# Properties

 Label 648.4.i.j 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: yes 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.

 $$f(q)$$ $$=$$ $$q + 5 \zeta_{6} q^{5} + ( -36 + 36 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + 5 \zeta_{6} q^{5} + ( -36 + 36 \zeta_{6} ) q^{7} + ( -64 + 64 \zeta_{6} ) q^{11} + 65 \zeta_{6} q^{13} + 59 q^{17} -28 q^{19} -160 \zeta_{6} q^{23} + ( 100 - 100 \zeta_{6} ) q^{25} + ( 57 - 57 \zeta_{6} ) q^{29} -164 \zeta_{6} q^{31} -180 q^{35} -321 q^{37} + 246 \zeta_{6} q^{41} + ( 8 - 8 \zeta_{6} ) q^{43} + ( -84 + 84 \zeta_{6} ) q^{47} -953 \zeta_{6} q^{49} + 478 q^{53} -320 q^{55} + 32 \zeta_{6} q^{59} + ( -415 + 415 \zeta_{6} ) q^{61} + ( -325 + 325 \zeta_{6} ) q^{65} + 220 \zeta_{6} q^{67} + 884 q^{71} -77 q^{73} -2304 \zeta_{6} q^{77} + ( 80 - 80 \zeta_{6} ) q^{79} + ( -1268 + 1268 \zeta_{6} ) q^{83} + 295 \zeta_{6} q^{85} + 123 q^{89} -2340 q^{91} -140 \zeta_{6} q^{95} + ( -1346 + 1346 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 5q^{5} - 36q^{7} + O(q^{10})$$ $$2q + 5q^{5} - 36q^{7} - 64q^{11} + 65q^{13} + 118q^{17} - 56q^{19} - 160q^{23} + 100q^{25} + 57q^{29} - 164q^{31} - 360q^{35} - 642q^{37} + 246q^{41} + 8q^{43} - 84q^{47} - 953q^{49} + 956q^{53} - 640q^{55} + 32q^{59} - 415q^{61} - 325q^{65} + 220q^{67} + 1768q^{71} - 154q^{73} - 2304q^{77} + 80q^{79} - 1268q^{83} + 295q^{85} + 246q^{89} - 4680q^{91} - 140q^{95} - 1346q^{97} + O(q^{100})$$

## 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 2.50000 + 4.33013i 0 −18.0000 + 31.1769i 0 0 0
433.1 0 0 0 2.50000 4.33013i 0 −18.0000 31.1769i 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.j 2
3.b odd 2 1 648.4.i.c 2
9.c even 3 1 648.4.a.a 1
9.c even 3 1 inner 648.4.i.j 2
9.d odd 6 1 648.4.a.b yes 1
9.d odd 6 1 648.4.i.c 2
36.f odd 6 1 1296.4.a.c 1
36.h even 6 1 1296.4.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.4.a.a 1 9.c even 3 1
648.4.a.b yes 1 9.d odd 6 1
648.4.i.c 2 3.b odd 2 1
648.4.i.c 2 9.d odd 6 1
648.4.i.j 2 1.a even 1 1 trivial
648.4.i.j 2 9.c even 3 1 inner
1296.4.a.c 1 36.f odd 6 1
1296.4.a.f 1 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 5 T_{5} + 25$$ acting on $$S_{4}^{\mathrm{new}}(648, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$25 - 5 T + T^{2}$$
$7$ $$1296 + 36 T + T^{2}$$
$11$ $$4096 + 64 T + T^{2}$$
$13$ $$4225 - 65 T + T^{2}$$
$17$ $$( -59 + T )^{2}$$
$19$ $$( 28 + T )^{2}$$
$23$ $$25600 + 160 T + T^{2}$$
$29$ $$3249 - 57 T + T^{2}$$
$31$ $$26896 + 164 T + T^{2}$$
$37$ $$( 321 + T )^{2}$$
$41$ $$60516 - 246 T + T^{2}$$
$43$ $$64 - 8 T + T^{2}$$
$47$ $$7056 + 84 T + T^{2}$$
$53$ $$( -478 + T )^{2}$$
$59$ $$1024 - 32 T + T^{2}$$
$61$ $$172225 + 415 T + T^{2}$$
$67$ $$48400 - 220 T + T^{2}$$
$71$ $$( -884 + T )^{2}$$
$73$ $$( 77 + T )^{2}$$
$79$ $$6400 - 80 T + T^{2}$$
$83$ $$1607824 + 1268 T + T^{2}$$
$89$ $$( -123 + T )^{2}$$
$97$ $$1811716 + 1346 T + T^{2}$$