# Properties

 Label 56.2.i.b Level $56$ Weight $2$ Character orbit 56.i Analytic conductor $0.447$ 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] = [56,2,Mod(9,56)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(56, 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("56.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$56 = 2^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 56.i (of order $$3$$, degree $$2$$, 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: $$0.447162251319$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{3} + ( - \zeta_{6} + 1) q^{5} + (2 \zeta_{6} - 3) q^{7} + ( - 2 \zeta_{6} + 2) q^{9}+O(q^{10})$$ q + z * q^3 + (-z + 1) * q^5 + (2*z - 3) * q^7 + (-2*z + 2) * q^9 $$q + \zeta_{6} q^{3} + ( - \zeta_{6} + 1) q^{5} + (2 \zeta_{6} - 3) q^{7} + ( - 2 \zeta_{6} + 2) q^{9} - 3 \zeta_{6} q^{11} - 6 q^{13} + q^{15} + 5 \zeta_{6} q^{17} + (\zeta_{6} - 1) q^{19} + ( - \zeta_{6} - 2) q^{21} + ( - 7 \zeta_{6} + 7) q^{23} + 4 \zeta_{6} q^{25} + 5 q^{27} + 2 q^{29} + 5 \zeta_{6} q^{31} + ( - 3 \zeta_{6} + 3) q^{33} + (3 \zeta_{6} - 1) q^{35} + (3 \zeta_{6} - 3) q^{37} - 6 \zeta_{6} q^{39} - 2 q^{41} - 4 q^{43} - 2 \zeta_{6} q^{45} + (5 \zeta_{6} - 5) q^{47} + ( - 8 \zeta_{6} + 5) q^{49} + (5 \zeta_{6} - 5) q^{51} + \zeta_{6} q^{53} - 3 q^{55} - q^{57} - 15 \zeta_{6} q^{59} + ( - 5 \zeta_{6} + 5) q^{61} + (6 \zeta_{6} - 2) q^{63} + (6 \zeta_{6} - 6) q^{65} + 9 \zeta_{6} q^{67} + 7 q^{69} - 7 \zeta_{6} q^{73} + (4 \zeta_{6} - 4) q^{75} + (3 \zeta_{6} + 6) q^{77} + (\zeta_{6} - 1) q^{79} - \zeta_{6} q^{81} + 12 q^{83} + 5 q^{85} + 2 \zeta_{6} q^{87} + (7 \zeta_{6} - 7) q^{89} + ( - 12 \zeta_{6} + 18) q^{91} + (5 \zeta_{6} - 5) q^{93} + \zeta_{6} q^{95} - 2 q^{97} - 6 q^{99} +O(q^{100})$$ q + z * q^3 + (-z + 1) * q^5 + (2*z - 3) * q^7 + (-2*z + 2) * q^9 - 3*z * q^11 - 6 * q^13 + q^15 + 5*z * q^17 + (z - 1) * q^19 + (-z - 2) * q^21 + (-7*z + 7) * q^23 + 4*z * q^25 + 5 * q^27 + 2 * q^29 + 5*z * q^31 + (-3*z + 3) * q^33 + (3*z - 1) * q^35 + (3*z - 3) * q^37 - 6*z * q^39 - 2 * q^41 - 4 * q^43 - 2*z * q^45 + (5*z - 5) * q^47 + (-8*z + 5) * q^49 + (5*z - 5) * q^51 + z * q^53 - 3 * q^55 - q^57 - 15*z * q^59 + (-5*z + 5) * q^61 + (6*z - 2) * q^63 + (6*z - 6) * q^65 + 9*z * q^67 + 7 * q^69 - 7*z * q^73 + (4*z - 4) * q^75 + (3*z + 6) * q^77 + (z - 1) * q^79 - z * q^81 + 12 * q^83 + 5 * q^85 + 2*z * q^87 + (7*z - 7) * q^89 + (-12*z + 18) * q^91 + (5*z - 5) * q^93 + z * q^95 - 2 * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + q^3 + q^5 - 4 * q^7 + 2 * q^9 $$2 q + q^{3} + q^{5} - 4 q^{7} + 2 q^{9} - 3 q^{11} - 12 q^{13} + 2 q^{15} + 5 q^{17} - q^{19} - 5 q^{21} + 7 q^{23} + 4 q^{25} + 10 q^{27} + 4 q^{29} + 5 q^{31} + 3 q^{33} + q^{35} - 3 q^{37} - 6 q^{39} - 4 q^{41} - 8 q^{43} - 2 q^{45} - 5 q^{47} + 2 q^{49} - 5 q^{51} + q^{53} - 6 q^{55} - 2 q^{57} - 15 q^{59} + 5 q^{61} + 2 q^{63} - 6 q^{65} + 9 q^{67} + 14 q^{69} - 7 q^{73} - 4 q^{75} + 15 q^{77} - q^{79} - q^{81} + 24 q^{83} + 10 q^{85} + 2 q^{87} - 7 