# Properties

 Label 1456.2.a.t Level $1456$ Weight $2$ Character orbit 1456.a Self dual yes Analytic conductor $11.626$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1456,2,Mod(1,1456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1456, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1456.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1456 = 2^{4} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1456.a (trivial)

## 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: yes Analytic conductor: $$11.6262185343$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - \beta_1 + 1) q^{3} + ( - \beta_1 + 1) q^{5} + q^{7} + ( - 2 \beta_1 + 3) q^{9}+O(q^{10})$$ q + (b2 - b1 + 1) * q^3 + (-b1 + 1) * q^5 + q^7 + (-2*b1 + 3) * q^9 $$q + (\beta_{2} - \beta_1 + 1) q^{3} + ( - \beta_1 + 1) q^{5} + q^{7} + ( - 2 \beta_1 + 3) q^{9} + ( - \beta_{2} + \beta_1 - 1) q^{11} + q^{13} + (\beta_{2} - 3 \beta_1 + 3) q^{15} + (\beta_{2} + \beta_1 + 1) q^{17} + (\beta_1 + 1) q^{19} + (\beta_{2} - \beta_1 + 1) q^{21} + (\beta_{2} + 2 \beta_1 - 4) q^{23} + (\beta_{2} - 2 \beta_1 - 1) q^{25} + ( - 4 \beta_1 + 4) q^{27} + (\beta_{2} + 8) q^{29} + ( - 2 \beta_{2} + \beta_1 + 1) q^{31} + (2 \beta_1 - 6) q^{33} + ( - \beta_1 + 1) q^{35} + (\beta_{2} + 3 \beta_1 - 1) q^{37} + (\beta_{2} - \beta_1 + 1) q^{39} + ( - 2 \beta_{2} + 2 \beta_1) q^{41} + (3 \beta_{2} + 2 \beta_1 - 4) q^{43} + (2 \beta_{2} - 5 \beta_1 + 9) q^{45} + (4 \beta_{2} - \beta_1 + 3) q^{47} + q^{49} + (2 \beta_1 + 2) q^{51} + ( - 3 \beta_{2} + 2 \beta_1 + 2) q^{53} + ( - \beta_{2} + 3 \beta_1 - 3) q^{55} + (\beta_{2} + \beta_1 - 1) q^{57} + ( - 4 \beta_{2} - 2 \beta_1 + 2) q^{59} - 2 q^{61} + ( - 2 \beta_1 + 3) q^{63} + ( - \beta_1 + 1) q^{65} + ( - 4 \beta_{2} + 6 \beta_1 + 2) q^{67} + ( - 5 \beta_{2} + 9 \beta_1 - 5) q^{69} + (\beta_{2} - 3 \beta_1 + 3) q^{71} + ( - 4 \beta_{2} - \beta_1 - 3) q^{73} + ( - 2 \beta_{2} - 2 \beta_1 + 6) q^{75} + ( - \beta_{2} + \beta_1 - 1) q^{77} + (\beta_{2} - 4 \beta_1 + 6) q^{79} + (4 \beta_{2} - 6 \beta_1 + 3) q^{81} + ( - 4 \beta_{2} + 9 \beta_1 + 1) q^{83} + ( - \beta_{2} - \beta_1 - 3) q^{85} + (7 \beta_{2} - 7 \beta_1 + 11) q^{87} + ( - 2 \beta_{2} + 5 \beta_1 - 1) q^{89} + q^{91} + (3 \beta_{2} - \beta_1 - 7) q^{93} + ( - \beta_{2} - 