# Properties

 Label 1575.2.a.j Level $1575$ Weight $2$ Character orbit 1575.a Self dual yes Analytic conductor $12.576$ 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] = [1575,2,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, 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("1575.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1575.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: $$12.5764383184$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 315) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{4} + q^{7} - 3 q^{8}+O(q^{10})$$ q + q^2 - q^4 + q^7 - 3 * q^8 $$q + q^{2} - q^{4} + q^{7} - 3 q^{8} - 4 q^{13} + q^{14} - q^{16} + 2 q^{17} - 4 q^{26} - q^{28} - 8 q^{29} - 4 q^{31} + 5 q^{32} + 2 q^{34} - 8 q^{37} - 4 q^{41} - 8 q^{43} - 12 q^{47} + q^{49} + 4 q^{52} + 6 q^{53} - 3 q^{56} - 8 q^{58} - 8 q^{59} + 10 q^{61} - 4 q^{62} + 7 q^{64} - 8 q^{67} - 2 q^{68} + 16 q^{71} - 12 q^{73} - 8 q^{74} - 8 q^{79} - 4 q^{82} + 16 q^{83} - 8 q^{86} + 12 q^{89} - 4 q^{91} - 12 q^{94} - 4 q^{97} + q^{98}+O(q^{100})$$ q + q^2 - q^4 + q^7 - 3 * q^8 - 4 * q^13 + q^14 - q^16 + 2 * q^17 - 4 * q^26 - q^28 - 8 * q^29 - 4 * q^31 + 5 * q^32 + 2 * q^34 - 8 * q^37 - 4 * q^41 - 8 * q^43 - 12 * q^47 + q^49 + 4 * q^52 + 6 * q^53 - 3 * q^56 - 8 * q^58 - 8 * q^59 + 10 * q^61 - 4 * q^62 + 7 * q^64 - 8 * q^67 - 2 * q^68 + 16 * q^71 - 12 * q^73 - 8 * q^74 - 8 * q^79 - 4 * q^82 + 16 * q^83 - 8 * q^86 + 12 * q^89 - 4 * q^91 - 12 * q^94 - 4 * q^97 + 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 1.00000 −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
$$3$$ $$+1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## 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 1575.2.a.j 1
3.b odd 2 1 1575.2.a.d 1
5.b even 2 1 1575.2.a.b 1
5.c odd 4 2 315.2.d.b 2
15.d odd 2 1 1575.2.a.g 1
15.e even 4 2 315.2.d.d yes 2
20.e even 4 2 5040.2.t.c 2
35.f even 4 2 2205.2.d.g 2
60.l odd 4 2 5040.2.t.r 2
105.k odd 4 2 2205.2.d.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.d.b 2 5.c odd 4 2
315.2.d.d yes 2 15.e even 4 2
1575.2.a.b 1 5.b even 2 1
1575.2.a.d 1 3.b odd 2 1
1575.2.a.g 1 15.d odd 2 1
1575.2.a.j 1 1.a even 1 1 trivial
2205.2.d.c 2 105.k odd 4 2
2205.2.d.g 2 35.f even 4 2
5040.2.t.c 2 20.e even 4 2
5040.2.t.r 2 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1575))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{11}$$ T11 $$T_{13} + 4$$ T13 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T$$
$13$ $$T + 4$$
$17$ $$T - 2$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T + 8$$
$31$ $$T + 4$$
$37$ $$T + 8$$
$41$ $$T + 4$$
$43$ $$T + 8$$
$47$ $$T + 12$$
$53$ $$T - 6$$
$59$ $$T + 8$$
$61$ $$T - 10$$
$67$ $$T + 8$$
$71$ $$T - 16$$
$73$ $$T + 12$$
$79$ $$T + 8$$
$83$ $$T - 16$$
$89$ $$T - 12$$
$97$ $$T + 4$$