Embedded newform 288.2.i.f.193.3

• Label: 288.2.i.f.193.3
• Level: $288$
• Weight: $2$
• Character: 288.193
• Analytic conductor: $2.300$
• Analytic rank: $0$
• Dimension: $8$
• 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:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	8.0.170772624.1
Defining polynomial:	\( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{2}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			193.3
Root			\(0.774115 - 1.18353i\) of defining polynomial
Character	\(\chi\)	\(=\)	288.193
Dual form			288.2.i.f.97.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.637910 - 1.61030i) q^{3} +(1.68614 - 2.92048i) q^{5} +(2.35143 + 4.07279i) q^{7} +(-2.18614 - 2.05446i) q^{9} +(0.437696 + 0.758112i) q^{11} +(0.686141 - 1.18843i) q^{13} +(-3.62725 - 4.57820i) q^{15} -2.37228 q^{17} -5.57825 q^{19} +(8.05842 - 1.18843i) q^{21} +(2.35143 - 4.07279i) q^{23} +(-3.18614 - 5.51856i) q^{25} +(-4.70285 + 2.20979i) q^{27} +(2.68614 + 4.65253i) q^{29} +(-3.22682 + 5.58902i) q^{31} +(1.50000 - 0.221215i) q^{33} +15.8593 q^{35} +4.00000 q^{37} +(-1.47603 - 1.86301i) q^{39} +(-0.500000 + 0.866025i) q^{41} +(-0.437696 - 0.758112i) q^{43} +(-9.68614 + 2.92048i) q^{45} +(2.35143 + 4.07279i) q^{47} +(-7.55842 + 13.0916i) q^{49} +(-1.51330 + 3.82009i) q^{51} -4.00000 q^{53} +2.95207 q^{55} +(-3.55842 + 8.98266i) q^{57} +(4.26516 - 7.38747i) q^{59} +(1.05842 + 1.83324i) q^{61} +(3.22682 - 13.7346i) q^{63} +(-2.31386 - 4.00772i) q^{65} +(-4.26516 + 7.38747i) q^{67} +(-5.05842 - 6.38458i) q^{69} -9.40571 q^{71} +10.3723 q^{73} +(-10.9190 + 1.61030i) q^{75} +(-2.05842 + 3.56529i) q^{77} +(3.22682 + 5.58902i) q^{79} +(0.558422 + 8.98266i) q^{81} +(-1.47603 - 2.55657i) q^{83} +(-4.00000 + 6.92820i) q^{85} +(9.20550 - 1.35760i) q^{87} +12.7446 q^{89} +6.45364 q^{91} +(6.94158 + 8.76144i) q^{93} +(-9.40571 + 16.2912i) q^{95} +(-4.50000 - 7.79423i) q^{97} +(0.600642 - 2.55657i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	8 * q + 2 * q^5 - 6 * q^9 - 6 * q^13 + 4 * q^17 + 30 * q^21 - 14 * q^25 + 10 * q^29 + 12 * q^33 + 32 * q^37 - 4 * q^41 - 66 * q^45 - 26 * q^49 - 32 * q^53 + 6 * q^57 - 26 * q^61 - 30 * q^65 - 6 * q^69 + 60 * q^73 + 18 * q^77 - 30 * q^81 - 32 * q^85 + 56 * q^89 + 90 * q^93 - 36 * q^97

