Embedded newform 288.2.i.e.193.1

• Label: 288.2.i.e.193.1
• Level: $288$
• Weight: $2$
• Character: 288.193
• Analytic conductor: $2.300$
• Analytic rank: $0$
• Dimension: $4$
• Inner twists: $2$

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:	\(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial:	\( x^{4} - 2x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			193.1
Root			\(1.22474 + 0.707107i\) of defining polynomial
Character	\(\chi\)	\(=\)	288.193
Dual form			288.2.i.e.97.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.41421i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-1.72474 - 2.98735i) q^{7} +(-1.00000 - 2.82843i) q^{9} +(-0.724745 - 1.25529i) q^{11} +(-2.94949 + 5.10867i) q^{13} +(-0.724745 - 1.57313i) q^{15} +4.89898 q^{17} +4.00000 q^{19} +(-5.94949 - 0.548188i) q^{21} +(2.72474 - 4.71940i) q^{23} +(2.00000 + 3.46410i) q^{25} +(-5.00000 - 1.41421i) q^{27} +(-0.0505103 - 0.0874863i) q^{29} +(-1.27526 + 2.20881i) q^{31} +(-2.50000 - 0.230351i) q^{33} -3.44949 q^{35} -0.898979 q^{37} +(4.27526 + 9.27987i) q^{39} +(5.94949 - 10.3048i) q^{41} +(1.17423 + 2.03383i) q^{43} +(-2.94949 - 0.548188i) q^{45} +(3.17423 + 5.49794i) q^{47} +(-2.44949 + 4.24264i) q^{49} +(4.89898 - 6.92820i) q^{51} +8.89898 q^{53} -1.44949 q^{55} +(4.00000 - 5.65685i) q^{57} +(-7.17423 + 12.4261i) q^{59} +(3.94949 + 6.84072i) q^{61} +(-6.72474 + 7.86566i) q^{63} +(2.94949 + 5.10867i) q^{65} +(-6.17423 + 10.6941i) q^{67} +(-3.94949 - 8.57277i) q^{69} +7.79796 q^{71} -4.89898 q^{73} +(6.89898 + 0.635674i) q^{75} +(-2.50000 + 4.33013i) q^{77} +(-6.72474 - 11.6476i) q^{79} +(-7.00000 + 5.65685i) q^{81} +(0.275255 + 0.476756i) q^{83} +(2.44949 - 4.24264i) q^{85} +(-0.174235 - 0.0160540i) q^{87} -12.8990 q^{89} +20.3485 q^{91} +(1.84847 + 4.01229i) q^{93} +(2.00000 - 3.46410i) q^{95} +(1.94949 + 3.37662i) q^{97} +(-2.82577 + 3.30518i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	4 * q + 4 * q^3 + 2 * q^5 - 2 * q^7 - 4 * q^9 + 2 * q^11 - 2 * q^13 + 2 * q^15 + 16 * q^19 - 14 * q^21 + 6 * q^23 + 8 * q^25 - 20 * q^27 - 10 * q^29 - 10 * q^31 - 10 * q^33 - 4 * q^35 + 16 * q^37 + 22 * q^39 + 14 * q^41 - 10 * q^43 - 2 * q^45 - 2 * q^47 + 16 * q^53 + 4 * q^55 + 16 * q^57 - 14 * q^59 + 6 * q^61 - 22 * q^63 + 2 * q^65 - 10 * q^67 - 6 * q^69 - 8 * q^71 + 8 * q^75 - 10 * q^77 - 22 * q^79 - 28 * q^81 + 6 * q^83 + 14 * q^87 - 32 * q^89 + 52 * q^91 - 22 * q^93 + 8 * q^95 - 2 * q^97 - 26 * q^99

