# Properties

 Label 1392.1.i.a Level $1392$ Weight $1$ Character orbit 1392.i Self dual yes Analytic conductor $0.695$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -87 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1392,1,Mod(1217,1392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1392, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1392.1217");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1392 = 2^{4} \cdot 3 \cdot 29$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1392.i (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: yes Analytic conductor: $$0.694698497585$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 87) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.87.1 Artin image: $D_6$ Artin field: Galois closure of 6.0.484416.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{7} + q^{9}+O(q^{10})$$ q - q^3 + q^7 + q^9 $$q - q^{3} + q^{7} + q^{9} + q^{11} - q^{13} - q^{17} - q^{21} + q^{25} - q^{27} + q^{29} - q^{33} + q^{39} + 2 q^{41} + q^{47} + q^{51} + q^{63} + q^{67} - q^{75} + q^{77} + q^{81} - q^{87} - q^{89} - q^{91} + q^{99}+O(q^{100})$$ q - q^3 + q^7 + q^9 + q^11 - q^13 - q^17 - q^21 + q^25 - q^27 + q^29 - q^33 + q^39 + 2 * q^41 + q^47 + q^51 + q^63 + q^67 - q^75 + q^77 + q^81 - q^87 - q^89 - q^91 + q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1392\mathbb{Z}\right)^\times$$.

 $$n$$ $$175$$ $$929$$ $$1045$$ $$1249$$ $$\chi(n)$$ $$0$$ $$1$$ $$0$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1217.1
 0
0 −1.00000 0 0 0 1.00000 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
87.d odd 2 1 CM by $$\Q(\sqrt{-87})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1392.1.i.a 1
3.b odd 2 1 1392.1.i.b 1
4.b odd 2 1 87.1.d.a 1
12.b even 2 1 87.1.d.b yes 1
20.d odd 2 1 2175.1.h.b 1
20.e even 4 2 2175.1.b.a 2
29.b even 2 1 1392.1.i.b 1
36.f odd 6 2 2349.1.h.b 2
36.h even 6 2 2349.1.h.a 2
60.h even 2 1 2175.1.h.a 1
60.l odd 4 2 2175.1.b.b 2
87.d odd 2 1 CM 1392.1.i.a 1
116.d odd 2 1 87.1.d.b yes 1
116.e even 4 2 2523.1.b.b 2
116.h odd 14 6 2523.1.h.a 6
116.j odd 14 6 2523.1.h.b 6
116.l even 28 12 2523.1.j.b 12
348.b even 2 1 87.1.d.a 1
348.k odd 4 2 2523.1.b.b 2
348.s even 14 6 2523.1.h.a 6
348.t even 14 6 2523.1.h.b 6
348.v odd 28 12 2523.1.j.b 12
580.e odd 2 1 2175.1.h.a 1
580.o even 4 2 2175.1.b.b 2
1044.o odd 6 2 2349.1.h.a 2
1044.t even 6 2 2349.1.h.b 2
1740.k even 2 1 2175.1.h.b 1
1740.v odd 4 2 2175.1.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.1.d.a 1 4.b odd 2 1
87.1.d.a 1 348.b even 2 1
87.1.d.b yes 1 12.b even 2 1
87.1.d.b yes 1 116.d odd 2 1
1392.1.i.a 1 1.a even 1 1 trivial
1392.1.i.a 1 87.d odd 2 1 CM
1392.1.i.b 1 3.b odd 2 1
1392.1.i.b 1 29.b even 2 1
2175.1.b.a 2 20.e even 4 2
2175.1.b.a 2 1740.v odd 4 2
2175.1.b.b 2 60.l odd 4 2
2175.1.b.b 2 580.o even 4 2
2175.1.h.a 1 60.h even 2 1
2175.1.h.a 1 580.e odd 2 1
2175.1.h.b 1 20.d odd 2 1
2175.1.h.b 1 1740.k even 2 1
2349.1.h.a 2 36.h even 6 2
2349.1.h.a 2 1044.o odd 6 2
2349.1.h.b 2 36.f odd 6 2
2349.1.h.b 2 1044.t even 6 2
2523.1.b.b 2 116.e even 4 2
2523.1.b.b 2 348.k odd 4 2
2523.1.h.a 6 116.h odd 14 6
2523.1.h.a 6 348.s even 14 6
2523.1.h.b 6 116.j odd 14 6
2523.1.h.b 6 348.t even 14 6
2523.1.j.b 12 116.l even 28 12
2523.1.j.b 12 348.v odd 28 12

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11} - 1$$ acting on $$S_{1}^{\mathrm{new}}(1392, [\chi])$$.

## Hecke characteristic polynomials

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