Embedded newform 16.10.e.a.5.17

• Label: 16.10.e.a.5.17
• Level: $16$
• Weight: $10$
• Character: 16.5
• Analytic conductor: $8.241$
• Analytic rank: $0$
• Dimension: $34$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	16.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.24057337862\)
Analytic rank:	\(0\)
Dimension:	\(34\)
Relative dimension:	\(17\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5.17
Character	\(\chi\)	\(=\)	16.5
Dual form			16.10.e.a.13.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(21.4373 + 7.24163i) q^{2} +(-33.5715 + 33.5715i) q^{3} +(407.118 + 310.482i) q^{4} +(315.639 + 315.639i) q^{5} +(-962.796 + 476.571i) q^{6} +3725.04i q^{7} +(6479.11 + 9604.11i) q^{8} +17428.9i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q - 2 q^{2} - 2 q^{3} + 168 q^{4} - 2 q^{5} - 2192 q^{6} - 716 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	21.4373	+	7.24163i	0.947405	+	0.320038i
							\(3\)	−33.5715	+	33.5715i	−0.239291	+	0.239291i	−0.816556	−	0.577266i	\(-0.804120\pi\)
0.577266	+	0.816556i	\(0.304120\pi\)
\(5\)	315.639	+	315.639i	0.225853	+	0.225853i	0.810958	−	0.585105i	\(-0.198947\pi\)
							−0.585105	+	0.810958i	\(0.698947\pi\)
\(7\)			3725.04i			0.586394i	0.956052	+	0.293197i	\(0.0947191\pi\)
							−0.956052	+	0.293197i	\(0.905281\pi\)
\(11\)	−6362.52	−	6362.52i	−0.131027	−	0.131027i	0.638552	−	0.769579i	\(-0.279534\pi\)
							−0.769579	+	0.638552i	\(0.779534\pi\)
\(13\)	16445.7	−	16445.7i	0.159701	−	0.159701i	−0.622733	−	0.782434i	\(-0.713978\pi\)
							0.782434	+	0.622733i	\(0.213978\pi\)
\(17\)	162975.			0.473261			0.236631	−	0.971600i	\(-0.423957\pi\)
							0.236631	+	0.971600i	\(0.423957\pi\)
\(19\)	183945.	−	183945.i	0.323816	−	0.323816i	−0.526413	−	0.850229i	\(-0.676464\pi\)
							0.850229	+	0.526413i	\(0.176464\pi\)
\(23\)		−	2.13295e6i		−	1.58930i	−0.607068	−	0.794650i	\(-0.707655\pi\)
							0.607068	−	0.794650i	\(-0.292345\pi\)
\(29\)	2.64765e6	−	2.64765e6i	0.695136	−	0.695136i	−0.268221	−	0.963357i	\(-0.586436\pi\)
							0.963357	+	0.268221i	\(0.0864358\pi\)
\(31\)	−1.47318e6			−0.286502			−0.143251	−	0.989686i	\(-0.545756\pi\)
							−0.143251	+	0.989686i	\(0.545756\pi\)
\(37\)	1.01299e7	+	1.01299e7i	0.888578	+	0.888578i	0.994387	−	0.105808i	\(-0.0337430\pi\)
							−0.105808	+	0.994387i	\(0.533743\pi\)
\(41\)			1.52822e7i			0.844613i	0.906453	+	0.422306i	\(0.138779\pi\)
							−0.906453	+	0.422306i	\(0.861221\pi\)
\(43\)	−7.77033e6	−	7.77033e6i	−0.346602	−	0.346602i	0.512240	−	0.858842i	\(-0.328816\pi\)
							−0.858842	+	0.512240i	\(0.828816\pi\)
\(47\)	−4.90714e7			−1.46686			−0.733429	−	0.679766i	\(-0.762081\pi\)
							−0.733429	+	0.679766i	\(0.762081\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.10.e.a.5.17	✓	34
4.3	odd	2		64.10.e.a.49.11		34
8.3	odd	2		128.10.e.a.97.7		34
8.5	even	2		128.10.e.b.97.11		34
16.3	odd	4		64.10.e.a.17.11		34
16.5	even	4		128.10.e.b.33.11		34
16.11	odd	4		128.10.e.a.33.7		34
16.13	even	4	inner	16.10.e.a.13.17	yes	34

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.10.e.a.5.17	✓	34	1.1	even	1	trivial
16.10.e.a.13.17	yes	34	16.13	even	4	inner
64.10.e.a.17.11		34	16.3	odd	4
64.10.e.a.49.11		34	4.3	odd	2
128.10.e.a.33.7		34	16.11	odd	4
128.10.e.a.97.7		34	8.3	odd	2
128.10.e.b.33.11		34	16.5	even	4
128.10.e.b.97.11		34	8.5	even	2