# Properties

 Label 576.2.bb.a Level $576$ Weight $2$ Character orbit 576.bb Analytic conductor $4.599$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 576.bb (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.59938315643$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 144) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{3} + (\zeta_{12}^{3} - \zeta_{12} - 1) q^{5} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12} - 4) q^{7} + 3 q^{9}+O(q^{10})$$ q + (z^3 - 2*z) * q^3 + (z^3 - z - 1) * q^5 + (z^3 + 2*z^2 - z - 4) * q^7 + 3 * q^9 $$q + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{3} + (\zeta_{12}^{3} - \zeta_{12} - 1) q^{5} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12} - 4) q^{7} + 3 q^{9} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + 1) q^{11} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{13} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{15} + 4 q^{17} + ( - 3 \zeta_{12}^{3} + 3) q^{19} + ( - 6 \zeta_{12}^{3} - \zeta_{12}^{2} + 6 \zeta_{12} + 2) q^{21} + (2 \zeta_{12}^{2} + 3 \zeta_{12} + 2) q^{23} + (3 \zeta_{12}^{3} - \zeta_{12}^{2} - 3 \zeta_{12} + 2) q^{25} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - \zeta_{12} + 1) q^{29} + ( - 6 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 3 \zeta_{12} + 4) q^{31} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12} - 1) q^{33} + ( - 5 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 5) q^{35} + ( - 7 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 7) q^{37} + (5 \zeta_{12}^{3} + \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{39} + ( - 3 \zeta_{12}^{2} + 6 \zeta_{12} - 3) q^{41} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 6 \zeta_{12} - 1) q^{43} + (3 \zeta_{12}^{3} - 3 \zeta_{12} - 3) q^{45} + (3 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 3 \zeta_{12}) q^{47} + ( - 8 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 4 \zeta_{12} + 6) q^{49} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{51} + (3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} + 3) q^{53} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{55} + (3 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6 \zeta_{12} - 3) q^{57} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}^{2}) q^{59} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + \zeta_{12} - 7) q^{61} + (3 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3 \zeta_{12} - 12) q^{63} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} + 2 \zeta_{12}) q^{65} + (3 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{67} + ( - 3 \zeta_{12}^{2} - 6 \zeta_{12} - 3) q^{69} + (4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{71} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{73} + (3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12} + 6) q^{75} + (3 