Embedded newform 287.3.y.a.10.16

• Label: 287.3.y.a.10.16
• Level: $287$
• Weight: $3$
• Character: 287.10
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.y (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(54\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			10.16
Character	\(\chi\)	\(=\)	287.10
Dual form			287.3.y.a.201.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.193036 + 1.83662i) q^{2} +(1.95836 + 1.13066i) q^{3} +(0.576688 + 0.122579i) q^{4} +(6.02254 + 5.42272i) q^{5} +(-2.45462 + 3.37850i) q^{6} +(-0.339639 + 6.99176i) q^{7} +(-2.61914 + 8.06089i) q^{8} +(-1.94322 - 3.36575i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 3 q^{2} - 24 q^{3} + 101 q^{4} - 9 q^{5} - 6 q^{7} - 16 q^{8} + 576 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/287\mathbb{Z}\right)^\times\).

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{5}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.193036	+	1.83662i	−0.0965182	+	0.918309i	0.833926	+	0.551876i	\(0.186088\pi\)
							−0.930444	+	0.366433i	\(0.880579\pi\)
\(3\)	1.95836	+	1.13066i	0.652787	+	0.376887i	0.789523	−	0.613721i	\(-0.210328\pi\)
							−0.136736	+	0.990607i	\(0.543661\pi\)
\(5\)	6.02254	+	5.42272i	1.20451	+	1.08454i	0.994270	+	0.106893i	\(0.0340904\pi\)
							0.210237	+	0.977650i	\(0.432576\pi\)
\(7\)	−0.339639	+	6.99176i	−0.0485198	+	0.998822i
							\(11\)	−5.57775	−	6.19471i	−0.507068	−	0.563156i	0.434200	−	0.900816i	\(-0.357031\pi\)
−0.941268	+	0.337661i	\(0.890364\pi\)
\(13\)	6.07416	−	8.36036i	0.467243	−	0.643105i	−0.508748	−	0.860915i	\(-0.669892\pi\)
							0.975991	+	0.217811i	\(0.0698916\pi\)
\(17\)	3.99190	−	3.59432i	0.234818	−	0.211431i	−0.543318	−	0.839527i	\(-0.682832\pi\)
							0.778136	+	0.628096i	\(0.216166\pi\)
\(19\)	11.8048	−	26.5140i	0.621305	−	1.39547i	−0.279032	−	0.960282i	\(-0.590014\pi\)
							0.900337	−	0.435192i	\(-0.143320\pi\)
\(23\)	−0.0838871	+	0.798132i	−0.00364726	+	0.0347014i	−0.996194	−	0.0871619i	\(-0.972220\pi\)
							0.992547	+	0.121863i	\(0.0388869\pi\)
\(29\)	−3.03156	−	9.33018i	−0.104536	−	0.321730i	0.885085	−	0.465430i	\(-0.154100\pi\)
							−0.989621	+	0.143699i	\(0.954100\pi\)
\(31\)	13.7122	−	12.3465i	0.442329	−	0.398274i	−0.417647	−	0.908609i	\(-0.637145\pi\)
							0.859976	+	0.510335i	\(0.170479\pi\)
\(37\)	−3.53488	+	3.92588i	−0.0955373	+	0.106105i	−0.789017	−	0.614371i	\(-0.789410\pi\)
							0.693480	+	0.720476i	\(0.256077\pi\)
\(41\)	−37.8283	−	15.8120i	−0.922642	−	0.385659i
							\(43\)	29.6775	+	21.5620i	0.690175	+	0.501441i	0.876718	−	0.481005i	\(-0.159728\pi\)
−0.186543	+	0.982447i	\(0.559728\pi\)
\(47\)	12.9459	+	1.36067i	0.275445	+	0.0289505i	0.241244	−	0.970464i	\(-0.422445\pi\)
							0.0342013	+	0.999415i	\(0.489111\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.y.a.10.16	✓	432
7.5	odd	6	inner	287.3.y.a.215.39	yes	432
41.37	even	5	inner	287.3.y.a.283.39	yes	432
287.201	odd	30	inner	287.3.y.a.201.16	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.y.a.10.16	✓	432	1.1	even	1	trivial
287.3.y.a.201.16	yes	432	287.201	odd	30	inner
287.3.y.a.215.39	yes	432	7.5	odd	6	inner
287.3.y.a.283.39	yes	432	41.37	even	5	inner