# Properties

 Label 392.2.i.e Level $392$ Weight $2$ Character orbit 392.i Analytic conductor $3.130$ Analytic rank $0$ Dimension $2$ 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: $$\Z[a_1, \ldots, a_{9}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{3} - 4 \zeta_{6} q^{5} - \zeta_{6} q^{9} +O(q^{10})$$ q + (-2*z + 2) * q^3 - 4*z * q^5 - z * q^9 $$q + ( - 2 \zeta_{6} + 2) q^{3} - 4 \zeta_{6} q^{5} - \zeta_{6} q^{9} - 8 q^{15} + (2 \zeta_{6} - 2) q^{17} - 2 \zeta_{6} q^{19} - 8 \zeta_{6} q^{23} + (11 \zeta_{6} - 11) q^{25} + 4 q^{27} + 2 q^{29} + ( - 4 \zeta_{6} + 4) q^{31} + 6 \zeta_{6} q^{37} + 2 q^{41} + 8 q^{43} + (4 \zeta_{6} - 4) q^{45} - 4 \zeta_{6} q^{47} + 4 \zeta_{6} q^{51} + ( - 10 \zeta_{6} + 10) q^{53} - 4 q^{57} + ( - 6 \zeta_{6} + 6) q^{59} + 4 \zeta_{6} q^{61} + ( - 12 \zeta_{6} + 12) q^{67} - 16 q^{69} + (14 \zeta_{6} - 14) q^{73} + 22 \zeta_{6} q^{75} + 8 \zeta_{6} q^{79} + ( - 11 \zeta_{6} + 11) q^{81} - 6 q^{83} + 8 q^{85} + ( - 4 \zeta_{6} + 4) q^{87} + 10 \zeta_{6} q^{89} - 8 \zeta_{6} q^{93} + (8 \zeta_{6} - 8) q^{95} + 2 q^{97} +O(q^{100})$$ q + (-2*z + 2) * q^3 - 4*z * q^5 - z * q^9 - 8 * q^15 + (2*z - 2) * q^17 - 2*z * q^19 - 8*z * q^23 + (11*z - 11) * q^25 + 4 * q^27 + 2 * q^29 + (-4*z + 4) * q^31 + 6*z * q^37 + 2 * q^41 + 8 * q^43 + (4*z - 4) * q^45 - 4*z * q^47 + 4*z * q^51 + (-10*z + 10) * q^53 - 4 * q^57 + (-6*z + 6) * q^59 + 4*z * q^61 + (-12*z + 12) * q^67 - 16 * q^69 + (14*z - 14) * q^73 + 22*z * q^75 + 8*z * q^79 + (-11*z + 11) * q^81 - 6 * q^83 + 8 * q^85 + (-4*z + 4) * q^87 + 10*z * q^89 - 8*z * q^93 + (8*z - 8) * q^95 + 2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 4 q^{5} - q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 4 * q^5 - q^9 $$2 q + 2 q^{3} - 4 q^{5} - q^{9} - 16 q^{15} - 2 q^{17} - 2 q^{19} - 8 q^{23} - 11 q^{25} + 8 q^{27} + 4 q^{29} + 4 q^{31} + 6 q^{37} + 4 q^{41} + 16 q^{43} - 4 q^{45} - 4 q^{47} + 4 q^{51} + 10 q^{53} - 8 q^{57} + 6 q^{59} + 4 q^{61} + 12 q^{67} - 32 q^{69} - 14 q^{73} + 22 q^{75} + 8 q^{79} + 11 q^{81} - 12 q^{83} + 16 q^{85} + 4 q^{87} + 10 q^{89} - 8 q^{93} - 8 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q + 2 * q^3 - 4 * q^5 - q^9 - 16 * q^15 - 2 * q^17 - 2 * q^19 - 8 * q^23 - 11 * q^25 + 8 * q^27 + 4 * q^29 + 4 * q^31 + 6 * q^37 + 4 * q^41 + 16 * q^43 - 4 * q^45 - 4 * q^47 + 4 * q^51 + 10 * q^53 - 8 * q^57 + 6 * q^59 + 4 * q^61 + 12 * q^67 - 32 * q^69 - 14 * q^73 + 22 * q^75 + 8 * q^79 + 11 * q^81 - 12 * q^83 + 16 * q^85 + 4 * q^87 + 10 * q^89 - 8 * q^93 - 8 * q^95 + 4 * q^97

