# Properties

 Label 216.3.h.d Level $216$ Weight $3$ Character orbit 216.h Analytic conductor $5.886$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 216.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.88557371018$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.121670000.1 Defining polynomial: $$x^{6} - x^{5} + 4 x^{4} - 6 x^{3} + 16 x^{2} - 16 x + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( \beta_{2} + \beta_{4} ) q^{5} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{7} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( \beta_{2} + \beta_{4} ) q^{5} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{7} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} ) q^{8} + ( \beta_{3} - \beta_{5} ) q^{10} + ( -2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{11} + ( -1 + 3 \beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{13} + ( 6 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} ) q^{14} + ( -3 + \beta_{1} - 3 \beta_{2} + \beta_{3} - 4 \beta_{4} + \beta_{5} ) q^{16} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{17} + ( 3 - 7 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{19} + ( 9 - 2 \beta_{1} - \beta_{2} - 4 \beta_{4} ) q^{20} + ( 6 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{22} + ( -3 + 5 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{23} + ( -9 - \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{25} + ( -6 - 2 \beta_{1} - 2 \beta_{2} - 8 \beta_{4} ) q^{26} + ( -5 + 8 \beta_{1} + \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} ) q^{28} + ( -17 + \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{29} + ( 2 + 6 \beta_{1} + 3 \beta_{2} + 3 \beta_{4} - 6 \beta_{5} ) q^{31} + ( -5 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} - 4 \beta_{4} + 5 \beta_{5} ) q^{32} + ( 6 - 4 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} - 8 \beta_{4} - 6 \beta_{5} ) q^{34} + ( -17 + 4 \beta_{1} - 4 \beta_{5} ) q^{35} + ( -3 + 13 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 7 \beta_{5} ) q^{37} + ( 22 - 6 \beta_{2} + 2 \beta_{3} - 8 \beta_{4} + 6 \beta_{5} ) q^{38} + ( 5 + 9 \beta_{1} + \beta_{2} - \beta_{3} + 7 \beta_{5} ) q^{40} + ( -1 - 3 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} - 5 \beta_{5} ) q^{41} + ( 4 - 16 \beta_{1} - 4 \beta_{2} + 4 \beta_{4} - 8 \beta_{5} ) q^{43} + ( 23 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 8 \beta_{4} + 2 \beta_{5} ) q^{44} + ( -10 - 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 8 \beta_{4} - 12 \beta_{5} ) q^{46} + ( 2 - 10 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} ) q^{47} + ( -9 - 9 \beta_{1} + 6 \beta_{2} + 6 \beta_{4} + 9 \beta_{5} ) q^{49} + ( -6 - 8 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{50} + ( 8 - 6 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 14 \beta_{5} ) q^{52} + ( 1 + 7 \beta_{1} + 5 \beta_{2} + 5 \beta_{4} - 7 \beta_{5} ) q^{53} + ( 31 + \beta_{1} - 7 \beta_{2} - 7 \beta_{4} - \beta_{5} ) q^{55} + ( -21 - 5 \beta_{1} - \beta_{2} + \beta_{3} - 8 \beta_{4} - 7 \beta_{5} ) q^{56} + ( 6 - 18 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{58} + ( -4 - 10 \beta_{1} + 12 \beta_{2} + 12 \beta_{4} + 10 \beta_{5} ) q^{59} + ( -14 + 34 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} + 8 \beta_{4} + 6 \beta_{5} ) q^{61} + ( 36 - 4 \beta_{1} + 12 