# Properties

 Label 1368.1.i.a Level $1368$ Weight $1$ Character orbit 1368.i Self dual yes Analytic conductor $0.683$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -152 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,1,Mod(37,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("1368.37");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1368 = 2^{3} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1368.i (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: $$0.682720937282$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.152.1 Artin image: $D_6$ Artin field: Galois closure of 6.0.623808.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 - q^7 - q^8 $$q - q^{2} + q^{4} - q^{7} - q^{8} - q^{13} + q^{14} + q^{16} + q^{17} + q^{19} + q^{23} + q^{25} + q^{26} - q^{28} + q^{29} - q^{32} - q^{34} + 2 q^{37} - q^{38} - q^{46} - 2 q^{47} - q^{50} - q^{52} + q^{53} + q^{56} - q^{58} + q^{59} + q^{64} - q^{67} + q^{68} - q^{73} - 2 q^{74} + q^{76} + q^{91} + q^{92} + 2 q^{94}+O(q^{100})$$ q - q^2 + q^4 - q^7 - q^8 - q^13 + q^14 + q^16 + q^17 + q^19 + q^23 + q^25 + q^26 - q^28 + q^29 - q^32 - q^34 + 2 * q^37 - q^38 - q^46 - 2 * q^47 - q^50 - q^52 + q^53 + q^56 - q^58 + q^59 + q^64 - q^67 + q^68 - q^73 - 2 * q^74 + q^76 + q^91 + q^92 + 2 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$$.

 $$n$$ $$343$$ $$685$$ $$1009$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0
−1.00000 0 1.00000 0 0 −1.00000 −1.00000 0 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
152.g odd 2 1 CM by $$\Q(\sqrt{-38})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.1.i.a 1
3.b odd 2 1 152.1.g.b yes 1
8.b even 2 1 1368.1.i.b 1
12.b even 2 1 608.1.g.b 1
15.d odd 2 1 3800.1.o.a 1
15.e even 4 2 3800.1.b.b 2
19.b odd 2 1 1368.1.i.b 1
24.f even 2 1 608.1.g.a 1
24.h odd 2 1 152.1.g.a 1
57.d even 2 1 152.1.g.a 1
57.f even 6 2 2888.1.l.b 2
57.h odd 6 2 2888.1.l.a 2
57.j even 18 6 2888.1.s.b 6
57.l odd 18 6 2888.1.s.a 6
120.i odd 2 1 3800.1.o.b 1
120.w even 4 2 3800.1.b.a 2
152.g odd 2 1 CM 1368.1.i.a 1
228.b odd 2 1 608.1.g.a 1
285.b even 2 1 3800.1.o.b 1
285.j odd 4 2 3800.1.b.a 2
456.l odd 2 1 608.1.g.b 1
456.p even 2 1 152.1.g.b yes 1
456.v even 6 2 2888.1.l.a 2
456.x odd 6 2 2888.1.l.b 2
456.bh odd 18 6 2888.1.s.b 6
456.bj even 18 6 2888.1.s.a 6
2280.m even 2 1 3800.1.o.a 1
2280.bw odd 4 2 3800.1.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.g.a 1 24.h odd 2 1
152.1.g.a 1 57.d even 2 1
152.1.g.b yes 1 3.b odd 2 1
152.1.g.b yes 1 456.p even 2 1
608.1.g.a 1 24.f even 2 1
608.1.g.a 1 228.b odd 2 1
608.1.g.b 1 12.b even 2 1
608.1.g.b 1 456.l odd 2 1
1368.1.i.a 1 1.a even 1 1 trivial
1368.1.i.a 1 152.g odd 2 1 CM
1368.1.i.b 1 8.b even 2 1
1368.1.i.b 1 19.b odd 2 1
2888.1.l.a 2 57.h odd 6 2
2888.1.l.a 2 456.v even 6 2
2888.1.l.b 2 57.f even 6 2
2888.1.l.b 2 456.x odd 6 2
2888.1.s.a 6 57.l odd 18 6
2888.1.s.a 6 456.bj even 18 6
2888.1.s.b 6 57.j even 18 6
2888.1.s.b 6 456.bh odd 18 6
3800.1.b.a 2 120.w even 4 2
3800.1.b.a 2 285.j odd 4 2
3800.1.b.b 2 15.e even 4 2
3800.1.b.b 2 2280.bw odd 4 2
3800.1.o.a 1 15.d odd 2 1
3800.1.o.a 1 2280.m even 2 1
3800.1.o.b 1 120.i odd 2 1
3800.1.o.b 1 285.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1368, [\chi])$$.

## Hecke characteristic polynomials

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