# Properties

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

# 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: $$\Q(\sqrt{-2}, \sqrt{-3})$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{5} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} +O(q^{10})$$ $$q -\beta_{1} q^{5} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{11} + ( -\beta_{1} + 2 \beta_{2} ) q^{13} + 2 \beta_{3} q^{17} -4 q^{19} + ( 3 \beta_{1} - \beta_{2} ) q^{23} + ( 4 - 4 \beta_{1} ) q^{25} + ( 5 - 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{29} + ( 5 \beta_{1} + \beta_{2} ) q^{31} + ( -1 + \beta_{3} ) q^{35} + ( 4 + 2 \beta_{3} ) q^{37} + ( -7 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 5 - 5 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{43} + ( -1 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{47} + 2 \beta_{2} q^{49} + ( -4 + 2 \beta_{3} ) q^{53} + ( -1 - \beta_{3} ) q^{55} + ( -7 \beta_{1} + 3 \beta_{2} ) q^{59} + ( 3 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{61} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{65} + ( 5 \beta_{1} - 3 \beta_{2} ) q^{67} + ( -2 - 4 \beta_{3} ) q^{71} + 2 \beta_{3} q^{73} + 5 \beta_{1} q^{77} + ( 11 - 11 \beta_{1} + \beta_{2} - \beta_{3} ) q^{79} + ( 3 - 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{83} -2 \beta_{2} q^{85} + ( 8 - 2 \beta_{3} ) q^{89} + ( -13 + 3 \beta_{3} ) q^{91} + 4 \beta_{1} q^{95} + ( -1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{5} + 2q^{7} + O(q^{10})$$ $$4q - 2q^{5} + 2q^{7} + 2q^{11} - 2q^{13} - 16q^{19} + 6q^{23} + 8q^{25} + 10q^{29} + 10q^{31} - 4q^{35} + 16q^{37} - 14q^{41} + 10q^{43} - 2q^{47} - 16q^{53} - 4q^{55} - 14q^{59} + 6q^{61} - 2q^{65} + 10q^{67} - 8q^{71} + 10q^{77} + 22q^{79} + 6q^{83} + 32q^{89} - 52q^{91} + 8q^{95} - 2q^{97} + O(q^{100})$$

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

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

## Character values

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

 $$n$$ $$325$$ $$353$$ $$703$$ $$\chi(n)$$ $$1$$ $$-\beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
0 0 0 −0.500000 0.866025i 0 −0.724745 + 1.25529i 0 0 0
289.2 0 0 0 −0.500000 0.866025i 0 1.72474 2.98735i 0 0 0
577.1 0 0 0 −0.500000 + 0.866025i 0 −0.724745 1.25529i 0 0 0
577.2 0 0 0 −0.500000 + 0.866025i 0 1.72474 + 2.98735i 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 864.2.i.e 4
3.b odd 2 1 288.2.i.c 4
4.b odd 2 1 864.2.i.c 4
8.b even 2 1 1728.2.i.m 4
8.d odd 2 1 1728.2.i.k 4
9.c even 3 1 inner 864.2.i.e 4
9.c even 3 1 2592.2.a.o 2
9.d odd 6 1 288.2.i.c 4
9.d odd 6 1 2592.2.a.j 2
12.b even 2 1 288.2.i.e yes 4
24.f even 2 1 576.2.i.i 4
24.h odd 2 1 576.2.i.m 4
36.f odd 6 1 864.2.i.c 4
36.f odd 6 1 2592.2.a.s 2
36.h even 6 1 288.2.i.e yes 4
36.h even 6 1 2592.2.a.n 2
72.j odd 6 1 576.2.i.m 4
72.j odd 6 1 5184.2.a.bu 2
72.l even 6 1 576.2.i.i 4
72.l even 6 1 5184.2.a.by 2
72.n even 6 1 1728.2.i.m 4
72.n even 6 1 5184.2.a.bj 2
72.p odd 6 1 1728.2.i.k 4
72.p odd 6 1 5184.2.a.bn 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.c 4 3.b odd 2 1
288.2.i.c 4 9.d odd 6 1
288.2.i.e yes 4 12.b even 2 1
288.2.i.e yes 4 36.h even 6 1
576.2.i.i 4 24.f even 2 1
576.2.i.i 4 72.l even 6 1
576.2.i.m 4 24.h odd 2 1
576.2.i.m 4 72.j odd 6 1
864.2.i.c 4 4.b odd 2 1
864.2.i.c 4 36.f odd 6 1
864.2.i.e 4 1.a even 1 1 trivial
864.2.i.e 4 9.c even 3 1 inner
1728.2.i.k 4 8.d odd 2 1
1728.2.i.k 4 72.p odd 6 1
1728.2.i.m 4 8.b even 2 1
1728.2.i.m 4 72.n even 6 1
2592.2.a.j 2 9.d odd 6 1
2592.2.a.n 2 36.h even 6 1
2592.2.a.o 2 9.c even 3 1
2592.2.a.s 2 36.f odd 6 1
5184.2.a.bj 2 72.n even 6 1
5184.2.a.bn 2 72.p odd 6 1
5184.2.a.bu 2 72.j odd 6 1
5184.2.a.by 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 $$S_{2}^{\mathrm{new}}(864, [\chi])$$:

