# Properties

 Label 1575.2.a.h 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 105) 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 $$q + q^{2} - q^{4} - q^{7} - 3 q^{8} + 6 q^{13} - q^{14} - q^{16} + 2 q^{17} - 8 q^{19} + 8 q^{23} + 6 q^{26} + q^{28} + 2 q^{29} + 4 q^{31} + 5 q^{32} + 2 q^{34} + 2 q^{37} - 8 q^{38} + 6 q^{41} - 4 q^{43} + 8 q^{46} + 8 q^{47} + q^{49} - 6 q^{52} + 10 q^{53} + 3 q^{56} + 2 q^{58} - 4 q^{59} - 2 q^{61} + 4 q^{62} + 7 q^{64} - 4 q^{67} - 2 q^{68} + 12 q^{71} + 2 q^{73} + 2 q^{74} + 8 q^{76} + 8 q^{79} + 6 q^{82} - 4 q^{83} - 4 q^{86} + 6 q^{89} - 6 q^{91} - 8 q^{92} + 8 q^{94} + 18 q^{97} + q^{98}+O(q^{100})$$ q + q^2 - q^4 - q^7 - 3 * q^8 + 6 * q^13 - q^14 - q^16 + 2 * q^17 - 8 * q^19 + 8 * q^23 + 6 * q^26 + q^28 + 2 * q^29 + 4 * q^31 + 5 * q^32 + 2 * q^34 + 2 * q^37 - 8 * q^38 + 6 * q^41 - 4 * q^43 + 8 * q^46 + 8 * q^47 + q^49 - 6 * q^52 + 10 * q^53 + 3 * q^56 + 2 * q^58 - 4 * q^59 - 2 * q^61 + 4 * q^62 + 7 * q^64 - 4 * q^67 - 2 * q^68 + 12 * q^71 + 2 * q^73 + 2 * q^74 + 8 * q^76 + 8 * q^79 + 6 * q^82 - 4 * q^83 - 4 * q^86 + 6 * q^89 - 6 * q^91 - 8 * q^92 + 8 * q^94 + 18 * 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.h 1
3.b odd 2 1 525.2.a.a 1
5.b even 2 1 315.2.a.a 1
5.c odd 4 2 1575.2.d.b 2
12.b even 2 1 8400.2.a.co 1
15.d odd 2 1 105.2.a.a 1
15.e even 4 2 525.2.d.b 2
20.d odd 2 1 5040.2.a.d 1
21.c even 2 1 3675.2.a.f 1
35.c odd 2 1 2205.2.a.b 1
60.h even 2 1 1680.2.a.f 1
105.g even 2 1 735.2.a.f 1
105.o odd 6 2 735.2.i.a 2
105.p even 6 2 735.2.i.b 2
120.i odd 2 1 6720.2.a.p 1
120.m even 2 1 6720.2.a.bk 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.a 1 15.d odd 2 1
315.2.a.a 1 5.b even 2 1
525.2.a.a 1 3.b odd 2 1
525.2.d.b 2 15.e even 4 2
735.2.a.f 1 105.g even 2 1
735.2.i.a 2 105.o odd 6 2
735.2.i.b 2 105.p even 6 2
1575.2.a.h 1 1.a even 1 1 trivial
1575.2.d.b 2 5.c odd 4 2
1680.2.a.f 1 60.h even 2 1
2205.2.a.b 1 35.c odd 2 1
3675.2.a.f 1 21.c even 2 1
5040.2.a.d 1 20.d odd 2 1
6720.2.a.p 1 120.i odd 2 1
6720.2.a.bk 1 120.m even 2 1
8400.2.a.co 1 12.b even 2 1

## Hecke kernels

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

 $$T_{2} - 1$$ T2 - 1 $$T_{11}$$ T11 $$T_{13} - 6$$ T13 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T$$
$13$ $$T - 6$$
$17$ $$T - 2$$
$19$ $$T + 8$$
$23$ $$T - 8$$
$29$ $$T - 2$$
$31$ $$T - 4$$
$37$ $$T - 2$$
$41$ $$T - 6$$
$43$ $$T + 4$$
$47$ $$T - 8$$
$53$ $$T - 10$$
$59$ $$T + 4$$
$61$ $$T + 2$$
$67$ $$T + 4$$
$71$ $$T - 12$$
$73$ $$T - 2$$
$79$ $$T - 8$$
$83$ $$T + 4$$
$89$ $$T - 6$$
$97$ $$T - 18$$