# Properties

 Label 3751.1.d.b Level $3751$ Weight $1$ Character orbit 3751.d Self dual yes Analytic conductor $1.872$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -31 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3751,1,Mod(1332,3751)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3751, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3751.1332");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3751 = 11^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3751.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: yes Analytic conductor: $$1.87199286239$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 31) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.31.1 Artin image: $D_6$ Artin field: Galois closure of 6.0.1279091.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{5} + q^{7} - q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^5 + q^7 - q^8 + q^9 $$q + q^{2} - q^{5} + q^{7} - q^{8} + q^{9} - q^{10} + q^{14} - q^{16} + q^{18} + q^{19} + q^{31} - q^{35} + q^{38} + q^{40} + q^{41} - q^{45} + 2 q^{47} - q^{56} - q^{59} + q^{62} + q^{63} + q^{64} + 2 q^{67} - q^{70} - q^{71} - q^{72} + q^{80} + q^{81} + q^{82} - q^{90} + 2 q^{94} - q^{95} - q^{97}+O(q^{100})$$ q + q^2 - q^5 + q^7 - q^8 + q^9 - q^10 + q^14 - q^16 + q^18 + q^19 + q^31 - q^35 + q^38 + q^40 + q^41 - q^45 + 2 * q^47 - q^56 - q^59 + q^62 + q^63 + q^64 + 2 * q^67 - q^70 - q^71 - q^72 + q^80 + q^81 + q^82 - q^90 + 2 * q^94 - q^95 - q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3751\mathbb{Z}\right)^\times$$.

 $$n$$ $$2421$$ $$2543$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1332.1
 0
1.00000 0 0 −1.00000 0 1.00000 −1.00000 1.00000 −1.00000
 $$n$$: 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
31.b odd 2 1 CM by $$\Q(\sqrt{-31})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3751.1.d.b 1
11.b odd 2 1 31.1.b.a 1
11.c even 5 4 3751.1.t.a 4
11.d odd 10 4 3751.1.t.c 4
31.b odd 2 1 CM 3751.1.d.b 1
33.d even 2 1 279.1.d.b 1
44.c even 2 1 496.1.e.a 1
55.d odd 2 1 775.1.d.b 1
55.e even 4 2 775.1.c.a 2
77.b even 2 1 1519.1.c.a 1
77.h odd 6 2 1519.1.n.b 2
77.i even 6 2 1519.1.n.a 2
88.b odd 2 1 1984.1.e.a 1
88.g even 2 1 1984.1.e.b 1
99.g even 6 2 2511.1.m.a 2
99.h odd 6 2 2511.1.m.e 2
341.b even 2 1 31.1.b.a 1
341.l odd 6 2 961.1.e.a 2
341.m even 6 2 961.1.e.a 2
341.t odd 10 4 3751.1.t.a 4
341.z odd 10 4 961.1.f.a 4
341.ba even 10 4 3751.1.t.c 4
341.bd even 10 4 961.1.f.a 4
341.bu even 30 8 961.1.h.a 8
341.by odd 30 8 961.1.h.a 8
1023.g odd 2 1 279.1.d.b 1
1364.h odd 2 1 496.1.e.a 1
1705.h even 2 1 775.1.d.b 1
1705.m odd 4 2 775.1.c.a 2
2387.d odd 2 1 1519.1.c.a 1
2387.bj odd 6 2 1519.1.n.a 2
2387.bm even 6 2 1519.1.n.b 2
2728.e odd 2 1 1984.1.e.b 1
2728.k even 2 1 1984.1.e.a 1
3069.ba odd 6 2 2511.1.m.a 2
3069.bh even 6 2 2511.1.m.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a 1 11.b odd 2 1
31.1.b.a 1 341.b even 2 1
279.1.d.b 1 33.d even 2 1
279.1.d.b 1 1023.g odd 2 1
496.1.e.a 1 44.c even 2 1
496.1.e.a 1 1364.h odd 2 1
775.1.c.a 2 55.e even 4 2
775.1.c.a 2 1705.m odd 4 2
775.1.d.b 1 55.d odd 2 1
775.1.d.b 1 1705.h even 2 1
961.1.e.a 2 341.l odd 6 2
961.1.e.a 2 341.m even 6 2
961.1.f.a 4 341.z odd 10 4
961.1.f.a 4 341.bd even 10 4
961.1.h.a 8 341.bu even 30 8
961.1.h.a 8 341.by odd 30 8
1519.1.c.a 1 77.b even 2 1
1519.1.c.a 1 2387.d odd 2 1
1519.1.n.a 2 77.i even 6 2
1519.1.n.a 2 2387.bj odd 6 2
1519.1.n.b 2 77.h odd 6 2
1519.1.n.b 2 2387.bm even 6 2
1984.1.e.a 1 88.b odd 2 1
1984.1.e.a 1 2728.k even 2 1
1984.1.e.b 1 88.g even 2 1
1984.1.e.b 1 2728.e odd 2 1
2511.1.m.a 2 99.g even 6 2
2511.1.m.a 2 3069.ba odd 6 2
2511.1.m.e 2 99.h odd 6 2
2511.1.m.e 2 3069.bh even 6 2
3751.1.d.b 1 1.a even 1 1 trivial
3751.1.d.b 1 31.b odd 2 1 CM
3751.1.t.a 4 11.c even 5 4
3751.1.t.a 4 341.t odd 10 4
3751.1.t.c 4 11.d odd 10 4
3751.1.t.c 4 341.ba even 10 4

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 1$$ acting on $$S_{1}^{\mathrm{new}}(3751, [\chi])$$.

## Hecke characteristic polynomials

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