# Properties

 Label 456.2.q.f Level $456$ Weight $2$ Character orbit 456.q Analytic conductor $3.641$ Analytic rank $0$ Dimension $6$ 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.q (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.64117833217$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{18}^{3} q^{3} + ( 2 \zeta_{18}^{2} + 2 \zeta_{18}^{4} ) q^{5} + ( 1 + 2 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{5} ) q^{7} + ( -1 + \zeta_{18}^{3} ) q^{9} +O(q^{10})$$ $$q + \zeta_{18}^{3} q^{3} + ( 2 \zeta_{18}^{2} + 2 \zeta_{18}^{4} ) q^{5} + ( 1 + 2 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{5} ) q^{7} + ( -1 + \zeta_{18}^{3} ) q^{9} + ( -2 - 2 \zeta_{18} - 2 \zeta_{18}^{2} + 4 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{11} + ( -1 + \zeta_{18}^{3} ) q^{13} + ( -2 \zeta_{18} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{15} + ( -4 \zeta_{18} - 4 \zeta_{18}^{5} ) q^{17} + ( 1 + 2 \zeta_{18} - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{19} + ( 2 \zeta_{18}^{2} + \zeta_{18}^{3} + 2 \zeta_{18}^{4} ) q^{21} + ( 2 \zeta_{18} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{23} + ( -3 - 4 \zeta_{18}^{2} + 3 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{25} - q^{27} + ( -4 - 4 \zeta_{18} + 4 \zeta_{18}^{2} + 4 \zeta_{18}^{3} ) q^{29} + ( 5 + 2 \zeta_{18} + 2 \zeta_{18}^{2} - 4 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{31} + ( -4 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{33} + ( 4 \zeta_{18} + 2 \zeta_{18}^{2} + 8 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{35} + ( -1 - 4 \zeta_{18} - 4 \zeta_{18}^{2} + 4 \zeta_{18}^{4} ) q^{37} - q^{39} + ( 4 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 4 \zeta_{18}^{4} ) q^{41} + ( 2 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 2 \zeta_{18}^{4} ) q^{43} + ( -2 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{5} ) q^{45} + ( 6 - 6 \zeta_{18}^{3} ) q^{47} + ( 2 + 8 \zeta_{18} + 8 \zeta_{18}^{2} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{49} + ( 4 \zeta_{18}^{2} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{51} + ( -6 \zeta_{18} + 4 \zeta_{18}^{2} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{53} + ( 4 \zeta_{18} - 12 \zeta_{18}^{2} - 12 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{55} + ( 2 + 2 \zeta_{18}^{2} - \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{57} + ( -6 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 6 \zeta_{18}^{4} ) q^{59} + ( 1 + 4 \zeta_{18} - \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{61} + ( -1 - 2 \zeta_{18} + \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{63} + ( -2 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{5} ) q^{65} + ( 1 + 2 \zeta_{18} - 4 \zeta_{18}^{2} - \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{67} + ( 2 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{5} ) q^{69} + ( 4 \zeta_{18} + 4 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{71} + ( -4 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 4 \zeta_{18}^{4} ) q^{73} + ( -3 - 4 \zeta_{18} - 4 \zeta_{18}^{2} + 4 \zeta_{18}^{4} ) q^{75} + ( -2 - 10 \zeta_{18} - 10 \zeta_{18}^{2} + 10 \zeta_{18}^{5} ) q^{77} + ( 4 \zeta_{18} - 6 \zeta_{18}^{2} - 9 \zeta_{18}^{3} - 6 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{79} -\zeta_{18}^{3} q^{81} + ( -4 + 4 \zeta_{18} + 4 \zeta_{18}^{2} + 4 \zeta_{18}^{4} - 8 \zeta_{18}^{5} ) q^{83} + ( 8 + 8 \zeta_{18} - 8 \zeta_{18}^{3} - 8 \zeta_{18}^{4} - 8 \zeta_{18}^{5} ) q^{85} + ( -4 - 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{87} + ( -6 + 2 \zeta_{18} - 4 \zeta_{18}^{2} + 6 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{89} + ( -1 - 2 \zeta_{18} + \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{91} + ( 4 \zeta_{18} - 2 \zeta_{18}^{2} + 5 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{93} + ( 