Embedded newform 288.2.i.d.97.1

• Label: 288.2.i.d.97.1
• Level: $288$
• Weight: $2$
• Character: 288.97
• Analytic conductor: $2.300$
• Analytic rank: $0$
• Dimension: $4$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	288.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29969157821\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			97.1
Root			\(0.866025 - 0.500000i\) of defining polynomial
Character	\(\chi\)	\(=\)	288.97
Dual form			288.2.i.d.193.1

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +(0.500000 + 0.866025i) q^{5} +(0.866025 - 1.50000i) q^{7} +3.00000 q^{9} +(-0.866025 + 1.50000i) q^{11} +(1.50000 + 2.59808i) q^{13} +(-0.866025 - 1.50000i) q^{15} +4.00000 q^{17} +6.92820 q^{19} +(-1.50000 + 2.59808i) q^{21} +(4.33013 + 7.50000i) q^{23} +(2.00000 - 3.46410i) q^{25} -5.19615 q^{27} +(-0.500000 + 0.866025i) q^{29} +(-2.59808 - 4.50000i) q^{31} +(1.50000 - 2.59808i) q^{33} +1.73205 q^{35} -8.00000 q^{37} +(-2.59808 - 4.50000i) q^{39} +(-2.50000 - 4.33013i) q^{41} +(4.33013 - 7.50000i) q^{43} +(1.50000 + 2.59808i) q^{45} +(-6.06218 + 10.5000i) q^{47} +(2.00000 + 3.46410i) q^{49} -6.92820 q^{51} -8.00000 q^{53} -1.73205 q^{55} -12.0000 q^{57} +(-0.866025 - 1.50000i) q^{59} +(3.50000 - 6.06218i) q^{61} +(2.59808 - 4.50000i) q^{63} +(-1.50000 + 2.59808i) q^{65} +(4.33013 + 7.50000i) q^{67} +(-7.50000 - 12.9904i) q^{69} +3.46410 q^{71} -12.0000 q^{73} +(-3.46410 + 6.00000i) q^{75} +(1.50000 + 2.59808i) q^{77} +(2.59808 - 4.50000i) q^{79} +9.00000 q^{81} +(4.33013 - 7.50000i) q^{83} +(2.00000 + 3.46410i) q^{85} +(0.866025 - 1.50000i) q^{87} -4.00000 q^{89} +5.19615 q^{91} +(4.50000 + 7.79423i) q^{93} +(3.46410 + 6.00000i) q^{95} +(1.50000 - 2.59808i) q^{97} +(-2.59808 + 4.50000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	4 * q + 2 * q^5 + 12 * q^9 + 6 * q^13 + 16 * q^17 - 6 * q^21 + 8 * q^25 - 2 * q^29 + 6 * q^33 - 32 * q^37 - 10 * q^41 + 6 * q^45 + 8 * q^49 - 32 * q^53 - 48 * q^57 + 14 * q^61 - 6 * q^65 - 30 * q^69 - 48 * q^73 + 6 * q^77 + 36 * q^81 + 8 * q^85 - 16 * q^89 + 18 * q^93 + 6 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(65\)	\(127\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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
							\(3\)	−1.73205			−1.00000
\(5\)	0.500000	+	0.866025i	0.223607	+	0.387298i								0.955901	−	0.293691i	\(-0.0948835\pi\)
							−0.732294	+	0.680989i	\(0.761550\pi\)
\(7\)	0.866025	−	1.50000i	0.327327	−	0.566947i	−0.654654	−	0.755929i	\(-0.727186\pi\)
							0.981981	+	0.188982i	\(0.0605189\pi\)
\(11\)	−0.866025	+	1.50000i	−0.261116	+	0.452267i	−0.966539	−	0.256520i	\(-0.917424\pi\)
							0.705422	+	0.708787i	\(0.250757\pi\)
\(13\)	1.50000	+	2.59808i	0.416025	+	0.720577i	0.995535	−	0.0943882i	\(-0.0300895\pi\)
							−0.579510	+	0.814965i	\(0.696756\pi\)
\(17\)	4.00000			0.970143			0.485071	−	0.874475i	\(-0.338794\pi\)
							0.485071	+	0.874475i	\(0.338794\pi\)
\(19\)	6.92820			1.58944			0.794719	−	0.606977i	\(-0.207618\pi\)
							0.794719	+	0.606977i	\(0.207618\pi\)
\(23\)	4.33013	+	7.50000i	0.902894	+	1.56386i	0.823720	+	0.566997i	\(0.191895\pi\)
							0.0791743	+	0.996861i	\(0.474772\pi\)
\(29\)	−0.500000	+	0.866025i	−0.0928477	+	0.160817i	−0.908708	−	0.417432i	\(-0.862930\pi\)
							0.815861	+	0.578249i	\(0.196264\pi\)
\(31\)	−2.59808	−	4.50000i	−0.466628	−	0.808224i	0.532645	−	0.846339i	\(-0.321198\pi\)
							−0.999273	+	0.0381148i	\(0.987865\pi\)
\(37\)	−8.00000			−1.31519			−0.657596	−	0.753371i	\(-0.728427\pi\)
							−0.657596	+	0.753371i	\(0.728427\pi\)
\(41\)	−2.50000	−	4.33013i	−0.390434	−	0.676252i	0.602072	−	0.798441i	\(-0.294342\pi\)
							−0.992507	+	0.122189i	\(0.961009\pi\)
\(43\)	4.33013	−	7.50000i	0.660338	−	1.14374i	−0.320189	−	0.947354i	\(-0.603746\pi\)
							0.980527	−	0.196385i	\(-0.0629204\pi\)
\(47\)	−6.06218	+	10.5000i	−0.884260	+	1.53158i	−0.0376995	+	0.999289i	\(0.512003\pi\)
							−0.846560	+	0.532293i	\(0.821330\pi\)
\(53\)	−8.00000			−1.09888			−0.549442	−	0.835532i	\(-0.685160\pi\)
							−0.549442	+	0.835532i	\(0.685160\pi\)
\(59\)	−0.866025	−	1.50000i	−0.112747	−	0.195283i	0.804130	−	0.594454i	\(-0.202632\pi\)
							−0.916877	+	0.399170i	\(0.869298\pi\)
\(61\)	3.50000	−	6.06218i	0.448129	−	0.776182i	−0.550135	−	0.835076i	\(-0.685424\pi\)
							0.998264	+	0.0588933i	\(0.0187572\pi\)
\(67\)	4.33013	+	7.50000i	0.529009	+	0.916271i	0.999428	+	0.0338274i	\(0.0107696\pi\)
							−0.470418	+	0.882443i	\(0.655897\pi\)
\(71\)	3.46410			0.411113			0.205557	−	0.978645i	\(-0.434100\pi\)
							0.205557	+	0.978645i	\(0.434100\pi\)
\(73\)	−12.0000			−1.40449			−0.702247	−	0.711934i	\(-0.747820\pi\)
							−0.702247	+	0.711934i	\(0.747820\pi\)
\(79\)	2.59808	−	4.50000i	0.292306	−	0.506290i	−0.682048	−	0.731307i	\(-0.738911\pi\)
							0.974355	+	0.225018i	\(0.0722440\pi\)
\(83\)	4.33013	−	7.50000i	0.475293	−	0.823232i	−0.524306	−	0.851530i	\(-0.675675\pi\)
							0.999600	+	0.0282978i	\(0.00900866\pi\)
\(89\)	−4.00000			−0.423999			−0.212000	−	0.977270i	\(-0.567998\pi\)
							−0.212000	+	0.977270i	\(0.567998\pi\)
\(97\)	1.50000	−	2.59808i	0.152302	−	0.263795i	−0.779771	−	0.626064i	\(-0.784665\pi\)
							0.932073	+	0.362270i	\(0.117998\pi\)

