Embedded newform 287.2.l.a.27.18

• Label: 287.2.l.a.27.18
• Level: $287$
• Weight: $2$
• Character: 287.27
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $104$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.l (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(26\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			27.18
Character	\(\chi\)	\(=\)	287.27
Dual form			287.2.l.a.202.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.652493 + 0.652493i) q^{2} +(1.09762 + 2.64989i) q^{3} -1.14851i q^{4} +(-0.202079 - 0.202079i) q^{5} +(-1.01284 + 2.44522i) q^{6} +(-1.80370 + 1.93563i) q^{7} +(2.05438 - 2.05438i) q^{8} +(-3.69581 + 3.69581i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 8 q^{2} - 4 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.652493	+	0.652493i	0.461382	+	0.461382i	0.899108	−	0.437726i	\(-0.144216\pi\)
							−0.437726	+	0.899108i	\(0.644216\pi\)
\(3\)	1.09762	+	2.64989i	0.633711	+	1.52991i	0.834924	+	0.550365i	\(0.185512\pi\)
							−0.201213	+	0.979547i	\(0.564488\pi\)
\(5\)	−0.202079	−	0.202079i	−0.0903725	−	0.0903725i	0.660475	−	0.750848i	\(-0.270355\pi\)
							−0.750848	+	0.660475i	\(0.770355\pi\)
\(7\)	−1.80370	+	1.93563i	−0.681733	+	0.731601i
							\(11\)	0.892321	+	2.15425i	0.269045	+	0.649532i	0.999439	−	0.0334944i	\(-0.0106636\pi\)
−0.730394	+	0.683026i	\(0.760664\pi\)
\(13\)	1.83770	+	4.43660i	0.509686	+	1.23049i	0.944065	+	0.329761i	\(0.106968\pi\)
							−0.434378	+	0.900730i	\(0.643032\pi\)
\(17\)	−0.748383	+	1.80676i	−0.181509	+	0.438203i	−0.988278	−	0.152665i	\(-0.951214\pi\)
							0.806769	+	0.590868i	\(0.201214\pi\)
\(19\)	2.76108	−	6.66583i	0.633435	−	1.52925i	−0.201842	−	0.979418i	\(-0.564693\pi\)
							0.835277	−	0.549829i	\(-0.185307\pi\)
\(23\)		−	6.75849i		−	1.40924i	−0.709583	−	0.704622i	\(-0.751117\pi\)
							0.709583	−	0.704622i	\(-0.248883\pi\)
\(29\)	4.62775	−	1.91688i	0.859352	−	0.355955i	0.0908982	−	0.995860i	\(-0.471026\pi\)
							0.768454	+	0.639905i	\(0.221026\pi\)
\(31\)	−4.80279			−0.862606			−0.431303	−	0.902207i	\(-0.641946\pi\)
							−0.431303	+	0.902207i	\(0.641946\pi\)
\(37\)	8.95411			1.47205			0.736023	−	0.676956i	\(-0.236701\pi\)
							0.736023	+	0.676956i	\(0.236701\pi\)
\(41\)	−3.54844	−	5.32997i	−0.554172	−	0.832402i
							\(43\)	−1.13799	−	1.13799i	−0.173542	−	0.173542i	0.614992	−	0.788534i	\(-0.289159\pi\)
−0.788534	+	0.614992i	\(0.789159\pi\)
\(47\)	−0.703429	+	1.69823i	−0.102606	+	0.247712i	−0.966843	−	0.255373i	\(-0.917802\pi\)
							0.864237	+	0.503085i	\(0.167802\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.l.a.27.18	yes	104
7.6	odd	2	inner	287.2.l.a.27.17	✓	104
41.38	odd	8	inner	287.2.l.a.202.17	yes	104
287.202	even	8	inner	287.2.l.a.202.18	yes	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.l.a.27.17	✓	104	7.6	odd	2	inner
287.2.l.a.27.18	yes	104	1.1	even	1	trivial
287.2.l.a.202.17	yes	104	41.38	odd	8	inner
287.2.l.a.202.18	yes	104	287.202	even	8	inner