# Properties

 Label 975.1.u.a Level $975$ Weight $1$ Character orbit 975.u Analytic conductor $0.487$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -3 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,1,Mod(632,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("975.632");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 975.u (of order $$4$$, degree $$2$$, 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: $$0.486588387317$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.164775.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - \zeta_{8}^{3} q^{3} - q^{4} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{7} - \zeta_{8}^{2} q^{9} +O(q^{10})$$ q - z^3 * q^3 - q^4 + (-z^3 + z) * q^7 - z^2 * q^9 $$q - \zeta_{8}^{3} q^{3} - q^{4} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{7} - \zeta_{8}^{2} q^{9} + \zeta_{8}^{3} q^{12} + \zeta_{8}^{3} q^{13} + q^{16} + ( - \zeta_{8}^{2} + 1) q^{19} + ( - \zeta_{8}^{2} + 1) q^{21} - \zeta_{8} q^{27} + (\zeta_{8}^{3} - \zeta_{8}) q^{28} + ( - \zeta_{8}^{2} - 1) q^{31} + \zeta_{8}^{2} q^{36} + (\zeta_{8}^{3} - \zeta_{8}) q^{37} + \zeta_{8}^{2} q^{39} + \zeta_{8} q^{43} - \zeta_{8}^{3} q^{48} + q^{49} - \zeta_{8}^{3} q^{52} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{57} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{63} - q^{64} + (\zeta_{8}^{3} + \zeta_{8}) q^{67} + (\zeta_{8}^{3} + \zeta_{8}) q^{73} + (\zeta_{8}^{2} - 1) q^{76} - q^{81} + (\zeta_{8}^{2} - 1) q^{84} + (\zeta_{8}^{2} - 1) q^{91} + (\zeta_{8}^{3} - \zeta_{8}) q^{93} + (\zeta_{8}^{3} + \zeta_{8}) q^{97} +O(q^{100})$$ q - z^3 * q^3 - q^4 + (-z^3 + z) * q^7 - z^2 * q^9 + z^3 * q^12 + z^3 * q^13 + q^16 + (-z^2 + 1) * q^19 + (-z^2 + 1) * q^21 - z * q^27 + (z^3 - z) * q^28 + (-z^2 - 1) * q^31 + z^2 * q^36 + (z^3 - z) * q^37 + z^2 * q^39 + z * q^43 - z^3 * q^48 + q^49 - z^3 * q^52 + (-z^3 - z) * q^57 + (-z^3 - z) * q^63 - q^64 + (z^3 + z) * q^67 + (z^3 + z) * q^73 + (z^2 - 1) * q^76 - q^81 + (z^2 - 1) * q^84 + (z^2 - 1) * q^91 + (z^3 - z) * q^93 + (z^3 + z) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4}+O(q^{10})$$ 4 * q - 4 * q^4 $$4 q - 4 q^{4} + 4 q^{16} + 4 q^{19} + 4 q^{21} - 4 q^{31} + 4 q^{49} - 4 q^{64} - 4 q^{76} - 4 q^{81} - 4 q^{84} - 4 q^{91}+O(q^{100})$$ 4 * q - 4 * q^4 + 4 * q^16 + 4 * q^19 + 4 * q^21 - 4 * q^31 + 4 * q^49 - 4 * q^64 - 4 * q^76 - 4 * q^81 - 4 * q^84 - 4 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$326$$ $$352$$ $$\chi(n)$$ $$\zeta_{8}^{2}$$ $$-1$$ $$\zeta_{8}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
632.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
0 −0.707107 + 0.707107i −1.00000 0 0 −1.41421 0 1.00000i 0
632.2 0 0.707107 0.707107i −1.00000 0 0 1.41421 0 1.00000i 0
668.1 0 −0.707107 0.707107i −1.00000 0 0 −1.41421 0 1.00000i 0
668.2 0 0.707107 + 0.707107i −1.00000 0 0 1.41421 0 1.00000i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
15.d odd 2 1 inner
65.f even 4 1 inner
65.k even 4 1 inner
195.j odd 4 1 inner
195.u odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.1.u.a yes 4
3.b odd 2 1 CM 975.1.u.a yes 4
5.b even 2 1 inner 975.1.u.a yes 4
5.c odd 4 2 975.1.j.a 4
13.d odd 4 1 975.1.j.a 4
15.d odd 2 1 inner 975.1.u.a yes 4
15.e even 4 2 975.1.j.a 4
39.f even 4 1 975.1.j.a 4
65.f even 4 1 inner 975.1.u.a yes 4
65.g odd 4 1 975.1.j.a 4
65.k even 4 1 inner 975.1.u.a yes 4
195.j odd 4 1 inner 975.1.u.a yes 4
195.n even 4 1 975.1.j.a 4
195.u odd 4 1 inner 975.1.u.a yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.1.j.a 4 5.c odd 4 2
975.1.j.a 4 13.d odd 4 1
975.1.j.a 4 15.e even 4 2
975.1.j.a 4 39.f even 4 1
975.1.j.a 4 65.g odd 4 1
975.1.j.a 4 195.n even 4 1
975.1.u.a yes 4 1.a even 1 1 trivial
975.1.u.a yes 4 3.b odd 2 1 CM
975.1.u.a yes 4 5.b even 2 1 inner
975.1.u.a yes 4 15.d odd 2 1 inner
975.1.u.a yes 4 65.f even 4 1 inner
975.1.u.a yes 4 65.k even 4 1 inner
975.1.u.a yes 4 195.j odd 4 1 inner
975.1.u.a yes 4 195.u odd 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(975, [\chi])$$.

## Hecke characteristic polynomials

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