4 2 2475.2.c.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.a.b 1 11.b odd 2 1
825.2.a.c yes 1 55.d odd 2 1
825.2.c.b 2 55.e even 4 2
2475.2.a.e 1 165.d even 2 1
2475.2.a.f 1 33.d even 2 1
2475.2.c.h 2 165.l odd 4 2
9075.2.a.i 1 1.a even 1 1 trivial
9075.2.a.l 1 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9075))$$:

 $$T_{2}$$ T2 $$T_{7} + 1$$ T7 + 1 $$T_{13} + 1$$ T13 + 1 $$T_{17} + 6$$ T17 + 6 $$T_{19} - 7$$ T19 - 7 $$T_{23} + 6$$ T23 + 6 $$T_{37} + 2$$ T37 + 2

## Hecke characteristic polynomials

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