# 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

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$11.6262185343$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 2$$ 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

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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$

## 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.

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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1456))$$:

 $$T_{3}^{3} - 2 T_{3}^{2} - 6 T_{3} + 8$$ $$T_{5}^{3} - 2 T_{5}^{2} - 3 T_{5} + 2$$ $$T_{11}^{3} + 2 T_{11}^{2} - 6 T_{11} - 8$$

## Hecke characteristic polynomials

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