Embedded newform 288.2.r.b.49.3

• Label: 288.2.r.b.49.3
• Level: $288$
• Weight: $2$
• Character: 288.49
• Analytic conductor: $2.300$
• Analytic rank: $0$
• Dimension: $16$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	288.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29969157821\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - x^15 + x^14 + 2*x^12 - 4*x^11 - 8*x^9 + 4*x^8 - 16*x^7 - 32*x^5 + 32*x^4 + 64*x^2 - 128*x + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{8}\cdot 3^{2} \)
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			49.3
Root			\(-1.12494 + 0.857038i\) of defining polynomial
Character	\(\chi\)	\(=\)	288.49
Dual form			288.2.r.b.241.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.986088 + 1.42395i) q^{3} +(-1.19115 - 0.687709i) q^{5} +(-1.80469 - 3.12581i) q^{7} +(-1.05526 - 2.80828i) q^{9} +(-1.83294 + 1.05825i) q^{11} +(-0.887751 - 0.512543i) q^{13} +(2.15384 - 1.01799i) q^{15} +0.808822 q^{17} -7.43122i q^{19} +(6.23057 + 0.512543i) q^{21} +(1.65498 - 2.86652i) q^{23} +(-1.55411 - 2.69180i) q^{25} +(5.03942 + 1.26658i) q^{27} +(-7.71083 + 4.45185i) q^{29} +(-3.26436 + 5.65403i) q^{31} +(0.300550 - 3.65354i) q^{33} +4.96439i q^{35} -4.01531i q^{37} +(1.60524 - 0.758698i) q^{39} +(-3.45852 + 5.99034i) q^{41} +(0.245957 - 0.142003i) q^{43} +(-0.674310 + 4.07078i) q^{45} +(3.61351 + 6.25878i) q^{47} +(-3.01378 + 5.22003i) q^{49} +(-0.797570 + 1.15172i) q^{51} -3.86330i q^{53} +2.91107 q^{55} +(10.5817 + 7.32784i) q^{57} +(-7.06904 - 4.08131i) q^{59} +(6.31237 - 3.64445i) q^{61} +(-6.87373 + 8.36660i) q^{63} +(0.704961 + 1.22103i) q^{65} +(2.43973 + 1.40858i) q^{67} +(2.44981 + 5.18325i) q^{69} -4.69830 q^{71} +0.409922 q^{73} +(5.36548 + 0.441379i) q^{75} +(6.61576 + 3.81961i) q^{77} +(0.0456121 + 0.0790024i) q^{79} +(-6.77286 + 5.92692i) q^{81} +(2.40891 - 1.39079i) q^{83} +(-0.963426 - 0.556234i) q^{85} +(1.26436 - 15.3697i) q^{87} +8.91934 q^{89} +3.69992i q^{91} +(-4.83210 - 10.2236i) q^{93} +(-5.11052 + 8.85168i) q^{95} +(-2.76022 - 4.78084i) q^{97} +(4.90608 + 4.03068i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	16 * q - 6 * q^7 + 2 * q^9 + 10 * q^15 - 28 * q^17 + 10 * q^23 + 2 * q^25 + 10 * q^31 - 2 * q^39 - 8 * q^41 - 6 * q^47 + 18 * q^49 + 4 * q^55 + 10 * q^57 - 2 * q^63 - 14 * q^65 - 72 * q^71 - 44 * q^73 + 30 * q^79 + 10 * q^81 - 42 * q^87 + 64 * q^89 - 44 * q^95

