# Properties

 Label 2475.2.a.f Level $2475$ Weight $2$ Character orbit 2475.a Self dual yes Analytic conductor $19.763$ 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] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.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: $$19.7629745003$$ Analytic rank: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{4} + q^{7}+O(q^{10})$$ q - 2 * q^4 + q^7 $$q - 2 q^{4} + q^{7} + q^{11} + q^{13} + 4 q^{16} - 6 q^{17} - 7 q^{19} + 6 q^{23} - 2 q^{28} + 6 q^{29} - 7 q^{31} - 2 q^{37} + 6 q^{41} + q^{43} - 2 q^{44} - 6 q^{49} - 2 q^{52} - 6 q^{53} + 5 q^{61} - 8 q^{64} - 5 q^{67} + 12 q^{68} + 12 q^{71} - 14 q^{73} + 14 q^{76} + q^{77} - 4 q^{79} - 6 q^{83} - 6 q^{89} + q^{91} - 12 q^{92} - 17 q^{97}+O(q^{100})$$ q - 2 * q^4 + q^7 + q^11 + q^13 + 4 * q^16 - 6 * q^17 - 7 * q^19 + 6 * q^23 - 2 * q^28 + 6 * q^29 - 7 * q^31 - 2 * q^37 + 6 * q^41 + q^43 - 2 * q^44 - 6 * q^49 - 2 * q^52 - 6 * q^53 + 5 * q^61 - 8 * q^64 - 5 * q^67 + 12 * q^68 + 12 * q^71 - 14 * q^73 + 14 * q^76 + q^77 - 4 * q^79 - 6 * q^83 - 6 * q^89 + q^91 - 12 * q^92 - 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 0 −2.00000 0 0 1.00000 0 0 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 2475.2.a.f 1
3.b odd 2 1 825.2.a.b 1
5.b even 2 1 2475.2.a.e 1
5.c odd 4 2 2475.2.c.h 2
15.d odd 2 1 825.2.a.c yes 1
15.e even 4 2 825.2.c.b 2
33.d even 2 1 9075.2.a.i 1
165.d even 2 1 9075.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.a.b 1 3.b odd 2 1
825.2.a.c yes 1 15.d odd 2 1
825.2.c.b 2 15.e even 4 2
2475.2.a.e 1 5.b even 2 1
2475.2.a.f 1 1.a even 1 1 trivial
2475.2.c.h 2 5.c odd 4 2
9075.2.a.i 1 33.d even 2 1
9075.2.a.l 1 165.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2475))$$:

 $$T_{2}$$ T2 $$T_{7} - 1$$ T7 - 1 $$T_{29} - 6$$ T29 - 6

## Hecke characteristic polynomials

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