Embedded newform 688.2.bi.a.27.19

• Label: 688.2.bi.a.27.19
• 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			27.19
Character	\(\chi\)	\(=\)	688.27
Dual form			688.2.bi.a.51.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16427 + 0.802788i) q^{2} +(0.219611 + 0.349509i) q^{3} +(0.711062 - 1.86933i) q^{4} +(0.0530869 + 0.00598146i) q^{5} +(-0.536268 - 0.230622i) q^{6} +2.39123i q^{7} +(0.672804 + 2.74724i) q^{8} +(1.22772 - 2.54939i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1032 q - 14 q^{2} - 14 q^{3} - 14 q^{4} - 14 q^{5} - 24 q^{6} - 14 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16427	+	0.802788i	−0.823265	+	0.567657i
							\(3\)	0.219611	+	0.349509i	0.126792	+	0.201789i	0.904088	−	0.427347i	\(-0.140552\pi\)
−0.777295	+	0.629136i	\(0.783409\pi\)
\(5\)	0.0530869	+	0.00598146i	0.0237412	+	0.00267499i	0.123827	−	0.992304i	\(-0.460483\pi\)
							−0.100086	+	0.994979i	\(0.531912\pi\)
\(7\)			2.39123i			0.903800i	0.892069	+	0.451900i	\(0.149254\pi\)
							−0.892069	+	0.451900i	\(0.850746\pi\)
\(11\)	−5.11890	+	1.79118i	−1.54341	+	0.540062i	−0.961882	−	0.273466i	\(-0.911830\pi\)
							−0.581526	+	0.813528i	\(0.697544\pi\)
\(13\)	−5.26946	−	0.593725i	−1.46148	−	0.164670i	−0.654867	−	0.755744i	\(-0.727275\pi\)
							−0.806617	+	0.591074i	\(0.798704\pi\)
\(17\)	2.16853	−	2.71926i	0.525947	−	0.659517i	−0.445913	−	0.895076i	\(-0.647121\pi\)
							0.971860	+	0.235560i	\(0.0756923\pi\)
\(19\)	−1.24213	+	3.54980i	−0.284964	+	0.814380i	0.709053	+	0.705155i	\(0.249123\pi\)
							−0.994017	+	0.109225i	\(0.965163\pi\)
\(23\)	6.35140	+	3.05867i	1.32436	+	0.637778i	0.956399	−	0.292064i	\(-0.0943420\pi\)
							0.367960	+	0.929842i	\(0.380056\pi\)
\(29\)	−3.50868	+	5.58404i	−0.651546	+	1.03693i	0.343656	+	0.939095i	\(0.388334\pi\)
							−0.995203	+	0.0978346i	\(0.968808\pi\)
\(31\)	−4.79589	+	1.09463i	−0.861367	+	0.196601i	−0.630315	−	0.776340i	\(-0.717074\pi\)
							−0.231053	+	0.972941i	\(0.574217\pi\)
\(37\)	−3.81774	+	3.81774i	−0.627633	+	0.627633i	−0.947472	−	0.319839i	\(-0.896371\pi\)
							0.319839	+	0.947472i	\(0.396371\pi\)
\(41\)	−5.56559	+	1.27031i	−0.869200	+	0.198389i	−0.633788	−	0.773507i	\(-0.718501\pi\)
							−0.235412	+	0.971896i	\(0.575644\pi\)
\(43\)	−5.16051	+	4.04588i	−0.786970	+	0.616991i
							\(47\)	−4.41035	−	9.15819i	−0.643316	−	1.33586i	−0.926319	−	0.376740i	\(-0.877045\pi\)
0.283003	−	0.959119i	\(-0.408669\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.bi.a.27.19	✓	1032
16.3	odd	4	inner	688.2.bi.a.371.7	yes	1032
43.8	odd	14	inner	688.2.bi.a.395.7	yes	1032
688.51	even	28	inner	688.2.bi.a.51.19	yes	1032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
688.2.bi.a.27.19	✓	1032	1.1	even	1	trivial
688.2.bi.a.51.19	yes	1032	688.51	even	28	inner
688.2.bi.a.371.7	yes	1032	16.3	odd	4	inner
688.2.bi.a.395.7	yes	1032	43.8	odd	14	inner