# Properties

 Label 648.4.i.f 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{5} + ( 9 - 9 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -\zeta_{6} q^{5} + ( 9 - 9 \zeta_{6} ) q^{7} + ( 17 - 17 \zeta_{6} ) q^{11} + 44 \zeta_{6} q^{13} + 56 q^{17} -94 q^{19} + 50 \zeta_{6} q^{23} + ( 124 - 124 \zeta_{6} ) q^{25} + ( 30 - 30 \zeta_{6} ) q^{29} + 139 \zeta_{6} q^{31} -9 q^{35} -174 q^{37} -318 \zeta_{6} q^{41} + ( 242 - 242 \zeta_{6} ) q^{43} + ( 630 - 630 \zeta_{6} ) q^{47} + 262 \zeta_{6} q^{49} + 547 q^{53} -17 q^{55} + 236 \zeta_{6} q^{59} + ( -328 + 328 \zeta_{6} ) q^{61} + ( 44 - 44 \zeta_{6} ) q^{65} -614 \zeta_{6} q^{67} + 296 q^{71} + 433 q^{73} -153 \zeta_{6} q^{77} + ( 56 - 56 \zeta_{6} ) q^{79} + ( 1225 - 1225 \zeta_{6} ) q^{83} -56 \zeta_{6} q^{85} + 1506 q^{89} + 396 q^{91} + 94 \zeta_{6} q^{95} + ( -1391 + 1391 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{5} + 9q^{7} + O(q^{10})$$ $$2q - q^{5} + 9q^{7} + 17q^{11} + 44q^{13} + 112q^{17} - 188q^{19} + 50q^{23} + 124q^{25} + 30q^{29} + 139q^{31} - 18q^{35} - 348q^{37} - 318q^{41} + 242q^{43} + 630q^{47} + 262q^{49} + 1094q^{53} - 34q^{55} + 236q^{59} - 328q^{61} + 44q^{65} - 614q^{67} + 592q^{71} + 866q^{73} - 153q^{77} + 56q^{79} + 1225q^{83} - 56q^{85} + 3012q^{89} + 792q^{91} + 94q^{95} - 1391q^{97} + O(q^{100})$$

## 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 −0.500000 0.866025i 0 4.50000 7.79423i 0 0 0
433.1 0 0 0 −0.500000 + 0.866025i 0 4.50000 + 7.79423i 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.f 2
3.b odd 2 1 648.4.i.g 2
9.c even 3 1 216.4.a.c yes 1
9.c even 3 1 inner 648.4.i.f 2
9.d odd 6 1 216.4.a.b 1
9.d odd 6 1 648.4.i.g 2
36.f odd 6 1 432.4.a.i 1
36.h even 6 1 432.4.a.f 1
72.j odd 6 1 1728.4.a.s 1
72.l even 6 1 1728.4.a.t 1
72.n even 6 1 1728.4.a.m 1
72.p odd 6 1 1728.4.a.n 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + T_{5} + 1$$ acting on $$S_{4}^{\mathrm{new}}(648, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$81 - 9 T + T^{2}$$
$11$ $$289 - 17 T + T^{2}$$
$13$ $$1936 - 44 T + T^{2}$$
$17$ $$( -56 + T )^{2}$$
$19$ $$( 94 + T )^{2}$$
$23$ $$2500 - 50 T + T^{2}$$
$29$ $$900 - 30 T + T^{2}$$
$31$ $$19321 - 139 T + T^{2}$$
$37$ $$( 174 + T )^{2}$$
$41$ $$101124 + 318 T + T^{2}$$
$43$ $$58564 - 242 T + T^{2}$$
$47$ $$396900 - 630 T + T^{2}$$
$53$ $$( -547 + T )^{2}$$
$59$ $$55696 - 236 T + T^{2}$$
$61$ $$107584 + 328 T + T^{2}$$
$67$ $$376996 + 614 T + T^{2}$$
$71$ $$( -296 + T )^{2}$$
$73$ $$( -433 + T )^{2}$$
$79$ $$3136 - 56 T + T^{2}$$
$83$ $$1500625 - 1225 T + T^{2}$$
$89$ $$( -1506 + T )^{2}$$
$97$ $$1934881 + 1391 T + T^{2}$$