# Properties

 Label 456.2.bf.a Level $456$ Weight $2$ Character orbit 456.bf Analytic conductor $3.641$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$456 = 2^{3} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 456.bf (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.64117833217$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -1 - \beta_{3} ) q^{5} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{7} + ( 3 \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -1 - \beta_{3} ) q^{5} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{7} + ( 3 \beta_{2} + \beta_{3} ) q^{9} + ( 2 - 4 \beta_{2} ) q^{11} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{13} + ( -3 - \beta_{3} ) q^{15} + ( -1 - \beta_{3} ) q^{17} + ( -5 + 2 \beta_{2} ) q^{19} + ( 3 + 3 \beta_{2} + \beta_{3} ) q^{21} + ( -2 + 5 \beta_{1} + 3 \beta_{2} ) q^{23} + ( -1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{25} + ( 3 + 2 \beta_{1} - 2 \beta_{3} ) q^{27} + ( -2 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{29} + ( -4 + \beta_{1} + 7 \beta_{2} - \beta_{3} ) q^{31} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{33} + ( -6 + 2 \beta_{2} - 2 \beta_{3} ) q^{35} + ( 6 - \beta_{1} - 11 \beta_{2} + \beta_{3} ) q^{37} + ( 6 \beta_{2} + \beta_{3} ) q^{39} + ( 1 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{41} + ( 2 - 4 \beta_{1} - 7 \beta_{2} + 2 \beta_{3} ) q^{43} + ( -3 - 2 \beta_{1} - \beta_{3} ) q^{45} + ( -2 + \beta_{1} - \beta_{2} ) q^{47} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{49} + ( -3 - \beta_{3} ) q^{51} + ( 8 + \beta_{1} - 9 \beta_{2} - 2 \beta_{3} ) q^{53} + ( -2 + 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{55} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{57} + ( -1 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{59} + ( 1 - \beta_{2} ) q^{61} + ( 3 + 5 \beta_{1} - 2 \beta_{3} ) q^{63} + ( -5 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{65} + ( 5 - 4 \beta_{1} + \beta_{2} ) q^{67} + ( \beta_{1} + 15 \beta_{2} + 2 \beta_{3} ) q^{69} + ( 1 - 2 \beta_{1} - 10 \beta_{2} + \beta_{3} ) q^{71} + ( 2 - 4 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{73} + ( 6 - \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{75} + ( 4 - 6 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} ) q^{77} + ( -2 - 3 \beta_{2} - 8 \beta_{3} ) q^{79} + ( -6 + 5 \beta_{1} + 6 \beta_{2} ) q^{81} + ( -2 + 4 \beta_{2} ) q^{83} + ( 4 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{85} + ( 6 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{87} + ( 4 + \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{89} + ( 5 + \beta_{1} + 6 \beta_{2} ) q^{91} + ( -3 + 4 \beta_{1} + 3 \beta_{2} - 7 \beta_{3} ) q^{93} + ( 5 - 2 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} ) q^{95} + ( 9 - 6 \beta_{2} - 3 \beta_{3} ) q^{97} + ( 12 - 4 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{3} - 3q^{5} + 2q^{7} + 5q^{9} + O(q^{10})$$ $$4q + q^{3} - 3q^{5} + 