Embedded newform 16.12.e.a.5.15

• Label: 16.12.e.a.5.15
• 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			5.15
Character	\(\chi\)	\(=\)	16.5
Dual form			16.12.e.a.13.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(26.1391 + 36.9425i) q^{2} +(255.917 - 255.917i) q^{3} +(-681.491 + 1931.29i) q^{4} +(-4214.49 - 4214.49i) q^{5} +(16143.6 + 2764.75i) q^{6} +80345.3i q^{7} +(-89160.1 + 25306.2i) q^{8} +46160.2i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{2} - 2 q^{3} + 3080 q^{4} - 2 q^{5} + 30352 q^{6} + 74836 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	26.1391	+	36.9425i	0.577599	+	0.816321i
							\(3\)	255.917	−	255.917i	0.608040	−	0.608040i	−0.334394	−	0.942433i	\(-0.608532\pi\)
0.942433	+	0.334394i	\(0.108532\pi\)
\(5\)	−4214.49	−	4214.49i	−0.603128	−	0.603128i	0.338013	−	0.941141i	\(-0.390245\pi\)
							−0.941141	+	0.338013i	\(0.890245\pi\)
\(7\)			80345.3i			1.80685i	0.428751	+	0.903423i	\(0.358954\pi\)
							−0.428751	+	0.903423i	\(0.641046\pi\)
\(11\)	611105.	+	611105.i	1.14408	+	1.14408i	0.987697	+	0.156382i	\(0.0499831\pi\)
							0.156382	+	0.987697i	\(0.450017\pi\)
\(13\)	370364.	−	370364.i	0.276656	−	0.276656i	−0.555116	−	0.831773i	\(-0.687326\pi\)
							0.831773	+	0.555116i	\(0.187326\pi\)
\(17\)	−725142.			−0.123867			−0.0619333	−	0.998080i	\(-0.519727\pi\)
							−0.0619333	+	0.998080i	\(0.519727\pi\)
\(19\)	−1.13100e7	+	1.13100e7i	−1.04789	+	1.04789i	−0.0490973	+	0.998794i	\(0.515634\pi\)
							−0.998794	+	0.0490973i	\(0.984366\pi\)
\(23\)		−	4.89011e7i		−	1.58422i	−0.610378	−	0.792110i	\(-0.708982\pi\)
							0.610378	−	0.792110i	\(-0.291018\pi\)
\(29\)	4.98291e7	−	4.98291e7i	0.451122	−	0.451122i	−0.444605	−	0.895727i	\(-0.646656\pi\)
							0.895727	+	0.444605i	\(0.146656\pi\)
\(31\)	4.20747e7			0.263956			0.131978	−	0.991253i	\(-0.457867\pi\)
							0.131978	+	0.991253i	\(0.457867\pi\)
\(37\)	−1.05666e8	−	1.05666e8i	−0.250510	−	0.250510i	0.570670	−	0.821180i	\(-0.306684\pi\)
							−0.821180	+	0.570670i	\(0.806684\pi\)
\(41\)			4.49063e8i			0.605335i	0.953096	+	0.302668i	\(0.0978772\pi\)
							−0.953096	+	0.302668i	\(0.902123\pi\)
\(43\)	7.89791e8	+	7.89791e8i	0.819286	+	0.819286i	0.986005	−	0.166718i	\(-0.0533170\pi\)
							−0.166718	+	0.986005i	\(0.553317\pi\)
\(47\)	1.32368e9			0.841870			0.420935	−	0.907091i	\(-0.361702\pi\)
							0.420935	+	0.907091i	\(0.361702\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.12.e.a.5.15	✓	42
4.3	odd	2		64.12.e.a.49.6		42
8.3	odd	2		128.12.e.a.97.16		42
8.5	even	2		128.12.e.b.97.6		42
16.3	odd	4		64.12.e.a.17.6		42
16.5	even	4		128.12.e.b.33.6		42
16.11	odd	4		128.12.e.a.33.16		42
16.13	even	4	inner	16.12.e.a.13.15	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.12.e.a.5.15	✓	42	1.1	even	1	trivial
16.12.e.a.13.15	yes	42	16.13	even	4	inner
64.12.e.a.17.6		42	16.3	odd	4
64.12.e.a.49.6		42	4.3	odd	2
128.12.e.a.33.16		42	16.11	odd	4
128.12.e.a.97.16		42	8.3	odd	2
128.12.e.b.33.6		42	16.5	even	4
128.12.e.b.97.6		42	8.5	even	2