Embedded newform 287.2.s.a.16.19

• Label: 287.2.s.a.16.19
• Level: $287$
• Weight: $2$
• Character: 287.16
• 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			16.19
Character	\(\chi\)	\(=\)	287.16
Dual form			287.2.s.a.18.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.863852 + 0.959405i) q^{2} +(1.34226 + 2.32487i) q^{3} +(0.0348396 - 0.331477i) q^{4} +(-2.70729 + 1.20536i) q^{5} +(-1.07097 + 3.29611i) q^{6} +(-0.897579 + 2.48885i) q^{7} +(2.43701 - 1.77059i) q^{8} +(-2.10333 + 3.64308i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 3 q^{2} - 8 q^{3} + 21 q^{4} - 7 q^{5} - 8 q^{6} - 6 q^{7} - 32 q^{8} - 92 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.863852	+	0.959405i	0.610836	+	0.678402i	0.966634	−	0.256161i	\(-0.0824578\pi\)
							−0.355799	+	0.934563i	\(0.615791\pi\)
\(3\)	1.34226	+	2.32487i	0.774955	+	1.34226i	0.934820	+	0.355123i	\(0.115561\pi\)
							−0.159864	+	0.987139i	\(0.551106\pi\)
\(5\)	−2.70729	+	1.20536i	−1.21074	+	0.539055i	−0.909982	−	0.414647i	\(-0.863905\pi\)
							−0.300755	+	0.953702i	\(0.597238\pi\)
\(7\)	−0.897579	+	2.48885i	−0.339253	+	0.940695i
							\(11\)	−2.61630	−	1.16485i	−0.788843	−	0.351215i	−0.0275473	−	0.999621i	\(-0.508770\pi\)
−0.761296	+	0.648405i	\(0.775436\pi\)
\(13\)	1.04883	−	3.22796i	0.290893	−	0.895275i	−0.693678	−	0.720286i	\(-0.744011\pi\)
							0.984570	−	0.174990i	\(-0.0559892\pi\)
\(17\)	6.16811	+	2.74622i	1.49599	+	0.666056i	0.981504	−	0.191444i	\(-0.0613171\pi\)
							0.514483	+	0.857500i	\(0.327984\pi\)
\(19\)	6.54965	+	1.39217i	1.50259	+	0.319386i	0.884433	−	0.466667i	\(-0.154545\pi\)
							0.618160	+	0.786053i	\(0.287878\pi\)
\(23\)	1.68263	+	1.86875i	0.350853	+	0.389661i	0.892577	−	0.450895i	\(-0.148895\pi\)
							−0.541724	+	0.840556i	\(0.682228\pi\)
\(29\)	−3.53260	−	2.56658i	−0.655987	−	0.476603i	0.209318	−	0.977848i	\(-0.432876\pi\)
							−0.865306	+	0.501245i	\(0.832876\pi\)
\(31\)	−4.39173	−	1.95532i	−0.788778	−	0.351186i	−0.0275078	−	0.999622i	\(-0.508757\pi\)
							−0.761270	+	0.648435i	\(0.775424\pi\)
\(37\)	−2.12456	+	0.945914i	−0.349275	+	0.155507i	−0.573873	−	0.818944i	\(-0.694560\pi\)
							0.224598	+	0.974452i	\(0.427893\pi\)
\(41\)	3.89825	+	5.07973i	0.608804	+	0.793320i
							\(43\)	2.67445	−	8.23110i	0.407849	−	1.25523i	−0.510643	−	0.859793i	\(-0.670593\pi\)
0.918492	−	0.395438i	\(-0.129407\pi\)
\(47\)	−6.27842	−	6.97289i	−0.915801	−	1.01710i	−0.999787	−	0.0206334i	\(-0.993432\pi\)
							0.0839858	−	0.996467i	\(-0.473235\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.s.a.16.19	✓	208
7.4	even	3	inner	287.2.s.a.221.8	yes	208
41.18	even	5	inner	287.2.s.a.100.8	yes	208
287.18	even	15	inner	287.2.s.a.18.19	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.s.a.16.19	✓	208	1.1	even	1	trivial
287.2.s.a.18.19	yes	208	287.18	even	15	inner
287.2.s.a.100.8	yes	208	41.18	even	5	inner
287.2.s.a.221.8	yes	208	7.4	even	3	inner