Embedded newform 16.12.e.a.5.11

• Label: 16.12.e.a.5.11
• 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			5.11
Character	\(\chi\)	\(=\)	16.5
Dual form			16.12.e.a.13.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.51469 + 45.2295i) q^{2} +(442.035 - 442.035i) q^{3} +(-2043.41 - 137.018i) q^{4} +(2497.66 + 2497.66i) q^{5} +(19323.5 + 20662.6i) q^{6} -39392.3i q^{7} +(9292.38 - 92214.9i) q^{8} -213643. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{2} - 2 q^{3} + 3080 q^{4} - 2 q^{5} + 30352 q^{6} + 74836 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.51469	+	45.2295i	−0.0334703	+	0.999440i
							\(3\)	442.035	−	442.035i	1.05024	−	1.05024i	0.0515741	−	0.998669i	\(-0.483576\pi\)
0.998669	−	0.0515741i	\(-0.0164238\pi\)
\(5\)	2497.66	+	2497.66i	0.357436	+	0.357436i	0.862867	−	0.505431i	\(-0.168666\pi\)
							−0.505431	+	0.862867i	\(0.668666\pi\)
\(7\)		−	39392.3i		−	0.885874i	−0.896553	−	0.442937i	\(-0.853936\pi\)
							0.896553	−	0.442937i	\(-0.146064\pi\)
\(11\)	−311395.	−	311395.i	−0.582977	−	0.582977i	0.352743	−	0.935720i	\(-0.385249\pi\)
							−0.935720	+	0.352743i	\(0.885249\pi\)
\(13\)	1.47077e6	−	1.47077e6i	1.09864	−	1.09864i	0.104072	−	0.994570i	\(-0.466813\pi\)
							0.994570	−	0.104072i	\(-0.0331874\pi\)
\(17\)	6.43972e6			1.10001			0.550007	−	0.835160i	\(-0.314625\pi\)
							0.550007	+	0.835160i	\(0.314625\pi\)
\(19\)	−3.94625e6	+	3.94625e6i	−0.365629	+	0.365629i	−0.865880	−	0.500252i	\(-0.833241\pi\)
							0.500252	+	0.865880i	\(0.333241\pi\)
\(23\)			2.40651e7i			0.779622i	0.920895	+	0.389811i	\(0.127460\pi\)
							−0.920895	+	0.389811i	\(0.872540\pi\)
\(29\)	1.10059e8	−	1.10059e8i	0.996403	−	0.996403i	−0.00359065	−	0.999994i	\(-0.501143\pi\)
							0.999994	+	0.00359065i	\(0.00114294\pi\)
\(31\)	−2.26680e8			−1.42208			−0.711039	−	0.703153i	\(-0.751775\pi\)
							−0.711039	+	0.703153i	\(0.751775\pi\)
\(37\)	−2.38575e8	−	2.38575e8i	−0.565608	−	0.565608i	0.365287	−	0.930895i	\(-0.380971\pi\)
							−0.930895	+	0.365287i	\(0.880971\pi\)
\(41\)			1.45913e9i			1.96691i	0.181160	+	0.983454i	\(0.442015\pi\)
							−0.181160	+	0.983454i	\(0.557985\pi\)
\(43\)	1.31591e8	+	1.31591e8i	0.136506	+	0.136506i	0.772058	−	0.635552i	\(-0.219227\pi\)
							−0.635552	+	0.772058i	\(0.719227\pi\)
\(47\)	−2.03828e9			−1.29636			−0.648180	−	0.761487i	\(-0.724469\pi\)
							−0.648180	+	0.761487i	\(0.724469\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.12.e.a.5.11	✓	42
4.3	odd	2		64.12.e.a.49.3		42
8.3	odd	2		128.12.e.a.97.19		42
8.5	even	2		128.12.e.b.97.3		42
16.3	odd	4		64.12.e.a.17.3		42
16.5	even	4		128.12.e.b.33.3		42
16.11	odd	4		128.12.e.a.33.19		42
16.13	even	4	inner	16.12.e.a.13.11	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.12.e.a.5.11	✓	42	1.1	even	1	trivial
16.12.e.a.13.11	yes	42	16.13	even	4	inner
64.12.e.a.17.3		42	16.3	odd	4
64.12.e.a.49.3		42	4.3	odd	2
128.12.e.a.33.19		42	16.11	odd	4
128.12.e.a.97.19		42	8.3	odd	2
128.12.e.b.33.3		42	16.5	even	4
128.12.e.b.97.3		42	8.5	even	2