Embedded newform 688.2.bi.a.27.1

• Label: 688.2.bi.a.27.1
• 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			27.1
Character	\(\chi\)	\(=\)	688.27
Dual form			688.2.bi.a.51.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41419 - 0.00818042i) q^{2} +(-0.335843 - 0.534490i) q^{3} +(1.99987 + 0.0231373i) q^{4} +(-1.88775 - 0.212698i) q^{5} +(0.470573 + 0.758618i) q^{6} +2.98976i q^{7} +(-2.82800 - 0.0490803i) q^{8} +(1.12876 - 2.34390i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1032 q - 14 q^{2} - 14 q^{3} - 14 q^{4} - 14 q^{5} - 24 q^{6} - 14 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41419	−	0.00818042i	−0.999983	−	0.00578443i
							\(3\)	−0.335843	−	0.534490i	−0.193899	−	0.308588i	0.735598	−	0.677418i	\(-0.236901\pi\)
−0.929497	+	0.368830i	\(0.879758\pi\)
\(5\)	−1.88775	−	0.212698i	−0.844227	−	0.0951215i	−0.320753	−	0.947163i	\(-0.603936\pi\)
							−0.523474	+	0.852042i	\(0.675364\pi\)
\(7\)			2.98976i			1.13002i	0.825084	+	0.565011i	\(0.191128\pi\)
							−0.825084	+	0.565011i	\(0.808872\pi\)
\(11\)	0.421455	−	0.147474i	0.127074	−	0.0444650i	−0.265993	−	0.963975i	\(-0.585700\pi\)
							0.393067	+	0.919510i	\(0.371414\pi\)
\(13\)	0.00359481		0.000405038i	0.000997022		0.000112337i	0.112463	−	0.993656i	\(-0.464126\pi\)
							−0.111466	+	0.993768i	\(0.535555\pi\)
\(17\)	0.346778	−	0.434846i	0.0841060	−	0.105466i	−0.737998	−	0.674803i	\(-0.764229\pi\)
							0.822104	+	0.569337i	\(0.192800\pi\)
\(19\)	−1.28476	+	3.67163i	−0.294744	+	0.842330i	0.697338	+	0.716742i	\(0.254368\pi\)
							−0.992082	+	0.125588i	\(0.959918\pi\)
\(23\)	−5.84975	−	2.81709i	−1.21976	−	0.587404i	−0.290513	−	0.956871i	\(-0.593826\pi\)
							−0.929244	+	0.369467i	\(0.879540\pi\)
\(29\)	−1.91307	+	3.04463i	−0.355248	+	0.565374i	−0.975766	−	0.218819i	\(-0.929780\pi\)
							0.620518	+	0.784192i	\(0.286922\pi\)
\(31\)	1.98386	−	0.452803i	0.356312	−	0.0813259i	−0.0406181	−	0.999175i	\(-0.512933\pi\)
							0.396930	+	0.917849i	\(0.370076\pi\)
\(37\)	−5.65164	+	5.65164i	−0.929123	+	0.929123i	−0.997649	−	0.0685262i	\(-0.978170\pi\)
							0.0685262	+	0.997649i	\(0.478170\pi\)
\(41\)	3.81880	−	0.871615i	0.596396	−	0.136123i	0.0863414	−	0.996266i	\(-0.472482\pi\)
							0.510054	+	0.860142i	\(0.329625\pi\)
\(43\)	2.89125	+	5.88563i	0.440912	+	0.897550i
							\(47\)	4.35304	+	9.03917i	0.634956	+	1.31850i	0.931591	+	0.363508i	\(0.118421\pi\)
−0.296635	+	0.954991i	\(0.595865\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.bi.a.27.1	✓	1032
16.3	odd	4	inner	688.2.bi.a.371.13	yes	1032
43.8	odd	14	inner	688.2.bi.a.395.13	yes	1032
688.51	even	28	inner	688.2.bi.a.51.1	yes	1032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
688.2.bi.a.27.1	✓	1032	1.1	even	1	trivial
688.2.bi.a.51.1	yes	1032	688.51	even	28	inner
688.2.bi.a.371.13	yes	1032	16.3	odd	4	inner
688.2.bi.a.395.13	yes	1032	43.8	odd	14	inner