# Properties

 Label 975.1.e.a Level $975$ Weight $1$ Character orbit 975.e Analytic conductor $0.487$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -3, -39, 13 Inner twists $8$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,1,Mod(974,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("975.974");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 975.e (of order $$2$$, degree $$1$$, not 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: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{13})$$ Artin image: $D_4:C_2$ Artin field: Galois closure of 8.0.213890625.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - i q^{3} + q^{4} - q^{9} +O(q^{10})$$ q - z * q^3 + q^4 - q^9 $$q - i q^{3} + q^{4} - q^{9} - i q^{12} - i q^{13} + q^{16} + i q^{27} - q^{36} - q^{39} + i q^{43} - i q^{48} - q^{49} - i q^{52} - q^{61} + q^{64} + q^{79} + q^{81} +O(q^{100})$$ q - z * q^3 + q^4 - q^9 - z * q^12 - z * q^13 + q^16 + z * q^27 - q^36 - q^39 + z * q^43 - z * q^48 - q^49 - z * q^52 - q^61 + q^64 + q^79 + q^81 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 - 2 * q^9 $$2 q + 2 q^{4} - 2 q^{9} + 2 q^{16} - 2 q^{36} - 2 q^{39} - 2 q^{49} - 4 q^{61} + 2 q^{64} + 4 q^{79} + 2 q^{81}+O(q^{100})$$ 2 * q + 2 * q^4 - 2 * q^9 + 2 * q^16 - 2 * q^36 - 2 * q^39 - 2 * q^49 - 4 * q^61 + 2 * q^64 + 4 * q^79 + 2 * q^81