2q^{7} + 5q^{9} - 11q^{15} - 3q^{17} - 16q^{19} + 17q^{21} + 3q^{23} - 3q^{25} + 16q^{27} - 5q^{29} - 6q^{33} - 18q^{35} + 11q^{39} - 7q^{41} - 12q^{43} - 13q^{45} - 9q^{47} + 6q^{49} - 11q^{51} + 17q^{53} + 6q^{55} - q^{57} - 9q^{59} + 2q^{61} + 19q^{63} - 22q^{65} + 18q^{67} + 29q^{69} - 19q^{71} + 18q^{75} - 6q^{79} - 7q^{81} + 7q^{85} + 19q^{87} + 9q^{89} + 33q^{91} + 5q^{93} + 9q^{95} + 27q^{97} + 30q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu + 3$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{3} + 2 \beta_{1} + 3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/456\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$229$$ $$305$$ $$343$$ $$\chi(n)$$ $$1 - \beta_{2}$$ $$1$$ $$-1$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 −1.18614 + 1.26217i 1.68614 − 0.396143i −1.18614 − 1.26217i 1.68614 + 0.396143i
0 −1.18614 + 1.26217i 0 0.686141 + 0.396143i 0 −2.37228 0 −0.186141 2.99422i 0
65.2 0 1.68614 0.396143i 0 −2.18614 1.26217i 0 3.37228 0 2.68614 1.33591i 0
449.1 0 −1.18614 1.26217i 0 0.686141 0.396143i 0 −2.37228 0 −0.186141 + 2.99422i 0
449.2 0 1.68614 + 0.396143i 0 −2.18614 + 1.26217i 0 3.37228 0 2.68614 + 1.33591i 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
57.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 456.2.bf.a 4
3.b odd 2 1 456.2.bf.b yes 4
4.b odd 2 1 912.2.bn.j 4
12.b even 2 1 912.2.bn.i 4
19.d odd 6 1 456.2.bf.b yes 4
57.f even 6 1 inner 456.2.bf.a 4
76.f even 6 1 912.2.bn.i 4
228.n odd 6 1 912.2.bn.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.bf.a 4 1.a even 1 1 trivial
456.2.bf.a 4 57.f even 6 1 inner
456.2.bf.b yes 4 3.b odd 2 1
456.2.bf.b yes 4 19.d odd 6 1
912.2.bn.i 4 12.b even 2 1
912.2.bn.i 4 76.f even 6 1
912.2.bn.j 4 4.b odd 2 1
912.2.bn.j 4 228.n odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 3 T_{5}^{3} + T_{5}^{2} - 6 T_{5} + 4$$ acting on $$S_{2}^{\mathrm{new}}(456, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 - 3 T - 2 T^{2} - T^{3} + T^{4}$$
$5$ $$4 - 6 T + T^{2} + 3 T^{3} + T^{4}$$
$7$ $$( -8 - T + T^{2} )^{2}$$
$11$ $$( 12 + T^{2} )^{2}$$
$13$ $$121 - 11 T^{2} + T^{4}$$
$17$ $$4 - 6 T + T^{2} + 3 T^{3} + T^{4}$$
$19$ $$( 19 + 8 T + T^{2} )^{2}$$
$23$ $$4624 + 204 T - 65 T^{2} - 3 T^{3} + T^{4}$$
$29$ $$4 - 10 T + 27 T^{2} + 5 T^{3} + T^{4}$$
$31$ $$1156 + 79 T^{2} + T^{4}$$
$37$ $$7744 + 187 T^{2} + T^{4}$$
$41$ $$16 + 28 T + 45 T^{2} + 7 T^{3} + T^{4}$$
$43$ $$9 + 36 T + 141 T^{2} + 12 T^{3} + T^{4}$$
$47$ $$16 + 36 T + 31 T^{2} + 9 T^{3} + T^{4}$$
$53$ $$4096 - 1088 T + 225 T^{2} - 17 T^{3} + T^{4}$$
$59$ $$144 + 108 T + 69 T^{2} + 9 T^{3} + T^{4}$$
$61$ $$( 1 - T + T^{2} )^{2}$$
$67$ $$289 + 306 T + 91 T^{2} - 18 T^{3} + T^{4}$$
$71$ $$6724 + 1558 T + 279 T^{2} + 19 T^{3} + T^{4}$$
$73$ $$1089 + 33 T^{2} + T^{4}$$
$79$ $$29929 - 1038 T - 161 T^{2} + 6 T^{3} + T^{4}$$
$83$ $$( 12 + T^{2} )^{2}$$
$89$ $$144 - 108 T + 69 T^{2} - 9 T^{3} + T^{4}$$
$97$ $$1296 - 972 T + 279 T^{2} - 27 T^{3} + T^{4}$$