# Properties

 Label 392.2.i.b Level $392$ Weight $2$ Character orbit 392.i Analytic conductor $3.130$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,2,Mod(177,392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("392.177");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$392 = 2^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 392.i (of order $$3$$, degree $$2$$, 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: $$3.13013575923$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{5} + 2 \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^3 - z * q^5 + 2*z * q^9 $$q + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{5} + 2 \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{11} + 6 q^{13} + q^{15} + (5 \zeta_{6} - 5) q^{17} + \zeta_{6} q^{19} + 7 \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} - 5 q^{27} + 2 q^{29} + (5 \zeta_{6} - 5) q^{31} - 3 \zeta_{6} q^{33} - 3 \zeta_{6} q^{37} + (6 \zeta_{6} - 6) q^{39} + 2 q^{41} - 4 q^{43} + ( - 2 \zeta_{6} + 2) q^{45} + 5 \zeta_{6} q^{47} - 5 \zeta_{6} q^{51} + ( - \zeta_{6} + 1) q^{53} + 3 q^{55} - q^{57} + ( - 15 \zeta_{6} + 15) q^{59} - 5 \zeta_{6} q^{61} - 6 \zeta_{6} q^{65} + ( - 9 \zeta_{6} + 9) q^{67} - 7 q^{69} + ( - 7 \zeta_{6} + 7) q^{73} + 4 \zeta_{6} q^{75} - \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} - 12 q^{83} + 5 q^{85} + (2 \zeta_{6} - 2) q^{87} + 7 \zeta_{6} q^{89} - 5 \zeta_{6} q^{93} + ( - \zeta_{6} + 1) q^{95} + 2 q^{97} - 6 q^{99} +O(q^{100})$$ q + (z - 1) * q^3 - z * q^5 + 2*z * q^9 + (3*z - 3) * q^11 + 6 * q^13 + q^15 + (5*z - 5) * q^17 + z * q^19 + 7*z * q^23 + (-4*z + 4) * q^25 - 5 * q^27 + 2 * q^29 + (5*z - 5) * q^31 - 3*z * q^33 - 3*z * q^37 + (6*z - 6) * q^39 + 2 * q^41 - 4 * q^43 + (-2*z + 2) * q^45 + 5*z * q^47 - 5*z * q^51 + (-z + 1) * q^53 + 3 * q^55 - q^57 + (-15*z + 15) * q^59 - 5*z * q^61 - 6*z * q^65 + (-9*z + 9) * q^67 - 7 * q^69 + (-7*z + 7) * q^73 + 4*z * q^75 - z * q^79 + (z - 1) * q^81 - 12 * q^83 + 5 * q^85 + (2*z - 2) * q^87 + 7*z * q^89 - 5*z * q^93 + (-z + 1) * q^95 + 2 * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q - q^3 - q^5 + 2 * q^9 $$2 q - q^{3} - q^{5} + 2 q^{9} - 3 q^{11} + 12 q^{13} + 2 q^{15} - 5 q^{17} + q^{19} + 7 q^{23} + 4 q^{25} - 10 q^{27} + 4 q^{29} - 5 q^{31} - 3 q^{33} - 3 q^{37} - 6 q^{39} + 4 q^{41} - 8 q^{43} + 2 q^{45} + 5 q^{47} - 5 q^{51} + q^{53} + 6 q^{55} - 2 q^{57} + 15 q^{59} - 5 q^{61} - 6 q^{65} + 9 q^{67} - 14 q^{69} + 7 q^{73} + 4 q^{75} - q^{79} - q^{81} - 24 q^{83} + 10 q^{85} - 2 q^{87} + 7 q^{89} - 5 q^{93} + q^{95} + 4 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q - q^3 - q^5 + 2 * q^9 - 3 * q^11 + 12 * q^13 + 2 * q^15 - 5 * q^17 + q^19 + 7 * q^23 + 4 * q^25 - 10 * q^27 + 4 * q^29 - 5 * q^31 - 3 * q^33 - 3 * q^37 - 6 * q^39 + 4 * q^41 - 8 * q^43 + 2 * q^45 + 5 * q^47 - 5 * q^51 + q^53 + 6 * q^55 - 2 * q^57 + 15 * q^59 - 5 * q^61 - 6 * q^65 + 9 * q^67 - 14 * q^69 + 7 * q^73 + 4 * q^75 - q^79 - q^81 - 24 * q^83 + 10 * q^85 - 2 * q^87 + 7 * q^89 - 5 * q^93 + q^95 + 4 * q^97 - 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/392\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$295$$ $$297$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
