Properties

 Label 864.2.i.d Level 864 Weight 2 Character orbit 864.i Analytic conductor 6.899 Analytic rank 0 Dimension 4 CM no Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ = $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 864.i (of order $$3$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$6.89907473464$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 288) 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{12}^{2} ) q^{5} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{7} +O(q^{10})$$ $$q + ( -1 + \zeta_{12}^{2} ) q^{5} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{7} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{11} + ( 3 - 3 \zeta_{12}^{2} ) q^{13} -4 q^{17} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{19} + ( 5 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{23} + 4 \zeta_{12}^{2} q^{25} + \zeta_{12}^{2} q^{29} + ( 3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{31} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{35} -8 q^{37} + ( 5 - 5 \zeta_{12}^{2} ) q^{41} + ( -5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{43} + ( -7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{47} + ( 4 - 4 \zeta_{12}^{2} ) q^{49} + 8 q^{53} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{55} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{59} + 7 \zeta_{12}^{2} q^{61} + 3 \zeta_{12}^{2} q^{65} + ( -5 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{67} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{71} -12 q^{73} + ( -3 + 3 \zeta_{12}^{2} ) q^{77} + ( -3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{79} + ( 5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{83} + ( 4 - 4 \zeta_{12}^{2} ) q^{85} + 4 q^{89} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{91} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{95} + 3 \zeta_{12}^{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{5} + O(q^{10})$$ $$4q - 2q^{5} + 6q^{13} - 16q^{17} + 8q^{25} + 2q^{29} - 32q^{37} + 10q^{41} + 8q^{49} + 32q^{53} + 14q^{61} + 6q^{65} - 48q^{73} - 6q^{77} + 8q^{85} + 16q^{89} + 6q^{97} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$353$$ $$703$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{12}$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 0 0 −0.500000 0.866025i 0 −0.866025 + 1.50000i 0 0 0
289.2 0 0 0 −0.500000 0.866025i 0 0.866025 1.50000i 0 0 0
577.1 0 0 0 −0.500000 + 0.866025i 0 −0.866025 1.50000i 0 0 0
577.2 0 0 0 −0.500000 + 0.866025i 0 0.866025 + 1.50000i 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
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.i.d 4
3.b odd 2 1 288.2.i.d 4
4.b odd 2 1 inner 864.2.i.d 4
8.b even 2 1 1728.2.i.l 4
8.d odd 2 1 1728.2.i.l 4
9.c even 3 1 inner 864.2.i.d 4
9.c even 3 1 2592.2.a.p 2
9.d odd 6 1 288.2.i.d 4
9.d odd 6 1 2592.2.a.l 2
12.b even 2 1 288.2.i.d 4
24.f even 2 1 576.2.i.k 4
24.h odd 2 1 576.2.i.k 4
36.f odd 6 1 inner 864.2.i.d 4
36.f odd 6 1 2592.2.a.p 2
36.h even 6 1 288.2.i.d 4
36.h even 6 1 2592.2.a.l 2
72.j odd 6 1 576.2.i.k 4
72.j odd 6 1 5184.2.a.bx 2
72.l even 6 1 576.2.i.k 4
72.l even 6 1 5184.2.a.bx 2
72.n even 6 1 1728.2.i.l 4
72.n even 6 1 5184.2.a.bl 2
72.p odd 6 1 1728.2.i.l 4
72.p odd 6 1 5184.2.a.bl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.d 4 3.b odd 2 1
288.2.i.d 4 9.d odd 6 1
288.2.i.d 4 12.b even 2 1
288.2.i.d 4 36.h even 6 1
576.2.i.k 4 24.f even 2 1
576.2.i.k 4 24.h odd 2 1
576.2.i.k 4 72.j odd 6 1
576.2.i.k 4 72.l even 6 1
864.2.i.d 4 1.a even 1 1 trivial
864.2.i.d 4 4.b odd 2 1 inner
864.2.i.d 4 9.c even 3 1 inner
864.2.i.d 4 36.f odd 6 1 inner
1728.2.i.l 4 8.b even 2 1
1728.2.i.l 4 8.d odd 2 1
1728.2.i.l 4 72.n even 6 1
1728.2.i.l 4 72.p odd 6 1
2592.2.a.l 2 9.d odd 6 1
2592.2.a.l 2 36.h even 6 1
2592.2.a.p 2 9.c even 3 1
2592.2.a.p 2 36.f odd 6 1
5184.2.a.bl 2 72.n even 6 1
5184.2.a.bl 2 72.p odd 6 1
5184.2.a.bx 2 72.j odd 6 1
5184.2.a.bx 2 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(864, [\chi])$$:

 $$T_{5}^{2} + T_{5} + 1$$ $$T_{7}^{4} + 3 T_{7}^{2} + 9$$

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ 
$5$ $$( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} )^{2}$$
$7$ $$( 1 - 13 T^{2} + 49 T^{4} )( 1 + 2 T^{2} + 49 T^{4} )$$
$11$ $$1 - 19 T^{2} + 240 T^{4} - 2299 T^{6} + 14641 T^{8}$$
$13$ $$( 1 - 3 T - 4 T^{2} - 39 T^{3} + 169 T^{4} )^{2}$$
$17$ $$( 1 + 4 T + 17 T^{2} )^{4}$$
$19$ $$( 1 - 10 T^{2} + 361 T^{4} )^{2}$$
$23$ $$1 + 29 T^{2} + 312 T^{4} + 15341 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} )^{2}$$
$31$ $$1 - 35 T^{2} + 264 T^{4} - 33635 T^{6} + 923521 T^{8}$$
$37$ $$( 1 + 8 T + 37 T^{2} )^{4}$$
$41$ $$( 1 - 5 T - 16 T^{2} - 205 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 - 11 T^{2} - 1728 T^{4} - 20339 T^{6} + 3418801 T^{8}$$
$47$ $$1 + 53 T^{2} + 600 T^{4} + 117077 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 - 8 T + 53 T^{2} )^{4}$$
$59$ $$1 - 115 T^{2} + 9744 T^{4} - 400315 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 - 7 T - 12 T^{2} - 427 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 - 59 T^{2} - 1008 T^{4} - 264851 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 + 130 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 12 T + 73 T^{2} )^{4}$$
$79$ $$( 1 - 142 T^{2} + 6241 T^{4} )( 1 + 11 T^{2} + 6241 T^{4} )$$
$83$ $$1 - 91 T^{2} + 1392 T^{4} - 626899 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 - 4 T + 89 T^{2} )^{4}$$
$97$ $$( 1 - 3 T - 88 T^{2} - 291 T^{3} + 9409 T^{4} )^{2}$$