Embedded newform 16.12.e.a.13.18

• Label: 16.12.e.a.13.18
• 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			13.18
Character	\(\chi\)	\(=\)	16.13
Dual form			16.12.e.a.5.18

$q$-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 \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	40.4837	−	20.2255i	0.894571	−	0.446925i
							\(3\)	−522.679	−	522.679i	−1.24185	−	1.24185i	−0.959234	−	0.282612i	\(-0.908799\pi\)
−0.282612	−	0.959234i	\(-0.591201\pi\)
\(5\)	−3202.37	+	3202.37i	−0.458287	+	0.458287i	−0.898093	−	0.439806i	\(-0.855047\pi\)
							0.439806	+	0.898093i	\(0.355047\pi\)
\(7\)		−	31977.4i		−	0.719123i	−0.933121	−	0.359562i	\(-0.882926\pi\)
							0.933121	−	0.359562i	\(-0.117074\pi\)
\(11\)	−406679.	+	406679.i	−0.761364	+	0.761364i	−0.976569	−	0.215205i	\(-0.930958\pi\)
							0.215205	+	0.976569i	\(0.430958\pi\)
\(13\)	1.12564e6	+	1.12564e6i	0.840835	+	0.840835i	0.988968	−	0.148132i	\(-0.0473262\pi\)
							−0.148132	+	0.988968i	\(0.547326\pi\)
\(17\)	−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	0.245687	−	0.969349i	\(-0.420986\pi\)
							−0.969349	+	0.245687i	\(0.920986\pi\)
\(23\)		−	3.53021e7i		−	1.14366i	−0.820371	−	0.571831i	\(-0.806233\pi\)
							0.820371	−	0.571831i	\(-0.193767\pi\)
\(29\)	−1.32950e8	−	1.32950e8i	−1.20364	−	1.20364i	−0.973050	−	0.230594i	\(-0.925933\pi\)
							−0.230594	−	0.973050i	\(-0.574067\pi\)
\(31\)	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	−0.779215	−	0.626756i	\(-0.784382\pi\)
							0.626756	+	0.779215i	\(0.284382\pi\)
\(41\)		−	1.19596e9i		−	1.61215i	−0.591817	−	0.806073i	\(-0.701589\pi\)
							0.591817	−	0.806073i	\(-0.298411\pi\)
\(43\)	−1.26260e9	+	1.26260e9i	−1.30975	+	1.30975i	−0.388154	+	0.921594i	\(0.626887\pi\)
							−0.921594	+	0.388154i	\(0.873113\pi\)
\(47\)	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.13.18	yes	42
4.3	odd	2		64.12.e.a.17.21		42
8.3	odd	2		128.12.e.a.33.1		42
8.5	even	2		128.12.e.b.33.21		42
16.3	odd	4		128.12.e.a.97.1		42
16.5	even	4	inner	16.12.e.a.5.18	✓	42
16.11	odd	4		64.12.e.a.49.21		42
16.13	even	4		128.12.e.b.97.21		42

    

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