Embedded newform 576.2.y.a.239.6

• Label: 576.2.y.a.239.6
• Level: $576$
• Weight: $2$
• Character: 576.239
• Analytic conductor: $4.599$
• Analytic rank: $0$
• Dimension: $88$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	576.y (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.59938315643\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(22\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			239.6
Character	\(\chi\)	\(=\)	576.239
Dual form			576.2.y.a.335.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.11494 - 1.32549i) q^{3} +(1.20583 - 0.323102i) q^{5} +(-0.140266 - 0.242948i) q^{7} +(-0.513832 + 2.95567i) q^{9} +(3.07444 + 0.823794i) q^{11} +(2.76539 - 0.740984i) q^{13} +(-1.77270 - 1.23808i) q^{15} -3.72031i q^{17} +(4.10860 - 4.10860i) q^{19} +(-0.165636 + 0.456792i) q^{21} +(1.57595 + 0.909876i) q^{23} +(-2.98049 + 1.72078i) q^{25} +(4.49059 - 2.61431i) q^{27} +(-3.83006 - 1.02626i) q^{29} +(-8.81101 - 5.08704i) q^{31} +(-2.33588 - 4.99361i) q^{33} +(-0.247635 - 0.247635i) q^{35} +(1.76964 - 1.76964i) q^{37} +(-4.06540 - 2.83934i) q^{39} +(2.66819 - 4.62144i) q^{41} +(1.84748 - 6.89490i) q^{43} +(0.335387 + 3.73007i) q^{45} +(5.48486 + 9.50006i) q^{47} +(3.46065 - 5.99402i) q^{49} +(-4.93123 + 4.14791i) q^{51} +(8.58403 + 8.58403i) q^{53} +3.97344 q^{55} +(-10.0267 - 0.865067i) q^{57} +(-1.44299 - 5.38532i) q^{59} +(1.66977 - 6.23168i) q^{61} +(0.790146 - 0.289745i) q^{63} +(3.09519 - 1.78701i) q^{65} +(1.00652 + 3.75640i) q^{67} +(-0.551057 - 3.10336i) q^{69} +10.6808i q^{71} -9.30419i q^{73} +(5.60393 + 2.03203i) q^{75} +(-0.231101 - 0.862479i) q^{77} +(-8.70990 + 5.02866i) q^{79} +(-8.47195 - 3.03744i) q^{81} +(-0.588691 + 2.19703i) q^{83} +(-1.20204 - 4.48608i) q^{85} +(2.90998 + 6.22091i) q^{87} -1.87637 q^{89} +(-0.567911 - 0.567911i) q^{91} +(3.08091 + 17.3506i) q^{93} +(3.62679 - 6.28179i) q^{95} +(9.19070 + 15.9188i) q^{97} +(-4.01461 + 8.66374i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	88 * q + 4 * q^3 - 6 * q^5 + 4 * q^7 + 6 * q^11 - 2 * q^13 + 8 * q^19 + 2 * q^21 + 12 * q^23 + 16 * q^27 - 6 * q^29 - 8 * q^33 - 8 * q^37 + 32 * q^39 + 2 * q^43 + 6 * q^45 - 24 * q^49 + 12 * q^51 + 16 * q^55 + 42 * q^59 - 2 * q^61 - 12 * q^65 + 2 * q^67 - 10 * q^69 + 56 * q^75 - 6 * q^77 - 8 * q^81 - 54 * q^83 + 8 * q^85 - 48 * q^87 - 20 * q^91 - 34 * q^93 - 4 * q^97 - 46 * q^99

