# Properties

 Label 576.2.k.c Level $576$ Weight $2$ Character orbit 576.k Analytic conductor $4.599$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.59938315643$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.629407744.1 Defining polynomial: $$x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 144) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} + ( \beta_{3} - \beta_{7} ) q^{7} +O(q^{10})$$ $$q + \beta_{2} q^{5} + ( \beta_{3} - \beta_{7} ) q^{7} + ( -\beta_{1} - \beta_{2} ) q^{11} + ( \beta_{6} - \beta_{7} ) q^{13} + ( \beta_{1} + \beta_{5} ) q^{17} + ( 1 - \beta_{3} + \beta_{6} - \beta_{7} ) q^{19} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{23} + ( \beta_{3} - 2 \beta_{7} ) q^{25} + ( -\beta_{4} - 2 \beta_{5} ) q^{29} + ( 3 + \beta_{6} ) q^{31} + ( -3 \beta_{4} + \beta_{5} ) q^{35} + ( 2 + 2 \beta_{3} - \beta_{6} - \beta_{7} ) q^{37} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{41} + ( 3 + 3 \beta_{3} - \beta_{6} - \beta_{7} ) q^{43} + ( \beta_{1} + 3 \beta_{2} + 3 \beta_{4} + \beta_{5} ) q^{47} + ( -1 + 2 \beta_{6} ) q^{49} + ( -2 \beta_{1} + \beta_{2} ) q^{53} -4 \beta_{3} q^{55} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( 2 - 2 \beta_{3} - \beta_{6} + \beta_{7} ) q^{61} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{65} + ( -4 + 4 \beta_{3} ) q^{67} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{71} + ( -2 \beta_{3} + 2 \beta_{7} ) q^{73} -2 \beta_{5} q^{77} + ( -7 - \beta_{6} ) q^{79} + ( \beta_{4} + \beta_{5} ) q^{83} + ( -2 - 2 \beta_{3} + 2 \beta_{6} + 2 \beta_{7} ) q^{85} + ( -2 \beta_{2} + 2 \beta_{4} ) q^{89} + ( -7 - 7 \beta_{3} + \beta_{6} + \beta_{7} ) q^{91} + ( \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} ) q^{95} -4 \beta_{6} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 8q^{19} + 24q^{31} + 16q^{37} + 24q^{43} - 8q^{49} + 16q^{61} - 32q^{67} - 56q^{79} - 16q^{85} - 56q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 2 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - 4 \nu^{5} + 10 \nu^{3} - 4 \nu$$$$)/12$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 2 \nu^{4} - 2 \nu^{2} - 4$$$$)/12$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{5} + 2 \nu^{3} - 2 \nu$$$$)/6$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} - 2 \nu^{3} + 12 \nu$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} + 2 \nu^{4} + 2 \nu^{2} + 4$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{6} - \nu^{4} + 4 \nu^{2} - 7$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} + \beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} - \beta_{3} + 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{4} + 2 \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} + 3 \beta_{3}$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{5} + \beta_{4} - \beta_{2} + 3 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$\beta_{7} - \beta_{6} + 5 \beta_{3} + 5$$ $$\nu^{7}$$ $$=$$ $$-\beta_{5} - 4 \beta_{4} + \beta_{2} + 2 \beta_{1}$$

