# Properties

 Label 648.4.i.h 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 8) 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{5} + ( -24 + 24 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{5} + ( -24 + 24 \zeta_{6} ) q^{7} + ( 44 - 44 \zeta_{6} ) q^{11} -22 \zeta_{6} q^{13} + 50 q^{17} + 44 q^{19} + 56 \zeta_{6} q^{23} + ( 121 - 121 \zeta_{6} ) q^{25} + ( -198 + 198 \zeta_{6} ) q^{29} + 160 \zeta_{6} q^{31} -48 q^{35} -162 q^{37} + 198 \zeta_{6} q^{41} + ( -52 + 52 \zeta_{6} ) q^{43} + ( -528 + 528 \zeta_{6} ) q^{47} -233 \zeta_{6} q^{49} -242 q^{53} + 88 q^{55} + 668 \zeta_{6} q^{59} + ( -550 + 550 \zeta_{6} ) q^{61} + ( 44 - 44 \zeta_{6} ) q^{65} -188 \zeta_{6} q^{67} + 728 q^{71} + 154 q^{73} + 1056 \zeta_{6} q^{77} + ( 656 - 656 \zeta_{6} ) q^{79} + ( -236 + 236 \zeta_{6} ) q^{83} + 100 \zeta_{6} q^{85} + 714 q^{89} + 528 q^{91} + 88 \zeta_{6} q^{95} + ( 478 - 478 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} - 24q^{7} + O(q^{10})$$ $$2q + 2q^{5} - 24q^{7} + 44q^{11} - 22q^{13} + 100q^{17} + 88q^{19} + 56q^{23} + 121q^{25} - 198q^{29} + 160q^{31} - 96q^{35} - 324q^{37} + 198q^{41} - 52q^{43} - 528q^{47} - 233q^{49} - 484q^{53} + 176q^{55} + 668q^{59} - 550q^{61} + 44q^{65} - 188q^{67} + 1456q^{71} + 308q^{73} + 1056q^{77} + 656q^{79} - 236q^{83} + 100q^{85} + 1428q^{89} + 1056q^{91} + 88q^{95} + 478q^{97} + O(q^{100})$$

## 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 1.00000 + 1.73205i 0 −12.0000 + 20.7846i 0 0 0
433.1 0 0 0 1.00000 1.73205i 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.h 2
3.b odd 2 1 648.4.i.e 2
9.c even 3 1 8.4.a.a 1
9.c even 3 1 inner 648.4.i.h 2
9.d odd 6 1 72.4.a.c 1
9.d odd 6 1 648.4.i.e 2
36.f odd 6 1 16.4.a.a 1
36.h even 6 1 144.4.a.e 1
45.h odd 6 1 1800.4.a.d 1
45.j even 6 1 200.4.a.g 1
45.k odd 12 2 200.4.c.e 2
45.l even 12 2 1800.4.f.u 2
63.g even 3 1 392.4.i.g 2
63.h even 3 1 392.4.i.g 2
63.k odd 6 1 392.4.i.b 2
63.l odd 6 1 392.4.a.e 1
63.t odd 6 1 392.4.i.b 2
72.j odd 6 1 576.4.a.k 1
72.l even 6 1 576.4.a.j 1
72.n even 6 1 64.4.a.d 1
72.p odd 6 1 64.4.a.b 1
99.h odd 6 1 968.4.a.a 1
117.t even 6 1 1352.4.a.a 1
144.v odd 12 2 256.4.b.g 2
144.x even 12 2 256.4.b.a 2
153.h even 6 1 2312.4.a.a 1
180.p odd 6 1 400.4.a.g 1
180.x even 12 2 400.4.c.i 2
252.bi even 6 1 784.4.a.e 1
360.z odd 6 1 1600.4.a.bm 1
360.bk even 6 1 1600.4.a.o 1
396.k even 6 1 1936.4.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.a.a 1 9.c even 3 1
16.4.a.a 1 36.f odd 6 1
64.4.a.b 1 72.p odd 6 1
64.4.a.d 1 72.n even 6 1
72.4.a.c 1 9.d odd 6 1
144.4.a.e 1 36.h even 6 1
200.4.a.g 1 45.j even 6 1
200.4.c.e 2 45.k odd 12 2
256.4.b.a 2 144.x even 12 2
256.4.b.g 2 144.v odd 12 2
392.4.a.e 1 63.l odd 6 1
392.4.i.b 2 63.k odd 6 1
392.4.i.b 2 63.t odd 6 1
392.4.i.g 2 63.g even 3 1
392.4.i.g 2 63.h even 3 1
400.4.a.g 1 180.p odd 6 1
400.4.c.i 2 180.x even 12 2
576.4.a.j 1 72.l even 6 1
576.4.a.k 1 72.j odd 6 1
648.4.i.e 2 3.b odd 2 1
648.4.i.e 2 9.d odd 6 1
648.4.i.h 2 1.a even 1 1 trivial
648.4.i.h 2 9.c even 3 1 inner
784.4.a.e 1 252.bi even 6 1
968.4.a.a 1 99.h odd 6 1
1352.4.a.a 1 117.t even 6 1
1600.4.a.o 1 360.bk even 6 1
1600.4.a.bm 1 360.z odd 6 1
1800.4.a.d 1 45.h odd 6 1
1800.4.f.u 2 45.l even 12 2
1936.4.a.l 1 396.k even 6 1
2312.4.a.a 1 153.h even 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$4 - 2 T + T^{2}$$
$7$ $$576 + 24 T + T^{2}$$
$11$ $$1936 - 44 T + T^{2}$$
$13$ $$484 + 22 T + T^{2}$$
$17$ $$( -50 + T )^{2}$$
$19$ $$( -44 + T )^{2}$$
$23$ $$3136 - 56 T + T^{2}$$
$29$ $$39204 + 198 T + T^{2}$$
$31$ $$25600 - 160 T + T^{2}$$
$37$ $$( 162 + T )^{2}$$
$41$ $$39204 - 198 T + T^{2}$$
$43$ $$2704 + 52 T + T^{2}$$
$47$ $$278784 + 528 T + T^{2}$$
$53$ $$( 242 + T )^{2}$$
$59$ $$446224 - 668 T + T^{2}$$
$61$ $$302500 + 550 T + T^{2}$$
$67$ $$35344 + 188 T + T^{2}$$
$71$ $$( -728 + T )^{2}$$
$73$ $$( -154 + T )^{2}$$
$79$ $$430336 - 656 T + T^{2}$$
$83$ $$55696 + 236 T + T^{2}$$
$89$ $$( -714 + T )^{2}$$
$97$ $$228484 - 478 T + T^{2}$$