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} ) q^{62} + ( -35 + 3 \beta_{1} + \beta_{2} - \beta_{3} + 12 \beta_{4} + 7 \beta_{5} ) q^{64} + ( -9 + 25 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 7 \beta_{5} ) q^{65} + ( 8 - 28 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} + 4 \beta_{4} - 12 \beta_{5} ) q^{67} + ( 44 + 2 \beta_{1} + 8 \beta_{2} - 6 \beta_{3} + 8 \beta_{4} + 10 \beta_{5} ) q^{68} + ( 24 - 21 \beta_{1} + 8 \beta_{2} ) q^{70} + ( -3 + 9 \beta_{1} + 6 \beta_{3} + 3 \beta_{5} ) q^{71} + ( 20 - 19 \beta_{1} - 14 \beta_{2} - 14 \beta_{4} + 19 \beta_{5} ) q^{73} + ( -58 + 6 \beta_{1} + 18 \beta_{2} + 4 \beta_{3} + 8 \beta_{4} + 12 \beta_{5} ) q^{74} + ( -36 + 26 \beta_{1} - 8 \beta_{2} - 6 \beta_{3} - 8 \beta_{4} + 10 \beta_{5} ) q^{76} + ( -10 + 2 \beta_{1} + 7 \beta_{2} + 7 \beta_{4} - 2 \beta_{5} ) q^{77} + ( -13 + 13 \beta_{1} + 14 \beta_{2} + 14 \beta_{4} - 13 \beta_{5} ) q^{79} + ( -59 + 13 \beta_{1} + 5 \beta_{2} + \beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{80} + ( 14 + 2 \beta_{2} - 6 \beta_{3} + 24 \beta_{4} - 18 \beta_{5} ) q^{82} + ( 14 + 17 \beta_{1} - 6 \beta_{2} - 6 \beta_{4} - 17 \beta_{5} ) q^{83} + ( -15 + 37 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} + 8 \beta_{4} + 7 \beta_{5} ) q^{85} + ( 64 - 4 \beta_{1} - 16 \beta_{2} - 4 \beta_{3} - 12 \beta_{5} ) q^{86} + ( -7 + 23 \beta_{1} + \beta_{2} - 3 \beta_{3} - 8 \beta_{4} + 13 \beta_{5} ) q^{88} + ( 10 - 22 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} - 2 \beta_{5} ) q^{89} + ( 19 - 43 \beta_{1} + 14 \beta_{2} - 6 \beta_{3} - 14 \beta_{4} - 5 \beta_{5} ) q^{91} + ( 72 - 18 \beta_{1} + 12 \beta_{2} + 2 \beta_{3} + 16 \beta_{4} - 14 \beta_{5} ) q^{92} + ( 52 - 8 \beta_{1} - 20 \beta_{2} - 4 \beta_{3} - 16 \beta_{4} - 12 \beta_{5} ) q^{94} + ( -7 + 25 \beta_{1} + 4 \beta_{2} - 10 \beta_{3} - 4 \beta_{4} + 11 \beta_{5} ) q^{95} + ( -25 - 8 \beta_{1} + 32 \beta_{2} + 32 \beta_{4} + 8 \beta_{5} ) q^{97} + ( -54 - 18 \beta_{2} + 6 \beta_{3} - 6 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + q^{2} - 7q^{4} + 2q^{5} - 10q^{7} + 7q^{8} + O(q^{10})$$ $$6q + q^{2} - 7q^{4} + 2q^{5} - 10q^{7} + 7q^{8} - 3q^{10} - 10q^{11} + 35q^{14} - 23q^{16} + 41q^{20} + 25q^{22} - 56q^{25} - 60q^{26} - 5q^{28} - 100q^{29} + 6q^{31} - 29q^{32} - 4q^{34} - 110q^{35} + 132q^{38} + 59q^{40} + 125q^{44} - 76q^{46} - 24q^{49} - 36q^{50} + 80q^{52} + 2q^{53} + 170q^{55} - 175q^{56} + 10q^{58} + 20q^{59} + 191q^{62} - 151q^{64} + 312q^{68} + 115q^{70} + 130q^{73} - 300q^{74} - 176q^{76} - 50q^{77} - 76q^{79} - 331q^{80} + 100q^{82} + 38q^{83} + 360q^{86} - 5q^{88} + 408q^{92} + 240q^{94} - 70q^{97} - 324q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 4 x^{4} - 6 x^{3} + 16 x^{2} - 16 x + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} + 4 \nu^{3} - 2 \nu^{2} - 8$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{4} + \nu^{3} - 4 \nu^{2} + 2 \nu - 8$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{5} - \nu^{4} + 4 \nu^{3} - 6 \nu^{2} + 8 \nu - 8$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{3} + \beta_{2} - \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{5} - 4 \beta_{4} + \beta_{3} - 3 \beta_{2} + \beta_{1} - 3$$ $$\nu^{5}$$ $$=$$ $$5 \beta_{5} - 4 \beta_{4} - 3 \beta_{3} - \beta_{2} - 3 \beta_{1} - 5$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$$.

