Embedded newform 16.12.e.a.5.14

• Label: 16.12.e.a.5.14
• 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			5.14
Character	\(\chi\)	\(=\)	16.5
Dual form			16.12.e.a.13.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(25.1148 - 37.6464i) q^{2} +(367.824 - 367.824i) q^{3} +(-786.499 - 1890.96i) q^{4} +(-502.344 - 502.344i) q^{5} +(-4609.43 - 23085.1i) q^{6} -33063.2i q^{7} +(-90940.4 - 17882.1i) q^{8} -93442.1i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{2} - 2 q^{3} + 3080 q^{4} - 2 q^{5} + 30352 q^{6} + 74836 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	25.1148	−	37.6464i	0.554963	−	0.831875i
							\(3\)	367.824	−	367.824i	0.873923	−	0.873923i	−0.118974	−	0.992897i	\(-0.537961\pi\)
0.992897	+	0.118974i	\(0.0379606\pi\)
\(5\)	−502.344	−	502.344i	−0.0718896	−	0.0718896i	0.670248	−	0.742137i	\(-0.266188\pi\)
							−0.742137	+	0.670248i	\(0.766188\pi\)
\(7\)		−	33063.2i		−	0.743542i	−0.928325	−	0.371771i	\(-0.878751\pi\)
							0.928325	−	0.371771i	\(-0.121249\pi\)
\(11\)	335681.	+	335681.i	0.628444	+	0.628444i	0.947677	−	0.319232i	\(-0.103425\pi\)
							−0.319232	+	0.947677i	\(0.603425\pi\)
\(13\)	568061.	−	568061.i	0.424333	−	0.424333i	−0.462360	−	0.886692i	\(-0.652997\pi\)
							0.886692	+	0.462360i	\(0.152997\pi\)
\(17\)	−6.78457e6			−1.15892			−0.579460	−	0.815001i	\(-0.696736\pi\)
							−0.579460	+	0.815001i	\(0.696736\pi\)
\(19\)	1.33879e6	−	1.33879e6i	0.124042	−	0.124042i	−0.642361	−	0.766402i	\(-0.722045\pi\)
							0.766402	+	0.642361i	\(0.222045\pi\)
\(23\)			1.07133e6i			0.0347074i	0.999849	+	0.0173537i	\(0.00552412\pi\)
							−0.999849	+	0.0173537i	\(0.994476\pi\)
\(29\)	5.01359e7	−	5.01359e7i	0.453899	−	0.453899i	−0.442747	−	0.896647i	\(-0.645996\pi\)
							0.896647	+	0.442747i	\(0.145996\pi\)
\(31\)	2.95969e8			1.85676			0.928382	−	0.371628i	\(-0.121200\pi\)
							0.928382	+	0.371628i	\(0.121200\pi\)
\(37\)	5.11883e8	+	5.11883e8i	1.21356	+	1.21356i	0.969848	+	0.243711i	\(0.0783649\pi\)
							0.243711	+	0.969848i	\(0.421635\pi\)
\(41\)		−	9.31803e8i		−	1.25607i	−0.778186	−	0.628034i	\(-0.783860\pi\)
							0.778186	−	0.628034i	\(-0.216140\pi\)
\(43\)	−1.27362e9	−	1.27362e9i	−1.32118	−	1.32118i	−0.912822	−	0.408358i	\(-0.866101\pi\)
							−0.408358	−	0.912822i	\(-0.633899\pi\)
\(47\)	−1.80739e7			−0.0114951			−0.00574756	−	0.999983i	\(-0.501830\pi\)
							−0.00574756	+	0.999983i	\(0.501830\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.12.e.a.5.14	✓	42
4.3	odd	2		64.12.e.a.49.5		42
8.3	odd	2		128.12.e.a.97.17		42
8.5	even	2		128.12.e.b.97.5		42
16.3	odd	4		64.12.e.a.17.5		42
16.5	even	4		128.12.e.b.33.5		42
16.11	odd	4		128.12.e.a.33.17		42
16.13	even	4	inner	16.12.e.a.13.14	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.12.e.a.5.14	✓	42	1.1	even	1	trivial
16.12.e.a.13.14	yes	42	16.13	even	4	inner
64.12.e.a.17.5		42	16.3	odd	4
64.12.e.a.49.5		42	4.3	odd	2
128.12.e.a.33.17		42	16.11	odd	4
128.12.e.a.97.17		42	8.3	odd	2
128.12.e.b.33.5		42	16.5	even	4
128.12.e.b.97.5		42	8.5	even	2