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	288.2.i.d.97.1	✓	4
3.2	odd	2		864.2.i.d.289.2		4
4.3	odd	2	inner	288.2.i.d.97.2	yes	4
8.3	odd	2		576.2.i.k.385.1		4
8.5	even	2		576.2.i.k.385.2		4
9.2	odd	6		2592.2.a.p.1.1		2
9.4	even	3	inner	288.2.i.d.193.1	yes	4
9.5	odd	6		864.2.i.d.577.2		4
9.7	even	3		2592.2.a.l.1.1		2
12.11	even	2		864.2.i.d.289.1		4
24.5	odd	2		1728.2.i.l.1153.2		4
24.11	even	2		1728.2.i.l.1153.1		4
36.7	odd	6		2592.2.a.l.1.2		2
36.11	even	6		2592.2.a.p.1.2		2
36.23	even	6		864.2.i.d.577.1		4
36.31	odd	6	inner	288.2.i.d.193.2	yes	4
72.5	odd	6		1728.2.i.l.577.2		4
72.11	even	6		5184.2.a.bl.1.2		2
72.13	even	6		576.2.i.k.193.2		4
72.29	odd	6		5184.2.a.bl.1.1		2
72.43	odd	6		5184.2.a.bx.1.2		2
72.59	even	6		1728.2.i.l.577.1		4
72.61	even	6		5184.2.a.bx.1.1		2
72.67	odd	6		576.2.i.k.193.1		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.2.i.d.97.1	✓	4	1.1	even	1	trivial
288.2.i.d.97.2	yes	4	4.3	odd	2	inner
288.2.i.d.193.1	yes	4	9.4	even	3	inner
288.2.i.d.193.2	yes	4	36.31	odd	6	inner
576.2.i.k.193.1		4	72.67	odd	6
576.2.i.k.193.2		4	72.13	even	6
576.2.i.k.385.1		4	8.3	odd	2
576.2.i.k.385.2		4	8.5	even	2
864.2.i.d.289.1		4	12.11	even	2
864.2.i.d.289.2		4	3.2	odd	2
864.2.i.d.577.1		4	36.23	even	6
864.2.i.d.577.2		4	9.5	odd	6
1728.2.i.l.577.1		4	72.59	even	6
1728.2.i.l.577.2		4	72.5	odd	6
1728.2.i.l.1153.1		4	24.11	even	2
1728.2.i.l.1153.2		4	24.5	odd	2
2592.2.a.l.1.1		2	9.7	even	3
2592.2.a.l.1.2		2	36.7	odd	6
2592.2.a.p.1.1		2	9.2	odd	6
2592.2.a.p.1.2		2	36.11	even	6
5184.2.a.bl.1.1		2	72.29	odd	6
5184.2.a.bl.1.2		2	72.11	even	6
5184.2.a.bx.1.1		2	72.61	even	6
5184.2.a.bx.1.2		2	72.43	odd	6