Embedded newform 576.2.i.j.385.1

• Label: 576.2.i.j.385.1
• Level: $576$
• Weight: $2$
• Character: 576.385
• Analytic conductor: $4.599$
• Analytic rank: $0$
• Dimension: $4$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	576.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.59938315643\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial:	\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			385.1
Root			\(1.68614 - 0.396143i\) of defining polynomial
Character	\(\chi\)	\(=\)	576.385
Dual form			576.2.i.j.193.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.68614 + 0.396143i) q^{3} +(1.18614 + 2.05446i) q^{5} +(-2.18614 + 3.78651i) q^{7} +(2.68614 - 1.33591i) q^{9} +(0.500000 - 0.866025i) q^{11} +(-0.186141 - 0.322405i) q^{13} +(-2.81386 - 2.99422i) q^{15} -5.37228 q^{17} -0.627719 q^{19} +(2.18614 - 7.25061i) q^{21} +(0.186141 + 0.322405i) q^{23} +(-0.313859 + 0.543620i) q^{25} +(-4.00000 + 3.31662i) q^{27} +(-2.18614 + 3.78651i) q^{29} +(-3.18614 - 5.51856i) q^{31} +(-0.500000 + 1.65831i) q^{33} -10.3723 q^{35} -8.74456 q^{37} +(0.441578 + 0.469882i) q^{39} +(5.87228 + 10.1711i) q^{41} +(-0.872281 + 1.51084i) q^{43} +(5.93070 + 3.93398i) q^{45} +(-2.18614 + 3.78651i) q^{47} +(-6.05842 - 10.4935i) q^{49} +(9.05842 - 2.12819i) q^{51} -0.744563 q^{53} +2.37228 q^{55} +(1.05842 - 0.248667i) q^{57} +(3.50000 + 6.06218i) q^{59} +(1.18614 - 2.05446i) q^{61} +(-0.813859 + 13.0916i) q^{63} +(0.441578 - 0.764836i) q^{65} +(-1.87228 - 3.24289i) q^{67} +(-0.441578 - 0.469882i) q^{69} -4.00000 q^{71} -12.1168 q^{73} +(0.313859 - 1.04095i) q^{75} +(2.18614 + 3.78651i) q^{77} +(3.18614 - 5.51856i) q^{79} +(5.43070 - 7.17687i) q^{81} +(4.81386 - 8.33785i) q^{83} +(-6.37228 - 11.0371i) q^{85} +(2.18614 - 7.25061i) q^{87} +6.00000 q^{89} +1.62772 q^{91} +(7.55842 + 8.04290i) q^{93} +(-0.744563 - 1.28962i) q^{95} +(-0.872281 + 1.51084i) q^{97} +(0.186141 - 2.99422i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	4 * q - q^3 - q^5 - 3 * q^7 + 5 * q^9 + 2 * q^11 + 5 * q^13 - 17 * q^15 - 10 * q^17 - 14 * q^19 + 3 * q^21 - 5 * q^23 - 7 * q^25 - 16 * q^27 - 3 * q^29 - 7 * q^31 - 2 * q^33 - 30 * q^35 - 12 * q^37 + 19 * q^39 + 12 * q^41 + 8 * q^43 - 5 * q^45 - 3 * q^47 - 7 * q^49 + 19 * q^51 + 20 * q^53 - 2 * q^55 - 13 * q^57 + 14 * q^59 - q^61 - 9 * q^63 + 19 * q^65 + 4 * q^67 - 19 * q^69 - 16 * q^71 - 14 * q^73 + 7 * q^75 + 3 * q^77 + 7 * q^79 - 7 * q^81 + 25 * q^83 - 14 * q^85 + 3 * q^87 + 24 * q^89 + 18 * q^91 + 13 * q^93 + 20 * q^95 + 8 * q^97 - 5 * q^99

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(1\)

