Embedded newform 576.2.i.h.193.1

• Label: 576.2.i.h.193.1
• Level: $576$
• Weight: $2$
• Character: 576.193
• Analytic conductor: $4.599$
• Analytic rank: $0$
• Dimension: $2$
• 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:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			193.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	576.193
Dual form			576.2.i.h.385.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.50000 - 0.866025i) q^{3} +(2.00000 - 3.46410i) q^{5} +(-1.00000 - 1.73205i) q^{7} +(1.50000 - 2.59808i) q^{9} +(2.50000 + 4.33013i) q^{11} +(-1.00000 + 1.73205i) q^{13} -6.92820i q^{15} -3.00000 q^{17} +1.00000 q^{19} +(-3.00000 - 1.73205i) q^{21} +(-3.00000 + 5.19615i) q^{23} +(-5.50000 - 9.52628i) q^{25} -5.19615i q^{27} +(-1.00000 - 1.73205i) q^{29} +(-2.00000 + 3.46410i) q^{31} +(7.50000 + 4.33013i) q^{33} -8.00000 q^{35} +8.00000 q^{37} +3.46410i q^{39} +(-0.500000 + 0.866025i) q^{41} +(3.50000 + 6.06218i) q^{43} +(-6.00000 - 10.3923i) q^{45} +(1.00000 + 1.73205i) q^{47} +(1.50000 - 2.59808i) q^{49} +(-4.50000 + 2.59808i) q^{51} +4.00000 q^{53} +20.0000 q^{55} +(1.50000 - 0.866025i) q^{57} +(-2.50000 + 4.33013i) q^{59} -6.00000 q^{63} +(4.00000 + 6.92820i) q^{65} +(6.50000 - 11.2583i) q^{67} +10.3923i q^{69} -8.00000 q^{71} +3.00000 q^{73} +(-16.5000 - 9.52628i) q^{75} +(5.00000 - 8.66025i) q^{77} +(4.00000 + 6.92820i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-6.00000 - 10.3923i) q^{83} +(-6.00000 + 10.3923i) q^{85} +(-3.00000 - 1.73205i) q^{87} -10.0000 q^{89} +4.00000 q^{91} +6.92820i q^{93} +(2.00000 - 3.46410i) q^{95} +(5.50000 + 9.52628i) q^{97} +15.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 3 * q^3 + 4 * q^5 - 2 * q^7 + 3 * q^9 + 5 * q^11 - 2 * q^13 - 6 * q^17 + 2 * q^19 - 6 * q^21 - 6 * q^23 - 11 * q^25 - 2 * q^29 - 4 * q^31 + 15 * q^33 - 16 * q^35 + 16 * q^37 - q^41 + 7 * q^43 - 12 * q^45 + 2 * q^47 + 3 * q^49 - 9 * q^51 + 8 * q^53 + 40 * q^55 + 3 * q^57 - 5 * q^59 - 12 * q^63 + 8 * q^65 + 13 * q^67 - 16 * q^71 + 6 * q^73 - 33 * q^75 + 10 * q^77 + 8 * q^79 - 9 * q^81 - 12 * q^83 - 12 * q^85 - 6 * q^87 - 20 * q^89 + 8 * q^91 + 4 * q^95 + 11 * q^97 + 30 * q^99

