Embedded newform 16.12.e.a.5.18

• Label: 16.12.e.a.5.18
• 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			5.18
Character	\(\chi\)	\(=\)	16.5
Dual form			16.12.e.a.13.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(40.4837 + 20.2255i) q^{2} +(-522.679 + 522.679i) q^{3} +(1229.86 + 1637.61i) q^{4} +(-3202.37 - 3202.37i) q^{5} +(-31731.4 + 10588.5i) q^{6} +31977.4i q^{7} +(16667.6 + 91170.9i) q^{8} -369239. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{2} - 2 q^{3} + 3080 q^{4} - 2 q^{5} + 30352 q^{6} + 74836 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	40.4837	+	20.2255i	0.894571	+	0.446925i
							\(3\)	−522.679	+	522.679i	−1.24185	+	1.24185i	−0.282612	+	0.959234i	\(0.591201\pi\)
−0.959234	+	0.282612i	\(0.908799\pi\)
\(5\)	−3202.37	−	3202.37i	−0.458287	−	0.458287i	0.439806	−	0.898093i	\(-0.355047\pi\)
							−0.898093	+	0.439806i	\(0.855047\pi\)
\(7\)			31977.4i			0.719123i	0.933121	+	0.359562i	\(0.117074\pi\)
							−0.933121	+	0.359562i	\(0.882926\pi\)
\(11\)	−406679.	−	406679.i	−0.761364	−	0.761364i	0.215205	−	0.976569i	\(-0.430958\pi\)
							−0.976569	+	0.215205i	\(0.930958\pi\)
\(13\)	1.12564e6	−	1.12564e6i	0.840835	−	0.840835i	−0.148132	−	0.988968i	\(-0.547326\pi\)
							0.988968	+	0.148132i	\(0.0473262\pi\)
\(17\)	−5.55545e6			−0.948965			−0.474483	−	0.880265i	\(-0.657365\pi\)
							−0.474483	+	0.880265i	\(0.657365\pi\)
\(19\)	−7.81054e6	+	7.81054e6i	−0.723662	+	0.723662i	−0.969349	−	0.245687i	\(-0.920986\pi\)
							0.245687	+	0.969349i	\(0.420986\pi\)
\(23\)			3.53021e7i			1.14366i	0.820371	+	0.571831i	\(0.193767\pi\)
							−0.820371	+	0.571831i	\(0.806233\pi\)
\(29\)	−1.32950e8	+	1.32950e8i	−1.20364	+	1.20364i	−0.230594	+	0.973050i	\(0.574067\pi\)
							−0.973050	+	0.230594i	\(0.925933\pi\)
\(31\)	7.64393e7			0.479543			0.239771	−	0.970829i	\(-0.422928\pi\)
							0.239771	+	0.970829i	\(0.422928\pi\)
\(37\)	−6.43077e7	−	6.43077e7i	−0.152459	−	0.152459i	0.626756	−	0.779215i	\(-0.284382\pi\)
							−0.779215	+	0.626756i	\(0.784382\pi\)
\(41\)			1.19596e9i			1.61215i	0.591817	+	0.806073i	\(0.298411\pi\)
							−0.591817	+	0.806073i	\(0.701589\pi\)
\(43\)	−1.26260e9	−	1.26260e9i	−1.30975	−	1.30975i	−0.921594	−	0.388154i	\(-0.873113\pi\)
							−0.388154	−	0.921594i	\(-0.626887\pi\)
\(47\)	5.29010e8			0.336454			0.168227	−	0.985748i	\(-0.446196\pi\)
							0.168227	+	0.985748i	\(0.446196\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.12.e.a.5.18	✓	42
4.3	odd	2		64.12.e.a.49.21		42
8.3	odd	2		128.12.e.a.97.1		42
8.5	even	2		128.12.e.b.97.21		42
16.3	odd	4		64.12.e.a.17.21		42
16.5	even	4		128.12.e.b.33.21		42
16.11	odd	4		128.12.e.a.33.1		42
16.13	even	4	inner	16.12.e.a.13.18	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.12.e.a.5.18	✓	42	1.1	even	1	trivial
16.12.e.a.13.18	yes	42	16.13	even	4	inner
64.12.e.a.17.21		42	16.3	odd	4
64.12.e.a.49.21		42	4.3	odd	2
128.12.e.a.33.1		42	16.11	odd	4
128.12.e.a.97.1		42	8.3	odd	2
128.12.e.b.33.21		42	16.5	even	4
128.12.e.b.97.21		42	8.5	even	2