Embedded newform 16.12.e.a.5.1

• Label: 16.12.e.a.5.1
• 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			5.1
Character	\(\chi\)	\(=\)	16.5
Dual form			16.12.e.a.13.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-45.0143 - 4.65973i) q^{2} +(13.6026 - 13.6026i) q^{3} +(2004.57 + 419.509i) q^{4} +(-3501.81 - 3501.81i) q^{5} +(-675.697 + 548.928i) q^{6} +37649.1i q^{7} +(-88279.7 - 28224.7i) q^{8} +176777. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{2} - 2 q^{3} + 3080 q^{4} - 2 q^{5} + 30352 q^{6} + 74836 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−45.0143	−	4.65973i	−0.994685	−	0.102966i
							\(3\)	13.6026	−	13.6026i	0.0323189	−	0.0323189i	−0.690763	−	0.723082i	\(-0.742725\pi\)
0.723082	+	0.690763i	\(0.242725\pi\)
\(5\)	−3501.81	−	3501.81i	−0.501138	−	0.501138i	0.410653	−	0.911791i	\(-0.365301\pi\)
							−0.911791	+	0.410653i	\(0.865301\pi\)
\(7\)			37649.1i			0.846673i	0.905972	+	0.423337i	\(0.139141\pi\)
							−0.905972	+	0.423337i	\(0.860859\pi\)
\(11\)	−283274.	−	283274.i	−0.530331	−	0.530331i	0.390340	−	0.920671i	\(-0.372357\pi\)
							−0.920671	+	0.390340i	\(0.872357\pi\)
\(13\)	1.70921e6	−	1.70921e6i	1.27676	−	1.27676i	0.334284	−	0.942472i	\(-0.391506\pi\)
							0.942472	−	0.334284i	\(-0.108494\pi\)
\(17\)	1.57559e6			0.269138			0.134569	−	0.990904i	\(-0.457035\pi\)
							0.134569	+	0.990904i	\(0.457035\pi\)
\(19\)	1.25182e7	−	1.25182e7i	1.15983	−	1.15983i	0.175323	−	0.984511i	\(-0.443903\pi\)
							0.984511	−	0.175323i	\(-0.0560971\pi\)
\(23\)			6.83457e6i			0.221416i	0.993853	+	0.110708i	\(0.0353118\pi\)
							−0.993853	+	0.110708i	\(0.964688\pi\)
\(29\)	5.00021e7	−	5.00021e7i	0.452688	−	0.452688i	−0.443558	−	0.896246i	\(-0.646284\pi\)
							0.896246	+	0.443558i	\(0.146284\pi\)
\(31\)	7.19392e7			0.451311			0.225656	−	0.974207i	\(-0.427548\pi\)
							0.225656	+	0.974207i	\(0.427548\pi\)
\(37\)	9.81694e7	+	9.81694e7i	0.232738	+	0.232738i	0.813834	−	0.581097i	\(-0.197376\pi\)
							−0.581097	+	0.813834i	\(0.697376\pi\)
\(41\)		−	7.24665e8i		−	0.976846i	−0.872607	−	0.488423i	\(-0.837572\pi\)
							0.872607	−	0.488423i	\(-0.162428\pi\)
\(43\)	1.93444e8	+	1.93444e8i	0.200668	+	0.200668i	0.800286	−	0.599618i	\(-0.204681\pi\)
							−0.599618	+	0.800286i	\(0.704681\pi\)
\(47\)	1.39935e9			0.889997			0.444999	−	0.895531i	\(-0.353204\pi\)
							0.444999	+	0.895531i	\(0.353204\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.12.e.a.5.1	✓	42
4.3	odd	2		64.12.e.a.49.10		42
8.3	odd	2		128.12.e.a.97.12		42
8.5	even	2		128.12.e.b.97.10		42
16.3	odd	4		64.12.e.a.17.10		42
16.5	even	4		128.12.e.b.33.10		42
16.11	odd	4		128.12.e.a.33.12		42
16.13	even	4	inner	16.12.e.a.13.1	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.12.e.a.5.1	✓	42	1.1	even	1	trivial
16.12.e.a.13.1	yes	42	16.13	even	4	inner
64.12.e.a.17.10		42	16.3	odd	4
64.12.e.a.49.10		42	4.3	odd	2
128.12.e.a.33.12		42	16.11	odd	4
128.12.e.a.97.12		42	8.3	odd	2
128.12.e.b.33.10		42	16.5	even	4
128.12.e.b.97.10		42	8.5	even	2