177.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 0 1.00000 1.73205i 0
361.1 0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 0 1.00000 + 1.73205i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.2.i.b 2
3.b odd 2 1 3528.2.s.q 2
4.b odd 2 1 784.2.i.h 2
7.b odd 2 1 56.2.i.b 2
7.c even 3 1 392.2.a.e 1
7.c even 3 1 inner 392.2.i.b 2
7.d odd 6 1 56.2.i.b 2
7.d odd 6 1 392.2.a.c 1
21.c even 2 1 504.2.s.c 2
21.g even 6 1 504.2.s.c 2
21.g even 6 1 3528.2.a.p 1
21.h odd 6 1 3528.2.a.j 1
21.h odd 6 1 3528.2.s.q 2
28.d even 2 1 112.2.i.a 2
28.f even 6 1 112.2.i.a 2
28.f even 6 1 784.2.a.h 1
28.g odd 6 1 784.2.a.c 1
28.g odd 6 1 784.2.i.h 2
35.c odd 2 1 1400.2.q.d 2
35.f even 4 2 1400.2.bh.a 4
35.i odd 6 1 1400.2.q.d 2
35.i odd 6 1 9800.2.a.be 1
35.j even 6 1 9800.2.a.s 1
35.k even 12 2 1400.2.bh.a 4
56.e even 2 1 448.2.i.d 2
56.h odd 2 1 448.2.i.b 2
56.j odd 6 1 448.2.i.b 2
56.j odd 6 1 3136.2.a.u 1
56.k odd 6 1 3136.2.a.t 1
56.m even 6 1 448.2.i.d 2
56.m even 6 1 3136.2.a.j 1
56.p even 6 1 3136.2.a.i 1
84.h odd 2 1 1008.2.s.g 2
84.j odd 6 1 1008.2.s.g 2
84.j odd 6 1 7056.2.a.bj 1
84.n even 6 1 7056.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.b 2 7.b odd 2 1
56.2.i.b 2 7.d odd 6 1
112.2.i.a 2 28.d even 2 1
112.2.i.a 2 28.f even 6 1
392.2.a.c 1 7.d odd 6 1
392.2.a.e 1 7.c even 3 1
392.2.i.b 2 1.a even 1 1 trivial
392.2.i.b 2 7.c even 3 1 inner
448.2.i.b 2 56.h odd 2 1
448.2.i.b 2 56.j odd 6 1
448.2.i.d 2 56.e even 2 1
448.2.i.d 2 56.m even 6 1
504.2.s.c 2 21.c even 2 1
504.2.s.c 2 21.g even 6 1
784.2.a.c 1 28.g odd 6 1
784.2.a.h 1 28.f even 6 1
784.2.i.h 2 4.b odd 2 1
784.2.i.h 2 28.g odd 6 1
1008.2.s.g 2 84.h odd 2 1
1008.2.s.g 2 84.j odd 6 1
1400.2.q.d 2 35.c odd 2 1
1400.2.q.d 2 35.i odd 6 1
1400.2.bh.a 4 35.f even 4 2
1400.2.bh.a 4 35.k even 12 2
3136.2.a.i 1 56.p even 6 1
3136.2.a.j 1 56.m even 6 1
3136.2.a.t 1 56.k odd 6 1
3136.2.a.u 1 56.j odd 6 1
3528.2.a.j 1 21.h odd 6 1
3528.2.a.p 1 21.g even 6 1
3528.2.s.q 2 3.b odd 2 1
3528.2.s.q 2 21.h odd 6 1
7056.2.a.u 1 84.n even 6 1
7056.2.a.bj 1 84.j odd 6 1
9800.2.a.s 1 35.j even 6 1
9800.2.a.be 1 35.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(392, [\chi])$$:

 $$T_{3}^{2} + T_{3} + 1$$ T3^2 + T3 + 1 $$T_{5}^{2} + T_{5} + 1$$ T5^2 + T5 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$(T - 6)^{2}$$
$17$ $$T^{2} + 5T + 25$$
$19$ $$T^{2} - T + 1$$
$23$ $$T^{2} - 7T + 49$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} + 5T + 25$$
$37$ $$T^{2} + 3T + 9$$
$41$ $$(T - 2)^{2}$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} - 5T + 25$$
$53$ $$T^{2} - T + 1$$
$59$ $$T^{2} - 15T + 225$$
$61$ $$T^{2} + 5T + 25$$
$67$ $$T^{2} - 9T + 81$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 7T + 49$$
$79$ $$T^{2} + T + 1$$
$83$ $$(T + 12)^{2}$$
$89$ $$T^{2} - 7T + 49$$
$97$ $$(T - 2)^{2}$$