Embedded newform 864.4.p.a.719.2

• Label: 864.4.p.a.719.2
• Level: $864$
• Weight: $4$
• Character: 864.719
• Analytic conductor: $50.978$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -8
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	864.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(50.9776502450\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial:	\( x^{4} - 2x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			719.2
Root			\(-1.22474 + 0.707107i\) of defining polynomial
Character	\(\chi\)	\(=\)	864.719
Dual form			864.4.p.a.143.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(17.1186 - 9.88344i) q^{11} -24.2022i q^{17} -163.227 q^{19} +(62.5000 + 108.253i) q^{25} +(367.005 + 211.890i) q^{41} +(-281.931 - 488.319i) q^{43} +(-171.500 + 297.047i) q^{49} +(-775.346 - 447.646i) q^{59} +(-456.476 + 790.640i) q^{67} -799.089 q^{73} +(-590.327 + 340.825i) q^{83} -1329.36i q^{89} +(455.455 + 788.870i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 54 q^{11} - 212 q^{19} + 250 q^{25} + 1566 q^{41} - 290 q^{43} - 686 q^{49} - 2538 q^{59} + 70 q^{67} + 860 q^{73} + 1910 q^{97}+O(q^{100}) \)

Character values

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

\(n\)	\(325\)	\(353\)	\(703\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		0			0
							\(3\)		0			0
\(5\)		0			0									−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(7\)		0			0		−0.500000	−	0.866025i	\(-0.666667\pi\)
							0.500000	+	0.866025i	\(0.333333\pi\)
\(11\)	17.1186	−	9.88344i	0.469224	−	0.270906i	−0.246691	−	0.969094i	\(-0.579343\pi\)
							0.715915	+	0.698188i	\(0.246010\pi\)
\(13\)		0			0		0.500000	−	0.866025i	\(-0.333333\pi\)
							−0.500000	+	0.866025i	\(0.666667\pi\)
\(17\)		−	24.2022i		−	0.345288i	−0.984984	−	0.172644i	\(-0.944769\pi\)
							0.984984	−	0.172644i	\(-0.0552309\pi\)
\(19\)	−163.227			−1.97089			−0.985443	−	0.170004i	\(-0.945622\pi\)
							−0.985443	+	0.170004i	\(0.945622\pi\)
\(23\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(29\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(31\)		0			0		0.500000	−	0.866025i	\(-0.333333\pi\)
							−0.500000	+	0.866025i	\(0.666667\pi\)
\(37\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(41\)	367.005	+	211.890i	1.39797	+	0.807116i	0.994179	−	0.107738i	\(-0.0343608\pi\)
							0.403786	+	0.914854i	\(0.367694\pi\)
\(43\)	−281.931	−	488.319i	−0.999864	−	1.73181i	−0.514239	−	0.857647i	\(-0.671926\pi\)
							−0.485624	−	0.874168i	\(-0.661408\pi\)
\(47\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(53\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(59\)	−775.346	−	447.646i	−1.71087	−	0.987772i	−0.933388	−	0.358868i	\(-0.883163\pi\)
							−0.777483	−	0.628904i	\(-0.783504\pi\)
\(61\)		0			0		−0.500000	−	0.866025i	\(-0.666667\pi\)
							0.500000	+	0.866025i	\(0.333333\pi\)
\(67\)	−456.476	+	790.640i	−0.832350	+	1.44167i	0.0638199	+	0.997961i	\(0.479672\pi\)
							−0.896170	+	0.443711i	\(0.853662\pi\)
\(71\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(73\)	−799.089			−1.28118			−0.640591	−	0.767882i	\(-0.721311\pi\)
							−0.640591	+	0.767882i	\(0.721311\pi\)
\(79\)		0			0		−0.500000	−	0.866025i	\(-0.666667\pi\)
							0.500000	+	0.866025i	\(0.333333\pi\)
\(83\)	−590.327	+	340.825i	−0.780684	+	0.450728i	−0.836673	−	0.547703i	\(-0.815502\pi\)
							0.0559884	+	0.998431i	\(0.482169\pi\)
\(89\)		−	1329.36i		−	1.58328i	−0.610988	−	0.791640i	\(-0.709227\pi\)
							0.610988	−	0.791640i	\(-0.290773\pi\)
\(97\)	455.455	+	788.870i	0.476746	+	0.825749i	0.999645	−	0.0266459i	\(-0.00848265\pi\)
							−0.522898	+	0.852395i	\(0.675149\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.4.p.a.719.2		4
3.2	odd	2		288.4.p.a.239.1		4
4.3	odd	2		216.4.l.a.179.1		4
8.3	odd	2	CM	864.4.p.a.719.2		4
8.5	even	2		216.4.l.a.179.1		4
9.2	odd	6	inner	864.4.p.a.143.2		4
9.7	even	3		288.4.p.a.47.1		4
12.11	even	2		72.4.l.a.59.2	yes	4
24.5	odd	2		72.4.l.a.59.2	yes	4
24.11	even	2		288.4.p.a.239.1		4
36.7	odd	6		72.4.l.a.11.2	✓	4
36.11	even	6		216.4.l.a.35.1		4
72.11	even	6	inner	864.4.p.a.143.2		4
72.29	odd	6		216.4.l.a.35.1		4
72.43	odd	6		288.4.p.a.47.1		4
72.61	even	6		72.4.l.a.11.2	✓	4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.4.l.a.11.2	✓	4	36.7	odd	6
72.4.l.a.11.2	✓	4	72.61	even	6
72.4.l.a.59.2	yes	4	12.11	even	2
72.4.l.a.59.2	yes	4	24.5	odd	2
216.4.l.a.35.1		4	36.11	even	6
216.4.l.a.35.1		4	72.29	odd	6
216.4.l.a.179.1		4	4.3	odd	2
216.4.l.a.179.1		4	8.5	even	2
288.4.p.a.47.1		4	9.7	even	3
288.4.p.a.47.1		4	72.43	odd	6
288.4.p.a.239.1		4	3.2	odd	2
288.4.p.a.239.1		4	24.11	even	2
864.4.p.a.143.2		4	9.2	odd	6	inner
864.4.p.a.143.2		4	72.11	even	6	inner
864.4.p.a.719.2		4	1.1	even	1	trivial
864.4.p.a.719.2		4	8.3	odd	2	CM