 $$n$$ $$55$$ $$109$$ $$137$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 −1.25395 − 1.55808i −1.25395 + 1.55808i 0.122180 − 1.99626i 0.122180 + 1.99626i 1.63177 − 1.15643i 1.63177 + 1.15643i
−1.25395 1.55808i 0 −0.855212 + 3.90751i 3.79748 0 −10.8133 7.16059 3.56734i 0 −4.76185 5.91677i
53.2 −1.25395 + 1.55808i 0 −0.855212 3.90751i 3.79748 0 −10.8133 7.16059 + 3.56734i 0 −4.76185 + 5.91677i
53.3 0.122180 1.99626i 0 −3.97014 0.487807i −5.18465 0 3.67337 −1.45886 + 7.86586i 0 −0.633460 + 10.3499i
53.4 0.122180 + 1.99626i 0 −3.97014 + 0.487807i −5.18465 0 3.67337 −1.45886 7.86586i 0 −0.633460 10.3499i
53.5 1.63177 1.15643i 0 1.32536 3.77405i 2.38717 0 2.13992 −2.20173 7.69106i 0 3.89531 2.76059i
53.6 1.63177 + 1.15643i 0 1.32536 + 3.77405i 2.38717 0 2.13992 −2.20173 + 7.69106i 0 3.89531 + 2.76059i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 53.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.3.h.d yes 6
3.b odd 2 1 216.3.h.c 6
4.b odd 2 1 864.3.h.d 6
8.b even 2 1 216.3.h.c 6
8.d odd 2 1 864.3.h.c 6
12.b even 2 1 864.3.h.c 6
24.f even 2 1 864.3.h.d 6
24.h odd 2 1 inner 216.3.h.d yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.h.c 6 3.b odd 2 1
216.3.h.c 6 8.b even 2 1
216.3.h.d yes 6 1.a even 1 1 trivial
216.3.h.d yes 6 24.h odd 2 1 inner
864.3.h.c 6 8.d odd 2 1
864.3.h.c 6 12.b even 2 1
864.3.h.d 6 4.b odd 2 1
864.3.h.d 6 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{3} - T_{5}^{2} - 23 T_{5} + 47$$ acting on $$S_{3}^{\mathrm{new}}(216, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$64 - 16 T + 16 T^{2} - 6 T^{3} + 4 T^{4} - T^{5} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$( 47 - 23 T - T^{2} + T^{3} )^{2}$$
$7$ $$( 85 - 55 T + 5 T^{2} + T^{3} )^{2}$$
$11$ $$( -45 - 105 T + 5 T^{2} + T^{3} )^{2}$$
$13$ $$5299200 + 115600 T^{2} + 680 T^{4} + T^{6}$$
$17$ $$17169408 + 393232 T^{2} + 1336 T^{4} + T^{6}$$
$19$ $$30523392 + 372112 T^{2} + 1304 T^{4} + T^{6}$$
$23$ $$245035008 + 1268752 T^{2} + 2056 T^{4} + T^{6}$$
$29$ $$( 2880 + 720 T + 50 T^{2} + T^{3} )^{2}$$
$31$ $$( -15971 - 1287 T - 3 T^{2} + T^{3} )^{2}$$
$37$ $$1716940800 + 8880400 T^{2} + 5960 T^{4} + T^{6}$$
$41$ $$529920000 + 5626000 T^{2} + 7000 T^{4} + T^{6}$$
$43$ $$339148800 + 9241600 T^{2} + 6080 T^{4} + T^{6}$$
$47$ $$6867763200 + 12960000 T^{2} + 7200 T^{4} + T^{6}$$
$53$ $$( -32553 - 1983 T - T^{2} + T^{3} )^{2}$$
$59$ $$( -190360 - 7460 T - 10 T^{2} + T^{3} )^{2}$$
$61$ $$89006211072 + 118132992 T^{2} + 21024 T^{4} + T^{6}$$
$67$ $$27471052800 + 101606400 T^{2} + 20160 T^{4} + T^{6}$$
$71$ $$3863116800 + 9363600 T^{2} + 6120 T^{4} + T^{6}$$
$73$ $$( 983025 - 13425 T - 65 T^{2} + T^{3} )^{2}$$
$79$ $$( -323744 - 8512 T + 38 T^{2} + T^{3} )^{2}$$
$83$ $$( 378683 - 11033 T - 19 T^{2} + T^{3} )^{2}$$
$89$ $$61809868800 + 50540800 T^{2} + 12640 T^{4} + T^{6}$$
$97$ $$( 193825 - 27325 T + 35 T^{2} + T^{3} )^{2}$$