# Properties

 Label 648.4.i.k 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 24) 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.

 $$f(q)$$ $$=$$ $$q + 14 \zeta_{6} q^{5} + ( 24 - 24 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + 14 \zeta_{6} q^{5} + ( 24 - 24 \zeta_{6} ) q^{7} + ( -28 + 28 \zeta_{6} ) q^{11} + 74 \zeta_{6} q^{13} -82 q^{17} + 92 q^{19} + 8 \zeta_{6} q^{23} + ( -71 + 71 \zeta_{6} ) q^{25} + ( -138 + 138 \zeta_{6} ) q^{29} -80 \zeta_{6} q^{31} + 336 q^{35} + 30 q^{37} + 282 \zeta_{6} q^{41} + ( -4 + 4 \zeta_{6} ) q^{43} + ( 240 - 240 \zeta_{6} ) q^{47} -233 \zeta_{6} q^{49} + 130 q^{53} -392 q^{55} + 596 \zeta_{6} q^{59} + ( 218 - 218 \zeta_{6} ) q^{61} + ( -1036 + 1036 \zeta_{6} ) q^{65} + 436 \zeta_{6} q^{67} -856 q^{71} -998 q^{73} + 672 \zeta_{6} q^{77} + ( 32 - 32 \zeta_{6} ) q^{79} + ( -1508 + 1508 \zeta_{6} ) q^{83} -1148 \zeta_{6} q^{85} + 246 q^{89} + 1776 q^{91} + 1288 \zeta_{6} q^{95} + ( -866 + 866 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 14q^{5} + 24q^{7} + O(q^{10})$$ $$2q + 14q^{5} + 24q^{7} - 28q^{11} + 74q^{13} - 164q^{17} + 184q^{19} + 8q^{23} - 71q^{25} - 138q^{29} - 80q^{31} + 672q^{35} + 60q^{37} + 282q^{41} - 4q^{43} + 240q^{47} - 233q^{49} + 260q^{53} - 784q^{55} + 596q^{59} + 218q^{61} - 1036q^{65} + 436q^{67} - 1712q^{71} - 1996q^{73} + 672q^{77} + 32q^{79} - 1508q^{83} - 1148q^{85} + 492q^{89} + 3552q^{91} + 1288q^{95} - 866q^{97} + O(q^{100})$$

## 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 7.00000 + 12.1244i 0 12.0000 20.7846i 0 0 0
433.1 0 0 0 7.00000 12.1244i 0 12.0000 + 20.7846i 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.k 2
3.b odd 2 1 648.4.i.b 2
9.c even 3 1 72.4.a.b 1
9.c even 3 1 inner 648.4.i.k 2
9.d odd 6 1 24.4.a.a 1
9.d odd 6 1 648.4.i.b 2
36.f odd 6 1 144.4.a.b 1
36.h even 6 1 48.4.a.b 1
45.h odd 6 1 600.4.a.h 1
45.j even 6 1 1800.4.a.bg 1
45.k odd 12 2 1800.4.f.q 2
45.l even 12 2 600.4.f.b 2
63.o even 6 1 1176.4.a.a 1
72.j odd 6 1 192.4.a.a 1
72.l even 6 1 192.4.a.g 1
72.n even 6 1 576.4.a.u 1
72.p odd 6 1 576.4.a.v 1
144.u even 12 2 768.4.d.b 2
144.w odd 12 2 768.4.d.o 2
180.n even 6 1 1200.4.a.u 1
180.v odd 12 2 1200.4.f.p 2
252.s odd 6 1 2352.4.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.a.a 1 9.d odd 6 1
48.4.a.b 1 36.h even 6 1
72.4.a.b 1 9.c even 3 1
144.4.a.b 1 36.f odd 6 1
192.4.a.a 1 72.j odd 6 1
192.4.a.g 1 72.l even 6 1
576.4.a.u 1 72.n even 6 1
576.4.a.v 1 72.p odd 6 1
600.4.a.h 1 45.h odd 6 1
600.4.f.b 2 45.l even 12 2
648.4.i.b 2 3.b odd 2 1
648.4.i.b 2 9.d odd 6 1
648.4.i.k 2 1.a even 1 1 trivial
648.4.i.k 2 9.c even 3 1 inner
768.4.d.b 2 144.u even 12 2
768.4.d.o 2 144.w odd 12 2
1176.4.a.a 1 63.o even 6 1
1200.4.a.u 1 180.n even 6 1
1200.4.f.p 2 180.v odd 12 2
1800.4.a.bg 1 45.j even 6 1
1800.4.f.q 2 45.k odd 12 2
2352.4.a.w 1 252.s odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 14 T_{5} + 196$$ acting on $$S_{4}^{\mathrm{new}}(648, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$196 - 14 T + T^{2}$$
$7$ $$576 - 24 T + T^{2}$$
$11$ $$784 + 28 T + T^{2}$$
$13$ $$5476 - 74 T + T^{2}$$
$17$ $$( 82 + T )^{2}$$
$19$ $$( -92 + T )^{2}$$
$23$ $$64 - 8 T + T^{2}$$
$29$ $$19044 + 138 T + T^{2}$$
$31$ $$6400 + 80 T + T^{2}$$
$37$ $$( -30 + T )^{2}$$
$41$ $$79524 - 282 T + T^{2}$$
$43$ $$16 + 4 T + T^{2}$$
$47$ $$57600 - 240 T + T^{2}$$
$53$ $$( -130 + T )^{2}$$
$59$ $$355216 - 596 T + T^{2}$$
$61$ $$47524 - 218 T + T^{2}$$
$67$ $$190096 - 436 T + T^{2}$$
$71$ $$( 856 + T )^{2}$$
$73$ $$( 998 + T )^{2}$$
$79$ $$1024 - 32 T + T^{2}$$
$83$ $$2274064 + 1508 T + T^{2}$$
$89$ $$( -246 + T )^{2}$$
$97$ $$749956 + 866 T + T^{2}$$