# Properties

 Label 648.2.i.h Level $648$ Weight $2$ Character orbit 648.i Analytic conductor $5.174$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.17430605098$$ 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.

 $$f(q)$$ $$=$$ $$q + 4 \zeta_{6} q^{5} + ( 3 - 3 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + 4 \zeta_{6} q^{5} + ( 3 - 3 \zeta_{6} ) q^{7} + ( 4 - 4 \zeta_{6} ) q^{11} -\zeta_{6} q^{13} + 4 q^{17} - q^{19} + 4 \zeta_{6} q^{23} + ( -11 + 11 \zeta_{6} ) q^{25} + 4 \zeta_{6} q^{31} + 12 q^{35} -9 q^{37} + ( 8 - 8 \zeta_{6} ) q^{43} + ( -12 + 12 \zeta_{6} ) q^{47} -2 \zeta_{6} q^{49} + 8 q^{53} + 16 q^{55} + 4 \zeta_{6} q^{59} + ( 5 - 5 \zeta_{6} ) q^{61} + ( 4 - 4 \zeta_{6} ) q^{65} -11 \zeta_{6} q^{67} -8 q^{71} + q^{73} -12 \zeta_{6} q^{77} + ( 5 - 5 \zeta_{6} ) q^{79} + ( 8 - 8 \zeta_{6} ) q^{83} + 16 \zeta_{6} q^{85} -12 q^{89} -3 q^{91} -4 \zeta_{6} q^{95} + ( -5 + 5 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5} + 3 q^{7} + O(q^{10})$$ $$2 q + 4 q^{5} + 3 q^{7} + 4 q^{11} - q^{13} + 8 q^{17} - 2 q^{19} + 4 q^{23} - 11 q^{25} + 4 q^{31} + 24 q^{35} - 18 q^{37} + 8 q^{43} - 12 q^{47} - 2 q^{49} + 16 q^{53} + 32 q^{55} + 4 q^{59} + 5 q^{61} + 4 q^{65} - 11 q^{67} - 16 q^{71} + 2 q^{73} - 12 q^{77} + 5 q^{79} + 8 q^{83} + 16 q^{85} - 24 q^{89} - 6 q^{91} - 4 q^{95} - 5 q^{97} + O(q^{100})$$

## 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 648.2.i.h 2
3.b odd 2 1 648.2.i.a 2
4.b odd 2 1 1296.2.i.q 2
9.c even 3 1 216.2.a.a 1
9.c even 3 1 inner 648.2.i.h 2
9.d odd 6 1 216.2.a.d yes 1
9.d odd 6 1 648.2.i.a 2
12.b even 2 1 1296.2.i.a 2
36.f odd 6 1 432.2.a.a 1
36.f odd 6 1 1296.2.i.q 2
36.h even 6 1 432.2.a.h 1
36.h even 6 1 1296.2.i.a 2
45.h odd 6 1 5400.2.a.bp 1
45.j even 6 1 5400.2.a.bn 1
45.k odd 12 2 5400.2.f.e 2
45.l even 12 2 5400.2.f.v 2
72.j odd 6 1 1728.2.a.a 1
72.l even 6 1 1728.2.a.b 1
72.n even 6 1 1728.2.a.ba 1
72.p odd 6 1 1728.2.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.a.a 1 9.c even 3 1
216.2.a.d yes 1 9.d odd 6 1
432.2.a.a 1 36.f odd 6 1
432.2.a.h 1 36.h even 6 1
648.2.i.a 2 3.b odd 2 1
648.2.i.a 2 9.d odd 6 1
648.2.i.h 2 1.a even 1 1 trivial
648.2.i.h 2 9.c even 3 1 inner
1296.2.i.a 2 12.b even 2 1
1296.2.i.a 2 36.h even 6 1
1296.2.i.q 2 4.b odd 2 1
1296.2.i.q 2 36.f odd 6 1
1728.2.a.a 1 72.j odd 6 1
1728.2.a.b 1 72.l even 6 1
1728.2.a.ba 1 72.n even 6 1
1728.2.a.bb 1 72.p odd 6 1
5400.2.a.bn 1 45.j even 6 1
5400.2.a.bp 1 45.h odd 6 1
5400.2.f.e 2 45.k odd 12 2
5400.2.f.v 2 45.l even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(648, [\chi])$$:

 $$T_{5}^{2} - 4 T_{5} + 16$$ $$T_{7}^{2} - 3 T_{7} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$16 - 4 T + T^{2}$$
$7$ $$9 - 3 T + T^{2}$$
$11$ $$16 - 4 T + T^{2}$$
$13$ $$1 + T + T^{2}$$
$17$ $$( -4 + T )^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$16 - 4 T + T^{2}$$
$29$ $$T^{2}$$
$31$ $$16 - 4 T + T^{2}$$
$37$ $$( 9 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$64 - 8 T + T^{2}$$
$47$ $$144 + 12 T + T^{2}$$
$53$ $$( -8 + T )^{2}$$
$59$ $$16 - 4 T + T^{2}$$
$61$ $$25 - 5 T + T^{2}$$
$67$ $$121 + 11 T + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$( -1 + T )^{2}$$
$79$ $$25 - 5 T + T^{2}$$
$83$ $$64 - 8 T + T^{2}$$
$89$ $$( 12 + T )^{2}$$
$97$ $$25 + 5 T + T^{2}$$