q^{89} + 24 q^{91} - 5 q^{93} + q^{95} - 4 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q + q^3 + q^5 - 4 * q^7 + 2 * q^9 - 3 * q^11 - 12 * q^13 + 2 * q^15 + 5 * q^17 - q^19 - 5 * q^21 + 7 * q^23 + 4 * q^25 + 10 * q^27 + 4 * q^29 + 5 * q^31 + 3 * q^33 + q^35 - 3 * q^37 - 6 * q^39 - 4 * q^41 - 8 * q^43 - 2 * q^45 - 5 * q^47 + 2 * q^49 - 5 * q^51 + q^53 - 6 * q^55 - 2 * q^57 - 15 * q^59 + 5 * q^61 + 2 * q^63 - 6 * q^65 + 9 * q^67 + 14 * q^69 - 7 * q^73 - 4 * q^75 + 15 * q^77 - q^79 - q^81 + 24 * q^83 + 10 * q^85 + 2 * q^87 - 7 * q^89 + 24 * q^91 - 5 * q^93 + q^95 - 4 * q^97 - 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/56\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$17$$ $$29$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 −2.00000 + 1.73205i 0 1.00000 1.73205i 0
25.1 0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 −2.00000 1.73205i 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 56.2.i.b 2
3.b odd 2 1 504.2.s.c 2
4.b odd 2 1 112.2.i.a 2
5.b even 2 1 1400.2.q.d 2
5.c odd 4 2 1400.2.bh.a 4
7.b odd 2 1 392.2.i.b 2
7.c even 3 1 inner 56.2.i.b 2
7.c even 3 1 392.2.a.c 1
7.d odd 6 1 392.2.a.e 1
7.d odd 6 1 392.2.i.b 2
8.b even 2 1 448.2.i.b 2
8.d odd 2 1 448.2.i.d 2
12.b even 2 1 1008.2.s.g 2
21.c even 2 1 3528.2.s.q 2
21.g even 6 1 3528.2.a.j 1
21.g even 6 1 3528.2.s.q 2
21.h odd 6 1 504.2.s.c 2
21.h odd 6 1 3528.2.a.p 1
28.d even 2 1 784.2.i.h 2
28.f even 6 1 784.2.a.c 1
28.f even 6 1 784.2.i.h 2
28.g odd 6 1 112.2.i.a 2
28.g odd 6 1 784.2.a.h 1
35.i odd 6 1 9800.2.a.s 1
35.j even 6 1 1400.2.q.d 2
35.j even 6 1 9800.2.a.be 1
35.l odd 12 2 1400.2.bh.a 4
56.j odd 6 1 3136.2.a.i 1
56.k odd 6 1 448.2.i.d 2
56.k odd 6 1 3136.2.a.j 1
56.m even 6 1 3136.2.a.t 1
56.p even 6 1 448.2.i.b 2
56.p even 6 1 3136.2.a.u 1
84.j odd 6 1 7056.2.a.u 1
84.n even 6 1 1008.2.s.g 2
84.n even 6 1 7056.2.a.bj 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.b 2 1.a even 1 1 trivial
56.2.i.b 2 7.c even 3 1 inner
112.2.i.a 2 4.b odd 2 1
112.2.i.a 2 28.g odd 6 1
392.2.a.c 1 7.c even 3 1
392.2.a.e 1 7.d odd 6 1
392.2.i.b 2 7.b odd 2 1
392.2.i.b 2 7.d odd 6 1
448.2.i.b 2 8.b even 2 1
448.2.i.b 2 56.p even 6 1
448.2.i.d 2 8.d odd 2 1
448.2.i.d 2 56.k odd 6 1
504.2.s.c 2 3.b odd 2 1
504.2.s.c 2 21.h odd 6 1
784.2.a.c 1 28.f even 6 1
784.2.a.h 1 28.g odd 6 1
784.2.i.h 2 28.d even 2 1
784.2.i.h 2 28.f even 6 1
1008.2.s.g 2 12.b even 2 1
1008.2.s.g 2 84.n even 6 1
1400.2.q.d 2 5.b even 2 1
1400.2.q.d 2 35.j even 6 1
1400.2.bh.a 4 5.c odd 4 2
1400.2.bh.a 4 35.l odd 12 2
3136.2.a.i 1 56.j odd 6 1
3136.2.a.j 1 56.k odd 6 1
3136.2.a.t 1 56.m even 6 1
3136.2.a.u 1 56.p even 6 1
3528.2.a.j 1 21.g even 6 1
3528.2.a.p 1 21.h odd 6 1
3528.2.s.q 2 21.c even 2 1
3528.2.s.q 2 21.g even 6 1
7056.2.a.u 1 84.j odd 6 1
7056.2.a.bj 1 84.n even 6 1
9800.2.a.s 1 35.i odd 6 1
9800.2.a.be 1 35.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(56, [\chi])$$.

## Hecke characteristic polynomials

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