## Character values

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

 $$n$$ $$301$$ $$326$$ $$352$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
974.1
 1.00000i − 1.00000i
0 1.00000i 1.00000 0 0 0 0 −1.00000 0
974.2 0 1.00000i 1.00000 0 0 0 0 −1.00000 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
13.b even 2 1 RM by $$\Q(\sqrt{13})$$
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$
5.b even 2 1 inner
15.d odd 2 1 inner
65.d even 2 1 inner
195.e odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.1.e.a 2
3.b odd 2 1 CM 975.1.e.a 2
5.b even 2 1 inner 975.1.e.a 2
5.c odd 4 1 39.1.d.a 1
5.c odd 4 1 975.1.g.a 1
13.b even 2 1 RM 975.1.e.a 2
15.d odd 2 1 inner 975.1.e.a 2
15.e even 4 1 39.1.d.a 1
15.e even 4 1 975.1.g.a 1
20.e even 4 1 624.1.l.a 1
35.f even 4 1 1911.1.h.a 1
35.k even 12 2 1911.1.w.a 2
35.l odd 12 2 1911.1.w.b 2
39.d odd 2 1 CM 975.1.e.a 2
40.i odd 4 1 2496.1.l.b 1
40.k even 4 1 2496.1.l.a 1
45.k odd 12 2 1053.1.n.b 2
45.l even 12 2 1053.1.n.b 2
60.l odd 4 1 624.1.l.a 1
65.d even 2 1 inner 975.1.e.a 2
65.f even 4 1 507.1.c.a 1
65.h odd 4 1 39.1.d.a 1
65.h odd 4 1 975.1.g.a 1
65.k even 4 1 507.1.c.a 1
65.o even 12 2 507.1.i.a 2
65.q odd 12 2 507.1.h.a 2
65.r odd 12 2 507.1.h.a 2
65.t even 12 2 507.1.i.a 2
105.k odd 4 1 1911.1.h.a 1
105.w odd 12 2 1911.1.w.a 2
105.x even 12 2 1911.1.w.b 2
120.q odd 4 1 2496.1.l.a 1
120.w even 4 1 2496.1.l.b 1
195.e odd 2 1 inner 975.1.e.a 2
195.j odd 4 1 507.1.c.a 1
195.s even 4 1 39.1.d.a 1
195.s even 4 1 975.1.g.a 1
195.u odd 4 1 507.1.c.a 1
195.bc odd 12 2 507.1.i.a 2
195.bf even 12 2 507.1.h.a 2
195.bl even 12 2 507.1.h.a 2
195.bn odd 12 2 507.1.i.a 2
260.p even 4 1 624.1.l.a 1
455.s even 4 1 1911.1.h.a 1
455.cv odd 12 2 1911.1.w.b 2
455.df even 12 2 1911.1.w.a 2
520.bc even 4 1 2496.1.l.a 1
520.bg odd 4 1 2496.1.l.b 1
585.cs even 12 2 1053.1.n.b 2
585.dm odd 12 2 1053.1.n.b 2
780.w odd 4 1 624.1.l.a 1
1365.bl odd 4 1 1911.1.h.a 1
1365.fn even 12 2 1911.1.w.b 2
1365.fr odd 12 2 1911.1.w.a 2
1560.bq even 4 1 2496.1.l.b 1
1560.cs odd 4 1 2496.1.l.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.1.d.a 1 5.c odd 4 1
39.1.d.a 1 15.e even 4 1
39.1.d.a 1 65.h odd 4 1
39.1.d.a 1 195.s even 4 1
507.1.c.a 1 65.f even 4 1
507.1.c.a 1 65.k even 4 1
507.1.c.a 1 195.j odd 4 1
507.1.c.a 1 195.u odd 4 1
507.1.h.a 2 65.q odd 12 2
507.1.h.a 2 65.r odd 12 2
507.1.h.a 2 195.bf even 12 2
507.1.h.a 2 195.bl even 12 2
507.1.i.a 2 65.o even 12 2
507.1.i.a 2 65.t even 12 2
507.1.i.a 2 195.bc odd 12 2
507.1.i.a 2 195.bn odd 12 2
624.1.l.a 1 20.e even 4 1
624.1.l.a 1 60.l odd 4 1
624.1.l.a 1 260.p even 4 1
624.1.l.a 1 780.w odd 4 1
975.1.e.a 2 1.a even 1 1 trivial
975.1.e.a 2 3.b odd 2 1 CM
975.1.e.a 2 5.b even 2 1 inner
975.1.e.a 2 13.b even 2 1 RM
975.1.e.a 2 15.d odd 2 1 inner
975.1.e.a 2 39.d odd 2 1 CM
975.1.e.a 2 65.d even 2 1 inner
975.1.e.a 2 195.e odd 2 1 inner
975.1.g.a 1 5.c odd 4 1
975.1.g.a 1 15.e even 4 1
975.1.g.a 1 65.h odd 4 1
975.1.g.a 1 195.s even 4 1
1053.1.n.b 2 45.k odd 12 2
1053.1.n.b 2 45.l even 12 2
1053.1.n.b 2 585.cs even 12 2
1053.1.n.b 2 585.dm odd 12 2
1911.1.h.a 1 35.f even 4 1
1911.1.h.a 1 105.k odd 4 1
1911.1.h.a 1 455.s even 4 1
1911.1.h.a 1 1365.bl odd 4 1
1911.1.w.a 2 35.k even 12 2
1911.1.w.a 2 105.w odd 12 2
1911.1.w.a 2 455.df even 12 2
1911.1.w.a 2 1365.fr odd 12 2
1911.1.w.b 2 35.l odd 12 2
1911.1.w.b 2 105.x even 12 2
1911.1.w.b 2 455.cv odd 12 2
1911.1.w.b 2 1365.fn even 12 2
2496.1.l.a 1 40.k even 4 1
2496.1.l.a 1 120.q odd 4 1
2496.1.l.a 1 520.bc even 4 1
2496.1.l.a 1 1560.cs odd 4 1
2496.1.l.b 1 40.i odd 4 1
2496.1.l.b 1 120.w even 4 1
2496.1.l.b 1 520.bg odd 4 1
2496.1.l.b 1 1560.bq even 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

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