Embedded newform 688.2.bi.a.371.28

• Label: 688.2.bi.a.371.28
• Level: $688$
• Weight: $2$
• Character: 688.371
• 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			371.28
Character	\(\chi\)	\(=\)	688.371
Dual form			688.2.bi.a.395.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.775205 + 1.18282i) q^{2} +(2.10385 - 1.32193i) q^{3} +(-0.798115 - 1.83385i) q^{4} +(-0.359149 + 3.18754i) q^{5} +(-0.0673043 + 3.51324i) q^{6} +3.24462i q^{7} +(2.78781 + 0.477585i) q^{8} +(1.37701 - 2.85939i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1032 q - 14 q^{2} - 14 q^{3} - 14 q^{4} - 14 q^{5} - 24 q^{6} - 14 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.775205	+	1.18282i	−0.548153	+	0.836378i
							\(3\)	2.10385	−	1.32193i	1.21466	−	0.763219i	0.236610	−	0.971605i	\(-0.423964\pi\)
0.978047	+	0.208386i	\(0.0668209\pi\)
\(5\)	−0.359149	+	3.18754i	−0.160616	+	1.42551i	0.612111	+	0.790771i	\(0.290320\pi\)
							−0.772728	+	0.634738i	\(0.781108\pi\)
\(7\)			3.24462i			1.22635i	0.789946	+	0.613176i	\(0.210108\pi\)
							−0.789946	+	0.613176i	\(0.789892\pi\)
\(11\)	−0.272048	−	0.777468i	−0.0820255	−	0.234415i	0.895565	−	0.444930i	\(-0.146772\pi\)
							−0.977591	+	0.210515i	\(0.932486\pi\)
\(13\)	−0.277416	+	2.46213i	−0.0769413	+	0.682873i	0.894742	+	0.446584i	\(0.147360\pi\)
							−0.971683	+	0.236289i	\(0.924069\pi\)
\(17\)	−2.07524	+	2.60227i	−0.503319	+	0.631142i	−0.966974	−	0.254873i	\(-0.917966\pi\)
							0.463655	+	0.886016i	\(0.346538\pi\)
\(19\)	−1.93692	−	0.677758i	−0.444360	−	0.155488i	0.0988186	−	0.995105i	\(-0.468494\pi\)
							−0.543179	+	0.839617i	\(0.682779\pi\)
\(23\)	−4.15338	−	2.00016i	−0.866040	−	0.417063i	−0.0525335	−	0.998619i	\(-0.516730\pi\)
							−0.813506	+	0.581557i	\(0.802444\pi\)
\(29\)	−4.56919	−	2.87101i	−0.848477	−	0.533133i	0.0361965	−	0.999345i	\(-0.488476\pi\)
							−0.884673	+	0.466211i	\(0.845619\pi\)
\(31\)	8.23582	−	1.87977i	1.47920	−	0.337617i	0.594618	−	0.804009i	\(-0.297303\pi\)
							0.884578	+	0.466392i	\(0.154446\pi\)
\(37\)	1.76475	+	1.76475i	0.290123	+	0.290123i	0.837129	−	0.547006i	\(-0.184233\pi\)
							−0.547006	+	0.837129i	\(0.684233\pi\)
\(41\)	−1.87000	+	0.426815i	−0.292044	+	0.0666572i	−0.366033	−	0.930602i	\(-0.619284\pi\)
							0.0739886	+	0.997259i	\(0.476427\pi\)
\(43\)	6.40094	−	1.42407i	0.976134	−	0.217169i
							\(47\)	5.86445	+	12.1777i	0.855418	+	1.77629i	0.579167	+	0.815209i	\(0.303378\pi\)
0.276251	+	0.961085i	\(0.410908\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.bi.a.371.28	yes	1032
16.11	odd	4	inner	688.2.bi.a.27.15	✓	1032
43.8	odd	14	inner	688.2.bi.a.51.15	yes	1032
688.395	even	28	inner	688.2.bi.a.395.28	yes	1032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
688.2.bi.a.27.15	✓	1032	16.11	odd	4	inner
688.2.bi.a.51.15	yes	1032	43.8	odd	14	inner
688.2.bi.a.371.28	yes	1032	1.1	even	1	trivial
688.2.bi.a.395.28	yes	1032	688.395	even	28	inner