# Properties

 Label 648.4.i.l 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: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) 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.

 $$f(q)$$ $$=$$ $$q + 16 \zeta_{6} q^{5} + ( 12 - 12 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + 16 \zeta_{6} q^{5} + ( 12 - 12 \zeta_{6} ) q^{7} + ( 64 - 64 \zeta_{6} ) q^{11} -58 \zeta_{6} q^{13} -32 q^{17} -136 q^{19} -128 \zeta_{6} q^{23} + ( -131 + 131 \zeta_{6} ) q^{25} + ( -144 + 144 \zeta_{6} ) q^{29} -20 \zeta_{6} q^{31} + 192 q^{35} -18 q^{37} -288 \zeta_{6} q^{41} + ( 200 - 200 \zeta_{6} ) q^{43} + ( 384 - 384 \zeta_{6} ) q^{47} + 199 \zeta_{6} q^{49} -496 q^{53} + 1024 q^{55} -128 \zeta_{6} q^{59} + ( 458 - 458 \zeta_{6} ) q^{61} + ( 928 - 928 \zeta_{6} ) q^{65} + 496 \zeta_{6} q^{67} -512 q^{71} -602 q^{73} -768 \zeta_{6} q^{77} + ( -1108 + 1108 \zeta_{6} ) q^{79} + ( 704 - 704 \zeta_{6} ) q^{83} -512 \zeta_{6} q^{85} + 960 q^{89} -696 q^{91} -2176 \zeta_{6} q^{95} + ( -206 + 206 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 16q^{5} + 12q^{7} + O(q^{10})$$ $$2q + 16q^{5} + 12q^{7} + 64q^{11} - 58q^{13} - 64q^{17} - 272q^{19} - 128q^{23} - 131q^{25} - 144q^{29} - 20q^{31} + 384q^{35} - 36q^{37} - 288q^{41} + 200q^{43} + 384q^{47} + 199q^{49} - 992q^{53} + 2048q^{55} - 128q^{59} + 458q^{61} + 928q^{65} + 496q^{67} - 1024q^{71} - 1204q^{73} - 768q^{77} - 1108q^{79} + 704q^{83} - 512q^{85} + 1920q^{89} - 1392q^{91} - 2176q^{95} - 206q^{97} + O(q^{100})$$

## 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 8.00000 + 13.8564i 0 6.00000 10.3923i 0 0 0
433.1 0 0 0 8.00000 13.8564i 0 6.00000 + 10.3923i 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.l 2
3.b odd 2 1 648.4.i.a 2
9.c even 3 1 72.4.a.a 1
9.c even 3 1 inner 648.4.i.l 2
9.d odd 6 1 72.4.a.d yes 1
9.d odd 6 1 648.4.i.a 2
36.f odd 6 1 144.4.a.a 1
36.h even 6 1 144.4.a.f 1
45.h odd 6 1 1800.4.a.ba 1
45.j even 6 1 1800.4.a.z 1
45.k odd 12 2 1800.4.f.b 2
45.l even 12 2 1800.4.f.x 2
72.j odd 6 1 576.4.a.c 1
72.l even 6 1 576.4.a.d 1
72.n even 6 1 576.4.a.w 1
72.p odd 6 1 576.4.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.a.a 1 9.c even 3 1
72.4.a.d yes 1 9.d odd 6 1
144.4.a.a 1 36.f odd 6 1
144.4.a.f 1 36.h even 6 1
576.4.a.c 1 72.j odd 6 1
576.4.a.d 1 72.l even 6 1
576.4.a.w 1 72.n even 6 1
576.4.a.x 1 72.p odd 6 1
648.4.i.a 2 3.b odd 2 1
648.4.i.a 2 9.d odd 6 1
648.4.i.l 2 1.a even 1 1 trivial
648.4.i.l 2 9.c even 3 1 inner
1800.4.a.z 1 45.j even 6 1
1800.4.a.ba 1 45.h odd 6 1
1800.4.f.b 2 45.k odd 12 2
1800.4.f.x 2 45.l even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 16 T_{5} + 256$$ acting on $$S_{4}^{\mathrm{new}}(648, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$256 - 16 T + T^{2}$$
$7$ $$144 - 12 T + T^{2}$$
$11$ $$4096 - 64 T + T^{2}$$
$13$ $$3364 + 58 T + T^{2}$$
$17$ $$( 32 + T )^{2}$$
$19$ $$( 136 + T )^{2}$$
$23$ $$16384 + 128 T + T^{2}$$
$29$ $$20736 + 144 T + T^{2}$$
$31$ $$400 + 20 T + T^{2}$$
$37$ $$( 18 + T )^{2}$$
$41$ $$82944 + 288 T + T^{2}$$
$43$ $$40000 - 200 T + T^{2}$$
$47$ $$147456 - 384 T + T^{2}$$
$53$ $$( 496 + T )^{2}$$
$59$ $$16384 + 128 T + T^{2}$$
$61$ $$209764 - 458 T + T^{2}$$
$67$ $$246016 - 496 T + T^{2}$$
$71$ $$( 512 + T )^{2}$$
$73$ $$( 602 + T )^{2}$$
$79$ $$1227664 + 1108 T + T^{2}$$
$83$ $$495616 - 704 T + T^{2}$$
$89$ $$( -960 + T )^{2}$$
$97$ $$42436 + 206 T + T^{2}$$