# Properties

 Label 9075.2.a.b Level $9075$ Weight $2$ Character orbit 9075.a Self dual yes Analytic conductor $72.464$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 363) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{6} + q^{7} + q^{9}+O(q^{10})$$ q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^6 + q^7 + q^9 $$q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{6} + q^{7} + q^{9} + 2 q^{12} - 2 q^{13} - 2 q^{14} - 4 q^{16} + 4 q^{17} - 2 q^{18} + 3 q^{19} + q^{21} - 2 q^{23} + 4 q^{26} + q^{27} + 2 q^{28} - 6 q^{29} - 5 q^{31} + 8 q^{32} - 8 q^{34} + 2 q^{36} - 3 q^{37} - 6 q^{38} - 2 q^{39} + 2 q^{41} - 2 q^{42} + 12 q^{43} + 4 q^{46} - 2 q^{47} - 4 q^{48} - 6 q^{49} + 4 q^{51} - 4 q^{52} - 6 q^{53} - 2 q^{54} + 3 q^{57} + 12 q^{58} - 10 q^{59} - 3 q^{61} + 10 q^{62} + q^{63} - 8 q^{64} + q^{67} + 8 q^{68} - 2 q^{69} - 11 q^{73} + 6 q^{74} + 6 q^{76} + 4 q^{78} - 11 q^{79} + q^{81} - 4 q^{82} + 6 q^{83} + 2 q^{84} - 24 q^{86} - 6 q^{87} + 12 q^{89} - 2 q^{91} - 4 q^{92} - 5 q^{93} + 4 q^{94} + 8 q^{96} - 5 q^{97} + 12 q^{98}+O(q^{100})$$ q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^6 + q^7 + q^9 + 2 * q^12 - 2 * q^13 - 2 * q^14 - 4 * q^16 + 4 * q^17 - 2 * q^18 + 3 * q^19 + q^21 - 2 * q^23 + 4 * q^26 + q^27 + 2 * q^28 - 6 * q^29 - 5 * q^31 + 8 * q^32 - 8 * q^34 + 2 * q^36 - 3 * q^37 - 6 * q^38 - 2 * q^39 + 2 * q^41 - 2 * q^42 + 12 * q^43 + 4 * q^46 - 2 * q^47 - 4 * q^48 - 6 * q^49 + 4 * q^51 - 4 * q^52 - 6 * q^53 - 2 * q^54 + 3 * q^57 + 12 * q^58 - 10 * q^59 - 3 * q^61 + 10 * q^62 + q^63 - 8 * q^64 + q^67 + 8 * q^68 - 2 * q^69 - 11 * q^73 + 6 * q^74 + 6 * q^76 + 4 * q^78 - 11 * q^79 + q^81 - 4 * q^82 + 6 * q^83 + 2 * q^84 - 24 * q^86 - 6 * q^87 + 12 * q^89 - 2 * q^91 - 4 * q^92 - 5 * q^93 + 4 * q^94 + 8 * q^96 - 5 * q^97 + 12 * 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
−2.00000 1.00000 2.00000 0 −2.00000 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.b 1
5.b even 2 1 363.2.a.c yes 1
11.b odd 2 1 9075.2.a.t 1
15.d odd 2 1 1089.2.a.a 1
20.d odd 2 1 5808.2.a.bi 1
55.d odd 2 1 363.2.a.a 1
55.h odd 10 4 363.2.e.i 4
55.j even 10 4 363.2.e.d 4
165.d even 2 1 1089.2.a.k 1
220.g even 2 1 5808.2.a.bh 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.a.a 1 55.d odd 2 1
363.2.a.c yes 1 5.b even 2 1
363.2.e.d 4 55.j even 10 4
363.2.e.i 4 55.h odd 10 4
1089.2.a.a 1 15.d odd 2 1
1089.2.a.k 1 165.d even 2 1
5808.2.a.bh 1 220.g even 2 1
5808.2.a.bi 1 20.d odd 2 1
9075.2.a.b 1 1.a even 1 1 trivial
9075.2.a.t 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9075))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{7} - 1$$ T7 - 1 $$T_{13} + 2$$ T13 + 2 $$T_{17} - 4$$ T17 - 4 $$T_{19} - 3$$ T19 - 3 $$T_{23} + 2$$ T23 + 2 $$T_{37} + 3$$ T37 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T$$
$13$ $$T + 2$$
$17$ $$T - 4$$
$19$ $$T - 3$$
$23$ $$T + 2$$
$29$ $$T + 6$$
$31$ $$T + 5$$
$37$ $$T + 3$$
$41$ $$T - 2$$
$43$ $$T - 12$$
$47$ $$T + 2$$
$53$ $$T + 6$$
$59$ $$T + 10$$
$61$ $$T + 3$$
$67$ $$T - 1$$
$71$ $$T$$
$73$ $$T + 11$$
$79$ $$T + 11$$
$83$ $$T - 6$$
$89$ $$T - 12$$
$97$ $$T + 5$$