# Properties

 Label 648.4.i.d Level $648$ Weight $4$ Character orbit 648.i Analytic conductor $38.233$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$648 = 2^{3} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 648.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$38.2332376837$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 216) 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -4 \zeta_{6} q^{5} + ( -3 + 3 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -4 \zeta_{6} q^{5} + ( -3 + 3 \zeta_{6} ) q^{7} + ( -28 + 28 \zeta_{6} ) q^{11} + 11 \zeta_{6} q^{13} + 44 q^{17} + 29 q^{19} -172 \zeta_{6} q^{23} + ( 109 - 109 \zeta_{6} ) q^{25} + ( -192 + 192 \zeta_{6} ) q^{29} -116 \zeta_{6} q^{31} + 12 q^{35} -69 q^{37} -384 \zeta_{6} q^{41} + ( -328 + 328 \zeta_{6} ) q^{43} + ( -156 + 156 \zeta_{6} ) q^{47} + 334 \zeta_{6} q^{49} -392 q^{53} + 112 q^{55} -412 \zeta_{6} q^{59} + ( 425 - 425 \zeta_{6} ) q^{61} + ( 44 - 44 \zeta_{6} ) q^{65} -257 \zeta_{6} q^{67} -1000 q^{71} -359 q^{73} -84 \zeta_{6} q^{77} + ( -877 + 877 \zeta_{6} ) q^{79} + ( 328 - 328 \zeta_{6} ) q^{83} -176 \zeta_{6} q^{85} -1572 q^{89} -33 q^{91} -116 \zeta_{6} q^{95} + ( 1483 - 1483 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{5} - 3q^{7} + O(q^{10})$$ $$2q - 4q^{5} - 3q^{7} - 28q^{11} + 11q^{13} + 88q^{17} + 58q^{19} - 172q^{23} + 109q^{25} - 192q^{29} - 116q^{31} + 24q^{35} - 138q^{37} - 384q^{41} - 328q^{43} - 156q^{47} + 334q^{49} - 784q^{53} + 224q^{55} - 412q^{59} + 425q^{61} + 44q^{65} - 257q^{67} - 2000q^{71} - 718q^{73} - 84q^{77} - 877q^{79} + 328q^{83} - 176q^{85} - 3144q^{89} - 66q^{91} - 116q^{95} + 1483q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$487$$ $$569$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
217.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −2.00000 3.46410i 0 −1.50000 + 2.59808i 0 0 0
433.1 0 0 0 −2.00000 + 3.46410i 0 −1.50000 2.59808i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.4.i.d 2
3.b odd 2 1 648.4.i.i 2
9.c even 3 1 216.4.a.d yes 1
9.c even 3 1 inner 648.4.i.d 2
9.d odd 6 1 216.4.a.a 1
9.d odd 6 1 648.4.i.i 2
36.f odd 6 1 432.4.a.k 1
36.h even 6 1 432.4.a.d 1
72.j odd 6 1 1728.4.a.x 1
72.l even 6 1 1728.4.a.w 1
72.n even 6 1 1728.4.a.j 1
72.p odd 6 1 1728.4.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.4.a.a 1 9.d odd 6 1
216.4.a.d yes 1 9.c even 3 1
432.4.a.d 1 36.h even 6 1
432.4.a.k 1 36.f odd 6 1
648.4.i.d 2 1.a even 1 1 trivial
648.4.i.d 2 9.c even 3 1 inner
648.4.i.i 2 3.b odd 2 1
648.4.i.i 2 9.d odd 6 1
1728.4.a.i 1 72.p odd 6 1
1728.4.a.j 1 72.n even 6 1
1728.4.a.w 1 72.l even 6 1
1728.4.a.x 1 72.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 4 T_{5} + 16$$ acting on $$S_{4}^{\mathrm{new}}(648, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$16 + 4 T + T^{2}$$
$7$ $$9 + 3 T + T^{2}$$
$11$ $$784 + 28 T + T^{2}$$
$13$ $$121 - 11 T + T^{2}$$
$17$ $$( -44 + T )^{2}$$
$19$ $$( -29 + T )^{2}$$
$23$ $$29584 + 172 T + T^{2}$$
$29$ $$36864 + 192 T + T^{2}$$
$31$ $$13456 + 116 T + T^{2}$$
$37$ $$( 69 + T )^{2}$$
$41$ $$147456 + 384 T + T^{2}$$
$43$ $$107584 + 328 T + T^{2}$$
$47$ $$24336 + 156 T + T^{2}$$
$53$ $$( 392 + T )^{2}$$
$59$ $$169744 + 412 T + T^{2}$$
$61$ $$180625 - 425 T + T^{2}$$
$67$ $$66049 + 257 T + T^{2}$$
$71$ $$( 1000 + T )^{2}$$
$73$ $$( 359 + T )^{2}$$
$79$ $$769129 + 877 T + T^{2}$$
$83$ $$107584 - 328 T + T^{2}$$
$89$ $$( 1572 + T )^{2}$$
$97$ $$2199289 - 1483 T + T^{2}$$