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
							\(3\)	−1.68614	+	0.396143i	−0.973494	+	0.228714i
\(5\)	1.18614	+	2.05446i	0.530458	+	0.918781i								0.999368	+	0.0355348i	\(0.0113134\pi\)
							−0.468910	+	0.883246i	\(0.655353\pi\)
\(7\)	−2.18614	+	3.78651i	−0.826284	+	1.43117i	0.0746509	+	0.997210i	\(0.476216\pi\)
							−0.900934	+	0.433955i	\(0.857118\pi\)
\(11\)	0.500000	−	0.866025i	0.150756	−	0.261116i	−0.780750	−	0.624844i	\(-0.785163\pi\)
							0.931505	+	0.363727i	\(0.118496\pi\)
\(13\)	−0.186141	−	0.322405i	−0.0516261	−	0.0894191i	0.839057	−	0.544043i	\(-0.183107\pi\)
							−0.890684	+	0.454624i	\(0.849774\pi\)
\(17\)	−5.37228			−1.30297			−0.651485	−	0.758662i	\(-0.725854\pi\)
							−0.651485	+	0.758662i	\(0.725854\pi\)
\(19\)	−0.627719			−0.144009			−0.0720043	−	0.997404i	\(-0.522940\pi\)
							−0.0720043	+	0.997404i	\(0.522940\pi\)
\(23\)	0.186141	+	0.322405i	0.0388130	+	0.0672261i	0.884779	−	0.466010i	\(-0.154309\pi\)
							−0.845966	+	0.533236i	\(0.820976\pi\)
\(29\)	−2.18614	+	3.78651i	−0.405956	+	0.703137i	−0.994432	−	0.105378i	\(-0.966395\pi\)
							0.588476	+	0.808515i	\(0.299728\pi\)
\(31\)	−3.18614	−	5.51856i	−0.572248	−	0.991162i	−0.996335	−	0.0855407i	\(-0.972738\pi\)
							0.424087	−	0.905621i	\(-0.360595\pi\)
\(37\)	−8.74456			−1.43760			−0.718799	−	0.695218i	\(-0.755308\pi\)
							−0.718799	+	0.695218i	\(0.755308\pi\)
\(41\)	5.87228	+	10.1711i	0.917096	+	1.58846i	0.803803	+	0.594896i	\(0.202807\pi\)
							0.113293	+	0.993562i	\(0.463860\pi\)
\(43\)	−0.872281	+	1.51084i	−0.133022	+	0.230400i	−0.924840	−	0.380356i	\(-0.875801\pi\)
							0.791818	+	0.610757i	\(0.209135\pi\)
\(47\)	−2.18614	+	3.78651i	−0.318881	+	0.552319i	−0.980255	−	0.197738i	\(-0.936640\pi\)
							0.661374	+	0.750057i	\(0.269974\pi\)
\(53\)	−0.744563			−0.102274			−0.0511368	−	0.998692i	\(-0.516284\pi\)
							−0.0511368	+	0.998692i	\(0.516284\pi\)
\(59\)	3.50000	+	6.06218i	0.455661	+	0.789228i	0.998726	−	0.0504625i	\(-0.0160695\pi\)
							−0.543065	+	0.839691i	\(0.682736\pi\)
\(61\)	1.18614	−	2.05446i	0.151870	−	0.263046i	−0.780045	−	0.625723i	\(-0.784804\pi\)
							0.931915	+	0.362677i	\(0.118137\pi\)
\(67\)	−1.87228	−	3.24289i	−0.228736	−	0.396182i	0.728698	−	0.684835i	\(-0.240126\pi\)
							−0.957434	+	0.288653i	\(0.906792\pi\)
\(71\)	−4.00000			−0.474713			−0.237356	−	0.971423i	\(-0.576281\pi\)
							−0.237356	+	0.971423i	\(0.576281\pi\)
\(73\)	−12.1168			−1.41817			−0.709085	−	0.705123i	\(-0.750892\pi\)
							−0.709085	+	0.705123i	\(0.750892\pi\)
\(79\)	3.18614	−	5.51856i	0.358469	−	0.620886i	−0.629236	−	0.777214i	\(-0.716632\pi\)
							0.987705	+	0.156328i	\(0.0499656\pi\)
\(83\)	4.81386	−	8.33785i	0.528390	−	0.915198i	−0.471062	−	0.882100i	\(-0.656129\pi\)
							0.999452	−	0.0330979i	\(-0.0105373\pi\)
\(89\)	6.00000			0.635999			0.317999	−	0.948091i	\(-0.396989\pi\)
							0.317999	+	0.948091i	\(0.396989\pi\)
\(97\)	−0.872281	+	1.51084i	−0.0885667	+	0.153402i	−0.906906	−	0.421334i	\(-0.861562\pi\)
							0.818339	+	0.574736i	\(0.194895\pi\)

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	576.2.i.j.385.1		4
3.2	odd	2		1728.2.i.i.1153.1		4
4.3	odd	2		576.2.i.l.385.2		4
8.3	odd	2		144.2.i.d.97.1		4
8.5	even	2		72.2.i.b.25.2	✓	4
9.2	odd	6		5184.2.a.bp.1.2		2
9.4	even	3	inner	576.2.i.j.193.1		4
9.5	odd	6		1728.2.i.i.577.1		4
9.7	even	3		5184.2.a.bt.1.1		2
12.11	even	2		1728.2.i.j.1153.1		4
24.5	odd	2		216.2.i.b.73.2		4
24.11	even	2		432.2.i.d.289.2		4
36.7	odd	6		5184.2.a.bs.1.1		2
36.11	even	6		5184.2.a.bo.1.2		2
36.23	even	6		1728.2.i.j.577.1		4
36.31	odd	6		576.2.i.l.193.2		4
72.5	odd	6		216.2.i.b.145.2		4
72.11	even	6		1296.2.a.p.1.1		2
72.13	even	6		72.2.i.b.49.2	yes	4
72.29	odd	6		648.2.a.g.1.1		2
72.43	odd	6		1296.2.a.n.1.2		2
72.59	even	6		432.2.i.d.145.2		4
72.61	even	6		648.2.a.f.1.2		2
72.67	odd	6		144.2.i.d.49.1		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.i.b.25.2	✓	4	8.5	even	2
72.2.i.b.49.2	yes	4	72.13	even	6
144.2.i.d.49.1		4	72.67	odd	6
144.2.i.d.97.1		4	8.3	odd	2
216.2.i.b.73.2		4	24.5	odd	2
216.2.i.b.145.2		4	72.5	odd	6
432.2.i.d.145.2		4	72.59	even	6
432.2.i.d.289.2		4	24.11	even	2
576.2.i.j.193.1		4	9.4	even	3	inner
576.2.i.j.385.1		4	1.1	even	1	trivial
576.2.i.l.193.2		4	36.31	odd	6
576.2.i.l.385.2		4	4.3	odd	2
648.2.a.f.1.2		2	72.61	even	6
648.2.a.g.1.1		2	72.29	odd	6
1296.2.a.n.1.2		2	72.43	odd	6
1296.2.a.p.1.1		2	72.11	even	6
1728.2.i.i.577.1		4	9.5	odd	6
1728.2.i.i.1153.1		4	3.2	odd	2
1728.2.i.j.577.1		4	36.23	even	6
1728.2.i.j.1153.1		4	12.11	even	2
5184.2.a.bo.1.2		2	36.11	even	6
5184.2.a.bp.1.2		2	9.2	odd	6
5184.2.a.bs.1.1		2	36.7	odd	6
5184.2.a.bt.1.1		2	9.7	even	3