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{77} + (\zeta_{12}^{3} + \zeta_{12}) q^{79} + 9 q^{81} + ( - 9 \zeta_{12}^{3} + 9 \zeta_{12}^{2} + 10 \zeta_{12} + 1) q^{83} + (4 \zeta_{12}^{3} - 4 \zeta_{12} - 4) q^{85} + (3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3) q^{87} + ( - 2 \zeta_{12}^{3} - 16 \zeta_{12}^{2} + 8) q^{89} + ( - 5 \zeta_{12}^{3} + 9 \zeta_{12}^{2} + 9 \zeta_{12} - 5) q^{91} + (8 \zeta_{12}^{3} + 9 \zeta_{12}^{2} - 4 \zeta_{12} - 9) q^{93} + (6 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{95} - \zeta_{12}^{2} q^{97} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3) q^{99} +O(q^{100})$$ q + (z^3 - 2*z) * q^3 + (z^3 - z - 1) * q^5 + (z^3 + 2*z^2 - z - 4) * q^7 + 3 * q^9 + (-z^3 - z^2 + 1) * q^11 + (z^3 - 2*z^2 - 3*z + 3) * q^13 + (-z^3 - z^2 + 2*z + 2) * q^15 + 4 * q^17 + (-3*z^3 + 3) * q^19 + (-6*z^3 - z^2 + 6*z + 2) * q^21 + (2*z^2 + 3*z + 2) * q^23 + (3*z^3 - z^2 - 3*z + 2) * q^25 + (3*z^3 - 6*z) * q^27 + (2*z^3 - 2*z^2 - z + 1) * q^29 + (-6*z^3 - 4*z^2 + 3*z + 4) * q^31 + (2*z^3 + 2*z^2 - z - 1) * q^33 + (-5*z^3 - 3*z^2 + 3*z + 5) * q^35 + (-7*z^3 + 2*z^2 + 2*z - 7) * q^37 + (5*z^3 + z^2 - 4*z + 4) * q^39 + (-3*z^2 + 6*z - 3) * q^41 + (-5*z^3 - 5*z^2 + 6*z - 1) * q^43 + (3*z^3 - 3*z - 3) * q^45 + (3*z^3 - 4*z^2 + 3*z) * q^47 + (-8*z^3 - 6*z^2 + 4*z + 6) * q^49 + (4*z^3 - 8*z) * q^51 + (3*z^3 + 2*z^2 + 2*z + 3) * q^53 + (2*z^3 + 2*z^2 - 1) * q^55 + (3*z^3 + 6*z^2 - 6*z - 3) * q^57 + (-3*z^3 + 3*z^2) * q^59 + (-8*z^3 + 8*z^2 + z - 7) * q^61 + (3*z^3 + 6*z^2 - 3*z - 12) * q^63 + (2*z^3 + z^2 + 2*z) * q^65 + (3*z^3 + 5*z^2 + 2*z - 2) * q^67 + (-3*z^2 - 6*z - 3) * q^69 + (4*z^3 - 8*z^2 + 4) * q^71 + (-4*z^3 - 4*z^2 + 2) * q^73 + (3*z^3 - 3*z^2 - 3*z + 6) * q^75 + (3*z^3 + 5*z^2 + 2*z - 2) * q^77 + (z^3 + z) * q^79 + 9 * q^81 + (-9*z^3 + 9*z^2 + 10*z + 1) * q^83 + (4*z^3 - 4*z - 4) * q^85 + (3*z^3 - 3*z^2 + 3) * q^87 + (-2*z^3 - 16*z^2 + 8) * q^89 + (-5*z^3 + 9*z^2 + 9*z - 5) * q^91 + (8*z^3 + 9*z^2 - 4*z - 9) * q^93 + (6*z^3 + 3*z^2 - 3*z - 3) * q^95 - z^2 * q^97 + (-3*z^3 - 3*z^2 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{5} - 12 q^{7} + 12 q^{9}+O(q^{10})$$ 4 * q - 4 * q^5 - 12 * q^7 + 12 * q^9 $$4 q - 4 q^{5} - 12 q^{7} + 12 q^{9} + 2 q^{11} + 8 q^{13} + 6 q^{15} + 16 q^{17} + 12 q^{19} + 6 q^{21} + 12 q^{23} + 6 q^{25} + 8 q^{31} + 14 q^{35} - 24 q^{37} + 18 q^{39} - 18 q^{41} - 14 q^{43} - 12 q^{45} - 8 q^{47} + 12 q^{49} + 16 q^{53} + 6 q^{59} - 12 q^{61} - 36 q^{63} + 2 q^{65} + 2 q^{67} - 18 q^{69} + 18 q^{75} + 2 q^{77} + 36 q^{81} + 22 q^{83} - 16 q^{85} + 6 q^{87} - 2 q^{91} - 18 q^{93} - 6 q^{95} - 2 q^{97} + 6 q^{99}+O(q^{100})$$ 4 * q - 4 * q^5 - 12 * q^7 + 12 * q^9 + 2 * q^11 + 8 * q^13 + 6 * q^15 + 16 * q^17 + 12 * q^19 + 6 * q^21 + 12 * q^23 + 6 * q^25 + 8 * q^31 + 14 * q^35 - 24 * q^37 + 18 * q^39 - 18 * q^41 - 14 * q^43 - 12 * q^45 - 8 * q^47 + 12 * q^49 + 16 * q^53 + 6 * q^59 - 12 * q^61 - 36 * q^63 + 2 * q^65 + 2 * q^67 - 18 * q^69 + 18 * q^75 + 2 * q^77 + 36 * q^81 + 22 * q^83 - 16 * q^85 + 6 * q^87 - 2 * q^91 - 18 * q^93 - 6 * q^95 - 2 * q^97 + 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$325$$ $$\chi(n)$$ $$-1 + \zeta_{12}^{2}$$ $$1$$ $$\zeta_{12}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i
