# Properties

 Label 864.2.d.a Level 864 Weight 2 Character orbit 864.d 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.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.89907473464$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{7})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 216) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{5} - q^{7} +O(q^{10})$$ $$q -\beta_{1} q^{5} - q^{7} + 3 \beta_{1} q^{11} -\beta_{3} q^{13} + \beta_{2} q^{17} -\beta_{3} q^{19} -\beta_{2} q^{23} + 4 q^{25} -6 \beta_{1} q^{29} + 7 q^{31} + \beta_{1} q^{35} -\beta_{3} q^{37} -\beta_{2} q^{41} -2 \beta_{3} q^{43} -6 q^{49} + 9 \beta_{1} q^{53} + 3 q^{55} -4 \beta_{1} q^{59} -\beta_{2} q^{65} + 3 \beta_{2} q^{71} + 3 q^{73} -3 \beta_{1} q^{77} -4 q^{79} -7 \beta_{1} q^{83} -\beta_{3} q^{85} + 2 \beta_{2} q^{89} + \beta_{3} q^{91} -\beta_{2} q^{95} + 7 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{7} + O(q^{10})$$ $$4q - 4q^{7} + 16q^{25} + 28q^{31} - 24q^{49} + 12q^{55} + 12q^{73} - 16q^{79} + 28q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$-\nu^{3} + 5 \nu$$ $$\beta_{3}$$ $$=$$ $$4 \nu^{2} - 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2 \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 6$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{2} + 10 \beta_{1}$$$$)/4$$

## Character values

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

 $$n$$ $$325$$ $$353$$ $$703$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
433.1
 1.32288 + 0.500000i −1.32288 + 0.500000i −1.32288 − 0.500000i 1.32288 − 0.500000i
0 0 0 1.00000i 0 −1.00000 0 0 0
433.2 0 0 0 1.00000i 0 −1.00000 0 0 0
433.3 0 0 0 1.00000i 0 −1.00000 0 0 0
433.4 0 0 0 1.00000i 0 −1.00000 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
3.b odd 2 1 inner
8.b even 2 1 inner
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 864.2.d.a 4
3.b odd 2 1 inner 864.2.d.a 4
4.b odd 2 1 216.2.d.b 4
8.b even 2 1 inner 864.2.d.a 4
8.d odd 2 1 216.2.d.b 4
9.c even 3 2 2592.2.r.p 8
9.d odd 6 2 2592.2.r.p 8
12.b even 2 1 216.2.d.b 4
16.e even 4 1 6912.2.a.bd 2
16.e even 4 1 6912.2.a.bv 2
16.f odd 4 1 6912.2.a.bc 2
16.f odd 4 1 6912.2.a.bu 2
24.f even 2 1 216.2.d.b 4
24.h odd 2 1 inner 864.2.d.a 4
36.f odd 6 2 648.2.n.n 8
36.h even 6 2 648.2.n.n 8
48.i odd 4 1 6912.2.a.bd 2
48.i odd 4 1 6912.2.a.bv 2
48.k even 4 1 6912.2.a.bc 2
48.k even 4 1 6912.2.a.bu 2
72.j odd 6 2 2592.2.r.p 8
72.l even 6 2 648.2.n.n 8
72.n even 6 2 2592.2.r.p 8
72.p odd 6 2 648.2.n.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.d.b 4 4.b odd 2 1
216.2.d.b 4 8.d odd 2 1
216.2.d.b 4 12.b even 2 1
216.2.d.b 4 24.f even 2 1
648.2.n.n 8 36.f odd 6 2
648.2.n.n 8 36.h even 6 2
648.2.n.n 8 72.l even 6 2
648.2.n.n 8 72.p odd 6 2
864.2.d.a 4 1.a even 1 1 trivial
864.2.d.a 4 3.b odd 2 1 inner
864.2.d.a 4 8.b even 2 1 inner
864.2.d.a 4 24.h odd 2 1 inner
2592.2.r.p 8 9.c even 3 2
2592.2.r.p 8 9.d odd 6 2
2592.2.r.p 8 72.j odd 6 2
2592.2.r.p 8 72.n even 6 2
6912.2.a.bc 2 16.f odd 4 1
6912.2.a.bc 2 48.k even 4 1
6912.2.a.bd 2 16.e even 4 1
6912.2.a.bd 2 48.i odd 4 1
6912.2.a.bu 2 16.f odd 4 1
6912.2.a.bu 2 48.k even 4 1
6912.2.a.bv 2 16.e even 4 1
6912.2.a.bv 2 48.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(864, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ 
$5$ $$( 1 - 9 T^{2} + 25 T^{4} )^{2}$$
$7$ $$( 1 + T + 7 T^{2} )^{4}$$
$11$ $$( 1 - 13 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 2 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 + 6 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 10 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 + 18 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 22 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 7 T + 31 T^{2} )^{4}$$
$37$ $$( 1 - 46 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 54 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 26 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 47 T^{2} )^{4}$$
$53$ $$( 1 - 25 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 - 102 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 61 T^{2} )^{4}$$
$67$ $$( 1 - 67 T^{2} )^{4}$$
$71$ $$( 1 - 110 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 3 T + 73 T^{2} )^{4}$$
$79$ $$( 1 + 4 T + 79 T^{2} )^{4}$$
$83$ $$( 1 - 117 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 66 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 7 T + 97 T^{2} )^{4}$$