Embedded newform 16.12.e.a.5.13

• Label: 16.12.e.a.5.13
• 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			5.13
Character	\(\chi\)	\(=\)	16.5
Dual form			16.12.e.a.13.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(16.6669 - 42.0739i) q^{2} +(-491.248 + 491.248i) q^{3} +(-1492.43 - 1402.49i) q^{4} +(6777.57 + 6777.57i) q^{5} +(12481.1 + 28856.3i) q^{6} -47511.7i q^{7} +(-83882.3 + 39417.0i) q^{8} -305503. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{2} - 2 q^{3} + 3080 q^{4} - 2 q^{5} + 30352 q^{6} + 74836 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	16.6669	−	42.0739i	0.368291	−	0.929711i
							\(3\)	−491.248	+	491.248i	−1.16717	+	1.16717i	−0.184301	+	0.982870i	\(0.559002\pi\)
−0.982870	+	0.184301i	\(0.940998\pi\)
\(5\)	6777.57	+	6777.57i	0.969927	+	0.969927i	0.999561	−	0.0296343i	\(-0.00943427\pi\)
							−0.0296343	+	0.999561i	\(0.509434\pi\)
\(7\)		−	47511.7i		−	1.06847i	−0.845337	−	0.534234i	\(-0.820600\pi\)
							0.845337	−	0.534234i	\(-0.179400\pi\)
\(11\)	−348627.	−	348627.i	−0.652681	−	0.652681i	0.300957	−	0.953638i	\(-0.402694\pi\)
							−0.953638	+	0.300957i	\(0.902694\pi\)
\(13\)	623712.	−	623712.i	0.465903	−	0.465903i	−0.434681	−	0.900584i	\(-0.643139\pi\)
							0.900584	+	0.434681i	\(0.143139\pi\)
\(17\)	−447898.			−0.0765085			−0.0382543	−	0.999268i	\(-0.512180\pi\)
							−0.0382543	+	0.999268i	\(0.512180\pi\)
\(19\)	1.39118e7	−	1.39118e7i	1.28895	−	1.28895i	0.353529	−	0.935423i	\(-0.384981\pi\)
							0.935423	−	0.353529i	\(-0.115019\pi\)
\(23\)		−	4.84106e7i		−	1.56833i	−0.620553	−	0.784165i	\(-0.713092\pi\)
							0.620553	−	0.784165i	\(-0.286908\pi\)
\(29\)	−4.44473e6	+	4.44473e6i	−0.0402399	+	0.0402399i	−0.726940	−	0.686701i	\(-0.759058\pi\)
							0.686701	+	0.726940i	\(0.259058\pi\)
\(31\)	−1.41505e8			−0.887731			−0.443865	−	0.896094i	\(-0.646393\pi\)
							−0.443865	+	0.896094i	\(0.646393\pi\)
\(37\)	4.51599e7	+	4.51599e7i	0.107064	+	0.107064i	0.758610	−	0.651546i	\(-0.225879\pi\)
							−0.651546	+	0.758610i	\(0.725879\pi\)
\(41\)			3.91782e8i			0.528121i	0.964506	+	0.264060i	\(0.0850618\pi\)
							−0.964506	+	0.264060i	\(0.914938\pi\)
\(43\)	8.76450e8	+	8.76450e8i	0.909182	+	0.909182i	0.996206	−	0.0870244i	\(-0.0277358\pi\)
							−0.0870244	+	0.996206i	\(0.527736\pi\)
\(47\)	−1.57121e9			−0.999300			−0.499650	−	0.866227i	\(-0.666538\pi\)
							−0.499650	+	0.866227i	\(0.666538\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.12.e.a.5.13	✓	42
4.3	odd	2		64.12.e.a.49.20		42
8.3	odd	2		128.12.e.a.97.2		42
8.5	even	2		128.12.e.b.97.20		42
16.3	odd	4		64.12.e.a.17.20		42
16.5	even	4		128.12.e.b.33.20		42
16.11	odd	4		128.12.e.a.33.2		42
16.13	even	4	inner	16.12.e.a.13.13	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.12.e.a.5.13	✓	42	1.1	even	1	trivial
16.12.e.a.13.13	yes	42	16.13	even	4	inner
64.12.e.a.17.20		42	16.3	odd	4
64.12.e.a.49.20		42	4.3	odd	2
128.12.e.a.33.2		42	16.11	odd	4
128.12.e.a.97.2		42	8.3	odd	2
128.12.e.b.33.20		42	16.5	even	4
128.12.e.b.97.20		42	8.5	even	2