Properties

 Label 1456.2.r.j Level $1456$ Weight $2$ Character orbit 1456.r Analytic conductor $11.626$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 2x^{2} + x + 1$$ x^4 - x^3 + 2*x^2 + x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + (\beta_{3} + \beta_{2}) q^{5} + (2 \beta_1 + 3) q^{7} + 2 \beta_1 q^{9}+O(q^{10})$$ q - b2 * q^3 + (b3 + b2) * q^5 + (2*b1 + 3) * q^7 + 2*b1 * q^9 $$q - \beta_{2} q^{3} + (\beta_{3} + \beta_{2}) q^{5} + (2 \beta_1 + 3) q^{7} + 2 \beta_1 q^{9} + ( - 3 \beta_1 - 3) q^{11} - q^{13} + 5 q^{15} + ( - 2 \beta_{2} + 3 \beta_1 + 3) q^{17} - 3 \beta_1 q^{19} + ( - 2 \beta_{3} - 3 \beta_{2}) q^{21} + (\beta_{3} + \beta_{2} + 6 \beta_1) q^{23} + \beta_{3} q^{27} - 2 \beta_{3} q^{29} + (5 \beta_1 + 5) q^{31} + (3 \beta_{3} + 3 \beta_{2}) q^{33} + (3 \beta_{3} + \beta_{2}) q^{35} + (3 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{37} + \beta_{2} q^{39} + 2 \beta_{3} q^{41} + 8 q^{43} - 2 \beta_{2} q^{45} + (2 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{47} + (8 \beta_1 + 5) q^{49} + ( - 3 \beta_{3} - 3 \beta_{2} + 10 \beta_1) q^{51} + (2 \beta_{2} + 3 \beta_1 + 3) q^{53} - 3 \beta_{3} q^{55} + 3 \beta_{3} q^{57} + ( - 2 \beta_{2} + 3 \beta_1 + 3) q^{59} + 3 \beta_1 q^{61} + (2 \beta_1 - 4) q^{63} + ( - \beta_{3} - \beta_{2}) q^{65} + ( - 3 \beta_1 - 3) q^{67} + ( - 6 \beta_{3} + 5) q^{69} - 4 \beta_{3} q^{71} + (3 \beta_{2} - 4 \beta_1 - 4) q^{73} + ( - 9 \beta_1 - 3) q^{77} + (3 \beta_{3} + 3 \beta_{2} - 4 \beta_1) q^{79} + (11 \beta_1 + 11) q^{81} + (3 \beta_{3} + 10) q^{85} + ( - 10 \beta_1 - 10) q^{87} + (\beta_{3} + \beta_{2}) q^{89} + ( - 2 \beta_1 - 3) q^{91} + ( - 5 \beta_{3} - 5 \beta_{2}) q^{93} + 3 \beta_{2} q^{95} + ( - 6 \beta_{3} - 4) q^{97} + 6 q^{99}+O(q^{100})$$ q - b2 * q^3 + (b3 + b2) * q^5 + (2*b1 + 3) * q^7 + 2*b1 * q^9 + (-3*b1 - 3) * q^11 - q^13 + 5 * q^15 + (-2*b2 + 3*b1 + 3) * q^17 - 3*b1 * q^19 + (-2*b3 - 3*b2) * q^21 + (b3 + b2 + 6*b1) * q^23 + b3 * q^27 - 2*b3 * q^29 + (5*b1 + 5) * q^31 + (3*b3 + 3*b2) * q^33 + (3*b3 + b2) * q^35 + (3*b3 + 3*b2 - 2*b1) * q^37 + b2 * q^39 + 2*b3 * q^41 + 8 * q^43 - 2*b2 * q^45 + (2*b3 + 2*b2 + 3*b1) * q^47 + (8*b1 + 5) * q^49 + (-3*b3 - 3*b2 + 10*b1) * q^51 + (2*b2 + 3*b1 + 3) * q^53 - 3*b3 * q^55 + 3*b3 * q^57 + (-2*b2 + 3*b1 + 3) * q^59 + 3*b1 * q^61 + (2*b1 - 4) * q^63 + (-b3 - b2) * q^65 + (-3*b1 - 3) * q^67 + (-6*b3 + 5) * q^69 - 4*b3 * q^71 + (3*b2 - 4*b1 - 4) * q^73 + (-9*b1 - 3) * q^77 + (3*b3 + 3*b2 - 4*b1) * q^79 + (11*b1 + 11) * q^81 + (3*b3 + 10) * q^85 + (-10*b1 - 10) * q^87 + (b3 + b2) * q^89 + (-2*b1 - 3) * q^91 + (-5*b3 - 5*b2) * q^93 + 3*b2 * q^95 + (-6*b3 - 4) * q^97 + 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{7} - 4 q^{9}+O(q^{10})$$ 4 * q + 8 * q^7 - 4 * q^9 $$4 q + 8 q^{7} - 4 q^{9} - 6 q^{11} - 4 q^{13} + 20 q^{15} + 6 q^{17} + 6 q^{19} - 12 q^{23} + 10 q^{31} + 4 q^{37} + 32 q^{43} - 6 q^{47} + 4 q^{49} - 20 q^{51} + 6 q^{53} + 6 q^{59} - 6 q^{61} - 20 q^{63} - 6 q^{67} + 20 q^{69} - 8 q^{73} + 6 q^{77} + 8 q^{79} + 22 q^{81} + 40 q^{85} - 20 q^{87} - 8 q^{91} - 16 q^{97} + 24 q^{99}+O(q^{100})$$ 4 * q + 8 * q^7 - 4 * q^9 - 6 * q^11 - 4 * q^13 + 20 * q^15 + 6 * q^17 + 6 * q^19 - 12 * q^23 + 10 * q^31 + 4 * q^37 + 32 * q^43 - 6 * q^47 + 4 * q^49 - 20 * q^51 + 6 * q^53 + 6 * q^59 - 6 * q^61 - 20 * q^63 - 6 * q^67 + 20 * q^69 - 8 * q^73 + 6 * q^77 + 8 * q^79 + 22 * q^81 + 40 * q^85 - 20 * q^87 - 8 * q^91 - 16 * q^97 + 24 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2x^{2} + x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2$$ (-v^3 + 2*v^2 - 2*v - 1) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 2\nu^{2} + 6\nu - 1 ) / 2$$ (v^3 - 2*v^2 + 6*v - 1) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2$$ v^3 + 2
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta _1 + 1 ) / 2$$ (b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 3\beta_1 ) / 2$$ (b3 + b2 + 3*b1) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2$$ b3 - 2

