# Properties

 Label 2475.2.a.h Level $2475$ Weight $2$ Character orbit 2475.a Self dual yes Analytic conductor $19.763$ 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] = [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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{4} - 3 q^{7} - 3 q^{8}+O(q^{10})$$ q + q^2 - q^4 - 3 * q^7 - 3 * q^8 $$q + q^{2} - q^{4} - 3 q^{7} - 3 q^{8} - q^{11} + 2 q^{13} - 3 q^{14} - q^{16} + 3 q^{17} - q^{19} - q^{22} + q^{23} + 2 q^{26} + 3 q^{28} - 6 q^{29} + 4 q^{31} + 5 q^{32} + 3 q^{34} + q^{37} - q^{38} + 5 q^{41} + 4 q^{43} + q^{44} + q^{46} + 3 q^{47} + 2 q^{49} - 2 q^{52} + 10 q^{53} + 9 q^{56} - 6 q^{58} - 11 q^{59} + 14 q^{61} + 4 q^{62} + 7 q^{64} + 2 q^{67} - 3 q^{68} + 5 q^{71} + 2 q^{73} + q^{74} + q^{76} + 3 q^{77} + 5 q^{79} + 5 q^{82} + 8 q^{83} + 4 q^{86} + 3 q^{88} + 10 q^{89} - 6 q^{91} - q^{92} + 3 q^{94} - 17 q^{97} + 2 q^{98}+O(q^{100})$$ q + q^2 - q^4 - 3 * q^7 - 3 * q^8 - q^11 + 2 * q^13 - 3 * q^14 - q^16 + 3 * q^17 - q^19 - q^22 + q^23 + 2 * q^26 + 3 * q^28 - 6 * q^29 + 4 * q^31 + 5 * q^32 + 3 * q^34 + q^37 - q^38 + 5 * q^41 + 4 * q^43 + q^44 + q^46 + 3 * q^47 + 2 * q^49 - 2 * q^52 + 10 * q^53 + 9 * q^56 - 6 * q^58 - 11 * q^59 + 14 * q^61 + 4 * q^62 + 7 * q^64 + 2 * q^67 - 3 * q^68 + 5 * q^71 + 2 * q^73 + q^74 + q^76 + 3 * q^77 + 5 * q^79 + 5 * q^82 + 8 * q^83 + 4 * q^86 + 3 * q^88 + 10 * q^89 - 6 * q^91 - q^92 + 3 * q^94 - 17 * q^97 + 2 * 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 0 −1.00000 0 0 −3.00000 −3.00000 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.h yes 1
3.b odd 2 1 2475.2.a.b 1
5.b even 2 1 2475.2.a.d yes 1
5.c odd 4 2 2475.2.c.c 2
15.d odd 2 1 2475.2.a.k yes 1
15.e even 4 2 2475.2.c.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2475.2.a.b 1 3.b odd 2 1
2475.2.a.d yes 1 5.b even 2 1
2475.2.a.h yes 1 1.a even 1 1 trivial
2475.2.a.k yes 1 15.d odd 2 1
2475.2.c.c 2 5.c odd 4 2
2475.2.c.e 2 15.e even 4 2

## Hecke kernels

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

 $$T_{2} - 1$$ T2 - 1 $$T_{7} + 3$$ T7 + 3 $$T_{29} + 6$$ T29 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 3$$
$11$ $$T + 1$$
$13$ $$T - 2$$
$17$ $$T - 3$$
$19$ $$T + 1$$
$23$ $$T - 1$$
$29$ $$T + 6$$
$31$ $$T - 4$$
$37$ $$T - 1$$
$41$ $$T - 5$$
$43$ $$T - 4$$
$47$ $$T - 3$$
$53$ $$T - 10$$
$59$ $$T + 11$$
$61$ $$T - 14$$
$67$ $$T - 2$$
$71$ $$T - 5$$
$73$ $$T - 2$$
$79$ $$T - 5$$
$83$ $$T - 8$$
$89$ $$T - 10$$
$97$ $$T + 17$$