2) q^{95} + ( - \beta_1 - 3) q^{97} + ( - 3 \beta_{2} + 7 \beta_1 - 7) q^{99}+O(q^{100})$$ q + (b2 - b1 + 1) * q^3 + (-b1 + 1) * q^5 + q^7 + (-2*b1 + 3) * q^9 + (-b2 + b1 - 1) * q^11 + q^13 + (b2 - 3*b1 + 3) * q^15 + (b2 + b1 + 1) * q^17 + (b1 + 1) * q^19 + (b2 - b1 + 1) * q^21 + (b2 + 2*b1 - 4) * q^23 + (b2 - 2*b1 - 1) * q^25 + (-4*b1 + 4) * q^27 + (b2 + 8) * q^29 + (-2*b2 + b1 + 1) * q^31 + (2*b1 - 6) * q^33 + (-b1 + 1) * q^35 + (b2 + 3*b1 - 1) * q^37 + (b2 - b1 + 1) * q^39 + (-2*b2 + 2*b1) * q^41 + (3*b2 + 2*b1 - 4) * q^43 + (2*b2 - 5*b1 + 9) * q^45 + (4*b2 - b1 + 3) * q^47 + q^49 + (2*b1 + 2) * q^51 + (-3*b2 + 2*b1 + 2) * q^53 + (-b2 + 3*b1 - 3) * q^55 + (b2 + b1 - 1) * q^57 + (-4*b2 - 2*b1 + 2) * q^59 - 2 * q^61 + (-2*b1 + 3) * q^63 + (-b1 + 1) * q^65 + (-4*b2 + 6*b1 + 2) * q^67 + (-5*b2 + 9*b1 - 5) * q^69 + (b2 - 3*b1 + 3) * q^71 + (-4*b2 - b1 - 3) * q^73 + (-2*b2 - 2*b1 + 6) * q^75 + (-b2 + b1 - 1) * q^77 + (b2 - 4*b1 + 6) * q^79 + (4*b2 - 6*b1 + 3) * q^81 + (-4*b2 + 9*b1 + 1) * q^83 + (-b2 - b1 - 3) * q^85 + (7*b2 - 7*b1 + 11) * q^87 + (-2*b2 + 5*b1 - 1) * q^89 + q^91 + (3*b2 - b1 - 7) * q^93 + (-b2 - 2) * q^95 + (-b1 - 3) * q^97 + (-3*b2 + 7*b1 - 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} + 2 q^{5} + 3 q^{7} + 7 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 + 2 * q^5 + 3 * q^7 + 7 * q^9 $$3 q + 2 q^{3} + 2 q^{5} + 3 q^{7} + 7 q^{9} - 2 q^{11} + 3 q^{13} + 6 q^{15} + 4 q^{17} + 4 q^{19} + 2 q^{21} - 10 q^{23} - 5 q^{25} + 8 q^{27} + 24 q^{29} + 4 q^{31} - 16 q^{33} + 2 q^{35} + 2 q^{39} + 2 q^{41} - 10 q^{43} + 22 q^{45} + 8 q^{47} + 3 q^{49} + 8 q^{51} + 8 q^{53} - 6 q^{55} - 2 q^{57} + 4 q^{59} - 6 q^{61} + 7 q^{63} + 2 q^{65} + 12 q^{67} - 6 q^{69} + 6 q^{71} - 10 q^{73} + 16 q^{75} - 2 q^{77} + 14 q^{79} + 3 q^{81} + 12 q^{83} - 10 q^{85} + 26 q^{87} + 2 q^{89} + 3 q^{91} - 22 q^{93} - 6 q^{95} - 10 q^{97} - 14 q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 + 2 * q^5 + 3 * q^7 + 7 * q^9 - 2 * q^11 + 3 * q^13 + 6 * q^15 + 4 * q^17 + 4 * q^19 + 2 * q^21 - 10 * q^23 - 5 * q^25 + 8 * q^27 + 24 * q^29 + 4 * q^31 - 16 * q^33 + 2 * q^35 + 2 * q^39 + 2 * q^41 - 10 * q^43 + 22 * q^45 + 8 * q^47 + 3 * q^49 + 8 * q^51 + 8 * q^53 - 6 * q^55 - 2 * q^57 + 4 * q^59 - 6 * q^61 + 7 * q^63 + 2 * q^65 + 12 * q^67 - 6 * q^69 + 6 * q^71 - 10 * q^73 + 16 * q^75 - 2 * q^77 + 14 * q^79 + 3 * q^81 + 12 * q^83 - 10 * q^85 + 26 * q^87 + 2 * q^89 + 3 * q^91 - 22 * q^93 - 6 * q^95 - 10 * q^97 - 14 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0.470683 2.34292 −1.81361
0 −2.24914 0 0.529317 0 1.00000 0 2.05863 0
