# Properties

 Label 216.4.a.h Level $216$ Weight $4$ Character orbit 216.a Self dual yes Analytic conductor $12.744$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 216.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.7444125612$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 3\sqrt{33}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 4 + \beta ) q^{5} + ( 12 + \beta ) q^{7} +O(q^{10})$$ $$q + ( 4 + \beta ) q^{5} + ( 12 + \beta ) q^{7} + q^{11} + ( 16 - 4 \beta ) q^{13} + ( -28 - 4 \beta ) q^{17} + ( 92 + 2 \beta ) q^{19} + ( 46 - 4 \beta ) q^{23} + ( 188 + 8 \beta ) q^{25} + ( -168 + 2 \beta ) q^{29} + ( 188 - 5 \beta ) q^{31} + ( 345 + 16 \beta ) q^{35} + ( 174 - 4 \beta ) q^{37} + ( -156 + 10 \beta ) q^{41} + ( 40 - 14 \beta ) q^{43} + ( -114 - 8 \beta ) q^{47} + ( 98 + 24 \beta ) q^{49} + ( 76 - 13 \beta ) q^{53} + ( 4 + \beta ) q^{55} + ( 340 - 24 \beta ) q^{59} + ( -56 - 32 \beta ) q^{61} -1124 q^{65} + ( 176 + 34 \beta ) q^{67} + ( 908 - 12 \beta ) q^{71} -287 q^{73} + ( 12 + \beta ) q^{77} + ( -680 + 32 \beta ) q^{79} + ( -391 + 32 \beta ) q^{83} + ( -1300 - 44 \beta ) q^{85} + ( 120 + 18 \beta ) q^{89} + ( -996 - 32 \beta ) q^{91} + ( 962 + 100 \beta ) q^{95} + ( -169 + 8 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 8q^{5} + 24q^{7} + O(q^{10})$$ $$2q + 8q^{5} + 24q^{7} + 2q^{11} + 32q^{13} - 56q^{17} + 184q^{19} + 92q^{23} + 376q^{25} - 336q^{29} + 376q^{31} + 690q^{35} + 348q^{37} - 312q^{41} + 80q^{43} - 228q^{47} + 196q^{49} + 152q^{53} + 8q^{55} + 680q^{59} - 112q^{61} - 2248q^{65} + 352q^{67} + 1816q^{71} - 574q^{73} + 24q^{77} - 1360q^{79} - 782q^{83} - 2600q^{85} + 240q^{89} - 1992q^{91} + 1924q^{95} - 338q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.37228 3.37228
0 0 0 −13.2337 0 −5.23369 0 0 0
1.2 0 0 0 21.2337 0 29.2337 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.4.a.h yes 2
3.b odd 2 1 216.4.a.e 2
4.b odd 2 1 432.4.a.s 2
8.b even 2 1 1728.4.a.bh 2
8.d odd 2 1 1728.4.a.bg 2
9.c even 3 2 648.4.i.m 4
9.d odd 6 2 648.4.i.s 4
12.b even 2 1 432.4.a.o 2
24.f even 2 1 1728.4.a.bs 2
24.h odd 2 1 1728.4.a.bt 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.4.a.e 2 3.b odd 2 1
216.4.a.h yes 2 1.a even 1 1 trivial
432.4.a.o 2 12.b even 2 1
432.4.a.s 2 4.b odd 2 1
648.4.i.m 4 9.c even 3 2
648.4.i.s 4 9.d odd 6 2
1728.4.a.bg 2 8.d odd 2 1
1728.4.a.bh 2 8.b even 2 1
1728.4.a.bs 2 24.f even 2 1
1728.4.a.bt 2 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 8 T_{5} - 281$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(216))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-281 - 8 T + T^{2}$$
$7$ $$-153 - 24 T + T^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$-4496 - 32 T + T^{2}$$
$17$ $$-3968 + 56 T + T^{2}$$
$19$ $$7276 - 184 T + T^{2}$$
$23$ $$-2636 - 92 T + T^{2}$$
$29$ $$27036 + 336 T + T^{2}$$
$31$ $$27919 - 376 T + T^{2}$$
$37$ $$25524 - 348 T + T^{2}$$
$41$ $$-5364 + 312 T + T^{2}$$
$43$ $$-56612 - 80 T + T^{2}$$
$47$ $$-6012 + 228 T + T^{2}$$
$53$ $$-44417 - 152 T + T^{2}$$
$59$ $$-55472 - 680 T + T^{2}$$
$61$ $$-300992 + 112 T + T^{2}$$
$67$ $$-312356 - 352 T + T^{2}$$
$71$ $$781696 - 1816 T + T^{2}$$
$73$ $$( 287 + T )^{2}$$
$79$ $$158272 + 1360 T + T^{2}$$
$83$ $$-151247 + 782 T + T^{2}$$
$89$ $$-81828 - 240 T + T^{2}$$
$97$ $$9553 + 338 T + T^{2}$$