# Properties

 Label 575.3.d.a Level $575$ Weight $3$ Character orbit 575.d Analytic conductor $15.668$ Analytic rank $0$ Dimension $2$ CM discriminant -115 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,3,Mod(551,575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("575.551");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$575 = 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 575.d (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$15.6676152007$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 115) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 4 q^{4} + 9 i q^{7} - 9 q^{9}+O(q^{10})$$ q - 4 * q^4 + 9*i * q^7 - 9 * q^9 $$q - 4 q^{4} + 9 i q^{7} - 9 q^{9} + 16 q^{16} - 11 i q^{17} - 23 i q^{23} - 36 i q^{28} + 57 q^{29} - 53 q^{31} + 36 q^{36} - 51 i q^{37} - 33 q^{41} - 6 i q^{43} - 32 q^{49} - 101 i q^{53} - 3 q^{59} - 81 i q^{63} - 64 q^{64} - 111 i q^{67} + 44 i q^{68} + 27 q^{71} + 81 q^{81} - 41 i q^{83} + 92 i q^{92} + 174 i q^{97} +O(q^{100})$$ q - 4 * q^4 + 9*i * q^7 - 9 * q^9 + 16 * q^16 - 11*i * q^17 - 23*i * q^23 - 36*i * q^28 + 57 * q^29 - 53 * q^31 + 36 * q^36 - 51*i * q^37 - 33 * q^41 - 6*i * q^43 - 32 * q^49 - 101*i * q^53 - 3 * q^59 - 81*i * q^63 - 64 * q^64 - 111*i * q^67 + 44*i * q^68 + 27 * q^71 + 81 * q^81 - 41*i * q^83 + 92*i * q^92 + 174*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4} - 18 q^{9}+O(q^{10})$$ 2 * q - 8 * q^4 - 18 * q^9 $$2 q - 8 q^{4} - 18 q^{9} + 32 q^{16} + 114 q^{29} - 106 q^{31} + 72 q^{36} - 66 q^{41} - 64 q^{49} - 6 q^{59} - 128 q^{64} + 54 q^{71} + 162 q^{81}+O(q^{100})$$ 2 * q - 8 * q^4 - 18 * q^9 + 32 * q^16 + 114 * q^29 - 106 * q^31 + 72 * q^36 - 66 * q^41 - 64 * q^49 - 6 * q^59 - 128 * q^64 + 54 * q^71 + 162 * q^81

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/575\mathbb{Z}\right)^\times$$.

 $$n$$ $$51$$ $$277$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
551.1
 − 1.00000i 1.00000i
0 0 −4.00000 0 0 9.00000i 0 −9.00000 0
551.2 0 0 −4.00000 0 0 9.00000i 0 −9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
115.c odd 2 1 CM by $$\Q(\sqrt{-115})$$
5.b even 2 1 inner
23.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.3.d.a 2
5.b even 2 1 inner 575.3.d.a 2
5.c odd 4 1 115.3.c.a 1
5.c odd 4 1 115.3.c.b yes 1
23.b odd 2 1 inner 575.3.d.a 2
115.c odd 2 1 CM 575.3.d.a 2
115.e even 4 1 115.3.c.a 1
115.e even 4 1 115.3.c.b yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.3.c.a 1 5.c odd 4 1
115.3.c.a 1 115.e even 4 1
115.3.c.b yes 1 5.c odd 4 1
115.3.c.b yes 1 115.e even 4 1
575.3.d.a 2 1.a even 1 1 trivial
575.3.d.a 2 5.b even 2 1 inner
575.3.d.a 2 23.b odd 2 1 inner
575.3.d.a 2 115.c odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{3}^{\mathrm{new}}(575, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 81$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 121$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 529$$
$29$ $$(T - 57)^{2}$$
$31$ $$(T + 53)^{2}$$
$37$ $$T^{2} + 2601$$
$41$ $$(T + 33)^{2}$$
$43$ $$T^{2} + 36$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 10201$$
$59$ $$(T + 3)^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 12321$$
$71$ $$(T - 27)^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 1681$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 30276$$