Embedded newform 16.12.e.a.5.6

• Label: 16.12.e.a.5.6
• Level: $16$
• Weight: $12$
• Character: 16.5
• Analytic conductor: $12.293$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	16.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.2934908890\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(21\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5.6
Character	\(\chi\)	\(=\)	16.5
Dual form			16.12.e.a.13.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-31.4901 + 32.5019i) q^{2} +(219.312 - 219.312i) q^{3} +(-64.7420 - 2046.98i) q^{4} +(-4575.66 - 4575.66i) q^{5} +(221.882 + 14034.2i) q^{6} +2778.29i q^{7} +(68569.3 + 62355.3i) q^{8} +80951.6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{2} - 2 q^{3} + 3080 q^{4} - 2 q^{5} + 30352 q^{6} + 74836 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−31.4901	+	32.5019i	−0.695840	+	0.718196i
							\(3\)	219.312	−	219.312i	0.521069	−	0.521069i	−0.396825	−	0.917894i	\(-0.629888\pi\)
0.917894	+	0.396825i	\(0.129888\pi\)
\(5\)	−4575.66	−	4575.66i	−0.654815	−	0.654815i	0.299333	−	0.954149i	\(-0.403236\pi\)
							−0.954149	+	0.299333i	\(0.903236\pi\)
\(7\)			2778.29i			0.0624796i	0.999512	+	0.0312398i	\(0.00994555\pi\)
							−0.999512	+	0.0312398i	\(0.990054\pi\)
\(11\)	17312.0	+	17312.0i	0.0324106	+	0.0324106i	0.723126	−	0.690716i	\(-0.242704\pi\)
							−0.690716	+	0.723126i	\(0.742704\pi\)
\(13\)	−1.16637e6	+	1.16637e6i	−0.871263	+	0.871263i	−0.992610	−	0.121347i	\(-0.961278\pi\)
							0.121347	+	0.992610i	\(0.461278\pi\)
\(17\)	−7.63947e6			−1.30495			−0.652475	−	0.757810i	\(-0.726269\pi\)
							−0.652475	+	0.757810i	\(0.726269\pi\)
\(19\)	1.62010e6	−	1.62010e6i	0.150106	−	0.150106i	−0.628060	−	0.778165i	\(-0.716151\pi\)
							0.778165	+	0.628060i	\(0.216151\pi\)
\(23\)			3.41251e7i			1.10553i	0.833337	+	0.552765i	\(0.186427\pi\)
							−0.833337	+	0.552765i	\(0.813573\pi\)
\(29\)	−5.22501e7	+	5.22501e7i	−0.473041	+	0.473041i	−0.902897	−	0.429856i	\(-0.858564\pi\)
							0.429856	+	0.902897i	\(0.358564\pi\)
\(31\)	−1.70466e8			−1.06942			−0.534710	−	0.845035i	\(-0.679579\pi\)
							−0.534710	+	0.845035i	\(0.679579\pi\)
\(37\)	1.36013e8	+	1.36013e8i	0.322455	+	0.322455i	0.849708	−	0.527253i	\(-0.176778\pi\)
							−0.527253	+	0.849708i	\(0.676778\pi\)
\(41\)			9.43574e8i			1.27193i	0.771716	+	0.635967i	\(0.219399\pi\)
							−0.771716	+	0.635967i	\(0.780601\pi\)
\(43\)	−1.75831e8	−	1.75831e8i	−0.182397	−	0.182397i	0.610002	−	0.792400i	\(-0.291169\pi\)
							−0.792400	+	0.610002i	\(0.791169\pi\)
\(47\)	−2.33824e9			−1.48714			−0.743569	−	0.668660i	\(-0.766868\pi\)
							−0.743569	+	0.668660i	\(0.766868\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.12.e.a.5.6	✓	42
4.3	odd	2		64.12.e.a.49.8		42
8.3	odd	2		128.12.e.a.97.14		42
8.5	even	2		128.12.e.b.97.8		42
16.3	odd	4		64.12.e.a.17.8		42
16.5	even	4		128.12.e.b.33.8		42
16.11	odd	4		128.12.e.a.33.14		42
16.13	even	4	inner	16.12.e.a.13.6	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.12.e.a.5.6	✓	42	1.1	even	1	trivial
16.12.e.a.13.6	yes	42	16.13	even	4	inner
64.12.e.a.17.8		42	16.3	odd	4
64.12.e.a.49.8		42	4.3	odd	2
128.12.e.a.33.14		42	16.11	odd	4
128.12.e.a.97.14		42	8.3	odd	2
128.12.e.b.33.8		42	16.5	even	4
128.12.e.b.97.8		42	8.5	even	2