Embedded newform 688.2.bi.a.27.7

• Label: 688.2.bi.a.27.7
• 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			27.7
Character	\(\chi\)	\(=\)	688.27
Dual form			688.2.bi.a.51.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37016 - 0.350224i) q^{2} +(-1.30119 - 2.07083i) q^{3} +(1.75469 + 0.959726i) q^{4} +(1.31865 + 0.148577i) q^{5} +(1.05759 + 3.29309i) q^{6} +3.77712i q^{7} +(-2.06809 - 1.92951i) q^{8} +(-1.29360 + 2.68619i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1032 q - 14 q^{2} - 14 q^{3} - 14 q^{4} - 14 q^{5} - 24 q^{6} - 14 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37016	−	0.350224i	−0.968851	−	0.247646i
							\(3\)	−1.30119	−	2.07083i	−0.751243	−	1.19560i	−0.975271	−	0.221014i	\(-0.929063\pi\)
0.224028	−	0.974583i	\(-0.428079\pi\)
\(5\)	1.31865	+	0.148577i	0.589720	+	0.0664454i	0.401782	−	0.915735i	\(-0.368391\pi\)
							0.187938	+	0.982181i	\(0.439820\pi\)
\(7\)			3.77712i			1.42762i	0.700341	+	0.713808i	\(0.253031\pi\)
							−0.700341	+	0.713808i	\(0.746969\pi\)
\(11\)	−2.69185	+	0.941918i	−0.811622	+	0.283999i	−0.704006	−	0.710194i	\(-0.748607\pi\)
							−0.107616	+	0.994193i	\(0.534322\pi\)
\(13\)	−2.58841	−	0.291644i	−0.717897	−	0.0808875i	−0.254544	−	0.967061i	\(-0.581925\pi\)
							−0.463353	+	0.886174i	\(0.653354\pi\)
\(17\)	3.62356	−	4.54380i	0.878843	−	1.10203i	−0.115232	−	0.993339i	\(-0.536761\pi\)
							0.994075	−	0.108695i	\(-0.0346673\pi\)
\(19\)	1.66134	−	4.74782i	0.381137	−	1.08923i	−0.580348	−	0.814368i	\(-0.697084\pi\)
							0.961485	−	0.274857i	\(-0.0886306\pi\)
\(23\)	2.72615	+	1.31285i	0.568442	+	0.273747i	0.695950	−	0.718090i	\(-0.254983\pi\)
							−0.127508	+	0.991838i	\(0.540698\pi\)
\(29\)	5.45164	−	8.67623i	1.01234	−	1.61114i	0.245399	−	0.969422i	\(-0.421081\pi\)
							0.766944	−	0.641714i	\(-0.221776\pi\)
\(31\)	1.66488	−	0.379998i	0.299022	−	0.0682497i	−0.0703779	−	0.997520i	\(-0.522421\pi\)
							0.369400	+	0.929271i	\(0.379563\pi\)
\(37\)	1.77537	−	1.77537i	0.291870	−	0.291870i	−0.545949	−	0.837819i	\(-0.683831\pi\)
							0.837819	+	0.545949i	\(0.183831\pi\)
\(41\)	8.00142	−	1.82627i	1.24961	−	0.285216i	0.453988	−	0.891008i	\(-0.350001\pi\)
							0.795623	+	0.605792i	\(0.207144\pi\)
\(43\)	−1.97776	−	6.25208i	−0.301605	−	0.953433i
							\(47\)	−3.65964	−	7.59931i	−0.533813	−	1.10847i	−0.977235	−	0.212161i	\(-0.931950\pi\)
0.443422	−	0.896313i	\(-0.353764\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.bi.a.27.7	✓	1032
16.3	odd	4	inner	688.2.bi.a.371.20	yes	1032
43.8	odd	14	inner	688.2.bi.a.395.20	yes	1032
688.51	even	28	inner	688.2.bi.a.51.7	yes	1032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
688.2.bi.a.27.7	✓	1032	1.1	even	1	trivial
688.2.bi.a.51.7	yes	1032	688.51	even	28	inner
688.2.bi.a.371.20	yes	1032	16.3	odd	4	inner
688.2.bi.a.395.20	yes	1032	43.8	odd	14	inner