# Properties

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

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## 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: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - \zeta_{12}^{2} ) q^{3} + ( -2 - 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{5} + ( 2 - \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{7} + ( 3 - 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( 2 - \zeta_{12}^{2} ) q^{3} + ( -2 - 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{5} + ( 2 - \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{7} + ( 3 - 3 \zeta_{12}^{2} ) q^{9} + ( 3 - 2 \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{11} + ( -1 + \zeta_{12} - 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{13} + ( -4 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{15} + ( -4 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{17} + ( 1 - \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{19} + ( 3 - \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{21} + ( -1 + 3 \zeta_{12} - \zeta_{12}^{2} ) q^{23} + ( 8 + 3 \zeta_{12} - 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{25} + ( 3 - 6 \zeta_{12}^{2} ) q^{27} + ( -2 - \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{29} + ( 4 + 2 \zeta_{12} - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{31} + ( 5 - 5 \zeta_{12} - 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{33} + ( -2 + 2 \zeta_{12}^{3} ) q^{35} + ( 4 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{37} + ( -5 - 2 \zeta_{12} - 2 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{39} -3 \zeta_{12} q^{41} + ( 3 - 5 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{43} + ( -6 + 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{45} + ( 5 \zeta_{12} + \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{47} + ( -3 - 2 \zeta_{12} + 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{49} + ( -8 + 3 \zeta_{12} + 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{51} + ( -6 + 4 \zeta_{12} + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{53} + ( -2 + 4 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{55} + ( 3 - 3 \zeta_{12} ) q^{57} + ( -4 - 4 \zeta_{12} + 5 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{59} + ( -6 + 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{61} + ( 3 - 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{63} + ( 6 \zeta_{12} + 14 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{65} + ( -3 + 3 \zeta_{12} - 2 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{67} + ( -3 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{69} + ( 4 - 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{71} + ( -1 + 2 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{73} + ( 12 + 3 \zeta_{12} - 12 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{75} + ( 7 - 7 \zeta_{12} - 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{77} + 12 \zeta_{12}^{2} q^{79} -9 \zeta_{12}^{2} q^{81} + ( \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{83} + ( 4 + 4 \zeta_{12} + 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{85} + ( -3 - 3 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{87} + 2 \zeta_{12}^{3} q^{89} + ( -6 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{91} + ( 4 - 8 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{93} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{95} + ( \zeta_{12} - 10 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{97} + ( 6 - 9 \zeta_{12} - 9 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{3} - 8q^{5} + 6q^{7} + 6q^{9} + O(q^{10})$$ $$4q + 6q^{3} - 8q^{5} + 6q^{7} + 6q^{9} + 10q^{11} - 10q^{13} - 12q^{15} - 16q^{17} + 6q^{19} + 6q^{21} - 6q^{23} + 24q^{25} - 6q^{29} + 8q^{31} + 12q^{33} - 8q^{35} + 12q^{37} - 24q^{39} + 16q^{43} - 12q^{45} + 2q^{47} - 6q^{49} - 24q^{51} - 16q^{53} + 12q^{57} - 6q^{59} - 12q^{61} + 28q^{65} - 16q^{67} - 12q^{69} + 24q^{75} + 22q^{77} + 24q^{79} - 18q^{81} + 2q^{83} + 20q^{85} - 6q^{87} - 20q^{91} - 20q^{97} + 6q^{99} + O(q^{100})$$

## 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 1.50000 0.866025i 0 −3.73205 + 1.00000i 0 0.633975 0.366025i 0 1.50000 2.59808i 0
241.1 0 1.50000 + 0.866025i 0 −0.267949 + 1.00000i 0 2.36603 + 1.36603i 0 1.50000 + 2.59808i 0
337.1 0 1.50000 0.866025i 0 −0.267949 1.00000i 0 2.36603 1.36603i 0 1.50000 2.59808i 0
529.1 0 1.50000 + 0.866025i 0 −3.73205 1.00000i 0 0.633975 + 0.366025i 0 1.50000 + 2.59808i 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.d 4
3.b odd 2 1 1728.2.bc.d 4
4.b odd 2 1 144.2.x.b 4
9.c even 3 1 576.2.bb.c 4
9.d odd 6 1 1728.2.bc.a 4
12.b even 2 1 432.2.y.c 4
16.e even 4 1 576.2.bb.c 4
16.f odd 4 1 144.2.x.c yes 4
36.f odd 6 1 144.2.x.c yes 4
36.h even 6 1 432.2.y.b 4
48.i odd 4 1 1728.2.bc.a 4
48.k even 4 1 432.2.y.b 4
144.u even 12 1 432.2.y.c 4
144.v odd 12 1 144.2.x.b 4
144.w odd 12 1 1728.2.bc.d 4
144.x even 12 1 inner 576.2.bb.d 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 8 T_{5}^{3} + 20 T_{5}^{2} + 16 T_{5} + 16$$ acting on $$S_{2}^{\mathrm{new}}(576, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 - 3 T + T^{2} )^{2}$$
$5$ $$16 + 16 T + 20 T^{2} + 8 T^{3} + T^{4}$$
$7$ $$4 - 12 T + 14 T^{2} - 6 T^{3} + T^{4}$$
$11$ $$169 - 104 T + 41 T^{2} - 10 T^{3} + T^{4}$$
$13$ $$484 + 308 T + 74 T^{2} + 10 T^{3} + T^{4}$$
$17$ $$( 13 + 8 T + T^{2} )^{2}$$
$19$ $$9 - 18 T + 18 T^{2} - 6 T^{3} + T^{4}$$
$23$ $$36 - 36 T + 6 T^{2} + 6 T^{3} + T^{4}$$
$29$ $$36 + 36 T + 18 T^{2} + 6 T^{3} + T^{4}$$
$31$ $$16 - 32 T + 60 T^{2} - 8 T^{3} + T^{4}$$
$37$ $$144 - 144 T + 72 T^{2} - 12 T^{3} + T^{4}$$
$41$ $$81 - 9 T^{2} + T^{4}$$
$43$ $$121 + 22 T + 65 T^{2} - 16 T^{3} + T^{4}$$
$47$ $$5476 + 148 T + 78 T^{2} - 2 T^{3} + T^{4}$$
$53$ $$64 + 128 T + 128 T^{2} + 16 T^{3} + T^{4}$$
$59$ $$1521 + 468 T + 45 T^{2} + 6 T^{3} + T^{4}$$
$61$ $$1296 + 864 T + 180 T^{2} + 12 T^{3} + T^{4}$$
$67$ $$1369 + 518 T + 113 T^{2} + 16 T^{3} + T^{4}$$
$71$ $$1024 + 128 T^{2} + T^{4}$$
$73$ $$3721 + 134 T^{2} + T^{4}$$
$79$ $$( 144 - 12 T + T^{2} )^{2}$$
$83$ $$4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4}$$
$89$ $$( 4 + T^{2} )^{2}$$
$97$ $$9409 + 1940 T + 303 T^{2} + 20 T^{3} + T^{4}$$
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