Embedded newform 688.2.bi.a.27.15

• Label: 688.2.bi.a.27.15
• 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			27.15
Character	\(\chi\)	\(=\)	688.27
Dual form			688.2.bi.a.51.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.21164 - 0.729333i) q^{2} +(1.32193 + 2.10385i) q^{3} +(0.936146 + 1.76738i) q^{4} +(-3.18754 - 0.359149i) q^{5} +(-0.0673043 - 3.51324i) q^{6} +3.24462i q^{7} +(0.154736 - 2.82419i) q^{8} +(-1.37701 + 2.85939i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1032 q - 14 q^{2} - 14 q^{3} - 14 q^{4} - 14 q^{5} - 24 q^{6} - 14 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.21164	−	0.729333i	−0.856759	−	0.515716i
							\(3\)	1.32193	+	2.10385i	0.763219	+	1.21466i	0.971605	+	0.236610i	\(0.0760362\pi\)
−0.208386	+	0.978047i	\(0.566821\pi\)
\(5\)	−3.18754	−	0.359149i	−1.42551	−	0.160616i	−0.634738	−	0.772728i	\(-0.718892\pi\)
							−0.790771	+	0.612111i	\(0.790320\pi\)
\(7\)			3.24462i			1.22635i	0.789946	+	0.613176i	\(0.210108\pi\)
							−0.789946	+	0.613176i	\(0.789892\pi\)
\(11\)	−0.777468	+	0.272048i	−0.234415	+	0.0820255i	−0.444930	−	0.895565i	\(-0.646772\pi\)
							0.210515	+	0.977591i	\(0.432486\pi\)
\(13\)	2.46213	+	0.277416i	0.682873	+	0.0769413i	0.446584	−	0.894742i	\(-0.352640\pi\)
							0.236289	+	0.971683i	\(0.424069\pi\)
\(17\)	−2.07524	+	2.60227i	−0.503319	+	0.631142i	−0.966974	−	0.254873i	\(-0.917966\pi\)
							0.463655	+	0.886016i	\(0.346538\pi\)
\(19\)	0.677758	−	1.93692i	0.155488	−	0.444360i	−0.839617	−	0.543179i	\(-0.817221\pi\)
							0.995105	+	0.0988186i	\(0.0315063\pi\)
\(23\)	−4.15338	−	2.00016i	−0.866040	−	0.417063i	−0.0525335	−	0.998619i	\(-0.516730\pi\)
							−0.813506	+	0.581557i	\(0.802444\pi\)
\(29\)	−2.87101	+	4.56919i	−0.533133	+	0.848477i	−0.999345	−	0.0361965i	\(-0.988476\pi\)
							0.466211	+	0.884673i	\(0.345619\pi\)
\(31\)	−8.23582	+	1.87977i	−1.47920	+	0.337617i	−0.884578	−	0.466392i	\(-0.845554\pi\)
							−0.594618	+	0.804009i	\(0.702697\pi\)
\(37\)	−1.76475	+	1.76475i	−0.290123	+	0.290123i	−0.837129	−	0.547006i	\(-0.815767\pi\)
							0.547006	+	0.837129i	\(0.315767\pi\)
\(41\)	1.87000	−	0.426815i	0.292044	−	0.0666572i	−0.0739886	−	0.997259i	\(-0.523573\pi\)
							0.366033	+	0.930602i	\(0.380716\pi\)
\(43\)	−1.42407	−	6.40094i	−0.217169	−	0.976134i
							\(47\)	−5.86445	−	12.1777i	−0.855418	−	1.77629i	−0.579167	−	0.815209i	\(-0.696622\pi\)
−0.276251	−	0.961085i	\(-0.589092\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
688.2.bi.a.27.15	✓	1032	1.1	even	1	trivial
688.2.bi.a.51.15	yes	1032	688.51	even	28	inner
688.2.bi.a.371.28	yes	1032	16.3	odd	4	inner
688.2.bi.a.395.28	yes	1032	43.8	odd	14	inner