## 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 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 0 0 0 −0.500000 + 0.866025i 0
361.1 0 1.00000 1.73205i 0 −2.00000 3.46410i 0 0 0 −0.500000 0.866025i 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.e 2
3.b odd 2 1 3528.2.s.ba 2
4.b odd 2 1 784.2.i.b 2
7.b odd 2 1 392.2.i.a 2
7.c even 3 1 392.2.a.b 1
7.c even 3 1 inner 392.2.i.e 2
7.d odd 6 1 56.2.a.b 1
7.d odd 6 1 392.2.i.a 2
21.c even 2 1 3528.2.s.a 2
21.g even 6 1 504.2.a.h 1
21.g even 6 1 3528.2.s.a 2
21.h odd 6 1 3528.2.a.b 1
21.h odd 6 1 3528.2.s.ba 2
28.d even 2 1 784.2.i.j 2
28.f even 6 1 112.2.a.a 1
28.f even 6 1 784.2.i.j 2
28.g odd 6 1 784.2.a.i 1
28.g odd 6 1 784.2.i.b 2
35.i odd 6 1 1400.2.a.a 1
35.j even 6 1 9800.2.a.bj 1
35.k even 12 2 1400.2.g.b 2
56.j odd 6 1 448.2.a.c 1
56.k odd 6 1 3136.2.a.c 1
56.m even 6 1 448.2.a.h 1
56.p even 6 1 3136.2.a.w 1
77.i even 6 1 6776.2.a.h 1
84.j odd 6 1 1008.2.a.m 1
84.n even 6 1 7056.2.a.c 1
91.s odd 6 1 9464.2.a.h 1
112.v even 12 2 1792.2.b.h 2
112.x odd 12 2 1792.2.b.a 2
140.s even 6 1 2800.2.a.bd 1
140.x odd 12 2 2800.2.g.g 2
168.ba even 6 1 4032.2.a.d 1
168.be odd 6 1 4032.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.b 1 7.d odd 6 1
112.2.a.a 1 28.f even 6 1
392.2.a.b 1 7.c even 3 1
392.2.i.a 2 7.b odd 2 1
392.2.i.a 2 7.d odd 6 1
392.2.i.e 2 1.a even 1 1 trivial
392.2.i.e 2 7.c even 3 1 inner
448.2.a.c 1 56.j odd 6 1
448.2.a.h 1 56.m even 6 1
504.2.a.h 1 21.g even 6 1
784.2.a.i 1 28.g odd 6 1
784.2.i.b 2 4.b odd 2 1
784.2.i.b 2 28.g odd 6 1
784.2.i.j 2 28.d even 2 1
784.2.i.j 2 28.f even 6 1
1008.2.a.m 1 84.j odd 6 1
1400.2.a.a 1 35.i odd 6 1
1400.2.g.b 2 35.k even 12 2
1792.2.b.a 2 112.x odd 12 2
1792.2.b.h 2 112.v even 12 2
2800.2.a.bd 1 140.s even 6 1
2800.2.g.g 2 140.x odd 12 2
3136.2.a.c 1 56.k odd 6 1
3136.2.a.w 1 56.p even 6 1
3528.2.a.b 1 21.h odd 6 1
3528.2.s.a 2 21.c even 2 1
3528.2.s.a 2 21.g even 6 1
3528.2.s.ba 2 3.b odd 2 1
3528.2.s.ba 2 21.h odd 6 1
4032.2.a.a 1 168.be odd 6 1
4032.2.a.d 1 168.ba even 6 1
6776.2.a.h 1 77.i even 6 1
7056.2.a.c 1 84.n even 6 1
9464.2.a.h 1 91.s odd 6 1
9800.2.a.bj 1 35.j even 6 1

## Hecke kernels

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

 $$T_{3}^{2} - 2T_{3} + 4$$ T3^2 - 2*T3 + 4 $$T_{5}^{2} + 4T_{5} + 16$$ T5^2 + 4*T5 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T + 4$$
$5$ $$T^{2} + 4T + 16$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$T^{2} + 2T + 4$$
$23$ $$T^{2} + 8T + 64$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$T^{2} - 6T + 36$$
$41$ $$(T - 2)^{2}$$
$43$ $$(T - 8)^{2}$$
$47$ $$T^{2} + 4T + 16$$
$53$ $$T^{2} - 10T + 100$$
$59$ $$T^{2} - 6T + 36$$
$61$ $$T^{2} - 4T + 16$$
$67$ $$T^{2} - 12T + 144$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 14T + 196$$
$79$ $$T^{2} - 8T + 64$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} - 10T + 100$$
$97$ $$(T - 2)^{2}$$