Embedded newform 17.12.d.a.15.6

• Label: 17.12.d.a.15.6
• Level: $17$
• Weight: $12$
• Character: 17.15
• Analytic conductor: $13.062$
• Analytic rank: $0$
• Dimension: $60$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	17.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.0618340695\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(15\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			15.6
Character	\(\chi\)	\(=\)	17.15
Dual form			17.12.d.a.8.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-14.6461 - 14.6461i) q^{2} +(-54.5226 - 131.629i) q^{3} -1618.98i q^{4} +(7213.60 - 2987.97i) q^{5} +(-1129.31 + 2726.39i) q^{6} +(26118.6 + 10818.7i) q^{7} +(-53706.9 + 53706.9i) q^{8} +(110908. - 110908. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 4 q^{2} - 4 q^{3} + 10120 q^{5} - 44716 q^{6} - 4 q^{7} + 8188 q^{8} + 621728 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−14.6461	−	14.6461i	−0.323636	−	0.323636i	0.526524	−	0.850160i	\(-0.323495\pi\)
							−0.850160	+	0.526524i	\(0.823495\pi\)
\(3\)	−54.5226	−	131.629i	−0.129542	−	0.312741i	0.845779	−	0.533533i	\(-0.179136\pi\)
							−0.975321	+	0.220792i	\(0.929136\pi\)
\(5\)	7213.60	−	2987.97i	1.03233	−	0.427604i	0.198775	−	0.980045i	\(-0.436304\pi\)
							0.833552	+	0.552441i	\(0.186304\pi\)
\(7\)	26118.6	+	10818.7i	0.587370	+	0.243296i	0.656519	−	0.754310i	\(-0.272028\pi\)
							−0.0691489	+	0.997606i	\(0.522028\pi\)
\(11\)	57777.1	−	139486.i	0.108167	−	0.261139i	−0.860523	−	0.509412i	\(-0.829863\pi\)
							0.968690	+	0.248273i	\(0.0798630\pi\)
\(13\)		−	492504.i		−	0.367893i	−0.982936	−	0.183946i	\(-0.941113\pi\)
							0.982936	−	0.183946i	\(-0.0588872\pi\)
\(17\)	−4.54872e6	−	3.68525e6i	−0.776998	−	0.629503i
							\(19\)	−5.78418e6	−	5.78418e6i	−0.535916	−	0.535916i	0.386411	−	0.922327i	\(-0.373715\pi\)
−0.922327	+	0.386411i	\(0.873715\pi\)
\(23\)	−1.39498e6	+	3.36777e6i	−0.0451922	+	0.109104i	−0.944864	−	0.327464i	\(-0.893806\pi\)
							0.899671	+	0.436568i	\(0.143806\pi\)
\(29\)	3.34996e7	−	1.38760e7i	0.303285	−	0.125625i	−0.225851	−	0.974162i	\(-0.572516\pi\)
							0.529136	+	0.848537i	\(0.322516\pi\)
\(31\)	7.58963e7	+	1.83230e8i	0.476137	+	1.14950i	0.961407	+	0.275131i	\(0.0887213\pi\)
							−0.485270	+	0.874364i	\(0.661279\pi\)
\(37\)	−9.74952e7	−	2.35374e8i	−0.231139	−	0.558019i	0.765173	−	0.643825i	\(-0.222654\pi\)
							−0.996312	+	0.0858057i	\(0.972654\pi\)
\(41\)	8.72540e8	+	3.61418e8i	1.17618	+	0.487190i	0.883231	−	0.468938i	\(-0.155363\pi\)
							0.292950	+	0.956128i	\(0.405363\pi\)
\(43\)	−1.77552e8	+	1.77552e8i	−0.184182	+	0.184182i	−0.793176	−	0.608993i	\(-0.791574\pi\)
							0.608993	+	0.793176i	\(0.291574\pi\)
\(47\)		−	1.46665e9i		−	0.932799i	−0.884574	−	0.466400i	\(-0.845551\pi\)
							0.884574	−	0.466400i	\(-0.154449\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.12.d.a.15.6	yes	60
17.8	even	8	inner	17.12.d.a.8.6	✓	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.12.d.a.8.6	✓	60	17.8	even	8	inner
17.12.d.a.15.6	yes	60	1.1	even	1	trivial