Embedded newform 17.6.d.a.15.5

• Label: 17.6.d.a.15.5
• Level: $17$
• Weight: $6$
• Character: 17.15
• Analytic conductor: $2.727$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	17.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.72652493682\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			15.5
Character	\(\chi\)	\(=\)	17.15
Dual form			17.6.d.a.8.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.46953 + 2.46953i) q^{2} +(-10.5837 - 25.5512i) q^{3} -19.8029i q^{4} +(-30.5219 + 12.6426i) q^{5} +(36.9628 - 89.2361i) q^{6} +(151.735 + 62.8506i) q^{7} +(127.929 - 127.929i) q^{8} +(-369.025 + 369.025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 4 q^{2} - 4 q^{3} + 40 q^{5} - 364 q^{6} - 4 q^{7} + 124 q^{8} - 136 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.46953	+	2.46953i	0.436555	+	0.436555i	0.890851	−	0.454296i	\(-0.150109\pi\)
							−0.454296	+	0.890851i	\(0.650109\pi\)
\(3\)	−10.5837	−	25.5512i	−0.678943	−	1.63911i	−0.765946	−	0.642905i	\(-0.777729\pi\)
							0.0870032	−	0.996208i	\(-0.472271\pi\)
\(5\)	−30.5219	+	12.6426i	−0.545993	+	0.226158i	−0.638592	−	0.769546i	\(-0.720483\pi\)
							0.0925986	+	0.995704i	\(0.470483\pi\)
\(7\)	151.735	+	62.8506i	1.17042	+	0.484802i	0.881330	−	0.472501i	\(-0.156649\pi\)
							0.289086	+	0.957303i	\(0.406649\pi\)
\(11\)	208.839	−	504.182i	0.520391	−	1.25634i	−0.417269	−	0.908783i	\(-0.637013\pi\)
							0.937661	−	0.347552i	\(-0.112987\pi\)
\(13\)			359.778i			0.590441i	0.955429	+	0.295221i	\(0.0953932\pi\)
							−0.955429	+	0.295221i	\(0.904607\pi\)
\(17\)	1168.94	−	231.139i	0.981006	−	0.193977i
							\(19\)	−92.6379	−	92.6379i	−0.0588714	−	0.0588714i	0.677058	−	0.735930i	\(-0.263255\pi\)
−0.735930	+	0.677058i	\(0.763255\pi\)
\(23\)	−23.8558	+	57.5930i	−0.00940317	+	0.0227013i	−0.928511	−	0.371304i	\(-0.878911\pi\)
							0.919108	+	0.394005i	\(0.128911\pi\)
\(29\)	3015.23	−	1248.95i	0.665772	−	0.275772i	−0.0240933	−	0.999710i	\(-0.507670\pi\)
							0.689865	+	0.723938i	\(0.257670\pi\)
\(31\)	3540.15	+	8546.68i	0.661634	+	1.59732i	0.795243	+	0.606291i	\(0.207343\pi\)
							−0.133609	+	0.991034i	\(0.542657\pi\)
\(37\)	−1574.90	−	3802.13i	−0.189124	−	0.456586i	0.800667	−	0.599109i	\(-0.204479\pi\)
							−0.989792	+	0.142523i	\(0.954479\pi\)
\(41\)	1722.10	+	713.317i	0.159992	+	0.0662709i	0.461242	−	0.887274i	\(-0.347404\pi\)
							−0.301250	+	0.953545i	\(0.597404\pi\)
\(43\)	−8771.98	+	8771.98i	−0.723480	+	0.723480i	−0.969312	−	0.245833i	\(-0.920939\pi\)
							0.245833	+	0.969312i	\(0.420939\pi\)
\(47\)		−	3152.42i		−	0.208161i	−0.994569	−	0.104081i	\(-0.966810\pi\)
							0.994569	−	0.104081i	\(-0.0331900\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.6.d.a.15.5	yes	28
17.5	odd	16		289.6.a.j.1.10		28
17.8	even	8	inner	17.6.d.a.8.5	✓	28
17.12	odd	16		289.6.a.j.1.9		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.6.d.a.8.5	✓	28	17.8	even	8	inner
17.6.d.a.15.5	yes	28	1.1	even	1	trivial
289.6.a.j.1.9		28	17.12	odd	16
289.6.a.j.1.10		28	17.5	odd	16