# Properties

 Label 456.2.a.f Level $456$ Weight $2$ Character orbit 456.a Self dual yes Analytic conductor $3.641$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$456 = 2^{3} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 456.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.64117833217$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + \beta q^{5} + \beta q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + \beta q^{5} + \beta q^{7} + q^{9} + ( 4 - \beta ) q^{11} -2 \beta q^{13} + \beta q^{15} + ( 2 - 3 \beta ) q^{17} + q^{19} + \beta q^{21} + ( 6 - 2 \beta ) q^{23} + ( -1 + \beta ) q^{25} + q^{27} + ( -2 + 4 \beta ) q^{29} -2 q^{31} + ( 4 - \beta ) q^{33} + ( 4 + \beta ) q^{35} -8 q^{37} -2 \beta q^{39} + ( -2 + 2 \beta ) q^{41} + \beta q^{43} + \beta q^{45} + ( -2 + 3 \beta ) q^{47} + ( -3 + \beta ) q^{49} + ( 2 - 3 \beta ) q^{51} + ( -2 - 4 \beta ) q^{53} + ( -4 + 3 \beta ) q^{55} + q^{57} + 12 q^{59} + ( 2 - 3 \beta ) q^{61} + \beta q^{63} + ( -8 - 2 \beta ) q^{65} -4 \beta q^{67} + ( 6 - 2 \beta ) q^{69} + ( -6 + 7 \beta ) q^{73} + ( -1 + \beta ) q^{75} + ( -4 + 3 \beta ) q^{77} + ( 2 - 6 \beta ) q^{79} + q^{81} + 4 q^{83} + ( -12 - \beta ) q^{85} + ( -2 + 4 \beta ) q^{87} -6 q^{89} + ( -8 - 2 \beta ) q^{91} -2 q^{93} + \beta q^{95} + ( -2 - 4 \beta ) q^{97} + ( 4 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + q^{5} + q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q + 2 q^{3} + q^{5} + q^{7} + 2 q^{9} + 7 q^{11} - 2 q^{13} + q^{15} + q^{17} + 2 q^{19} + q^{21} + 10 q^{23} - q^{25} + 2 q^{27} - 4 q^{31} + 7 q^{33} + 9 q^{35} - 16 q^{37} - 2 q^{39} - 2 q^{41} + q^{43} + q^{45} - q^{47} - 5 q^{49} + q^{51} - 8 q^{53} - 5 q^{55} + 2 q^{57} + 24 q^{59} + q^{61} + q^{63} - 18 q^{65} - 4 q^{67} + 10 q^{69} - 5 q^{73} - q^{75} - 5 q^{77} - 2 q^{79} + 2 q^{81} + 8 q^{83} - 25 q^{85} - 12 q^{89} - 18 q^{91} - 4 q^{93} + q^{95} - 8 q^{97} + 7 q^{99} + O(q^{100})$$

## 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
 −1.56155 2.56155
0 1.00000 0 −1.56155 0 −1.56155 0 1.00000 0
1.2 0 1.00000 0 2.56155 0 2.56155 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$19$$ $$-1$$

## 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 456.2.a.f 2
3.b odd 2 1 1368.2.a.k 2
4.b odd 2 1 912.2.a.m 2
8.b even 2 1 3648.2.a.bl 2
8.d odd 2 1 3648.2.a.br 2
12.b even 2 1 2736.2.a.z 2
19.b odd 2 1 8664.2.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.a.f 2 1.a even 1 1 trivial
912.2.a.m 2 4.b odd 2 1
1368.2.a.k 2 3.b odd 2 1
2736.2.a.z 2 12.b even 2 1
3648.2.a.bl 2 8.b even 2 1
3648.2.a.br 2 8.d odd 2 1
8664.2.a.r 2 19.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - T_{5} - 4$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(456))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$-4 - T + T^{2}$$
$7$ $$-4 - T + T^{2}$$
$11$ $$8 - 7 T + T^{2}$$
$13$ $$-16 + 2 T + T^{2}$$
$17$ $$-38 - T + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$8 - 10 T + T^{2}$$
$29$ $$-68 + T^{2}$$
$31$ $$( 2 + T )^{2}$$
$37$ $$( 8 + T )^{2}$$
$41$ $$-16 + 2 T + T^{2}$$
$43$ $$-4 - T + T^{2}$$
$47$ $$-38 + T + T^{2}$$
$53$ $$-52 + 8 T + T^{2}$$
$59$ $$( -12 + T )^{2}$$
$61$ $$-38 - T + T^{2}$$
$67$ $$-64 + 4 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$-202 + 5 T + T^{2}$$
$79$ $$-152 + 2 T + T^{2}$$
$83$ $$( -4 + T )^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$-52 + 8 T + T^{2}$$