# Properties

 Label 1575.4.a.f Level $1575$ Weight $4$ Character orbit 1575.a Self dual yes Analytic conductor $92.928$ 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] = [1575,4,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, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$92.9280082590$$ Analytic rank: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - 8 q^{4} - 7 q^{7}+O(q^{10})$$ q - 8 * q^4 - 7 * q^7 $$q - 8 q^{4} - 7 q^{7} - 42 q^{11} - 20 q^{13} + 64 q^{16} + 66 q^{17} + 38 q^{19} + 12 q^{23} + 56 q^{28} + 258 q^{29} + 146 q^{31} - 434 q^{37} + 282 q^{41} - 20 q^{43} + 336 q^{44} - 72 q^{47} + 49 q^{49} + 160 q^{52} + 336 q^{53} + 360 q^{59} - 682 q^{61} - 512 q^{64} - 812 q^{67} - 528 q^{68} - 810 q^{71} + 124 q^{73} - 304 q^{76} + 294 q^{77} + 1136 q^{79} + 156 q^{83} + 1038 q^{89} + 140 q^{91} - 96 q^{92} - 1208 q^{97}+O(q^{100})$$ q - 8 * q^4 - 7 * q^7 - 42 * q^11 - 20 * q^13 + 64 * q^16 + 66 * q^17 + 38 * q^19 + 12 * q^23 + 56 * q^28 + 258 * q^29 + 146 * q^31 - 434 * q^37 + 282 * q^41 - 20 * q^43 + 336 * q^44 - 72 * q^47 + 49 * q^49 + 160 * q^52 + 336 * q^53 + 360 * q^59 - 682 * q^61 - 512 * q^64 - 812 * q^67 - 528 * q^68 - 810 * q^71 + 124 * q^73 - 304 * q^76 + 294 * q^77 + 1136 * q^79 + 156 * q^83 + 1038 * q^89 + 140 * q^91 - 96 * q^92 - 1208 * 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 −8.00000 0 0 −7.00000 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 1575.4.a.f 1
3.b odd 2 1 525.4.a.e 1
5.b even 2 1 315.4.a.d 1
15.d odd 2 1 105.4.a.a 1
15.e even 4 2 525.4.d.f 2
35.c odd 2 1 2205.4.a.o 1
60.h even 2 1 1680.4.a.s 1
105.g even 2 1 735.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.a 1 15.d odd 2 1
315.4.a.d 1 5.b even 2 1
525.4.a.e 1 3.b odd 2 1
525.4.d.f 2 15.e even 4 2
735.4.a.c 1 105.g even 2 1
1575.4.a.f 1 1.a even 1 1 trivial
1680.4.a.s 1 60.h even 2 1
2205.4.a.o 1 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1575))$$:

 $$T_{2}$$ T2 $$T_{11} + 42$$ T11 + 42 $$T_{13} + 20$$ T13 + 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 7$$
$11$ $$T + 42$$
$13$ $$T + 20$$
$17$ $$T - 66$$
$19$ $$T - 38$$
$23$ $$T - 12$$
$29$ $$T - 258$$
$31$ $$T - 146$$
$37$ $$T + 434$$
$41$ $$T - 282$$
$43$ $$T + 20$$
$47$ $$T + 72$$
$53$ $$T - 336$$
$59$ $$T - 360$$
$61$ $$T + 682$$
$67$ $$T + 812$$
$71$ $$T + 810$$
$73$ $$T - 124$$
$79$ $$T - 1136$$
$83$ $$T - 156$$
$89$ $$T - 1038$$
$97$ $$T + 1208$$