1.2 0 1.14637 0 −1.34292 0 1.00000 0 −1.68585 0
1.3 0 3.10278 0 2.81361 0 1.00000 0 6.62721 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1456.2.a.t 3
4.b odd 2 1 91.2.a.d 3
8.b even 2 1 5824.2.a.bs 3
8.d odd 2 1 5824.2.a.by 3
12.b even 2 1 819.2.a.i 3
20.d odd 2 1 2275.2.a.m 3
28.d even 2 1 637.2.a.j 3
28.f even 6 2 637.2.e.i 6
28.g odd 6 2 637.2.e.j 6
52.b odd 2 1 1183.2.a.i 3
52.f even 4 2 1183.2.c.f 6
84.h odd 2 1 5733.2.a.x 3
364.h even 2 1 8281.2.a.bg 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.d 3 4.b odd 2 1
637.2.a.j 3 28.d even 2 1
637.2.e.i 6 28.f even 6 2
637.2.e.j 6 28.g odd 6 2
819.2.a.i 3 12.b even 2 1
1183.2.a.i 3 52.b odd 2 1
1183.2.c.f 6 52.f even 4 2
1456.2.a.t 3 1.a even 1 1 trivial
2275.2.a.m 3 20.d odd 2 1
5733.2.a.x 3 84.h odd 2 1
5824.2.a.bs 3 8.b even 2 1
5824.2.a.by 3 8.d odd 2 1
8281.2.a.bg 3 364.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1456))$$:

 $$T_{3}^{3} - 2T_{3}^{2} - 6T_{3} + 8$$ T3^3 - 2*T3^2 - 6*T3 + 8 $$T_{5}^{3} - 2T_{5}^{2} - 3T_{5} + 2$$ T5^3 - 2*T5^2 - 3*T5 + 2 $$T_{11}^{3} + 2T_{11}^{2} - 6T_{11} - 8$$ T11^3 + 2*T11^2 - 6*T11 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 2 T^{2} - 6 T + 8$$
$5$ $$T^{3} - 2 T^{2} - 3 T + 2$$
$7$ $$(T - 1)^{3}$$
$11$ $$T^{3} + 2 T^{2} - 6 T - 8$$
$13$ $$(T - 1)^{3}$$
$17$ $$T^{3} - 4 T^{2} - 10 T - 4$$
$19$ $$T^{3} - 4T^{2} + T + 4$$
$23$ $$T^{3} + 10 T^{2} + T - 136$$
$29$ $$T^{3} - 24 T^{2} + 185 T - 454$$
$31$ $$T^{3} - 4 T^{2} - 19 T - 16$$
$37$ $$T^{3} - 58T - 124$$
$41$ $$T^{3} - 2 T^{2} - 28 T - 8$$
$43$ $$T^{3} + 10 T^{2} - 71 T - 628$$
$47$ $$T^{3} - 8 T^{2} - 79 T + 544$$
$53$ $$T^{3} - 8 T^{2} - 35 T - 22$$
$59$ $$T^{3} - 4 T^{2} - 156 T + 688$$
$61$ $$(T + 2)^{3}$$
$67$ $$T^{3} - 12 T^{2} - 124 T + 976$$
$71$ $$T^{3} - 6 T^{2} - 22 T - 16$$
$73$ $$T^{3} + 10 T^{2} - 99 T - 274$$
$79$ $$T^{3} - 14 T^{2} + 5 T + 16$$
$83$ $$T^{3} - 12 T^{2} - 271 T + 3268$$
$89$ $$T^{3} - 2 T^{2} - 95 T + 422$$
$97$ $$T^{3} + 10 T^{2} + 29 T + 22$$