Embedded newform 688.2.bi.a.27.10

• Label: 688.2.bi.a.27.10
• 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			27.10
Character	\(\chi\)	\(=\)	688.27
Dual form			688.2.bi.a.51.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30271 + 0.550405i) q^{2} +(-1.21176 - 1.92851i) q^{3} +(1.39411 - 1.43404i) q^{4} +(-1.58240 - 0.178294i) q^{5} +(2.64004 + 1.84533i) q^{6} -3.53511i q^{7} +(-1.02682 + 2.63546i) q^{8} +(-0.949134 + 1.97090i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1032 q - 14 q^{2} - 14 q^{3} - 14 q^{4} - 14 q^{5} - 24 q^{6} - 14 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30271	+	0.550405i	−0.921155	+	0.389195i
							\(3\)	−1.21176	−	1.92851i	−0.699612	−	1.11343i	−0.987781	−	0.155847i	\(-0.950189\pi\)
0.288169	−	0.957580i	\(-0.406954\pi\)
\(5\)	−1.58240	−	0.178294i	−0.707673	−	0.0797355i	−0.249210	−	0.968449i	\(-0.580171\pi\)
							−0.458462	+	0.888714i	\(0.651600\pi\)
\(7\)		−	3.53511i		−	1.33615i	−0.744096	−	0.668073i	\(-0.767119\pi\)
							0.744096	−	0.668073i	\(-0.232881\pi\)
\(11\)	5.33521	−	1.86687i	1.60863	−	0.562883i	0.631470	−	0.775400i	\(-0.282452\pi\)
							0.977157	+	0.212517i	\(0.0681662\pi\)
\(13\)	−1.21056	−	0.136397i	−0.335749	−	0.0378298i	−0.0575191	−	0.998344i	\(-0.518319\pi\)
							−0.278230	+	0.960515i	\(0.589748\pi\)
\(17\)	4.87379	−	6.11154i	1.18207	−	1.48227i	0.342071	−	0.939674i	\(-0.388872\pi\)
							0.839997	−	0.542591i	\(-0.182557\pi\)
\(19\)	1.28404	−	3.66957i	0.294578	−	0.841856i	−0.697539	−	0.716547i	\(-0.745722\pi\)
							0.992118	−	0.125310i	\(-0.0399925\pi\)
\(23\)	3.48419	+	1.67790i	0.726504	+	0.349866i	0.760308	−	0.649562i	\(-0.225048\pi\)
							−0.0338039	+	0.999428i	\(0.510762\pi\)
\(29\)	1.09374	−	1.74067i	0.203102	−	0.323234i	−0.729606	−	0.683868i	\(-0.760297\pi\)
							0.932708	+	0.360633i	\(0.117439\pi\)
\(31\)	1.72255	−	0.393161i	0.309379	−	0.0706138i	−0.0650125	−	0.997884i	\(-0.520709\pi\)
							0.374392	+	0.927271i	\(0.377852\pi\)
\(37\)	−6.67679	+	6.67679i	−1.09766	+	1.09766i	−0.102974	+	0.994684i	\(0.532836\pi\)
							−0.994684	+	0.102974i	\(0.967164\pi\)
\(41\)	2.25218	−	0.514046i	0.351732	−	0.0802806i	−0.0430045	−	0.999075i	\(-0.513693\pi\)
							0.394737	+	0.918794i	\(0.370836\pi\)
\(43\)	0.917162	+	6.49298i	0.139866	+	0.990170i
							\(47\)	−2.40090	−	4.98552i	−0.350207	−	0.727213i	0.649235	−	0.760587i	\(-0.275089\pi\)
−0.999443	+	0.0333742i	\(0.989375\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.bi.a.27.10	✓	1032
16.3	odd	4	inner	688.2.bi.a.371.3	yes	1032
43.8	odd	14	inner	688.2.bi.a.395.3	yes	1032
688.51	even	28	inner	688.2.bi.a.51.10	yes	1032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
688.2.bi.a.27.10	✓	1032	1.1	even	1	trivial
688.2.bi.a.51.10	yes	1032	688.51	even	28	inner
688.2.bi.a.371.3	yes	1032	16.3	odd	4	inner
688.2.bi.a.395.3	yes	1032	43.8	odd	14	inner