# Properties

 Label 9075.2.a.o 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 1815) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} - q^{4} - q^{6} + 4 q^{7} - 3 q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 - q^4 - q^6 + 4 * q^7 - 3 * q^8 + q^9 $$q + q^{2} - q^{3} - q^{4} - q^{6} + 4 q^{7} - 3 q^{8} + q^{9} + q^{12} + 2 q^{13} + 4 q^{14} - q^{16} + 2 q^{17} + q^{18} + 8 q^{19} - 4 q^{21} + 4 q^{23} + 3 q^{24} + 2 q^{26} - q^{27} - 4 q^{28} + 4 q^{29} - 8 q^{31} + 5 q^{32} + 2 q^{34} - q^{36} + 8 q^{37} + 8 q^{38} - 2 q^{39} + 12 q^{41} - 4 q^{42} - 8 q^{43} + 4 q^{46} - 4 q^{47} + q^{48} + 9 q^{49} - 2 q^{51} - 2 q^{52} - 4 q^{53} - q^{54} - 12 q^{56} - 8 q^{57} + 4 q^{58} - 8 q^{59} - 8 q^{62} + 4 q^{63} + 7 q^{64} - 4 q^{67} - 2 q^{68} - 4 q^{69} - 12 q^{71} - 3 q^{72} + 2 q^{73} + 8 q^{74} - 8 q^{76} - 2 q^{78} + 8 q^{79} + q^{81} + 12 q^{82} + 4 q^{83} + 4 q^{84} - 8 q^{86} - 4 q^{87} + 6 q^{89} + 8 q^{91} - 4 q^{92} + 8 q^{93} - 4 q^{94} - 5 q^{96} - 8 q^{97} + 9 q^{98}+O(q^{100})$$ q + q^2 - q^3 - q^4 - q^6 + 4 * q^7 - 3 * q^8 + q^9 + q^12 + 2 * q^13 + 4 * q^14 - q^16 + 2 * q^17 + q^18 + 8 * q^19 - 4 * q^21 + 4 * q^23 + 3 * q^24 + 2 * q^26 - q^27 - 4 * q^28 + 4 * q^29 - 8 * q^31 + 5 * q^32 + 2 * q^34 - q^36 + 8 * q^37 + 8 * q^38 - 2 * q^39 + 12 * q^41 - 4 * q^42 - 8 * q^43 + 4 * q^46 - 4 * q^47 + q^48 + 9 * q^49 - 2 * q^51 - 2 * q^52 - 4 * q^53 - q^54 - 12 * q^56 - 8 * q^57 + 4 * q^58 - 8 * q^59 - 8 * q^62 + 4 * q^63 + 7 * q^64 - 4 * q^67 - 2 * q^68 - 4 * q^69 - 12 * q^71 - 3 * q^72 + 2 * q^73 + 8 * q^74 - 8 * q^76 - 2 * q^78 + 8 * q^79 + q^81 + 12 * q^82 + 4 * q^83 + 4 * q^84 - 8 * q^86 - 4 * q^87 + 6 * q^89 + 8 * q^91 - 4 * q^92 + 8 * q^93 - 4 * q^94 - 5 * q^96 - 8 * q^97 + 9 * 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
 0
1.00000 −1.00000 −1.00000 0 −1.00000 4.00000 −3.00000 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.o 1
5.b even 2 1 9075.2.a.f 1
5.c odd 4 2 1815.2.c.a 2
11.b odd 2 1 9075.2.a.c 1
55.d odd 2 1 9075.2.a.r 1
55.e even 4 2 1815.2.c.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.c.a 2 5.c odd 4 2
1815.2.c.b yes 2 55.e even 4 2
9075.2.a.c 1 11.b odd 2 1
9075.2.a.f 1 5.b even 2 1
9075.2.a.o 1 1.a even 1 1 trivial
9075.2.a.r 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9075))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{7} - 4$$ T7 - 4 $$T_{13} - 2$$ T13 - 2 $$T_{17} - 2$$ T17 - 2 $$T_{19} - 8$$ T19 - 8 $$T_{23} - 4$$ T23 - 4 $$T_{37} - 8$$ T37 - 8

## Hecke characteristic polynomials

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