Embedded newform 287.2.s.a.100.4

• Label: 287.2.s.a.100.4
• Level: $287$
• Weight: $2$
• Character: 287.100
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $208$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.s (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(208\)
Relative dimension:	\(26\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			100.4
Character	\(\chi\)	\(=\)	287.100
Dual form			287.2.s.a.221.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.16271 - 0.459699i) q^{2} +(0.943374 + 1.63397i) q^{3} +(2.63892 + 1.17492i) q^{4} +(0.323505 + 3.07794i) q^{5} +(-1.28911 - 3.96748i) q^{6} +(-1.43035 + 2.22578i) q^{7} +(-1.58959 - 1.15490i) q^{8} +(-0.279910 + 0.484819i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 3 q^{2} - 8 q^{3} + 21 q^{4} - 7 q^{5} - 8 q^{6} - 6 q^{7} - 32 q^{8} - 92 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.16271	−	0.459699i	−1.52927	−	0.325056i	−0.634974	−	0.772533i	\(-0.718989\pi\)
							−0.894295	+	0.447477i	\(0.852323\pi\)
\(3\)	0.943374	+	1.63397i	0.544657	+	0.943374i	0.998628	+	0.0523578i	\(0.0166736\pi\)
							−0.453971	+	0.891016i	\(0.649993\pi\)
\(5\)	0.323505	+	3.07794i	0.144676	+	1.37650i	0.790243	+	0.612794i	\(0.209954\pi\)
							−0.645567	+	0.763704i	\(0.723379\pi\)
\(7\)	−1.43035	+	2.22578i	−0.540623	+	0.841265i
							\(11\)	−0.475254	+	4.52174i	−0.143295	+	1.36336i	0.652499	+	0.757789i	\(0.273721\pi\)
−0.795794	+	0.605568i	\(0.792946\pi\)
\(13\)	−0.814483	−	2.50672i	−0.225897	−	0.695239i	−0.998199	−	0.0599842i	\(-0.980895\pi\)
							0.772302	−	0.635255i	\(-0.219105\pi\)
\(17\)	0.558021	−	5.30922i	0.135340	−	1.28767i	−0.690319	−	0.723505i	\(-0.742530\pi\)
							0.825659	−	0.564169i	\(-0.190804\pi\)
\(19\)	−3.08521	−	3.42647i	−0.707795	−	0.786086i	0.276801	−	0.960927i	\(-0.410726\pi\)
							−0.984597	+	0.174841i	\(0.944059\pi\)
\(23\)	4.77633	+	1.01524i	0.995933	+	0.211692i	0.676919	−	0.736058i	\(-0.263315\pi\)
							0.319014	+	0.947750i	\(0.396648\pi\)
\(29\)	−5.24965	+	3.81409i	−0.974835	+	0.708259i	−0.956548	−	0.291574i	\(-0.905821\pi\)
							−0.0182864	+	0.999833i	\(0.505821\pi\)
\(31\)	−0.888691	+	8.45533i	−0.159614	+	1.51862i	0.562471	+	0.826817i	\(0.309851\pi\)
							−0.722084	+	0.691805i	\(0.756816\pi\)
\(37\)	−0.502681	−	4.78269i	−0.0826403	−	0.786270i	−0.954840	−	0.297120i	\(-0.903974\pi\)
							0.872200	−	0.489150i	\(-0.162693\pi\)
\(41\)	−3.02789	+	5.64197i	−0.472877	+	0.881128i
							\(43\)	−0.546187	−	1.68099i	−0.0832927	−	0.256349i	0.900734	−	0.434372i	\(-0.143030\pi\)
−0.984026	+	0.178023i	\(0.943030\pi\)
\(47\)	8.64912	+	1.83843i	1.26160	+	0.268162i	0.789703	−	0.613489i	\(-0.210235\pi\)
							0.471901	+	0.881652i	\(0.343568\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.s.a.100.4	yes	208
7.4	even	3	inner	287.2.s.a.18.23	yes	208
41.16	even	5	inner	287.2.s.a.16.23	✓	208
287.221	even	15	inner	287.2.s.a.221.4	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.s.a.16.23	✓	208	41.16	even	5	inner
287.2.s.a.18.23	yes	208	7.4	even	3	inner
287.2.s.a.100.4	yes	208	1.1	even	1	trivial
287.2.s.a.221.4	yes	208	287.221	even	15	inner