Character values

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

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

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
							\(3\)	0.637910	−	1.61030i	0.368298	−	0.929708i
\(5\)	1.68614	−	2.92048i	0.754065	−	1.30608i								−0.191773	−	0.981439i	\(-0.561424\pi\)
							0.945838	−	0.324640i	\(-0.105243\pi\)
\(7\)	2.35143	+	4.07279i	0.888756	+	1.53937i	0.841347	+	0.540495i	\(0.181763\pi\)
							0.0474088	+	0.998876i	\(0.484904\pi\)
\(11\)	0.437696	+	0.758112i	0.131970	+	0.228579i	0.924436	−	0.381337i	\(-0.124536\pi\)
							−0.792466	+	0.609917i	\(0.791203\pi\)
\(13\)	0.686141	−	1.18843i	0.190301	−	0.329611i	−0.755049	−	0.655669i	\(-0.772387\pi\)
							0.945350	+	0.326057i	\(0.105720\pi\)
\(17\)	−2.37228			−0.575363			−0.287681	−	0.957726i	\(-0.592884\pi\)
							−0.287681	+	0.957726i	\(0.592884\pi\)
\(19\)	−5.57825			−1.27974			−0.639869	−	0.768484i	\(-0.721011\pi\)
							−0.639869	+	0.768484i	\(0.721011\pi\)
\(23\)	2.35143	−	4.07279i	0.490307	−	0.849236i	−0.509631	−	0.860393i	\(-0.670218\pi\)
							0.999938	+	0.0111571i	\(0.00355150\pi\)
\(29\)	2.68614	+	4.65253i	0.498804	+	0.863954i	0.999999	−	0.00138070i	\(-0.000439492\pi\)
							−0.501195	+	0.865334i	\(0.667106\pi\)
\(31\)	−3.22682	+	5.58902i	−0.579554	+	1.00382i	0.415976	+	0.909375i	\(0.363440\pi\)
							−0.995530	+	0.0944415i	\(0.969893\pi\)
\(37\)	4.00000			0.657596			0.328798	−	0.944400i	\(-0.393356\pi\)
							0.328798	+	0.944400i	\(0.393356\pi\)
\(41\)	−0.500000	+	0.866025i	−0.0780869	+	0.135250i	−0.902424	−	0.430848i	\(-0.858214\pi\)
							0.824338	+	0.566099i	\(0.191548\pi\)
\(43\)	−0.437696	−	0.758112i	−0.0667481	−	0.115611i	0.830720	−	0.556690i	\(-0.187929\pi\)
							−0.897468	+	0.441079i	\(0.854596\pi\)
\(47\)	2.35143	+	4.07279i	0.342991	+	0.594078i	0.984987	−	0.172630i	\(-0.0552266\pi\)
							−0.641996	+	0.766708i	\(0.721893\pi\)
\(53\)	−4.00000			−0.549442			−0.274721	−	0.961524i	\(-0.588586\pi\)
							−0.274721	+	0.961524i	\(0.588586\pi\)
\(59\)	4.26516	−	7.38747i	0.555276	−	0.961767i	−0.442606	−	0.896716i	\(-0.645946\pi\)
							0.997882	−	0.0650505i	\(-0.0207208\pi\)
\(61\)	1.05842	+	1.83324i	0.135517	+	0.234722i	0.925795	−	0.378026i	\(-0.123397\pi\)
							−0.790278	+	0.612749i	\(0.790064\pi\)
\(67\)	−4.26516	+	7.38747i	−0.521072	+	0.902523i	0.478628	+	0.878018i	\(0.341134\pi\)
							−0.999700	+	0.0245053i	\(0.992199\pi\)
\(71\)	−9.40571			−1.11625			−0.558126	−	0.829756i	\(-0.688480\pi\)
							−0.558126	+	0.829756i	\(0.688480\pi\)
\(73\)	10.3723			1.21398			0.606992	−	0.794708i	\(-0.292376\pi\)
							0.606992	+	0.794708i	\(0.292376\pi\)
\(79\)	3.22682	+	5.58902i	0.363046	+	0.628813i	0.988460	−	0.151479i	\(-0.0484036\pi\)
							−0.625415	+	0.780292i	\(0.715070\pi\)
\(83\)	−1.47603	−	2.55657i	−0.162016	−	0.280620i	0.773576	−	0.633704i	\(-0.218466\pi\)
							−0.935592	+	0.353084i	\(0.885133\pi\)
\(89\)	12.7446			1.35092			0.675460	−	0.737396i	\(-0.263945\pi\)
							0.675460	+	0.737396i	\(0.263945\pi\)
\(97\)	−4.50000	−	7.79423i	−0.456906	−	0.791384i	0.541890	−	0.840450i	\(-0.317709\pi\)
							−0.998796	+	0.0490655i	\(0.984376\pi\)

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.f.193.3	yes	8
3.2	odd	2		864.2.i.f.577.2		8
4.3	odd	2	inner	288.2.i.f.193.2	yes	8
8.3	odd	2		576.2.i.n.193.3		8
8.5	even	2		576.2.i.n.193.2		8
9.2	odd	6		864.2.i.f.289.2		8
9.4	even	3		2592.2.a.u.1.1		4
9.5	odd	6		2592.2.a.x.1.3		4
9.7	even	3	inner	288.2.i.f.97.3	yes	8
12.11	even	2		864.2.i.f.577.1		8
24.5	odd	2		1728.2.i.n.577.4		8
24.11	even	2		1728.2.i.n.577.3		8
36.7	odd	6	inner	288.2.i.f.97.2	✓	8
36.11	even	6		864.2.i.f.289.1		8
36.23	even	6		2592.2.a.x.1.4		4
36.31	odd	6		2592.2.a.u.1.2		4
72.5	odd	6		5184.2.a.cc.1.1		4
72.11	even	6		1728.2.i.n.1153.3		8
72.13	even	6		5184.2.a.cf.1.3		4
72.29	odd	6		1728.2.i.n.1153.4		8
72.43	odd	6		576.2.i.n.385.3		8
72.59	even	6		5184.2.a.cc.1.2		4
72.61	even	6		576.2.i.n.385.2		8
72.67	odd	6		5184.2.a.cf.1.4		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.2.i.f.97.2	✓	8	36.7	odd	6	inner
288.2.i.f.97.3	yes	8	9.7	even	3	inner
288.2.i.f.193.2	yes	8	4.3	odd	2	inner
288.2.i.f.193.3	yes	8	1.1	even	1	trivial
576.2.i.n.193.2		8	8.5	even	2
576.2.i.n.193.3		8	8.3	odd	2
576.2.i.n.385.2		8	72.61	even	6
576.2.i.n.385.3		8	72.43	odd	6
864.2.i.f.289.1		8	36.11	even	6
864.2.i.f.289.2		8	9.2	odd	6
864.2.i.f.577.1		8	12.11	even	2
864.2.i.f.577.2		8	3.2	odd	2
1728.2.i.n.577.3		8	24.11	even	2
1728.2.i.n.577.4		8	24.5	odd	2
1728.2.i.n.1153.3		8	72.11	even	6
1728.2.i.n.1153.4		8	72.29	odd	6
2592.2.a.u.1.1		4	9.4	even	3
2592.2.a.u.1.2		4	36.31	odd	6
2592.2.a.x.1.3		4	9.5	odd	6
2592.2.a.x.1.4		4	36.23	even	6
5184.2.a.cc.1.1		4	72.5	odd	6
5184.2.a.cc.1.2		4	72.59	even	6
5184.2.a.cf.1.3		4	72.13	even	6
5184.2.a.cf.1.4		4	72.67	odd	6