Character values

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

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

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
							\(3\)	1.50000	−	0.866025i	0.866025	−	0.500000i
\(5\)	2.00000	−	3.46410i	0.894427	−	1.54919i								0.0599153	−	0.998203i	\(-0.480917\pi\)
							0.834512	−	0.550990i	\(-0.185750\pi\)
\(7\)	−1.00000	−	1.73205i	−0.377964	−	0.654654i	0.612801	−	0.790237i	\(-0.290043\pi\)
							−0.990766	+	0.135583i	\(0.956709\pi\)
\(11\)	2.50000	+	4.33013i	0.753778	+	1.30558i	0.945979	+	0.324227i	\(0.105104\pi\)
							−0.192201	+	0.981356i	\(0.561563\pi\)
\(13\)	−1.00000	+	1.73205i	−0.277350	+	0.480384i	−0.970725	−	0.240192i	\(-0.922790\pi\)
							0.693375	+	0.720577i	\(0.256123\pi\)
\(17\)	−3.00000			−0.727607			−0.363803	−	0.931476i	\(-0.618522\pi\)
							−0.363803	+	0.931476i	\(0.618522\pi\)
\(19\)	1.00000			0.229416			0.114708	−	0.993399i	\(-0.463407\pi\)
							0.114708	+	0.993399i	\(0.463407\pi\)
\(23\)	−3.00000	+	5.19615i	−0.625543	+	1.08347i	0.362892	+	0.931831i	\(0.381789\pi\)
							−0.988436	+	0.151642i	\(0.951544\pi\)
\(29\)	−1.00000	−	1.73205i	−0.185695	−	0.321634i	0.758115	−	0.652121i	\(-0.226120\pi\)
							−0.943811	+	0.330487i	\(0.892787\pi\)
\(31\)	−2.00000	+	3.46410i	−0.359211	+	0.622171i	−0.987829	−	0.155543i	\(-0.950287\pi\)
							0.628619	+	0.777714i	\(0.283621\pi\)
\(37\)	8.00000			1.31519			0.657596	−	0.753371i	\(-0.271573\pi\)
							0.657596	+	0.753371i	\(0.271573\pi\)
\(41\)	−0.500000	+	0.866025i	−0.0780869	+	0.135250i	−0.902424	−	0.430848i	\(-0.858214\pi\)
							0.824338	+	0.566099i	\(0.191548\pi\)
\(43\)	3.50000	+	6.06218i	0.533745	+	0.924473i	0.999223	+	0.0394140i	\(0.0125491\pi\)
							−0.465478	+	0.885059i	\(0.654118\pi\)
\(47\)	1.00000	+	1.73205i	0.145865	+	0.252646i	0.929695	−	0.368329i	\(-0.120070\pi\)
							−0.783830	+	0.620975i	\(0.786737\pi\)
\(53\)	4.00000			0.549442			0.274721	−	0.961524i	\(-0.411414\pi\)
							0.274721	+	0.961524i	\(0.411414\pi\)
\(59\)	−2.50000	+	4.33013i	−0.325472	+	0.563735i	−0.981608	−	0.190909i	\(-0.938857\pi\)
							0.656136	+	0.754643i	\(0.272190\pi\)
\(61\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(67\)	6.50000	−	11.2583i	0.794101	−	1.37542i	−0.129307	−	0.991605i	\(-0.541275\pi\)
							0.923408	−	0.383819i	\(-0.125391\pi\)
\(71\)	−8.00000			−0.949425			−0.474713	−	0.880141i	\(-0.657448\pi\)
							−0.474713	+	0.880141i	\(0.657448\pi\)
\(73\)	3.00000			0.351123			0.175562	−	0.984468i	\(-0.443826\pi\)
							0.175562	+	0.984468i	\(0.443826\pi\)
\(79\)	4.00000	+	6.92820i	0.450035	+	0.779484i	0.998388	−	0.0567635i	\(-0.0180781\pi\)
							−0.548352	+	0.836247i	\(0.684745\pi\)
\(83\)	−6.00000	−	10.3923i	−0.658586	−	1.14070i	−0.980982	−	0.194099i	\(-0.937822\pi\)
							0.322396	−	0.946605i	\(-0.395512\pi\)
\(89\)	−10.0000			−1.06000			−0.529999	−	0.847998i	\(-0.677808\pi\)
							−0.529999	+	0.847998i	\(0.677808\pi\)
\(97\)	5.50000	+	9.52628i	0.558440	+	0.967247i	0.997627	+	0.0688512i	\(0.0219334\pi\)
							−0.439187	+	0.898396i	\(0.644733\pi\)

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.h.193.1		2
3.2	odd	2		1728.2.i.a.577.1		2
4.3	odd	2		576.2.i.b.193.1		2
8.3	odd	2		288.2.i.b.193.1	yes	2
8.5	even	2		288.2.i.a.193.1	yes	2
9.2	odd	6		1728.2.i.a.1153.1		2
9.4	even	3		5184.2.a.b.1.1		1
9.5	odd	6		5184.2.a.bf.1.1		1
9.7	even	3	inner	576.2.i.h.385.1		2
12.11	even	2		1728.2.i.b.577.1		2
24.5	odd	2		864.2.i.a.577.1		2
24.11	even	2		864.2.i.b.577.1		2
36.7	odd	6		576.2.i.b.385.1		2
36.11	even	6		1728.2.i.b.1153.1		2
36.23	even	6		5184.2.a.be.1.1		1
36.31	odd	6		5184.2.a.a.1.1		1
72.5	odd	6		2592.2.a.b.1.1		1
72.11	even	6		864.2.i.b.289.1		2
72.13	even	6		2592.2.a.h.1.1		1
72.29	odd	6		864.2.i.a.289.1		2
72.43	odd	6		288.2.i.b.97.1	yes	2
72.59	even	6		2592.2.a.a.1.1		1
72.61	even	6		288.2.i.a.97.1	✓	2
72.67	odd	6		2592.2.a.g.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.2.i.a.97.1	✓	2	72.61	even	6
288.2.i.a.193.1	yes	2	8.5	even	2
288.2.i.b.97.1	yes	2	72.43	odd	6
288.2.i.b.193.1	yes	2	8.3	odd	2
576.2.i.b.193.1		2	4.3	odd	2
576.2.i.b.385.1		2	36.7	odd	6
576.2.i.h.193.1		2	1.1	even	1	trivial
576.2.i.h.385.1		2	9.7	even	3	inner
864.2.i.a.289.1		2	72.29	odd	6
864.2.i.a.577.1		2	24.5	odd	2
864.2.i.b.289.1		2	72.11	even	6
864.2.i.b.577.1		2	24.11	even	2
1728.2.i.a.577.1		2	3.2	odd	2
1728.2.i.a.1153.1		2	9.2	odd	6
1728.2.i.b.577.1		2	12.11	even	2
1728.2.i.b.1153.1		2	36.11	even	6
2592.2.a.a.1.1		1	72.59	even	6
2592.2.a.b.1.1		1	72.5	odd	6
2592.2.a.g.1.1		1	72.67	odd	6
2592.2.a.h.1.1		1	72.13	even	6
5184.2.a.a.1.1		1	36.31	odd	6
5184.2.a.b.1.1		1	9.4	even	3
5184.2.a.be.1.1		1	36.23	even	6
5184.2.a.bf.1.1		1	9.5	odd	6