Embedded newform 16.12.e.a.13.19

• Label: 16.12.e.a.13.19
• Level: $16$
• Weight: $12$
• Character: 16.13
• 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			13.19
Character	\(\chi\)	\(=\)	16.13
Dual form			16.12.e.a.5.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(43.9461 - 10.8046i) q^{2} +(221.481 + 221.481i) q^{3} +(1814.52 - 949.643i) q^{4} +(-7918.20 + 7918.20i) q^{5} +(12126.3 + 7340.22i) q^{6} +76836.1i q^{7} +(69480.6 - 61338.3i) q^{8} -79039.1i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{2} - 2 q^{3} + 3080 q^{4} - 2 q^{5} + 30352 q^{6} + 74836 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	43.9461	−	10.8046i	0.971081	−	0.238751i
							\(3\)	221.481	+	221.481i	0.526223	+	0.526223i	0.919444	−	0.393221i	\(-0.128639\pi\)
−0.393221	+	0.919444i	\(0.628639\pi\)
\(5\)	−7918.20	+	7918.20i	−1.13316	+	1.13316i	−0.143512	+	0.989649i	\(0.545839\pi\)
							−0.989649	+	0.143512i	\(0.954161\pi\)
\(7\)			76836.1i			1.72793i	0.503552	+	0.863965i	\(0.332026\pi\)
							−0.503552	+	0.863965i	\(0.667974\pi\)
\(11\)	−197951.	+	197951.i	−0.370593	+	0.370593i	−0.867693	−	0.497100i	\(-0.834398\pi\)
							0.497100	+	0.867693i	\(0.334398\pi\)
\(13\)	−46856.2	−	46856.2i	−0.0350008	−	0.0350008i	0.689390	−	0.724391i	\(-0.257879\pi\)
							−0.724391	+	0.689390i	\(0.757879\pi\)
\(17\)	9.74414e6			1.66446			0.832232	−	0.554428i	\(-0.187063\pi\)
							0.832232	+	0.554428i	\(0.187063\pi\)
\(19\)	2.85298e6	+	2.85298e6i	0.264334	+	0.264334i	0.826812	−	0.562478i	\(-0.190152\pi\)
							−0.562478	+	0.826812i	\(0.690152\pi\)
\(23\)			3.12758e7i			1.01322i	0.862174	+	0.506611i	\(0.169102\pi\)
							−0.862174	+	0.506611i	\(0.830898\pi\)
\(29\)	−1.26920e8	−	1.26920e8i	−1.14905	−	1.14905i	−0.986739	−	0.162315i	\(-0.948104\pi\)
							−0.162315	−	0.986739i	\(-0.551896\pi\)
\(31\)	3.46657e7			0.217475			0.108738	−	0.994070i	\(-0.465319\pi\)
							0.108738	+	0.994070i	\(0.465319\pi\)
\(37\)	2.25685e8	−	2.25685e8i	0.535048	−	0.535048i	−0.387022	−	0.922070i	\(-0.626496\pi\)
							0.922070	+	0.387022i	\(0.126496\pi\)
\(41\)			4.55938e8i			0.614603i	0.951612	+	0.307302i	\(0.0994260\pi\)
							−0.951612	+	0.307302i	\(0.900574\pi\)
\(43\)	7.03132e8	−	7.03132e8i	0.729390	−	0.729390i	−0.241108	−	0.970498i	\(-0.577511\pi\)
							0.970498	+	0.241108i	\(0.0775108\pi\)
\(47\)	6.32523e8			0.402289			0.201144	−	0.979562i	\(-0.435534\pi\)
							0.201144	+	0.979562i	\(0.435534\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.12.e.a.13.19	yes	42
4.3	odd	2		64.12.e.a.17.7		42
8.3	odd	2		128.12.e.a.33.15		42
8.5	even	2		128.12.e.b.33.7		42
16.3	odd	4		128.12.e.a.97.15		42
16.5	even	4	inner	16.12.e.a.5.19	✓	42
16.11	odd	4		64.12.e.a.49.7		42
16.13	even	4		128.12.e.b.97.7		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.12.e.a.5.19	✓	42	16.5	even	4	inner
16.12.e.a.13.19	yes	42	1.1	even	1	trivial
64.12.e.a.17.7		42	4.3	odd	2
64.12.e.a.49.7		42	16.11	odd	4
128.12.e.a.33.15		42	8.3	odd	2
128.12.e.a.97.15		42	16.3	odd	4
128.12.e.b.33.7		42	8.5	even	2
128.12.e.b.97.7		42	16.13	even	4