Character values

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

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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
							\(3\)	−1.11494	−	1.32549i	−0.643709	−	0.765270i
\(5\)	1.20583	−	0.323102i	0.539266	−	0.144496i								0.0211020	−	0.999777i	\(-0.493283\pi\)
							0.518164	+	0.855282i	\(0.326616\pi\)
\(7\)	−0.140266	−	0.242948i	−0.0530156	−	0.0918256i	0.838300	−	0.545210i	\(-0.183550\pi\)
							−0.891315	+	0.453384i	\(0.850217\pi\)
\(11\)	3.07444	+	0.823794i	0.926979	+	0.248383i	0.690566	−	0.723269i	\(-0.257361\pi\)
							0.236413	+	0.971653i	\(0.424028\pi\)
\(13\)	2.76539	−	0.740984i	0.766982	−	0.205512i	0.145944	−	0.989293i	\(-0.453378\pi\)
							0.621038	+	0.783781i	\(0.286711\pi\)
\(17\)		−	3.72031i		−	0.902309i	−0.892446	−	0.451154i	\(-0.851012\pi\)
							0.892446	−	0.451154i	\(-0.148988\pi\)
\(19\)	4.10860	−	4.10860i	0.942577	−	0.942577i	−0.0558614	−	0.998439i	\(-0.517791\pi\)
							0.998439	+	0.0558614i	\(0.0177905\pi\)
\(23\)	1.57595	+	0.909876i	0.328609	+	0.189722i	0.655223	−	0.755435i	\(-0.272575\pi\)
							−0.326615	+	0.945158i	\(0.605908\pi\)
\(29\)	−3.83006	−	1.02626i	−0.711224	−	0.190572i	−0.114972	−	0.993369i	\(-0.536678\pi\)
							−0.596253	+	0.802797i	\(0.703344\pi\)
\(31\)	−8.81101	−	5.08704i	−1.58250	−	0.913659i	−0.994493	−	0.104807i	\(-0.966578\pi\)
							−0.588012	−	0.808852i	\(-0.700089\pi\)
\(37\)	1.76964	−	1.76964i	0.290928	−	0.290928i	−0.546519	−	0.837447i	\(-0.684047\pi\)
							0.837447	+	0.546519i	\(0.184047\pi\)
\(41\)	2.66819	−	4.62144i	0.416701	−	0.721747i	−0.578904	−	0.815395i	\(-0.696520\pi\)
							0.995605	+	0.0936482i	\(0.0298529\pi\)
\(43\)	1.84748	−	6.89490i	0.281738	−	1.05146i	−0.669452	−	0.742856i	\(-0.733471\pi\)
							0.951190	−	0.308606i	\(-0.0998625\pi\)
\(47\)	5.48486	+	9.50006i	0.800049	+	1.38573i	0.919583	+	0.392896i	\(0.128527\pi\)
							−0.119534	+	0.992830i	\(0.538140\pi\)
\(53\)	8.58403	+	8.58403i	1.17911	+	1.17911i	0.979972	+	0.199135i	\(0.0638133\pi\)
							0.199135	+	0.979972i	\(0.436187\pi\)
\(59\)	−1.44299	−	5.38532i	−0.187862	−	0.701109i	−0.994000	−	0.109382i	\(-0.965113\pi\)
							0.806138	−	0.591727i	\(-0.201554\pi\)
\(61\)	1.66977	−	6.23168i	0.213793	−	0.797885i	−0.772796	−	0.634655i	\(-0.781142\pi\)
							0.986588	−	0.163230i	\(-0.0521911\pi\)
\(67\)	1.00652	+	3.75640i	0.122967	+	0.458918i	0.999759	−	0.0219514i	\(-0.00698792\pi\)
							−0.876792	+	0.480869i	\(0.840321\pi\)
\(71\)			10.6808i			1.26758i	0.773506	+	0.633789i	\(0.218501\pi\)
							−0.773506	+	0.633789i	\(0.781499\pi\)
\(73\)		−	9.30419i		−	1.08897i	−0.838770	−	0.544487i	\(-0.816725\pi\)
							0.838770	−	0.544487i	\(-0.183275\pi\)
\(79\)	−8.70990	+	5.02866i	−0.979940	+	0.565769i	−0.902252	−	0.431209i	\(-0.858087\pi\)
							−0.0776882	+	0.996978i	\(0.524754\pi\)
\(83\)	−0.588691	+	2.19703i	−0.0646173	+	0.241155i	−0.990679	−	0.136217i	\(-0.956506\pi\)
							0.926062	+	0.377372i	\(0.123172\pi\)
\(89\)	−1.87637			−0.198895			−0.0994475	−	0.995043i	\(-0.531708\pi\)
							−0.0994475	+	0.995043i	\(0.531708\pi\)
\(97\)	9.19070	+	15.9188i	0.933175	+	1.61631i	0.777857	+	0.628441i	\(0.216307\pi\)
							0.155317	+	0.987865i	\(0.450360\pi\)

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.y.a.239.6		88
3.2	odd	2		1728.2.z.a.1583.7		88
4.3	odd	2		144.2.u.a.131.16	yes	88
9.2	odd	6	inner	576.2.y.a.47.17		88
9.7	even	3		1728.2.z.a.1007.7		88
12.11	even	2		432.2.v.a.179.7		88
16.5	even	4		144.2.u.a.59.22	yes	88
16.11	odd	4	inner	576.2.y.a.527.17		88
36.7	odd	6		432.2.v.a.35.1		88
36.11	even	6		144.2.u.a.83.22	yes	88
48.5	odd	4		432.2.v.a.395.1		88
48.11	even	4		1728.2.z.a.719.7		88
144.11	even	12	inner	576.2.y.a.335.6		88
144.43	odd	12		1728.2.z.a.143.7		88
144.101	odd	12		144.2.u.a.11.16	✓	88
144.133	even	12		432.2.v.a.251.7		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.u.a.11.16	✓	88	144.101	odd	12
144.2.u.a.59.22	yes	88	16.5	even	4
144.2.u.a.83.22	yes	88	36.11	even	6
144.2.u.a.131.16	yes	88	4.3	odd	2
432.2.v.a.35.1		88	36.7	odd	6
432.2.v.a.179.7		88	12.11	even	2
432.2.v.a.251.7		88	144.133	even	12
432.2.v.a.395.1		88	48.5	odd	4
576.2.y.a.47.17		88	9.2	odd	6	inner
576.2.y.a.239.6		88	1.1	even	1	trivial
576.2.y.a.335.6		88	144.11	even	12	inner
576.2.y.a.527.17		88	16.11	odd	4	inner
1728.2.z.a.143.7		88	144.43	odd	12
1728.2.z.a.719.7		88	48.11	even	4
1728.2.z.a.1007.7		88	9.7	even	3
1728.2.z.a.1583.7		88	3.2	odd	2