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
							\(3\)	−0.986088	+	1.42395i	−0.569318	+	0.822117i
\(5\)	−1.19115	−	0.687709i	−0.532697	−	0.307553i								0.209417	−	0.977826i	\(-0.432843\pi\)
							−0.742114	+	0.670274i	\(0.766177\pi\)
\(7\)	−1.80469	−	3.12581i	−0.682107	−	1.18144i	−0.974337	−	0.225096i	\(-0.927730\pi\)
							0.292229	−	0.956348i	\(-0.405603\pi\)
\(11\)	−1.83294	+	1.05825i	−0.552652	+	0.319074i	−0.750191	−	0.661221i	\(-0.770039\pi\)
							0.197539	+	0.980295i	\(0.436705\pi\)
\(13\)	−0.887751	−	0.512543i	−0.246218	−	0.142154i	0.371813	−	0.928307i	\(-0.378736\pi\)
							−0.618031	+	0.786154i	\(0.712069\pi\)
\(17\)	0.808822			0.196168			0.0980841	−	0.995178i	\(-0.468729\pi\)
							0.0980841	+	0.995178i	\(0.468729\pi\)
\(19\)		−	7.43122i		−	1.70484i	−0.522858	−	0.852420i	\(-0.675134\pi\)
							0.522858	−	0.852420i	\(-0.324866\pi\)
\(23\)	1.65498	−	2.86652i	0.345088	−	0.597710i	−0.640282	−	0.768140i	\(-0.721182\pi\)
							0.985370	+	0.170430i	\(0.0545157\pi\)
\(29\)	−7.71083	+	4.45185i	−1.43187	+	0.826688i	−0.997263	−	0.0739344i	\(-0.976444\pi\)
							−0.434603	+	0.900622i	\(0.643111\pi\)
\(31\)	−3.26436	+	5.65403i	−0.586296	+	1.01549i	0.408417	+	0.912796i	\(0.366081\pi\)
							−0.994713	+	0.102698i	\(0.967252\pi\)
\(37\)		−	4.01531i		−	0.660113i	−0.943961	−	0.330057i	\(-0.892932\pi\)
							0.943961	−	0.330057i	\(-0.107068\pi\)
\(41\)	−3.45852	+	5.99034i	−0.540131	+	0.935534i	0.458765	+	0.888557i	\(0.348292\pi\)
							−0.998896	+	0.0469764i	\(0.985041\pi\)
\(43\)	0.245957	−	0.142003i	0.0375081	−	0.0216553i	−0.481129	−	0.876650i	\(-0.659773\pi\)
							0.518637	+	0.854995i	\(0.326440\pi\)
\(47\)	3.61351	+	6.25878i	0.527084	+	0.912937i	0.999502	+	0.0315619i	\(0.0100481\pi\)
							−0.472417	+	0.881375i	\(0.656619\pi\)
\(53\)		−	3.86330i		−	0.530666i	−0.964157	−	0.265333i	\(-0.914518\pi\)
							0.964157	−	0.265333i	\(-0.0854818\pi\)
\(59\)	−7.06904	−	4.08131i	−0.920310	−	0.531341i	−0.0365764	−	0.999331i	\(-0.511645\pi\)
							−0.883734	+	0.467989i	\(0.844979\pi\)
\(61\)	6.31237	−	3.64445i	0.808216	−	0.466624i	−0.0381201	−	0.999273i	\(-0.512137\pi\)
							0.846336	+	0.532650i	\(0.178804\pi\)
\(67\)	2.43973	+	1.40858i	0.298061	+	0.172085i	0.641571	−	0.767063i	\(-0.278283\pi\)
							−0.343511	+	0.939149i	\(0.611616\pi\)
\(71\)	−4.69830			−0.557586			−0.278793	−	0.960351i	\(-0.589934\pi\)
							−0.278793	+	0.960351i	\(0.589934\pi\)
\(73\)	0.409922			0.0479777			0.0239889	−	0.999712i	\(-0.492363\pi\)
							0.0239889	+	0.999712i	\(0.492363\pi\)
\(79\)	0.0456121	+	0.0790024i	0.00513176	+	0.00888847i	0.868580	−	0.495549i	\(-0.165033\pi\)
							−0.863448	+	0.504438i	\(0.831700\pi\)
\(83\)	2.40891	−	1.39079i	0.264412	−	0.152659i	−0.361933	−	0.932204i	\(-0.617883\pi\)
							0.626346	+	0.779545i	\(0.284550\pi\)
\(89\)	8.91934			0.945448			0.472724	−	0.881210i	\(-0.343271\pi\)
							0.472724	+	0.881210i	\(0.343271\pi\)
\(97\)	−2.76022	−	4.78084i	−0.280258	−	0.485421i	0.691190	−	0.722673i	\(-0.257087\pi\)
							−0.971448	+	0.237252i	\(0.923753\pi\)

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.r.b.49.3		16
3.2	odd	2		864.2.r.b.145.6		16
4.3	odd	2		72.2.n.b.13.4	yes	16
8.3	odd	2		72.2.n.b.13.2	✓	16
8.5	even	2	inner	288.2.r.b.49.6		16
9.2	odd	6		864.2.r.b.721.3		16
9.4	even	3		2592.2.d.j.1297.6		8
9.5	odd	6		2592.2.d.k.1297.3		8
9.7	even	3	inner	288.2.r.b.241.6		16
12.11	even	2		216.2.n.b.37.5		16
24.5	odd	2		864.2.r.b.145.3		16
24.11	even	2		216.2.n.b.37.7		16
36.7	odd	6		72.2.n.b.61.2	yes	16
36.11	even	6		216.2.n.b.181.7		16
36.23	even	6		648.2.d.k.325.1		8
36.31	odd	6		648.2.d.j.325.8		8
72.5	odd	6		2592.2.d.k.1297.6		8
72.11	even	6		216.2.n.b.181.5		16
72.13	even	6		2592.2.d.j.1297.3		8
72.29	odd	6		864.2.r.b.721.6		16
72.43	odd	6		72.2.n.b.61.4	yes	16
72.59	even	6		648.2.d.k.325.2		8
72.61	even	6	inner	288.2.r.b.241.3		16
72.67	odd	6		648.2.d.j.325.7		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.n.b.13.2	✓	16	8.3	odd	2
72.2.n.b.13.4	yes	16	4.3	odd	2
72.2.n.b.61.2	yes	16	36.7	odd	6
72.2.n.b.61.4	yes	16	72.43	odd	6
216.2.n.b.37.5		16	12.11	even	2
216.2.n.b.37.7		16	24.11	even	2
216.2.n.b.181.5		16	72.11	even	6
216.2.n.b.181.7		16	36.11	even	6
288.2.r.b.49.3		16	1.1	even	1	trivial
288.2.r.b.49.6		16	8.5	even	2	inner
288.2.r.b.241.3		16	72.61	even	6	inner
288.2.r.b.241.6		16	9.7	even	3	inner
648.2.d.j.325.7		8	72.67	odd	6
648.2.d.j.325.8		8	36.31	odd	6
648.2.d.k.325.1		8	36.23	even	6
648.2.d.k.325.2		8	72.59	even	6
864.2.r.b.145.3		16	24.5	odd	2
864.2.r.b.145.6		16	3.2	odd	2
864.2.r.b.721.3		16	9.2	odd	6
864.2.r.b.721.6		16	72.29	odd	6
2592.2.d.j.1297.3		8	72.13	even	6
2592.2.d.j.1297.6		8	9.4	even	3
2592.2.d.k.1297.3		8	9.5	odd	6
2592.2.d.k.1297.6		8	72.5	odd	6