Embedded newform 688.2.bi.a.371.13

• Label: 688.2.bi.a.371.13
• 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			371.13
Character	\(\chi\)	\(=\)	688.371
Dual form			688.2.bi.a.395.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.27059 + 0.620964i) q^{2} +(-0.534490 + 0.335843i) q^{3} +(1.22881 - 1.57798i) q^{4} +(-0.212698 + 1.88775i) q^{5} +(0.470573 - 0.758618i) q^{6} +2.98976i q^{7} +(-0.581440 + 2.76802i) q^{8} +(-1.12876 + 2.34390i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1032 q - 14 q^{2} - 14 q^{3} - 14 q^{4} - 14 q^{5} - 24 q^{6} - 14 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.27059	+	0.620964i	−0.898444	+	0.439088i
							\(3\)	−0.534490	+	0.335843i	−0.308588	+	0.193899i	−0.677418	−	0.735598i	\(-0.736901\pi\)
0.368830	+	0.929497i	\(0.379758\pi\)
\(5\)	−0.212698	+	1.88775i	−0.0951215	+	0.844227i	0.852042	+	0.523474i	\(0.175364\pi\)
							−0.947163	+	0.320753i	\(0.896064\pi\)
\(7\)			2.98976i			1.13002i	0.825084	+	0.565011i	\(0.191128\pi\)
							−0.825084	+	0.565011i	\(0.808872\pi\)
\(11\)	0.147474	+	0.421455i	0.0444650	+	0.127074i	0.963975	−	0.265993i	\(-0.0856999\pi\)
							−0.919510	+	0.393067i	\(0.871414\pi\)
\(13\)	−0.000405038		0.00359481i	−0.000112337		0.000997022i	−0.993768	−	0.111466i	\(-0.964445\pi\)
							0.993656	+	0.112463i	\(0.0358740\pi\)
\(17\)	0.346778	−	0.434846i	0.0841060	−	0.105466i	−0.737998	−	0.674803i	\(-0.764229\pi\)
							0.822104	+	0.569337i	\(0.192800\pi\)
\(19\)	3.67163	+	1.28476i	0.842330	+	0.294744i	0.716742	−	0.697338i	\(-0.245632\pi\)
							0.125588	+	0.992082i	\(0.459918\pi\)
\(23\)	−5.84975	−	2.81709i	−1.21976	−	0.587404i	−0.290513	−	0.956871i	\(-0.593826\pi\)
							−0.929244	+	0.369467i	\(0.879540\pi\)
\(29\)	−3.04463	−	1.91307i	−0.565374	−	0.355248i	0.218819	−	0.975766i	\(-0.429780\pi\)
							−0.784192	+	0.620518i	\(0.786922\pi\)
\(31\)	−1.98386	+	0.452803i	−0.356312	+	0.0813259i	−0.396930	−	0.917849i	\(-0.629924\pi\)
							0.0406181	+	0.999175i	\(0.487067\pi\)
\(37\)	5.65164	+	5.65164i	0.929123	+	0.929123i	0.997649	−	0.0685262i	\(-0.0218297\pi\)
							−0.0685262	+	0.997649i	\(0.521830\pi\)
\(41\)	−3.81880	+	0.871615i	−0.596396	+	0.136123i	−0.510054	−	0.860142i	\(-0.670375\pi\)
							−0.0863414	+	0.996266i	\(0.527518\pi\)
\(43\)	−5.88563	+	2.89125i	−0.897550	+	0.440912i
							\(47\)	−4.35304	−	9.03917i	−0.634956	−	1.31850i	−0.931591	−	0.363508i	\(-0.881579\pi\)
0.296635	−	0.954991i	\(-0.404135\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
688.2.bi.a.27.1	✓	1032	16.11	odd	4	inner
688.2.bi.a.51.1	yes	1032	43.8	odd	14	inner
688.2.bi.a.371.13	yes	1032	1.1	even	1	trivial
688.2.bi.a.395.13	yes	1032	688.395	even	28	inner