Embedded newform 287.2.l.a.27.1

• Label: 287.2.l.a.27.1
• 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			27.1
Character	\(\chi\)	\(=\)	287.27
Dual form			287.2.l.a.202.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.82241 - 1.82241i) q^{2} +(-1.24228 - 2.99913i) q^{3} +4.64235i q^{4} +(-1.81451 - 1.81451i) q^{5} +(-3.20170 + 7.72960i) q^{6} +(-0.372244 - 2.61943i) q^{7} +(4.81545 - 4.81545i) q^{8} +(-5.33022 + 5.33022i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 8 q^{2} - 4 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.82241	−	1.82241i	−1.28864	−	1.28864i	−0.935614	−	0.353024i	\(-0.885153\pi\)
							−0.353024	−	0.935614i	\(-0.614847\pi\)
\(3\)	−1.24228	−	2.99913i	−0.717232	−	1.73155i	−0.681085	−	0.732205i	\(-0.738491\pi\)
							−0.0361473	−	0.999346i	\(-0.511509\pi\)
\(5\)	−1.81451	−	1.81451i	−0.811474	−	0.811474i	0.173381	−	0.984855i	\(-0.444531\pi\)
							−0.984855	+	0.173381i	\(0.944531\pi\)
\(7\)	−0.372244	−	2.61943i	−0.140695	−	0.990053i
							\(11\)	−1.05974	−	2.55845i	−0.319525	−	0.771401i	−0.999279	−	0.0379612i	\(-0.987914\pi\)
0.679755	−	0.733440i	\(-0.262086\pi\)
\(13\)	0.906014	+	2.18731i	0.251283	+	0.606651i	0.998308	−	0.0581449i	\(-0.0185185\pi\)
							−0.747025	+	0.664796i	\(0.768519\pi\)
\(17\)	0.0827984	−	0.199893i	0.0200816	−	0.0484812i	−0.913521	−	0.406792i	\(-0.866647\pi\)
							0.933602	+	0.358311i	\(0.116647\pi\)
\(19\)	1.23704	−	2.98648i	0.283797	−	0.685146i	−0.716121	−	0.697976i	\(-0.754084\pi\)
							0.999918	+	0.0128304i	\(0.00408417\pi\)
\(23\)		−	6.85267i		−	1.42888i	−0.699696	−	0.714441i	\(-0.746681\pi\)
							0.699696	−	0.714441i	\(-0.253319\pi\)
\(29\)	8.03627	−	3.32873i	1.49230	−	0.618130i	0.520483	−	0.853872i	\(-0.325752\pi\)
							0.971816	+	0.235742i	\(0.0757520\pi\)
\(31\)	0.304196			0.0546353			0.0273176	−	0.999627i	\(-0.491303\pi\)
							0.0273176	+	0.999627i	\(0.491303\pi\)
\(37\)	−0.712359			−0.117111			−0.0585556	−	0.998284i	\(-0.518649\pi\)
							−0.0585556	+	0.998284i	\(0.518649\pi\)
\(41\)	−6.30628	+	1.10944i	−0.984875	+	0.173265i
							\(43\)	4.17852	+	4.17852i	0.637218	+	0.637218i	0.949868	−	0.312650i	\(-0.101217\pi\)
−0.312650	+	0.949868i	\(0.601217\pi\)
\(47\)	0.224739	−	0.542568i	0.0327815	−	0.0791416i	−0.906641	−	0.421903i	\(-0.861362\pi\)
							0.939422	+	0.342761i	\(0.111362\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.l.a.27.1	✓	104
7.6	odd	2	inner	287.2.l.a.27.2	yes	104
41.38	odd	8	inner	287.2.l.a.202.2	yes	104
287.202	even	8	inner	287.2.l.a.202.1	yes	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.l.a.27.1	✓	104	1.1	even	1	trivial
287.2.l.a.27.2	yes	104	7.6	odd	2	inner
287.2.l.a.202.1	yes	104	287.202	even	8	inner
287.2.l.a.202.2	yes	104	41.38	odd	8	inner