Embedded newform 17.5.e.a.7.4

• Label: 17.5.e.a.7.4
• Level: $17$
• Weight: $5$
• Character: 17.7
• Analytic conductor: $1.757$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	17.e (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.75728937243\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(5\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			7.4
Character	\(\chi\)	\(=\)	17.7
Dual form			17.5.e.a.5.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.12676 - 2.72024i) q^{2} +(2.50076 - 3.74265i) q^{3} +(5.18357 + 5.18357i) q^{4} +(-7.34972 - 36.9495i) q^{5} +(-7.36316 - 11.0198i) q^{6} +(-15.5535 + 78.1927i) q^{7} +(63.4651 - 26.2881i) q^{8} +(23.2437 + 56.1153i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{11}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.12676	−	2.72024i	0.281690	−	0.680061i	−0.718185	−	0.695852i	\(-0.755027\pi\)
							0.999875	+	0.0157914i	\(0.00502678\pi\)
\(3\)	2.50076	−	3.74265i	0.277862	−	0.415850i	−0.666121	−	0.745844i	\(-0.732047\pi\)
							0.943983	+	0.329994i	\(0.107047\pi\)
\(5\)	−7.34972	−	36.9495i	−0.293989	−	1.47798i	−0.791845	−	0.610722i	\(-0.790879\pi\)
							0.497857	−	0.867259i	\(-0.334121\pi\)
\(7\)	−15.5535	+	78.1927i	−0.317418	+	1.59577i	0.411659	+	0.911338i	\(0.364949\pi\)
							−0.729077	+	0.684431i	\(0.760051\pi\)
\(11\)	−96.4952	+	64.4760i	−0.797481	+	0.532860i	−0.886258	−	0.463192i	\(-0.846704\pi\)
							0.0887770	+	0.996052i	\(0.471704\pi\)
\(13\)	33.3223	−	33.3223i	0.197173	−	0.197173i	−0.601614	−	0.798787i	\(-0.705475\pi\)
							0.798787	+	0.601614i	\(0.205475\pi\)
\(17\)	−199.511	−	209.084i	−0.690351	−	0.723475i
							\(19\)	108.966	−	263.067i	0.301845	−	0.728717i	−0.698075	−	0.716025i	\(-0.745960\pi\)
0.999919	−	0.0126924i	\(-0.00404023\pi\)
\(23\)	36.8747	+	55.1869i	0.0697064	+	0.104323i	0.864685	−	0.502314i	\(-0.167518\pi\)
							−0.794979	+	0.606637i	\(0.792518\pi\)
\(29\)	−793.411	+	157.819i	−0.943414	+	0.187657i	−0.642752	−	0.766074i	\(-0.722208\pi\)
							−0.300662	+	0.953731i	\(0.597208\pi\)
\(31\)	−775.397	−	518.103i	−0.806864	−	0.539129i	0.0823591	−	0.996603i	\(-0.473755\pi\)
							−0.889223	+	0.457473i	\(0.848755\pi\)
\(37\)	811.877	−	1215.06i	0.593044	−	0.887553i	−0.406617	−	0.913599i	\(-0.633292\pi\)
							0.999661	+	0.0260457i	\(0.00829155\pi\)
\(41\)	−14.5413	+	73.1039i	−0.00865037	+	0.0434884i	−0.984869	−	0.173298i	\(-0.944558\pi\)
							0.976219	+	0.216787i	\(0.0695576\pi\)
\(43\)	848.620	+	2048.75i	0.458962	+	1.10803i	0.968818	+	0.247772i	\(0.0796984\pi\)
							−0.509857	+	0.860259i	\(0.670302\pi\)
\(47\)	−903.329	+	903.329i	−0.408931	+	0.408931i	−0.881366	−	0.472434i	\(-0.843375\pi\)
							0.472434	+	0.881366i	\(0.343375\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.5.e.a.7.4	yes	40
3.2	odd	2		153.5.p.a.109.2		40
17.5	odd	16	inner	17.5.e.a.5.4	✓	40
51.5	even	16		153.5.p.a.73.2		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.5.e.a.5.4	✓	40	17.5	odd	16	inner
17.5.e.a.7.4	yes	40	1.1	even	1	trivial
153.5.p.a.73.2		40	51.5	even	16
153.5.p.a.109.2		40	3.2	odd	2