Embedded newform 287.3.m.a.85.6

• Label: 287.3.m.a.85.6
• Level: $287$
• Weight: $3$
• Character: 287.85
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.m (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(42\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			85.6
Character	\(\chi\)	\(=\)	287.85
Dual form			287.3.m.a.260.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.24689 + 2.24689i) q^{2} +(1.13818 - 2.74781i) q^{3} -6.09703i q^{4} +(6.89996 + 6.89996i) q^{5} +(3.61666 + 8.73139i) q^{6} +(-1.01249 + 2.44436i) q^{7} +(4.71181 + 4.71181i) q^{8} +(0.108960 + 0.108960i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 8 q^{2} - 16 q^{3} + 24 q^{6} - 48 q^{8} - 48 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.24689	+	2.24689i	−1.12345	+	1.12345i	−0.132226	+	0.991220i	\(0.542212\pi\)
							−0.991220	+	0.132226i	\(0.957788\pi\)
\(3\)	1.13818	−	2.74781i	0.379393	−	0.915936i	−0.612686	−	0.790326i	\(-0.709911\pi\)
							0.992080	−	0.125610i	\(-0.0400889\pi\)
\(5\)	6.89996	+	6.89996i	1.37999	+	1.37999i	0.844611	+	0.535381i	\(0.179832\pi\)
							0.535381	+	0.844611i	\(0.320168\pi\)
\(7\)	−1.01249	+	2.44436i	−0.144641	+	0.349194i
							\(11\)	4.90163	+	2.03032i	0.445603	+	0.184575i	0.594190	−	0.804325i	\(-0.297473\pi\)
−0.148588	+	0.988899i	\(0.547473\pi\)
\(13\)	−1.52615	+	3.68446i	−0.117396	+	0.283420i	−0.971645	−	0.236445i	\(-0.924018\pi\)
							0.854249	+	0.519865i	\(0.174018\pi\)
\(17\)	−8.50523	−	20.5334i	−0.500308	−	1.20785i	−0.949316	−	0.314322i	\(-0.898223\pi\)
							0.449009	−	0.893527i	\(-0.351777\pi\)
\(19\)	2.60179	+	6.28128i	0.136936	+	0.330594i	0.977440	−	0.211212i	\(-0.0677409\pi\)
							−0.840504	+	0.541806i	\(0.817741\pi\)
\(23\)			13.5854i			0.590671i	0.955394	+	0.295336i	\(0.0954315\pi\)
							−0.955394	+	0.295336i	\(0.904569\pi\)
\(29\)	11.1582	−	26.9382i	0.384765	−	0.928905i	−0.606265	−	0.795263i	\(-0.707333\pi\)
							0.991030	−	0.133642i	\(-0.0426673\pi\)
\(31\)			36.2518i			1.16941i	0.811246	+	0.584706i	\(0.198790\pi\)
							−0.811246	+	0.584706i	\(0.801210\pi\)
\(37\)	−21.0007			−0.567586			−0.283793	−	0.958886i	\(-0.591593\pi\)
							−0.283793	+	0.958886i	\(0.591593\pi\)
\(41\)	−3.31298	+	40.8659i	−0.0808044	+	0.996730i
							\(43\)	15.1986	−	15.1986i	0.353455	−	0.353455i	−0.507938	−	0.861394i	\(-0.669592\pi\)
0.861394	+	0.507938i	\(0.169592\pi\)
\(47\)	−0.0370444	−	0.0894332i	−0.000788179	−	0.00190283i	0.923485	−	0.383635i	\(-0.125328\pi\)
							−0.924273	+	0.381732i	\(0.875328\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.m.a.85.6	✓	168
41.14	odd	8	inner	287.3.m.a.260.6	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.m.a.85.6	✓	168	1.1	even	1	trivial
287.3.m.a.260.6	yes	168	41.14	odd	8	inner