Character values

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

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

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
							\(3\)	1.00000	−	1.41421i	0.577350	−	0.816497i
\(5\)	0.500000	−	0.866025i	0.223607	−	0.387298i								−0.732294	−	0.680989i	\(-0.761550\pi\)
							0.955901	+	0.293691i	\(0.0948835\pi\)
\(7\)	−1.72474	−	2.98735i	−0.651892	−	1.12911i	−0.982663	−	0.185399i	\(-0.940642\pi\)
							0.330771	−	0.943711i	\(-0.392691\pi\)
\(11\)	−0.724745	−	1.25529i	−0.218519	−	0.378486i	0.735837	−	0.677159i	\(-0.236789\pi\)
							−0.954355	+	0.298674i	\(0.903456\pi\)
\(13\)	−2.94949	+	5.10867i	−0.818041	+	1.41689i	0.0890821	+	0.996024i	\(0.471607\pi\)
							−0.907123	+	0.420865i	\(0.861727\pi\)
\(17\)	4.89898			1.18818			0.594089	−	0.804400i	\(-0.297513\pi\)
							0.594089	+	0.804400i	\(0.297513\pi\)
\(19\)	4.00000			0.917663			0.458831	−	0.888523i	\(-0.348268\pi\)
							0.458831	+	0.888523i	\(0.348268\pi\)
\(23\)	2.72474	−	4.71940i	0.568149	−	0.984062i	−0.428601	−	0.903494i	\(-0.640993\pi\)
							0.996749	−	0.0805681i	\(-0.0256735\pi\)
\(29\)	−0.0505103	−	0.0874863i	−0.00937952	−	0.0162458i	0.861298	−	0.508101i	\(-0.169652\pi\)
							−0.870677	+	0.491855i	\(0.836319\pi\)
\(31\)	−1.27526	+	2.20881i	−0.229043	+	0.396713i	−0.957525	−	0.288352i	\(-0.906893\pi\)
							0.728482	+	0.685065i	\(0.240226\pi\)
\(37\)	−0.898979			−0.147791			−0.0738957	−	0.997266i	\(-0.523543\pi\)
							−0.0738957	+	0.997266i	\(0.523543\pi\)
\(41\)	5.94949	−	10.3048i	0.929154	−	1.60934i	0.144414	−	0.989517i	\(-0.453870\pi\)
							0.784740	−	0.619825i	\(-0.212796\pi\)
\(43\)	1.17423	+	2.03383i	0.179069	+	0.310157i	0.941562	−	0.336840i	\(-0.109358\pi\)
							−0.762493	+	0.646997i	\(0.776025\pi\)
\(47\)	3.17423	+	5.49794i	0.463010	+	0.801956i	0.999109	−	0.0421984i	\(-0.0134362\pi\)
							−0.536100	+	0.844155i	\(0.680103\pi\)
\(53\)	8.89898			1.22237			0.611184	−	0.791488i	\(-0.290693\pi\)
							0.611184	+	0.791488i	\(0.290693\pi\)
\(59\)	−7.17423	+	12.4261i	−0.934006	+	1.61775i	−0.157609	+	0.987502i	\(0.550378\pi\)
							−0.776397	+	0.630244i	\(0.782955\pi\)
\(61\)	3.94949	+	6.84072i	0.505680	+	0.875864i	0.999978	+	0.00657156i	\(0.00209181\pi\)
							−0.494298	+	0.869292i	\(0.664575\pi\)
\(67\)	−6.17423	+	10.6941i	−0.754303	+	1.30649i	0.191417	+	0.981509i	\(0.438692\pi\)
							−0.945720	+	0.324982i	\(0.894642\pi\)
\(71\)	7.79796			0.925447			0.462724	−	0.886503i	\(-0.346872\pi\)
							0.462724	+	0.886503i	\(0.346872\pi\)
\(73\)	−4.89898			−0.573382			−0.286691	−	0.958023i	\(-0.592555\pi\)
							−0.286691	+	0.958023i	\(0.592555\pi\)
\(79\)	−6.72474	−	11.6476i	−0.756593	−	1.31046i	−0.944579	−	0.328286i	\(-0.893529\pi\)
							0.187986	−	0.982172i	\(-0.439804\pi\)
\(83\)	0.275255	+	0.476756i	0.0302132	+	0.0523308i	0.880737	−	0.473606i	\(-0.157048\pi\)
							−0.850523	+	0.525937i	\(0.823715\pi\)
\(89\)	−12.8990			−1.36729			−0.683645	−	0.729815i	\(-0.739606\pi\)
							−0.683645	+	0.729815i	\(0.739606\pi\)
\(97\)	1.94949	+	3.37662i	0.197941	+	0.342843i	0.947861	−	0.318685i	\(-0.103241\pi\)
							−0.749920	+	0.661529i	\(0.769908\pi\)

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.e.193.1	yes	4
3.2	odd	2		864.2.i.c.577.1		4
4.3	odd	2		288.2.i.c.193.2	yes	4
8.3	odd	2		576.2.i.m.193.1		4
8.5	even	2		576.2.i.i.193.2		4
9.2	odd	6		864.2.i.c.289.1		4
9.4	even	3		2592.2.a.n.1.2		2
9.5	odd	6		2592.2.a.s.1.2		2
9.7	even	3	inner	288.2.i.e.97.2	yes	4
12.11	even	2		864.2.i.e.577.2		4
24.5	odd	2		1728.2.i.k.577.1		4
24.11	even	2		1728.2.i.m.577.2		4
36.7	odd	6		288.2.i.c.97.1	✓	4
36.11	even	6		864.2.i.e.289.2		4
36.23	even	6		2592.2.a.o.1.1		2
36.31	odd	6		2592.2.a.j.1.1		2
72.5	odd	6		5184.2.a.bn.1.2		2
72.11	even	6		1728.2.i.m.1153.2		4
72.13	even	6		5184.2.a.by.1.2		2
72.29	odd	6		1728.2.i.k.1153.1		4
72.43	odd	6		576.2.i.m.385.2		4
72.59	even	6		5184.2.a.bj.1.1		2
72.61	even	6		576.2.i.i.385.1		4
72.67	odd	6		5184.2.a.bu.1.1		2

    

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