Embedded newform 688.2.bi.a.51.9

• Label: 688.2.bi.a.51.9
• Level: $688$
• Weight: $2$
• Character: 688.51
• Analytic conductor: $5.494$
• Analytic rank: $0$
• Dimension: $1032$
• 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			51.9
Character	\(\chi\)	\(=\)	688.51
Dual form			688.2.bi.a.27.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35485 - 0.405438i) q^{2} +(0.624444 - 0.993797i) q^{3} +(1.67124 + 1.09861i) q^{4} +(0.629114 - 0.0708842i) q^{5} +(-1.24895 + 1.09327i) q^{6} +3.36367i q^{7} +(-1.81886 - 2.16604i) q^{8} +(0.703949 + 1.46176i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1032 q - 14 q^{2} - 14 q^{3} - 14 q^{4} - 14 q^{5} - 24 q^{6} - 14 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(431\)	\(433\)	\(517\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35485	−	0.405438i	−0.958024	−	0.286688i
							\(3\)	0.624444	−	0.993797i	0.360523	−	0.573769i	−0.616381	−	0.787448i	\(-0.711402\pi\)
0.976904	+	0.213679i	\(0.0685447\pi\)
\(5\)	0.629114	−	0.0708842i	0.281348	−	0.0317004i	0.0298374	−	0.999555i	\(-0.490501\pi\)
							0.251511	+	0.967854i	\(0.419072\pi\)
\(7\)			3.36367i			1.27135i	0.771958	+	0.635674i	\(0.219278\pi\)
							−0.771958	+	0.635674i	\(0.780722\pi\)
\(11\)	−3.59701	−	1.25865i	−1.08454	−	0.379497i	−0.271937	−	0.962315i	\(-0.587664\pi\)
							−0.812602	+	0.582818i	\(0.801950\pi\)
\(13\)	1.64835	−	0.185724i	0.457169	−	0.0515106i	0.119623	−	0.992819i	\(-0.461832\pi\)
							0.337547	+	0.941309i	\(0.390403\pi\)
\(17\)	2.12697	+	2.66714i	0.515867	+	0.646876i	0.969726	−	0.244197i	\(-0.0785245\pi\)
							−0.453859	+	0.891074i	\(0.649953\pi\)
\(19\)	2.06935	+	5.91387i	0.474742	+	1.35673i	0.893674	+	0.448717i	\(0.148119\pi\)
							−0.418932	+	0.908017i	\(0.637596\pi\)
\(23\)	−7.03544	+	3.38809i	−1.46699	+	0.706466i	−0.985451	−	0.169958i	\(-0.945637\pi\)
							−0.481540	+	0.876424i	\(0.659922\pi\)
\(29\)	−1.11972	−	1.78203i	−0.207927	−	0.330915i	0.726437	−	0.687233i	\(-0.241175\pi\)
							−0.934365	+	0.356318i	\(0.884032\pi\)
\(31\)	−2.31780	−	0.529022i	−0.416289	−	0.0950152i	0.00924592	−	0.999957i	\(-0.497057\pi\)
							−0.425535	+	0.904942i	\(0.639914\pi\)
\(37\)	0.0864539	+	0.0864539i	0.0142129	+	0.0142129i	0.714178	−	0.699965i	\(-0.246801\pi\)
							−0.699965	+	0.714178i	\(0.746801\pi\)
\(41\)	3.66814	+	0.837230i	0.572868	+	0.130753i	0.499134	−	0.866525i	\(-0.333651\pi\)
							0.0737336	+	0.997278i	\(0.476509\pi\)
\(43\)	4.19027	+	5.04397i	0.639010	+	0.769198i
							\(47\)	1.15176	−	2.39166i	0.168002	−	0.348859i	−0.799922	−	0.600104i	\(-0.795126\pi\)
0.967924	+	0.251245i	\(0.0808400\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.bi.a.51.9	yes	1032
16.11	odd	4	inner	688.2.bi.a.395.8	yes	1032
43.27	odd	14	inner	688.2.bi.a.371.8	yes	1032
688.27	even	28	inner	688.2.bi.a.27.9	✓	1032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
688.2.bi.a.27.9	✓	1032	688.27	even	28	inner
688.2.bi.a.51.9	yes	1032	1.1	even	1	trivial
688.2.bi.a.371.8	yes	1032	43.27	odd	14	inner
688.2.bi.a.395.8	yes	1032	16.11	odd	4	inner