Embedded newform 688.2.bi.a.27.3

• Label: 688.2.bi.a.27.3
• Level: $688$
• Weight: $2$
• Character: 688.27
• Analytic conductor: $5.494$
• Analytic rank: $0$
• Dimension: $1032$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	688.bi (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.49370765906\)
Analytic rank:	\(0\)
Dimension:	\(1032\)
Relative dimension:	\(86\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			27.3
Character	\(\chi\)	\(=\)	688.27
Dual form			688.2.bi.a.51.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40726 + 0.140039i) q^{2} +(1.06241 + 1.69081i) q^{3} +(1.96078 - 0.394144i) q^{4} +(-3.99512 - 0.450141i) q^{5} +(-1.73187 - 2.23064i) q^{6} -2.81641i q^{7} +(-2.70413 + 0.829249i) q^{8} +(-0.428483 + 0.889754i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1032 q - 14 q^{2} - 14 q^{3} - 14 q^{4} - 14 q^{5} - 24 q^{6} - 14 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/688\mathbb{Z}\right)^\times\).

\(n\)	\(431\)	\(433\)	\(517\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40726	+	0.140039i	−0.995085	+	0.0990226i
							\(3\)	1.06241	+	1.69081i	0.613381	+	0.976190i	0.998576	+	0.0533551i	\(0.0169915\pi\)
−0.385194	+	0.922835i	\(0.625866\pi\)
\(5\)	−3.99512	−	0.450141i	−1.78667	−	0.201309i	−0.844298	−	0.535875i	\(-0.819982\pi\)
							−0.942373	+	0.334565i	\(0.891411\pi\)
\(7\)		−	2.81641i		−	1.06450i	−0.846586	−	0.532252i	\(-0.821346\pi\)
							0.846586	−	0.532252i	\(-0.178654\pi\)
\(11\)	−1.18168	+	0.413489i	−0.356291	+	0.124672i	−0.502489	−	0.864584i	\(-0.667582\pi\)
							0.146197	+	0.989255i	\(0.453297\pi\)
\(13\)	3.08772	+	0.347902i	0.856378	+	0.0964907i	0.529226	−	0.848481i	\(-0.322482\pi\)
							0.327152	+	0.944972i	\(0.393911\pi\)
\(17\)	0.906099	−	1.13621i	0.219761	−	0.275572i	−0.659714	−	0.751517i	\(-0.729322\pi\)
							0.879475	+	0.475945i	\(0.157894\pi\)
\(19\)	−2.70269	+	7.72384i	−0.620040	+	1.77197i	0.0154722	+	0.999880i	\(0.495075\pi\)
							−0.635512	+	0.772091i	\(0.719211\pi\)
\(23\)	8.24613	+	3.97113i	1.71944	+	0.828037i	0.989502	+	0.144522i	\(0.0461644\pi\)
							0.729936	+	0.683516i	\(0.239550\pi\)
\(29\)	2.15259	−	3.42583i	0.399726	−	0.636161i	−0.584845	−	0.811145i	\(-0.698845\pi\)
							0.984571	+	0.174984i	\(0.0559875\pi\)
\(31\)	9.00754	−	2.05591i	1.61780	−	0.369253i	0.684689	−	0.728835i	\(-0.259938\pi\)
							0.933113	+	0.359583i	\(0.117081\pi\)
\(37\)	−0.321787	+	0.321787i	−0.0529015	+	0.0529015i	−0.733063	−	0.680161i	\(-0.761910\pi\)
							0.680161	+	0.733063i	\(0.261910\pi\)
\(41\)	−0.496208	+	0.113256i	−0.0774947	+	0.0176877i	−0.261092	−	0.965314i	\(-0.584083\pi\)
							0.183598	+	0.983001i	\(0.441226\pi\)
\(43\)	−6.55026	+	0.306742i	−0.998905	+	0.0467777i
							\(47\)	0.209970	+	0.436007i	0.0306273	+	0.0635982i	0.915721	−	0.401814i	\(-0.131620\pi\)
−0.885094	+	0.465413i	\(0.845906\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.bi.a.27.3	✓	1032
16.3	odd	4	inner	688.2.bi.a.371.11	yes	1032
43.8	odd	14	inner	688.2.bi.a.395.11	yes	1032
688.51	even	28	inner	688.2.bi.a.51.3	yes	1032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
688.2.bi.a.27.3	✓	1032	1.1	even	1	trivial
688.2.bi.a.51.3	yes	1032	688.51	even	28	inner
688.2.bi.a.371.11	yes	1032	16.3	odd	4	inner
688.2.bi.a.395.11	yes	1032	43.8	odd	14	inner