0 −1.73205 0 −1.86603 + 0.500000i 0 −3.86603 + 2.23205i 0 3.00000 0
241.1 0 1.73205 0 −0.133975 + 0.500000i 0 −2.13397 1.23205i 0 3.00000 0
337.1 0 1.73205 0 −0.133975 0.500000i 0 −2.13397 + 1.23205i 0 3.00000 0
529.1 0 −1.73205 0 −1.86603 0.500000i 0 −3.86603 2.23205i 0 3.00000 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
144.x even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.bb.a 4
3.b odd 2 1 1728.2.bc.c 4
4.b odd 2 1 144.2.x.a 4
9.c even 3 1 576.2.bb.b 4
9.d odd 6 1 1728.2.bc.b 4
12.b even 2 1 432.2.y.d 4
16.e even 4 1 576.2.bb.b 4
16.f odd 4 1 144.2.x.d yes 4
36.f odd 6 1 144.2.x.d yes 4
36.h even 6 1 432.2.y.a 4
48.i odd 4 1 1728.2.bc.b 4
48.k even 4 1 432.2.y.a 4
144.u even 12 1 432.2.y.d 4
144.v odd 12 1 144.2.x.a 4
144.w odd 12 1 1728.2.bc.c 4
144.x even 12 1 inner 576.2.bb.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.x.a 4 4.b odd 2 1
144.2.x.a 4 144.v odd 12 1
144.2.x.d yes 4 16.f odd 4 1
144.2.x.d yes 4 36.f odd 6 1
432.2.y.a 4 36.h even 6 1
432.2.y.a 4 48.k even 4 1
432.2.y.d 4 12.b even 2 1
432.2.y.d 4 144.u even 12 1
576.2.bb.a 4 1.a even 1 1 trivial
576.2.bb.a 4 144.x even 12 1 inner
576.2.bb.b 4 9.c even 3 1
576.2.bb.b 4 16.e even 4 1
1728.2.bc.b 4 9.d odd 6 1
1728.2.bc.b 4 48.i odd 4 1
1728.2.bc.c 4 3.b odd 2 1
1728.2.bc.c 4 144.w odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 4T_{5}^{3} + 5T_{5}^{2} + 2T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(576, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 3)^{2}$$
$5$ $$T^{4} + 4 T^{3} + 5 T^{2} + 2 T + 1$$
$7$ $$T^{4} + 12 T^{3} + 59 T^{2} + \cdots + 121$$
$11$ $$T^{4} - 2 T^{3} + 5 T^{2} - 4 T + 1$$
$13$ $$T^{4} - 8 T^{3} + 17 T^{2} - 22 T + 121$$
$17$ $$(T - 4)^{4}$$
$19$ $$(T^{2} - 6 T + 18)^{2}$$
$23$ $$T^{4} - 12 T^{3} + 51 T^{2} - 36 T + 9$$
$29$ $$T^{4} + 9 T^{2} + 18 T + 9$$
$31$ $$T^{4} - 8 T^{3} + 75 T^{2} + 88 T + 121$$
$37$ $$T^{4} + 24 T^{3} + 288 T^{2} + \cdots + 4356$$
$41$ $$T^{4} + 18 T^{3} + 99 T^{2} - 162 T + 81$$
$43$ $$T^{4} + 14 T^{3} + 65 T^{2} + \cdots + 3481$$
$47$ $$T^{4} + 8 T^{3} + 75 T^{2} - 88 T + 121$$
$53$ $$T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 676$$
$59$ $$T^{4} - 6 T^{3} + 45 T^{2} - 108 T + 81$$
$61$ $$T^{4} + 12 T^{3} + 261 T^{2} + \cdots + 1089$$
$67$ $$T^{4} - 2 T^{3} + 65 T^{2} + 176 T + 121$$
$71$ $$T^{4} + 128T^{2} + 1024$$
$73$ $$T^{4} + 56T^{2} + 16$$
$79$ $$T^{4} + 3T^{2} + 9$$
$83$ $$T^{4} - 22 T^{3} + 185 T^{2} + \cdots + 32041$$
$89$ $$T^{4} + 392 T^{2} + 35344$$
$97$ $$(T^{2} + T + 1)^{2}$$