Character values

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

 $$n$$ $$561$$ $$911$$ $$1093$$ $$1249$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$\beta_{1}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
417.1
 0.809017 − 1.40126i −0.309017 + 0.535233i 0.809017 + 1.40126i −0.309017 − 0.535233i
0 −1.11803 + 1.93649i 0 −1.11803 1.93649i 0 2.00000 1.73205i 0 −1.00000 1.73205i 0
417.2 0 1.11803 1.93649i 0 1.11803 + 1.93649i 0 2.00000 1.73205i 0 −1.00000 1.73205i 0
625.1 0 −1.11803 1.93649i 0 −1.11803 + 1.93649i 0 2.00000 + 1.73205i 0 −1.00000 + 1.73205i 0
625.2 0 1.11803 + 1.93649i 0 1.11803 1.93649i 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 1456.2.r.j 4
4.b odd 2 1 91.2.e.b 4
7.c even 3 1 inner 1456.2.r.j 4
12.b even 2 1 819.2.j.c 4
28.d even 2 1 637.2.e.h 4
28.f even 6 1 637.2.a.e 2
28.f even 6 1 637.2.e.h 4
28.g odd 6 1 91.2.e.b 4
28.g odd 6 1 637.2.a.f 2
52.b odd 2 1 1183.2.e.d 4
84.j odd 6 1 5733.2.a.w 2
84.n even 6 1 819.2.j.c 4
84.n even 6 1 5733.2.a.v 2
364.x even 6 1 8281.2.a.ba 2
364.bl odd 6 1 1183.2.e.d 4
364.bl odd 6 1 8281.2.a.z 2

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

Hecke kernels

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

 $$T_{3}^{4} + 5T_{3}^{2} + 25$$ T3^4 + 5*T3^2 + 25 $$T_{5}^{4} + 5T_{5}^{2} + 25$$ T5^4 + 5*T5^2 + 25 $$T_{11}^{2} + 3T_{11} + 9$$ T11^2 + 3*T11 + 9

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 5T^{2} + 25$$
$5$ $$T^{4} + 5T^{2} + 25$$
$7$ $$(T^{2} - 4 T + 7)^{2}$$
$11$ $$(T^{2} + 3 T + 9)^{2}$$
$13$ $$(T + 1)^{4}$$
$17$ $$T^{4} - 6 T^{3} + \cdots + 121$$
$19$ $$(T^{2} - 3 T + 9)^{2}$$
$23$ $$T^{4} + 12 T^{3} + \cdots + 961$$
$29$ $$(T^{2} - 20)^{2}$$
$31$ $$(T^{2} - 5 T + 25)^{2}$$
$37$ $$T^{4} - 4 T^{3} + \cdots + 1681$$
$41$ $$(T^{2} - 20)^{2}$$
$43$ $$(T - 8)^{4}$$
$47$ $$T^{4} + 6 T^{3} + \cdots + 121$$
$53$ $$T^{4} - 6 T^{3} + \cdots + 121$$
$59$ $$T^{4} - 6 T^{3} + \cdots + 121$$
$61$ $$(T^{2} + 3 T + 9)^{2}$$
$67$ $$(T^{2} + 3 T + 9)^{2}$$
$71$ $$(T^{2} - 80)^{2}$$
$73$ $$T^{4} + 8 T^{3} + \cdots + 841$$
$79$ $$T^{4} - 8 T^{3} + \cdots + 841$$
$83$ $$T^{4}$$
$89$ $$T^{4} + 5T^{2} + 25$$
$97$ $$(T^{2} + 8 T - 164)^{2}$$