## Character values

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

 $$n$$ $$65$$ $$127$$ $$325$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 0.767178 + 1.18804i −1.38255 + 0.297594i 1.38255 − 0.297594i −0.767178 − 1.18804i 0.767178 − 1.18804i −1.38255 − 0.297594i 1.38255 + 0.297594i −0.767178 + 1.18804i
0 0 0 −2.37608 + 2.37608i 0 3.64575i 0 0 0
145.2 0 0 0 −0.595188 + 0.595188i 0 1.64575i 0 0 0
145.3 0 0 0 0.595188 0.595188i 0 1.64575i 0 0 0
145.4 0 0 0 2.37608 2.37608i 0 3.64575i 0 0 0
433.1 0 0 0 −2.37608 2.37608i 0 3.64575i 0 0 0
433.2 0 0 0 −0.595188 0.595188i 0 1.64575i 0 0 0
433.3 0 0 0 0.595188 + 0.595188i 0 1.64575i 0 0 0
433.4 0 0 0 2.37608 + 2.37608i 0 3.64575i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 433.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.k.c 8
3.b odd 2 1 inner 576.2.k.c 8
4.b odd 2 1 144.2.k.c 8
8.b even 2 1 1152.2.k.d 8
8.d odd 2 1 1152.2.k.e 8
12.b even 2 1 144.2.k.c 8
16.e even 4 1 inner 576.2.k.c 8
16.e even 4 1 1152.2.k.d 8
16.f odd 4 1 144.2.k.c 8
16.f odd 4 1 1152.2.k.e 8
24.f even 2 1 1152.2.k.e 8
24.h odd 2 1 1152.2.k.d 8
32.g even 8 2 9216.2.a.bt 8
32.h odd 8 2 9216.2.a.bq 8
48.i odd 4 1 inner 576.2.k.c 8
48.i odd 4 1 1152.2.k.d 8
48.k even 4 1 144.2.k.c 8
48.k even 4 1 1152.2.k.e 8
96.o even 8 2 9216.2.a.bq 8
96.p odd 8 2 9216.2.a.bt 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.k.c 8 4.b odd 2 1
144.2.k.c 8 12.b even 2 1
144.2.k.c 8 16.f odd 4 1
144.2.k.c 8 48.k even 4 1
576.2.k.c 8 1.a even 1 1 trivial
576.2.k.c 8 3.b odd 2 1 inner
576.2.k.c 8 16.e even 4 1 inner
576.2.k.c 8 48.i odd 4 1 inner
1152.2.k.d 8 8.b even 2 1
1152.2.k.d 8 16.e even 4 1
1152.2.k.d 8 24.h odd 2 1
1152.2.k.d 8 48.i odd 4 1
1152.2.k.e 8 8.d odd 2 1
1152.2.k.e 8 16.f odd 4 1
1152.2.k.e 8 24.f even 2 1
1152.2.k.e 8 48.k even 4 1
9216.2.a.bq 8 32.h odd 8 2
9216.2.a.bq 8 96.o even 8 2
9216.2.a.bt 8 32.g even 8 2
9216.2.a.bt 8 96.p odd 8 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 128 T_{5}^{4} + 64$$ acting on $$S_{2}^{\mathrm{new}}(576, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$64 + 128 T^{4} + T^{8}$$
$7$ $$( 36 + 16 T^{2} + T^{4} )^{2}$$
$11$ $$1024 + 512 T^{4} + T^{8}$$
$13$ $$( 196 + T^{4} )^{2}$$
$17$ $$( 288 - 40 T^{2} + T^{4} )^{2}$$
$19$ $$( 144 + 48 T + 8 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$23$ $$( 1152 + 80 T^{2} + T^{4} )^{2}$$
$29$ $$5184 + 5632 T^{4} + T^{8}$$
$31$ $$( 2 - 6 T + T^{2} )^{4}$$
$37$ $$( 36 + 48 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$41$ $$( 2592 + 104 T^{2} + T^{4} )^{2}$$
$43$ $$( 16 - 48 T + 72 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$47$ $$( 10368 - 208 T^{2} + T^{4} )^{2}$$
$53$ $$8340544 + 5888 T^{4} + T^{8}$$
$59$ $$21233664 + 16384 T^{4} + T^{8}$$
$61$ $$( 36 + 48 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$67$ $$( 32 + 8 T + T^{2} )^{4}$$
$71$ $$( 2048 + 192 T^{2} + T^{4} )^{2}$$
$73$ $$( 576 + 64 T^{2} + T^{4} )^{2}$$
$79$ $$( 42 + 14 T + T^{2} )^{4}$$
$83$ $$1024 + 512 T^{4} + T^{8}$$
$89$ $$( 512 + 96 T^{2} + T^{4} )^{2}$$
$97$ $$( -112 + T^{2} )^{4}$$