Embedded newform 16.12.e.a.13.13

• Label: 16.12.e.a.13.13
• 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			13.13
Character	\(\chi\)	\(=\)	16.13
Dual form			16.12.e.a.5.13

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	16.6669	+	42.0739i	0.368291	+	0.929711i
							\(3\)	−491.248	−	491.248i	−1.16717	−	1.16717i	−0.982870	−	0.184301i	\(-0.940998\pi\)
−0.184301	−	0.982870i	\(-0.559002\pi\)
\(5\)	6777.57	−	6777.57i	0.969927	−	0.969927i	−0.0296343	−	0.999561i	\(-0.509434\pi\)
							0.999561	+	0.0296343i	\(0.00943427\pi\)
\(7\)			47511.7i			1.06847i	0.845337	+	0.534234i	\(0.179400\pi\)
							−0.845337	+	0.534234i	\(0.820600\pi\)
\(11\)	−348627.	+	348627.i	−0.652681	+	0.652681i	−0.953638	−	0.300957i	\(-0.902694\pi\)
							0.300957	+	0.953638i	\(0.402694\pi\)
\(13\)	623712.	+	623712.i	0.465903	+	0.465903i	0.900584	−	0.434681i	\(-0.143139\pi\)
							−0.434681	+	0.900584i	\(0.643139\pi\)
\(17\)	−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	0.935423	+	0.353529i	\(0.115019\pi\)
							0.353529	+	0.935423i	\(0.384981\pi\)
\(23\)			4.84106e7i			1.56833i	0.620553	+	0.784165i	\(0.286908\pi\)
							−0.620553	+	0.784165i	\(0.713092\pi\)
\(29\)	−4.44473e6	−	4.44473e6i	−0.0402399	−	0.0402399i	0.686701	−	0.726940i	\(-0.259058\pi\)
							−0.726940	+	0.686701i	\(0.759058\pi\)
\(31\)	−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	−0.651546	−	0.758610i	\(-0.725879\pi\)
							0.758610	+	0.651546i	\(0.225879\pi\)
\(41\)		−	3.91782e8i		−	0.528121i	−0.964506	−	0.264060i	\(-0.914938\pi\)
							0.964506	−	0.264060i	\(-0.0850618\pi\)
\(43\)	8.76450e8	−	8.76450e8i	0.909182	−	0.909182i	−0.0870244	−	0.996206i	\(-0.527736\pi\)
							0.996206	+	0.0870244i	\(0.0277358\pi\)
\(47\)	−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.13.13	yes	42
4.3	odd	2		64.12.e.a.17.20		42
8.3	odd	2		128.12.e.a.33.2		42
8.5	even	2		128.12.e.b.33.20		42
16.3	odd	4		128.12.e.a.97.2		42
16.5	even	4	inner	16.12.e.a.5.13	✓	42
16.11	odd	4		64.12.e.a.49.20		42
16.13	even	4		128.12.e.b.97.20		42

    

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