 $$T_{5}^{2} + T_{5} + 1$$ $$T_{7}^{4} - 2 T_{7}^{3} + 9 T_{7}^{2} + 10 T_{7} + 25$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ 
$5$ $$( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} )^{2}$$
$7$ $$1 - 2 T - 5 T^{2} + 10 T^{3} + 4 T^{4} + 70 T^{5} - 245 T^{6} - 686 T^{7} + 2401 T^{8}$$
$11$ $$1 - 2 T - 13 T^{2} + 10 T^{3} + 124 T^{4} + 110 T^{5} - 1573 T^{6} - 2662 T^{7} + 14641 T^{8}$$
$13$ $$1 + 2 T + T^{2} - 46 T^{3} - 212 T^{4} - 598 T^{5} + 169 T^{6} + 4394 T^{7} + 28561 T^{8}$$
$17$ $$( 1 + 10 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 + 4 T + 19 T^{2} )^{4}$$
$23$ $$1 - 6 T - 13 T^{2} - 18 T^{3} + 1044 T^{4} - 414 T^{5} - 6877 T^{6} - 73002 T^{7} + 279841 T^{8}$$
$29$ $$1 - 10 T + 41 T^{2} - 10 T^{3} - 260 T^{4} - 290 T^{5} + 34481 T^{6} - 243890 T^{7} + 707281 T^{8}$$
$31$ $$1 - 10 T + 19 T^{2} - 190 T^{3} + 2500 T^{4} - 5890 T^{5} + 18259 T^{6} - 297910 T^{7} + 923521 T^{8}$$
$37$ $$( 1 - 8 T + 66 T^{2} - 296 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$1 + 14 T + 89 T^{2} + 350 T^{3} + 1732 T^{4} + 14350 T^{5} + 149609 T^{6} + 964894 T^{7} + 2825761 T^{8}$$
$43$ $$( 1 - 10 T + 43 T^{2} )^{2}( 1 + 10 T + 57 T^{2} + 430 T^{3} + 1849 T^{4} )$$
$47$ $$1 + 2 T - 37 T^{2} - 106 T^{3} - 716 T^{4} - 4982 T^{5} - 81733 T^{6} + 207646 T^{7} + 4879681 T^{8}$$
$53$ $$( 1 + 8 T + 98 T^{2} + 424 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$1 + 14 T + 83 T^{2} - 70 T^{3} - 2276 T^{4} - 4130 T^{5} + 288923 T^{6} + 2875306 T^{7} + 12117361 T^{8}$$
$61$ $$1 - 6 T - 71 T^{2} + 90 T^{3} + 5532 T^{4} + 5490 T^{5} - 264191 T^{6} - 1361886 T^{7} + 13845841 T^{8}$$
$67$ $$1 - 10 T - 5 T^{2} + 290 T^{3} - 164 T^{4} + 19430 T^{5} - 22445 T^{6} - 3007630 T^{7} + 20151121 T^{8}$$
$71$ $$( 1 + 4 T + 50 T^{2} + 284 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 122 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$1 - 22 T + 211 T^{2} - 2530 T^{3} + 30052 T^{4} - 199870 T^{5} + 1316851 T^{6} - 10846858 T^{7} + 38950081 T^{8}$$
$83$ $$1 - 6 T - 133 T^{2} - 18 T^{3} + 18684 T^{4} - 1494 T^{5} - 916237 T^{6} - 3430722 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 - 16 T + 218 T^{2} - 1424 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 + 2 T - 167 T^{2} - 46 T^{3} + 19444 T^{4} - 4462 T^{5} - 1571303 T^{6} + 1825346 T^{7} + 88529281 T^{8}$$