# Properties

 Label 1456.2.r.g Level $1456$ Weight $2$ Character orbit 1456.r Analytic conductor $11.626$ 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] = [1456,2,Mod(417,1456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1456, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4, 0]))

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

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

chi := DirichletCharacter("1456.417");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1456 = 2^{4} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1456.r (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: $$11.6262185343$$ 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: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) 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.

 $$f(q)$$ $$=$$ $$q + (3 \zeta_{6} - 2) q^{7} + 3 \zeta_{6} q^{9}+O(q^{10})$$ q + (3*z - 2) * q^7 + 3*z * q^9 $$q + (3 \zeta_{6} - 2) q^{7} + 3 \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{11} - q^{13} + (7 \zeta_{6} - 7) q^{17} - 7 \zeta_{6} q^{19} - 6 \zeta_{6} q^{23} + ( - 5 \zeta_{6} + 5) q^{25} - 5 q^{29} - 8 \zeta_{6} q^{37} - 2 q^{43} + 7 \zeta_{6} q^{47} + ( - 3 \zeta_{6} - 5) q^{49} + ( - 3 \zeta_{6} + 3) q^{53} + (7 \zeta_{6} - 7) q^{59} + 7 \zeta_{6} q^{61} + (3 \zeta_{6} - 9) q^{63} + (3 \zeta_{6} - 3) q^{67} + 5 q^{71} + (14 \zeta_{6} - 14) q^{73} + ( - 6 \zeta_{6} - 3) q^{77} - 6 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} + ( - 3 \zeta_{6} + 2) q^{91} - 14 q^{97} - 9 q^{99} +O(q^{100})$$ q + (3*z - 2) * q^7 + 3*z * q^9 + (3*z - 3) * q^11 - q^13 + (7*z - 7) * q^17 - 7*z * q^19 - 6*z * q^23 + (-5*z + 5) * q^25 - 5 * q^29 - 8*z * q^37 - 2 * q^43 + 7*z * q^47 + (-3*z - 5) * q^49 + (-3*z + 3) * q^53 + (7*z - 7) * q^59 + 7*z * q^61 + (3*z - 9) * q^63 + (3*z - 3) * q^67 + 5 * q^71 + (14*z - 14) * q^73 + (-6*z - 3) * q^77 - 6*z * q^79 + (9*z - 9) * q^81 + (-3*z + 2) * q^91 - 14 * q^97 - 9 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q - q^7 + 3 * q^9 $$2 q - q^{7} + 3 q^{9} - 3 q^{11} - 2 q^{13} - 7 q^{17} - 7 q^{19} - 6 q^{23} + 5 q^{25} - 10 q^{29} - 8 q^{37} - 4 q^{43} + 7 q^{47} - 13 q^{49} + 3 q^{53} - 7 q^{59} + 7 q^{61} - 15 q^{63} - 3 q^{67} + 10 q^{71} - 14 q^{73} - 12 q^{77} - 6 q^{79} - 9 q^{81} + q^{91} - 28 q^{97} - 18 q^{99}+O(q^{100})$$ 2 * q - q^7 + 3 * q^9 - 3 * q^11 - 2 * q^13 - 7 * q^17 - 7 * q^19 - 6 * q^23 + 5 * q^25 - 10 * q^29 - 8 * q^37 - 4 * q^43 + 7 * q^47 - 13 * q^49 + 3 * q^53 - 7 * q^59 + 7 * q^61 - 15 * q^63 - 3 * q^67 + 10 * q^71 - 14 * q^73 - 12 * q^77 - 6 * q^79 - 9 * q^81 + q^91 - 28 * q^97 - 18 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times$$.

 $$n$$ $$561$$ $$911$$ $$1093$$ $$1249$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
417.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 0 0 −0.500000 + 2.59808i 0 1.50000 + 2.59808i 0
625.1 0 0 0 0 0 −0.500000 2.59808i 0 1.50000 2.59808i 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 1456.2.r.g 2
4.b odd 2 1 91.2.e.a 2
7.c even 3 1 inner 1456.2.r.g 2
12.b even 2 1 819.2.j.b 2
28.d even 2 1 637.2.e.a 2
28.f even 6 1 637.2.a.d 1
28.f even 6 1 637.2.e.a 2
28.g odd 6 1 91.2.e.a 2
28.g odd 6 1 637.2.a.c 1
52.b odd 2 1 1183.2.e.b 2
84.j odd 6 1 5733.2.a.d 1
84.n even 6 1 819.2.j.b 2
84.n even 6 1 5733.2.a.c 1
364.x even 6 1 8281.2.a.e 1
364.bl odd 6 1 1183.2.e.b 2
364.bl odd 6 1 8281.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.a 2 4.b odd 2 1
91.2.e.a 2 28.g odd 6 1
637.2.a.c 1 28.g odd 6 1
637.2.a.d 1 28.f even 6 1
637.2.e.a 2 28.d even 2 1
637.2.e.a 2 28.f even 6 1
819.2.j.b 2 12.b even 2 1
819.2.j.b 2 84.n even 6 1
1183.2.e.b 2 52.b odd 2 1
1183.2.e.b 2 364.bl odd 6 1
1456.2.r.g 2 1.a even 1 1 trivial
1456.2.r.g 2 7.c even 3 1 inner
5733.2.a.c 1 84.n even 6 1
5733.2.a.d 1 84.j odd 6 1
8281.2.a.e 1 364.x even 6 1
8281.2.a.f 1 364.bl odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1456, [\chi])$$:

 $$T_{3}$$ T3 $$T_{5}$$ T5 $$T_{11}^{2} + 3T_{11} + 9$$ T11^2 + 3*T11 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + T + 7$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + 7T + 49$$
$19$ $$T^{2} + 7T + 49$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$(T + 5)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 8T + 64$$
$41$ $$T^{2}$$
$43$ $$(T + 2)^{2}$$
$47$ $$T^{2} - 7T + 49$$
$53$ $$T^{2} - 3T + 9$$
$59$ $$T^{2} + 7T + 49$$
$61$ $$T^{2} - 7T + 49$$
$67$ $$T^{2} + 3T + 9$$
$71$ $$(T - 5)^{2}$$
$73$ $$T^{2} + 14T + 196$$
$79$ $$T^{2} + 6T + 36$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$(T + 14)^{2}$$