# Properties

 Label 576.7.g.h Level $576$ Weight $7$ Character orbit 576.g Analytic conductor $132.511$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 576.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$132.511152165$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-15})$$ Defining polynomial: $$x^{2} - x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 4) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 8\sqrt{-15}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 10 q^{5} + 10 \beta q^{7} +O(q^{10})$$ $$q + 10 q^{5} + 10 \beta q^{7} -31 \beta q^{11} -1466 q^{13} + 4766 q^{17} -243 \beta q^{19} + 338 \beta q^{23} -15525 q^{25} + 25498 q^{29} + 1352 \beta q^{31} + 100 \beta q^{35} -1994 q^{37} -29362 q^{41} + 695 \beta q^{43} + 244 \beta q^{47} + 21649 q^{49} -192854 q^{53} -310 \beta q^{55} -2531 \beta q^{59} + 10918 q^{61} -14660 q^{65} + 12721 \beta q^{67} -17178 \beta q^{71} + 288626 q^{73} + 297600 q^{77} -10028 \beta q^{79} -6589 \beta q^{83} + 47660 q^{85} -310738 q^{89} -14660 \beta q^{91} -2430 \beta q^{95} -1457086 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 20q^{5} + O(q^{10})$$ $$2q + 20q^{5} - 2932q^{13} + 9532q^{17} - 31050q^{25} + 50996q^{29} - 3988q^{37} - 58724q^{41} + 43298q^{49} - 385708q^{53} + 21836q^{61} - 29320q^{65} + 577252q^{73} + 595200q^{77} + 95320q^{85} - 621476q^{89} - 2914172q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$65$$ $$127$$ $$325$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 0.5 − 1.93649i 0.5 + 1.93649i
0 0 0 10.0000 0 309.839i 0 0 0
127.2 0 0 0 10.0000 0 309.839i 0 0 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.7.g.h 2
3.b odd 2 1 64.7.c.c 2
4.b odd 2 1 inner 576.7.g.h 2
8.b even 2 1 36.7.d.c 2
8.d odd 2 1 36.7.d.c 2
12.b even 2 1 64.7.c.c 2
24.f even 2 1 4.7.b.a 2
24.h odd 2 1 4.7.b.a 2
48.i odd 4 2 256.7.d.f 4
48.k even 4 2 256.7.d.f 4
120.i odd 2 1 100.7.b.c 2
120.m even 2 1 100.7.b.c 2
120.q odd 4 2 100.7.d.a 4
120.w even 4 2 100.7.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.7.b.a 2 24.f even 2 1
4.7.b.a 2 24.h odd 2 1
36.7.d.c 2 8.b even 2 1
36.7.d.c 2 8.d odd 2 1
64.7.c.c 2 3.b odd 2 1
64.7.c.c 2 12.b even 2 1
100.7.b.c 2 120.i odd 2 1
100.7.b.c 2 120.m even 2 1
100.7.d.a 4 120.q odd 4 2
100.7.d.a 4 120.w even 4 2
256.7.d.f 4 48.i odd 4 2
256.7.d.f 4 48.k even 4 2
576.7.g.h 2 1.a even 1 1 trivial
576.7.g.h 2 4.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 10$$ acting on $$S_{7}^{\mathrm{new}}(576, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -10 + T )^{2}$$
$7$ $$96000 + T^{2}$$
$11$ $$922560 + T^{2}$$
$13$ $$( 1466 + T )^{2}$$
$17$ $$( -4766 + T )^{2}$$
$19$ $$56687040 + T^{2}$$
$23$ $$109674240 + T^{2}$$
$29$ $$( -25498 + T )^{2}$$
$31$ $$1754787840 + T^{2}$$
$37$ $$( 1994 + T )^{2}$$
$41$ $$( 29362 + T )^{2}$$
$43$ $$463704000 + T^{2}$$
$47$ $$57154560 + T^{2}$$
$53$ $$( 192854 + T )^{2}$$
$59$ $$6149722560 + T^{2}$$
$61$ $$( -10918 + T )^{2}$$
$67$ $$155350887360 + T^{2}$$
$71$ $$283280336640 + T^{2}$$
$73$ $$( -288626 + T )^{2}$$
$79$ $$96538352640 + T^{2}$$
$83$ $$41678324160 + T^{2}$$
$89$ $$( 310738 + T )^{2}$$
$97$ $$( 1457086 + T )^{2}$$