# Properties

 Label 1575.2.a.f Level $1575$ Weight $2$ Character orbit 1575.a Self dual yes Analytic conductor $12.576$ 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] = [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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) 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 $$q - 2 q^{4} - q^{7} + 3 q^{11} - 5 q^{13} + 4 q^{16} + 3 q^{17} + 2 q^{19} - 6 q^{23} + 2 q^{28} - 3 q^{29} - 4 q^{31} - 2 q^{37} + 12 q^{41} + 10 q^{43} - 6 q^{44} + 9 q^{47} + q^{49} + 10 q^{52} + 12 q^{53} + 8 q^{61} - 8 q^{64} + 4 q^{67} - 6 q^{68} - 2 q^{73} - 4 q^{76} - 3 q^{77} - q^{79} + 12 q^{83} + 12 q^{89} + 5 q^{91} + 12 q^{92} + q^{97}+O(q^{100})$$ q - 2 * q^4 - q^7 + 3 * q^11 - 5 * q^13 + 4 * q^16 + 3 * q^17 + 2 * q^19 - 6 * q^23 + 2 * q^28 - 3 * q^29 - 4 * q^31 - 2 * q^37 + 12 * q^41 + 10 * q^43 - 6 * q^44 + 9 * q^47 + q^49 + 10 * q^52 + 12 * q^53 + 8 * q^61 - 8 * q^64 + 4 * q^67 - 6 * q^68 - 2 * q^73 - 4 * q^76 - 3 * q^77 - q^79 + 12 * q^83 + 12 * q^89 + 5 * q^91 + 12 * q^92 + 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
$$3$$ $$-1$$
$$5$$ $$+1$$
$$7$$ $$+1$$

## 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.f 1
3.b odd 2 1 175.2.a.b 1
5.b even 2 1 315.2.a.b 1
5.c odd 4 2 1575.2.d.c 2
12.b even 2 1 2800.2.a.z 1
15.d odd 2 1 35.2.a.a 1
15.e even 4 2 175.2.b.a 2
20.d odd 2 1 5040.2.a.v 1
21.c even 2 1 1225.2.a.e 1
35.c odd 2 1 2205.2.a.e 1
60.h even 2 1 560.2.a.b 1
60.l odd 4 2 2800.2.g.l 2
105.g even 2 1 245.2.a.c 1
105.k odd 4 2 1225.2.b.d 2
105.o odd 6 2 245.2.e.a 2
105.p even 6 2 245.2.e.b 2
120.i odd 2 1 2240.2.a.k 1
120.m even 2 1 2240.2.a.u 1
165.d even 2 1 4235.2.a.c 1
195.e odd 2 1 5915.2.a.f 1
420.o odd 2 1 3920.2.a.ba 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.a 1 15.d odd 2 1
175.2.a.b 1 3.b odd 2 1
175.2.b.a 2 15.e even 4 2
245.2.a.c 1 105.g even 2 1
245.2.e.a 2 105.o odd 6 2
245.2.e.b 2 105.p even 6 2
315.2.a.b 1 5.b even 2 1
560.2.a.b 1 60.h even 2 1
1225.2.a.e 1 21.c even 2 1
1225.2.b.d 2 105.k odd 4 2
1575.2.a.f 1 1.a even 1 1 trivial
1575.2.d.c 2 5.c odd 4 2
2205.2.a.e 1 35.c odd 2 1
2240.2.a.k 1 120.i odd 2 1
2240.2.a.u 1 120.m even 2 1
2800.2.a.z 1 12.b even 2 1
2800.2.g.l 2 60.l odd 4 2
3920.2.a.ba 1 420.o odd 2 1
4235.2.a.c 1 165.d even 2 1
5040.2.a.v 1 20.d odd 2 1
5915.2.a.f 1 195.e odd 2 1

## Hecke kernels

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

 $$T_{2}$$ T2 $$T_{11} - 3$$ T11 - 3 $$T_{13} + 5$$ T13 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T - 3$$
$13$ $$T + 5$$
$17$ $$T - 3$$
$19$ $$T - 2$$
$23$ $$T + 6$$
$29$ $$T + 3$$
$31$ $$T + 4$$
$37$ $$T + 2$$
$41$ $$T - 12$$
$43$ $$T - 10$$
$47$ $$T - 9$$
$53$ $$T - 12$$
$59$ $$T$$
$61$ $$T - 8$$
$67$ $$T - 4$$
$71$ $$T$$
$73$ $$T + 2$$
$79$ $$T + 1$$
$83$ $$T - 12$$
$89$ $$T - 12$$
$97$ $$T - 1$$