8 + 8 \zeta_{18} + 2 \zeta_{18}^{2} - 10 \zeta_{18}^{4} ) q^{95} + ( 4 \zeta_{18} + 4 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{97} + ( 2 - 2 \zeta_{18} + 4 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{3} + 6 q^{7} - 3 q^{9} + O(q^{10})$$ $$6 q + 3 q^{3} + 6 q^{7} - 3 q^{9} - 12 q^{11} - 3 q^{13} + 3 q^{21} - 9 q^{25} - 6 q^{27} - 12 q^{29} + 30 q^{31} - 6 q^{33} + 24 q^{35} - 6 q^{37} - 6 q^{39} - 12 q^{41} - 9 q^{43} + 18 q^{47} + 12 q^{49} + 9 q^{57} + 6 q^{59} + 3 q^{61} - 3 q^{63} + 3 q^{67} - 6 q^{71} + 9 q^{73} - 18 q^{75} - 12 q^{77} - 27 q^{79} - 3 q^{81} - 24 q^{83} + 24 q^{85} - 24 q^{87} - 18 q^{89} - 3 q^{91} + 15 q^{93} + 48 q^{95} - 6 q^{97} + 6 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$97$$ $$229$$ $$305$$ $$343$$ $$\chi(n)$$ $$-1 + \zeta_{18}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.766044 − 0.642788i −0.173648 + 0.984808i 0.939693 − 0.342020i −0.766044 + 0.642788i −0.173648 − 0.984808i 0.939693 + 0.342020i
0 0.500000 0.866025i 0 −1.53209 + 2.65366i 0 −2.06418 0 −0.500000 0.866025i 0
49.2 0 0.500000 0.866025i 0 −0.347296 + 0.601535i 0 0.305407 0 −0.500000 0.866025i 0
49.3 0 0.500000 0.866025i 0 1.87939 3.25519i 0 4.75877 0 −0.500000 0.866025i 0
121.1 0 0.500000 + 0.866025i 0 −1.53209 2.65366i 0 −2.06418 0 −0.500000 + 0.866025i 0
121.2 0 0.500000 + 0.866025i 0 −0.347296 0.601535i 0 0.305407 0 −0.500000 + 0.866025i 0
121.3 0 0.500000 + 0.866025i 0 1.87939 + 3.25519i 0 4.75877 0 −0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 121.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 456.2.q.f 6
3.b odd 2 1 1368.2.s.j 6
4.b odd 2 1 912.2.q.k 6
12.b even 2 1 2736.2.s.x 6
19.c even 3 1 inner 456.2.q.f 6
19.c even 3 1 8664.2.a.x 3
19.d odd 6 1 8664.2.a.z 3
57.h odd 6 1 1368.2.s.j 6
76.g odd 6 1 912.2.q.k 6
228.m even 6 1 2736.2.s.x 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.q.f 6 1.a even 1 1 trivial
456.2.q.f 6 19.c even 3 1 inner
912.2.q.k 6 4.b odd 2 1
912.2.q.k 6 76.g odd 6 1
1368.2.s.j 6 3.b odd 2 1
1368.2.s.j 6 57.h odd 6 1
2736.2.s.x 6 12.b even 2 1
2736.2.s.x 6 228.m even 6 1
8664.2.a.x 3 19.c even 3 1
8664.2.a.z 3 19.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 12 T_{5}^{4} + 16 T_{5}^{3} + 144 T_{5}^{2} + 96 T_{5} + 64$$ acting on $$S_{2}^{\mathrm{new}}(456, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$( 1 - T + T^{2} )^{3}$$
$5$ $$64 + 96 T + 144 T^{2} + 16 T^{3} + 12 T^{4} + T^{6}$$
$7$ $$( 3 - 9 T - 3 T^{2} + T^{3} )^{2}$$
$11$ $$( -136 - 24 T + 6 T^{2} + T^{3} )^{2}$$
$13$ $$( 1 + T + T^{2} )^{3}$$
$17$ $$4096 + 3072 T + 2304 T^{2} + 128 T^{3} + 48 T^{4} + T^{6}$$
$19$ $$6859 + 171 T^{2} + 64 T^{3} + 9 T^{4} + T^{6}$$
$23$ $$64 + 96 T + 144 T^{2} + 16 T^{3} + 12 T^{4} + T^{6}$$
$29$ $$36864 + 2304 T^{2} + 384 T^{3} + 144 T^{4} + 12 T^{5} + T^{6}$$
$31$ $$( 127 + 39 T - 15 T^{2} + T^{3} )^{2}$$
$37$ $$( -111 - 45 T + 3 T^{2} + T^{3} )^{2}$$
$41$ $$36864 + 2304 T^{2} + 384 T^{3} + 144 T^{4} + 12 T^{5} + T^{6}$$
$43$ $$289 - 255 T + 378 T^{2} + 169 T^{3} + 66 T^{4} + 9 T^{5} + T^{6}$$
$47$ $$( 36 - 6 T + T^{2} )^{3}$$
$53$ $$87616 + 24864 T + 7056 T^{2} + 592 T^{3} + 84 T^{4} + T^{6}$$
$59$ $$179776 - 40704 T + 11760 T^{2} - 272 T^{3} + 132 T^{4} - 6 T^{5} + T^{6}$$
$61$ $$289 + 765 T + 1974 T^{2} + 169 T^{3} + 54 T^{4} - 3 T^{5} + T^{6}$$
$67$ $$1369 + 1221 T + 978 T^{2} + 173 T^{3} + 42 T^{4} - 3 T^{5} + T^{6}$$
$71$ $$732736 + 112992 T + 22560 T^{2} + 920 T^{3} + 168 T^{4} + 6 T^{5} + T^{6}$$
$73$ $$32761 - 3801 T + 2070 T^{2} - 173 T^{3} + 102 T^{4} - 9 T^{5} + T^{6}$$
$79$ $$104329 - 51357 T + 34002 T^{2} + 4939 T^{3} + 570 T^{4} + 27 T^{5} + T^{6}$$
$83$ $$( 64 - 96 T + 12 T^{2} + T^{3} )^{2}$$
$89$ $$5184 - 5184 T + 6480 T^{2} + 1440 T^{3} + 252 T^{4} + 18 T^{5} + T^{6}$$
$97$ $$732736 + 112992 T + 22560 T^{2} + 920 T^{3} + 168 T^{4} + 6 T^{5} + T^{6}$$